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## **Meet the editor**

Alejandro Salcido is currently Lead Scientist in the Alternative Energies Division (División de Energías Alternas) at the Institute of Electrical Research (Instituto de Investigaciones Eléctricas) in Cuernavaca, Mexico. Dr. Salcido received his PhD in Physics from the Physics Department of the Faculty of Sciences at the National University of Mexico (Universidad Nacional Autónoma

de México, UNAM) in 1992. Up to 1991, Dr. Salcido was Staff Scientist (Statistical Physics and Lattice Gas Models) and Faculty Professor (Thermodynamics, Fluid Mechanics and Electrodynamics) at the Physics Department of UNAM. Dr. Salcido has published close to a hundred scholarly articles in international journals, proceedings of conferences and book chapters, where his main research topics include micrometeorology, air pollution modeling, wind taxonomy, traffic cellular automata, and non-equilibrium thermodynamics. Recently, Dr. Salcido has edited two books on cellular automata published by In Tech.

Contents

**Preface VII**

H. Fort

**Automata Model 23**

Chapter 3 **Cellular Automata for Pattern Recognition 53**

Sartra Wongthanavasu and Jetsada Ponkaew

Chapter 5 **Cellular Learning Automata and Its Applications 85**

**Statistical Mechanics of Particle Models 113** Jeffrey Zheng, Christian Zheng and Tosiyasu Kunii

Chapter 6 **Interactive Maps on Variant Phase Spaces– From**

Chapter 4 **Using Cellular Automata and Global Sensitivity Analysis to**

Advait A. Apte, Stephen S. Fong and Ryan S. Senger

Amir Hosein Fathy Navid and Amir Bagheri Aghababa

**Study the Regulatory Network of the L-Arabinose Operon 69**

**Measurements - Micro Ensembles to Ensemble Matrices on**

Chapter 1 **Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA 1**

Chapter 2 **Validating Spatial Patterns of Urban Growth from a Cellular**

Khalid Al-Ahmadi, Linda See and Alison Heppenstall

## Contents

#### **Preface XI**



Preface

many others.

sity of complex systems.

as spatial multi-resolution validation.

The first investigations on cellular automata were carried out by John von Neumann (with some important contributions from Stanislaw Ulam) in the early 1950's. Von Neumann's pri‐ mary aim was to devise a simple dynamical system capable of reproducing itself in the manner of a living organism. In 1970, almost twenty years later, the mathematician John Horton Con‐ way invented "the game of life", a cellular automaton he devised as an attempt to drastically simplify the ideas of von Neumann. As the name suggests, the Conway´s model also has a biological aspect: cells are born, live or die depending on the local population density. The "game of life" opened up a whole new field of mathematical research, the field of cellular au‐ tomata. Cellular automata make up a very important class of completely discrete dynamical systems, and nowadays this field of research is a core subject in the sciences of complexity.

The physical environment of cellular automata is constituted of a finite-dimensional lattice, with each site having a finite number of discrete states. The evolution in time of a cellular automaton goes on in discrete steps, and its dynamics is specified by some local transition rule, fixed and definite. In spite of their conceptual simplicity, cellular automata systems are able to exhibit a wide variety of amazingly complex behavior. This feature of cellular automa‐ ta has attracted the researchers' attention from a wide range of divergent fields of physical sciences and engineering, and also from disciplines as the biological and social ones, and

In this book, six outstanding emerging cellular automata applications have been compiled. These contributions underline the versatility of cellular automata as models for a wide diver‐

In Chapter 1, it is considered the application of cellular automata to two different important problems in ecology where the space introduces important information or it simply cannot be neglected. The second problem represents a major challenge in ecology: to understand and predict the organization and spatial distribution of biodiversity using mechanistic models. Chapter 2 presents the validation process of a fuzzy cellular urban growth model, which has been applied to the city of Riyadh, Saudi Arabia. A number of different validation methods are used on three model simulations during three periods of urban growth: 1987-1997, 1997-2005 and the combined time period of 1987-2005. These methods include: visual inspec‐ tion, accuracy and spatial statistics, metrics for spatial patterns and district structures as well

Chapter 3 presents a non-uniform cellular automata-based algorithm with binary classifier, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artificial point, for pattern recognition. The model is built around the simple structure of evolving

## Preface

The first investigations on cellular automata were carried out by John von Neumann (with some important contributions from Stanislaw Ulam) in the early 1950's. Von Neumann's pri‐ mary aim was to devise a simple dynamical system capable of reproducing itself in the manner of a living organism. In 1970, almost twenty years later, the mathematician John Horton Con‐ way invented "the game of life", a cellular automaton he devised as an attempt to drastically simplify the ideas of von Neumann. As the name suggests, the Conway´s model also has a biological aspect: cells are born, live or die depending on the local population density. The "game of life" opened up a whole new field of mathematical research, the field of cellular au‐ tomata. Cellular automata make up a very important class of completely discrete dynamical systems, and nowadays this field of research is a core subject in the sciences of complexity.

The physical environment of cellular automata is constituted of a finite-dimensional lattice, with each site having a finite number of discrete states. The evolution in time of a cellular automaton goes on in discrete steps, and its dynamics is specified by some local transition rule, fixed and definite. In spite of their conceptual simplicity, cellular automata systems are able to exhibit a wide variety of amazingly complex behavior. This feature of cellular automa‐ ta has attracted the researchers' attention from a wide range of divergent fields of physical sciences and engineering, and also from disciplines as the biological and social ones, and many others.

In this book, six outstanding emerging cellular automata applications have been compiled. These contributions underline the versatility of cellular automata as models for a wide diver‐ sity of complex systems.

In Chapter 1, it is considered the application of cellular automata to two different important problems in ecology where the space introduces important information or it simply cannot be neglected. The second problem represents a major challenge in ecology: to understand and predict the organization and spatial distribution of biodiversity using mechanistic models.

Chapter 2 presents the validation process of a fuzzy cellular urban growth model, which has been applied to the city of Riyadh, Saudi Arabia. A number of different validation methods are used on three model simulations during three periods of urban growth: 1987-1997, 1997-2005 and the combined time period of 1987-2005. These methods include: visual inspec‐ tion, accuracy and spatial statistics, metrics for spatial patterns and district structures as well as spatial multi-resolution validation.

Chapter 3 presents a non-uniform cellular automata-based algorithm with binary classifier, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artificial point, for pattern recognition. The model is built around the simple structure of evolving non-uniform cellular automata called attractor basin, and classify the patterns on the basis of two-class classifier architecture similar to support vector machines.

Chapter 4 presents cellular automata models are useful tools to simulate biological systems. One of the prominent issues in cellular automata modeling is choosing correct parameter values in the absence of experimental results. This chapter addresses this issue with global sensitivity analysis. By means of "first-order" and "total effect" indices, this Monte Carlo based method explains the influence individual parameters have on the model output.

Chapter 5 proposes routing misbehavior detection in wireless sensor networks using irregular cellular learning automata. The proposed protocol uses the concept of cellular learning autom‐ ata to detect suspect nodes based on behavior and energy level of each node in packet forward‐ ing in order to achieve an energy aware intrusion detection system. Simulation results show that the system has good detection capabilities in finding malicious nodes in network.

Chapter 6 provides a brief investigation for variant phase spaces construction. Ten proposi‐ tions and four predictions are established in the chapter as the main results to provide a foundation for further exploration on quantum interpretation, statistical mechanics, complex dynamic systems and cellular automata.

We hope that, after the reading of the outstanding contributions compiled in this book, we will have succeeded in bringing across what engineers and scientists are now doing about the application of cellular automata for solving practical problems in diverse disciplines. We also hope that this book will have been to your interest and liking.

Lastly, we would like to thank all the authors for their excellent contributions in the different topics of the field of cellular automata covered in this book.

> **Alejandro Salcido** Instituto de Investigaciones Eléctricas Cuernavaca, Mexico

**Chapter 1**

**Two Cellular Automata Designed for Ecological**

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52068

state, especially in novel conditions [1].

ly reflect any particular system since they lack realism.

**1. Introduction**

H. Fort

**Problems: Mendota CA and Barro Colorado Island CA**

Our world is changing at an unprecedented rate and we need to understand the nature of the change and to make predictions about the way in which it might affect systems of inter‐ est. In order to address questions about the impact of environmental change, and to under‐ stand what might be done to mitigate the predicted negative effects, ecologists need to develop the ability to project models into novel, future conditions. This will require the de‐ velopment of models based on understanding the (dynamical) processes that result in a sys‐ tem behaving the way it does. The majority of current ecological models are excellent at describing the way in which a system has behaved, but they are poor at predicting its future

One fruitful model strategy in ecology has been the "biology-as-physics" way to approach ecosystems, i.e. setting up simple equations from which they could obtain precise answers [2]. This tradition of modelling has a long history that can be traced back to the work of the mathematical physicist Vito Volterra, who used simple differential equations to examine trends in prey and predatory fish populations in the Adriatic [3]. Such models have the ad‐ vantage of mathematical tractability; often they can be solved analytically to give precise an‐ swers and can be easily interrogated to determine the sensitivity of the model to its parameters [1]. On the other hand, it seems obvious that simple models will never accurate‐

A common simplification of these mathematical ecological models is that they rely on the well-mixed assumption or, in the physics parlance, the mean-field (MF) approximation. It is well known that the MF assumption can simplify a complex *n*-species system by replacing all interactions for any one species with the average or effective interaction strength. While the MF assumption seems reasonable in the case of, for example, plankton dynamics, it

> © 2013 Fort; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Fort; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA**

H. Fort

non-uniform cellular automata called attractor basin, and classify the patterns on the basis of

Chapter 4 presents cellular automata models are useful tools to simulate biological systems. One of the prominent issues in cellular automata modeling is choosing correct parameter values in the absence of experimental results. This chapter addresses this issue with global sensitivity analysis. By means of "first-order" and "total effect" indices, this Monte Carlo based method explains the influence individual parameters have on the model output.

Chapter 5 proposes routing misbehavior detection in wireless sensor networks using irregular cellular learning automata. The proposed protocol uses the concept of cellular learning autom‐ ata to detect suspect nodes based on behavior and energy level of each node in packet forward‐ ing in order to achieve an energy aware intrusion detection system. Simulation results show

Chapter 6 provides a brief investigation for variant phase spaces construction. Ten proposi‐ tions and four predictions are established in the chapter as the main results to provide a foundation for further exploration on quantum interpretation, statistical mechanics, complex

We hope that, after the reading of the outstanding contributions compiled in this book, we will have succeeded in bringing across what engineers and scientists are now doing about the application of cellular automata for solving practical problems in diverse disciplines. We also

Lastly, we would like to thank all the authors for their excellent contributions in the different

**Alejandro Salcido**

Cuernavaca, Mexico

Instituto de Investigaciones Eléctricas

that the system has good detection capabilities in finding malicious nodes in network.

two-class classifier architecture similar to support vector machines.

dynamic systems and cellular automata.

VIII Preface

hope that this book will have been to your interest and liking.

topics of the field of cellular automata covered in this book.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52068

#### **1. Introduction**

Our world is changing at an unprecedented rate and we need to understand the nature of the change and to make predictions about the way in which it might affect systems of inter‐ est. In order to address questions about the impact of environmental change, and to under‐ stand what might be done to mitigate the predicted negative effects, ecologists need to develop the ability to project models into novel, future conditions. This will require the de‐ velopment of models based on understanding the (dynamical) processes that result in a sys‐ tem behaving the way it does. The majority of current ecological models are excellent at describing the way in which a system has behaved, but they are poor at predicting its future state, especially in novel conditions [1].

One fruitful model strategy in ecology has been the "biology-as-physics" way to approach ecosystems, i.e. setting up simple equations from which they could obtain precise answers [2]. This tradition of modelling has a long history that can be traced back to the work of the mathematical physicist Vito Volterra, who used simple differential equations to examine trends in prey and predatory fish populations in the Adriatic [3]. Such models have the ad‐ vantage of mathematical tractability; often they can be solved analytically to give precise an‐ swers and can be easily interrogated to determine the sensitivity of the model to its parameters [1]. On the other hand, it seems obvious that simple models will never accurate‐ ly reflect any particular system since they lack realism.

A common simplification of these mathematical ecological models is that they rely on the well-mixed assumption or, in the physics parlance, the mean-field (MF) approximation. It is well known that the MF assumption can simplify a complex *n*-species system by replacing all interactions for any one species with the average or effective interaction strength. While the MF assumption seems reasonable in the case of, for example, plankton dynamics, it

seems hardly appropriate for other situations. One of such case is when there is spatial het‐ erogeneity, produced for instance by a gradient in some parameter that controls the dynam‐ ics. This is the situation in the first problem that we will analyze here. Another case if in which MF breaks down is for example for sessile species whose individuals by definition may only interact with others in a limited neighborhood provided that their niches overlap. The latter is the case for forest trees (the second system we will analyze here). Relaxing the MF assumption requires us to take into account the extent to which the strength of interac‐ tions among many species changes with the relative distances between individuals in space [4]. Many features of ecological dynamics such as the patterns of diversity and spatial distri‐ butions of species can be fundamentally changed when abandoning the MF assumption [4, 5]. A common way of relaxing the MF assumption is by formulating a spatially explicit indi‐ vidual-based model (IBM), or multi-agent system, whose straightforward implementation is by means of a cellular automaton.

dress the biodiversity distribution and spatial patterns observed in natural communities. This model, in terms of local competitive interactions between sessile species (for example

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

**2.** *Mendota* **cellular automaton: catastrophic shift in lakes and its spatial**

The *Mendota* cellular automaton (MCA) described in this section is proposed to analyze the catastrophic transition from clear to turbid water and discuss possible early warning signals to this shift. The model will be a spatial version of a mean field model introduced by Car‐ penter [16] consisting of the three differential equations for phosphorus densities in soil (*U*), in surface sediment (*M*) and in water (*P*). In fact *P* and *M*, who are attached to the lake, are local spatial variables, while *U*, which describes the surroundings of the lake is taken as a global (non-spatial) variable. The evolution equations for *U*(*t*), *P*(*x,y;t*) and *M*(*x,y;t*) are:

Parameters of the model are defined in Table 1. We have also included diffusion with a dif‐ fusion coefficient *D* = 0.1. Another modification, in order to incorporate the effect of mechan‐ ical stirring of the lake waters (wind, currents, animals) is that we consider that at each time *t*, *a*(*t*) ≡ *cU*(*t*) *a*(*x,y;t*) fluctuates locally, from point to point, around its average global value *a*(*t*) in the interval [*a*(*t*) - Δ, *a*(*t*) + Δ ] where we have taken Δ = 0.125 and have verified that

The lake is represented by a square lattice of *L*×*L* cells each one identified by its integer coor‐ dinates (*i,j*). Of course lakes of arbitrary shape could be studied by embedding them into a square lattice like the one above, with appropriate boundary conditions. Another approxi‐ mation is that the system is two-dimensional, there is no depth. That is, on each cell there are two local variables assigned: *P*(*i,j*) and *M*(*i,j*). Therefore, equations (1) to (3) lead to the following CA synchronous update rules in discrete time, where now *t* represent the time

*dP*(*x*, *y*;*t*) / *dt* =*cU* (*t*)−(*s* + *h* )*P*(*x*, *y*;*t*) + *rM* (*x*, *y*;*t*) *f* (*P*) + *DÑ* <sup>2</sup>

*dP*(*x*, *y*;*t*) / *dt* =*cU* (*t*)−(*s* + *h* )*P*(*x*, *y*;*t*) + *rM* (*x*, *y*;*t*) *f* (*P*) + *DÑ* <sup>2</sup>

the results do not depend much on this value.

*dU* (*t*) / *dt* =*W* + *F* −*H* −*cU* (*t*) (1)

*f* (*P*)=*P <sup>q</sup>* /(*P <sup>q</sup>* + *mq*) (4)

*P*(*x*, *y*;*t*) (2)

http://dx.doi.org/10.5772/52068

3

*P*(*x*, *y*;*t*) (3)

trees), requires both niche overlap and spatial proximity.

**early warnings**

where

measured in years:

**2.1. The Cellular Automaton**

Therefore, in order to go beyond MF and taking into account spatial heterogeneity, we will consider the application of cellular automata (CA) to two different important problems in Ecology where the space introduces important information or it simply cannot be neglected.

The first problem is about getting spatio-temporal early warnings of *catastrophic regime shifts* in ecosystems [6]. There is increasing evidence that ecosystems can pass thresholds and go through regime shifts where sudden and large changes in their functions take place. An ex‐ ample of such a regime shift is lakes that suddenly switch from clear to turbid water due to algae blooms. These blooms are connected to *eutrophication* i.e. the overenrichment of aquat‐ ic ecosystems with nutrients, principally phosphorous [7]. This is a widespread environmen‐ tal problem because when it occurs, many of the ecosystem services which humans derive from these systems, such as fisheries and places for recreation, can be lost. Furthermore, it is often difficult, costly and impossible to reverse these changes once a certain threshold has been crossed. This is why early warnings of these shifts are so important to ecosystem man‐ agement. In the case of MF models the rising of the temporal variance for the nutrient con‐ centration was shown that it works as an early warning signal [8]. Later on it was shown that in many cases if one takes into accounts explicitly the space the spatial variance pro‐ vides an even earlier early warning [9,10]. Thus, in section 2, I will present a cellular autom‐ aton that models a lake as a square lattice with the phosphorous concentration as the dynamical variable defined on lattice cells.

The second problem represents a major challenge in ecology: to understand and predict the organization and spatial distribution of biodiversity using mechanistic models. Ecologists have long strived to understand the distribution of relative species abundance (RSA) as well as the species–area relationships (SAR) in different communities [11, 12]. These metrics pro‐ vide critical information that together can help uncover the forces that structure and main‐ tain ecological diversity [13, 14]. Competition between species is one of the main mechanisms proposed to explain the observed RSA and SAR in different communities. In its basic form, the dynamics of competition-driven communities result from the degree to which species have overlapping niches because of their sharing of similar resource needs [15]. Hence in section 3 I will introduce a simple microscopic spatially explicit model to ad‐ dress the biodiversity distribution and spatial patterns observed in natural communities. This model, in terms of local competitive interactions between sessile species (for example trees), requires both niche overlap and spatial proximity.

#### **2.** *Mendota* **cellular automaton: catastrophic shift in lakes and its spatial early warnings**

#### **2.1. The Cellular Automaton**

The *Mendota* cellular automaton (MCA) described in this section is proposed to analyze the catastrophic transition from clear to turbid water and discuss possible early warning signals to this shift. The model will be a spatial version of a mean field model introduced by Car‐ penter [16] consisting of the three differential equations for phosphorus densities in soil (*U*), in surface sediment (*M*) and in water (*P*). In fact *P* and *M*, who are attached to the lake, are local spatial variables, while *U*, which describes the surroundings of the lake is taken as a global (non-spatial) variable. The evolution equations for *U*(*t*), *P*(*x,y;t*) and *M*(*x,y;t*) are:

$$d\mathcal{U}\{t\}/dt = \mathcal{W} + \mathcal{F} - H - c\mathcal{U}\{t\} \tag{1}$$

$$dP(\mathbf{x}\_{\prime}, y; t) / dt = c\mathcal{U}(t) - (\mathbf{s} + \mathbf{h})\mathcal{P}(\mathbf{x}\_{\prime}, y; t) + r\mathcal{M}(\mathbf{x}\_{\prime}, y; t)f(\mathcal{P}) + D\tilde{\mathcal{N}}^2\mathcal{P}(\mathbf{x}, y; t) \tag{2}$$

$$dP(\mathbf{x},\ \mathbf{y};t)/dt = c\mathcal{U}(t) - (\mathbf{s} + \mathbf{h})\mathcal{P}(\mathbf{x},\ \mathbf{y};t) + r\mathcal{M}(\mathbf{x},\ \mathbf{y};t)f(\mathcal{P}) + D\tilde{\mathcal{N}}^2\mathcal{P}(\mathbf{x},\ \mathbf{y};t) \tag{3}$$

where

seems hardly appropriate for other situations. One of such case is when there is spatial het‐ erogeneity, produced for instance by a gradient in some parameter that controls the dynam‐ ics. This is the situation in the first problem that we will analyze here. Another case if in which MF breaks down is for example for sessile species whose individuals by definition may only interact with others in a limited neighborhood provided that their niches overlap. The latter is the case for forest trees (the second system we will analyze here). Relaxing the MF assumption requires us to take into account the extent to which the strength of interac‐ tions among many species changes with the relative distances between individuals in space [4]. Many features of ecological dynamics such as the patterns of diversity and spatial distri‐ butions of species can be fundamentally changed when abandoning the MF assumption [4, 5]. A common way of relaxing the MF assumption is by formulating a spatially explicit indi‐ vidual-based model (IBM), or multi-agent system, whose straightforward implementation is

Therefore, in order to go beyond MF and taking into account spatial heterogeneity, we will consider the application of cellular automata (CA) to two different important problems in Ecology where the space introduces important information or it simply cannot be neglected.

The first problem is about getting spatio-temporal early warnings of *catastrophic regime shifts* in ecosystems [6]. There is increasing evidence that ecosystems can pass thresholds and go through regime shifts where sudden and large changes in their functions take place. An ex‐ ample of such a regime shift is lakes that suddenly switch from clear to turbid water due to algae blooms. These blooms are connected to *eutrophication* i.e. the overenrichment of aquat‐ ic ecosystems with nutrients, principally phosphorous [7]. This is a widespread environmen‐ tal problem because when it occurs, many of the ecosystem services which humans derive from these systems, such as fisheries and places for recreation, can be lost. Furthermore, it is often difficult, costly and impossible to reverse these changes once a certain threshold has been crossed. This is why early warnings of these shifts are so important to ecosystem man‐ agement. In the case of MF models the rising of the temporal variance for the nutrient con‐ centration was shown that it works as an early warning signal [8]. Later on it was shown that in many cases if one takes into accounts explicitly the space the spatial variance pro‐ vides an even earlier early warning [9,10]. Thus, in section 2, I will present a cellular autom‐ aton that models a lake as a square lattice with the phosphorous concentration as the

The second problem represents a major challenge in ecology: to understand and predict the organization and spatial distribution of biodiversity using mechanistic models. Ecologists have long strived to understand the distribution of relative species abundance (RSA) as well as the species–area relationships (SAR) in different communities [11, 12]. These metrics pro‐ vide critical information that together can help uncover the forces that structure and main‐ tain ecological diversity [13, 14]. Competition between species is one of the main mechanisms proposed to explain the observed RSA and SAR in different communities. In its basic form, the dynamics of competition-driven communities result from the degree to which species have overlapping niches because of their sharing of similar resource needs [15]. Hence in section 3 I will introduce a simple microscopic spatially explicit model to ad‐

by means of a cellular automaton.

2 Emerging Applications of Cellular Automata

dynamical variable defined on lattice cells.

$$f\left(P\right) = P^{\left(q\right)}\left(P^{\left(q\right)} + m^{\left(q\right)}\right) \tag{4}$$

Parameters of the model are defined in Table 1. We have also included diffusion with a dif‐ fusion coefficient *D* = 0.1. Another modification, in order to incorporate the effect of mechan‐ ical stirring of the lake waters (wind, currents, animals) is that we consider that at each time *t*, *a*(*t*) ≡ *cU*(*t*) *a*(*x,y;t*) fluctuates locally, from point to point, around its average global value *a*(*t*) in the interval [*a*(*t*) - Δ, *a*(*t*) + Δ ] where we have taken Δ = 0.125 and have verified that the results do not depend much on this value.

The lake is represented by a square lattice of *L*×*L* cells each one identified by its integer coor‐ dinates (*i,j*). Of course lakes of arbitrary shape could be studied by embedding them into a square lattice like the one above, with appropriate boundary conditions. Another approxi‐ mation is that the system is two-dimensional, there is no depth. That is, on each cell there are two local variables assigned: *P*(*i,j*) and *M*(*i,j*). Therefore, equations (1) to (3) lead to the following CA synchronous update rules in discrete time, where now *t* represent the time measured in years:

$$
\mathcal{L}I(t+1) = \mathcal{U}I(t) + \mathcal{W}(t) + F(t) - H(t) - c\mathcal{U}(t) \tag{5}
$$

2. The *patchiness or cluster structure*.

defined as for temporal bins of size τ.

**2.3. Simulations and Results**

policies for soil phosphorus.

*σt* 2 =

That is, four different stages are considered:

**3.** years 201-250 *W*=1.55, *F*-*H* = 31.6-18.6=13*.*

**1.** years 0-100 *W=W <sup>0</sup>*=0.147, *F*-*H =* 0*.* **2.** years 101-200 *W*=1.55, *F*-*H* = 0*.*

**4.** after year 250 *W*=1.55, *F*-*H* = 0*.*

maintained at *W* D.

3. Similarly to the mean free model, the *temporal variance σ <sup>t</sup>*

∑ *t*'=*t*−*τ*+1 *t*

*P*(0, 0;*t* ')

*P*(*x,y;t*) at a given time.

The reason leading to the rise in *σ <sup>s</sup>* is the onset of fluctuations in the spatial dependence of

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

Brock and Carpenter [8] as an early warning signal. At an arbitrary point, say (*x,y*) = (0,0), is

<sup>2</sup> −( ∑ *t*'=*t*−*τ*+1 *t*

Different simulations were run for 1500 years to illustrate changes of the system over time. The first 250 years of each simulation are a highly simplified representation of the history of Lake Mendota. The remaining years of each simulation illustrate contrasting management

Following [16], simulations were initiated at stable equilibrium values calculated with *F* = *H* = 0 and *W* for undisturbed conditions. These represent presettlement conditions, and were maintained for years 0–100. For years 100–200, *W* was changed to the value for disturbed conditions (*W*D, Table 1), representing the advent of agriculture in the region. For years 200–250, *F* and *H* were increased to the values estimated for a period of intensive industrial‐ ized agriculture in the Lake Mendota watershed, and *W* was maintained at *W* <sup>D</sup> (Table 1).

After year 250, the simulations were different. Simulation 1 represents management to bal‐ ance the phosphorus budget of agriculture. In this simulation, after year 250 *F* = *H* and *W* is

As one can see from Figure 1, the 1500 years history of lake Mendota produced by the Men‐ dota CA is similar to the Carpenter's 2005 non spatial model [16]: a gradual first shift for *P* that start at year 250 and a second more drastic shift between years 440 and 485. Notice that the *σ <sup>s</sup>* produced by the BCI CA reaches at year 425 three times the constant value it had along the first 400 years while it reaches its maximum value at year 467. The anticipation of the early warning depends on the convention chosen to establish when the shift actually happens. If we adopt the *Maxwell convention* [17] i.e. the shift coincides with the time when the variance reaches its maximum value [9]. If, on the other hand, we chose the Delay Con‐ vention the shift occurs when the system stabilizes in its final attractor i.e. at year 485 (see

*τ*

*2*

2

*P*(0, 0;*t* '))

, which has been suggested by

http://dx.doi.org/10.5772/52068

(9)

5

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( ) q q <sup>q</sup> ,; 1 ,; , ,; ,; ,; / P ,; 0.25 ( 1, ; 1, ; , 1; , 1; 4 , ; ) *P i j t P i j t a i j s h P i j t rM i j t P i j t i j t h DPi jt Pi jt Pi j t Pi j t Pi jt* += + -+ + + + - + + + -+ +- (6)

$$M\begin{pmatrix} i,j;t+1 \end{pmatrix} = M\begin{pmatrix} i,j;t \end{pmatrix} \ + sP\begin{pmatrix} i,j;t \end{pmatrix} \ -bM\begin{pmatrix} i,j;t \end{pmatrix} \ + rM\begin{pmatrix} i,j;t \end{pmatrix} P\begin{pmatrix} i,j;t \end{pmatrix}^q / \left( \mathbf{P}\begin{pmatrix} i,j;t \end{pmatrix}^q + h^q \right) \tag{7}$$


**Table 1.** Model parameters

#### **2.2. Observables**

Catastrophic shifts have characteristic fingerprints or 'wave flags'. Some of the standard

catastrophe flags are: modality (at least two well defined attractors), sudden jumps and a large or anomalous variance [17]. Therefore we will calculate the following corresponding observable quantities for the phosphorous density in water *P,* which is the relevant variable in our case.

1. The *spatial variance* of *P*(*x,y;t*), *σ <sup>s</sup> 2* , defined as

$$\sigma\_s^2 \equiv  \dots < P>^2 = \frac{\sum\_{\substack{\mathbf{x}, y = 1}}^L P\{\mathbf{x}, \ y; t\}^2 - \left(\sum\_{\substack{\mathbf{x}, y = 1}}^L P\{\mathbf{x}, \ y; t\}\right)^2}{L^{-2}} \tag{8}$$

#### 2. The *patchiness or cluster structure*.

*U* (*t* + 1)=*U* (*t*) + *W* (*t*) + *F* (*t*)−*H* (*t*)−*cU* (*t*) (5)

y\_1 y\_1 0.001 0.00115

g\_m\_2y\_1 0 or 13

g\_m\_2\_y\_1 0.147

g\_m\_2\_y\_1 1.55

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) )

*P i j t P i j t a i j s h P i j t rM i j t P i j t i j t h*

**Symbol Definition Units value**

*h* Outflow rate of P y\_1 0.15 *m* P density in the lake when recycling is 0.5 *r* g\_m\_2 2.4 *r* Maximum recycling rate of P g\_m\_2y\_1 0.019 *q* Parameter for steepness of *f*(P) near *m* Unitless 8 *s* Sedimentation rate of P y\_1 0.7

Catastrophic shifts have characteristic fingerprints or 'wave flags'. Some of the standard

*2*

, defined as

∑ *x*,*y*=1 *L*

catastrophe flags are: modality (at least two well defined attractors), sudden jumps and a large or anomalous variance [17]. Therefore we will calculate the following corresponding observable quantities for the phosphorous density in water *P,* which is the relevant variable

*P*(*x*, *y*;*t*)

<sup>2</sup> −( ∑ *x*,*y*=1 *L*

*L* <sup>2</sup>

*P*(*x*, *y*;*t*))

2

(8)

q q <sup>q</sup> ,; 1 ,; , ,; ,; ,; / P ,;

+= + -+ + + +

( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) q q <sup>q</sup> *M i j t M i j t sP i j t bM i j t rM i j t P i j t i j t h* ,; 1 ,; ,; ,; ,; ,; / P ,; += + - + <sup>+</sup> (7)


( ) ( ) ( ) ( ) ( )

0.25 ( 1, ; 1, ; , 1; , 1; 4 , ; )

Permanent burial rate of sediment P

F-H Annual agricultural import minus export of P per unit lake area to the watershed

W0 Nonagricultural inputs of P to the watershed before disturbance, per unit lake area

WD Nonagricultural inputs of P to the watershed after disturbance, per unit lake area

**Table 1.** Model parameters

**2.2. Observables**

in our case.

1. The *spatial variance* of *P*(*x,y;t*), *σ <sup>s</sup>*

*σs*

<sup>2</sup> ≡ <*P* <sup>2</sup> > − <*P*><sup>2</sup> =

P runoff coefficient

4 Emerging Applications of Cellular Automata

b c

*DPi jt Pi jt Pi j t Pi j t Pi jt*

The reason leading to the rise in *σ <sup>s</sup>* is the onset of fluctuations in the spatial dependence of *P*(*x,y;t*) at a given time.

3. Similarly to the mean free model, the *temporal variance σ <sup>t</sup> 2* , which has been suggested by Brock and Carpenter [8] as an early warning signal. At an arbitrary point, say (*x,y*) = (0,0), is defined as for temporal bins of size τ.

$$\sigma\_t^2 = \frac{\sum\_{t'=t-\tau+1}^t P(0, \ 0; t')^2 - \left(\sum\_{t'=t-\tau+1}^t P(0, \ 0; t')\right)^2}{\tau} \tag{9}$$

#### **2.3. Simulations and Results**

Different simulations were run for 1500 years to illustrate changes of the system over time. The first 250 years of each simulation are a highly simplified representation of the history of Lake Mendota. The remaining years of each simulation illustrate contrasting management policies for soil phosphorus.

Following [16], simulations were initiated at stable equilibrium values calculated with *F* = *H* = 0 and *W* for undisturbed conditions. These represent presettlement conditions, and were maintained for years 0–100. For years 100–200, *W* was changed to the value for disturbed conditions (*W*D, Table 1), representing the advent of agriculture in the region. For years 200–250, *F* and *H* were increased to the values estimated for a period of intensive industrial‐ ized agriculture in the Lake Mendota watershed, and *W* was maintained at *W* <sup>D</sup> (Table 1). That is, four different stages are considered:


After year 250, the simulations were different. Simulation 1 represents management to bal‐ ance the phosphorus budget of agriculture. In this simulation, after year 250 *F* = *H* and *W* is maintained at *W* D.

As one can see from Figure 1, the 1500 years history of lake Mendota produced by the Men‐ dota CA is similar to the Carpenter's 2005 non spatial model [16]: a gradual first shift for *P* that start at year 250 and a second more drastic shift between years 440 and 485. Notice that the *σ <sup>s</sup>* produced by the BCI CA reaches at year 425 three times the constant value it had along the first 400 years while it reaches its maximum value at year 467. The anticipation of the early warning depends on the convention chosen to establish when the shift actually happens. If we adopt the *Maxwell convention* [17] i.e. the shift coincides with the time when the variance reaches its maximum value [9]. If, on the other hand, we chose the Delay Con‐ vention the shift occurs when the system stabilizes in its final attractor i.e. at year 485 (see Figure 1). Therefore *σ <sup>s</sup>* provides an early warning of the coming shift that varies between 40 and 60 years.

number of points involved in the statistics, the sooner will be this early signal, until, for grids above 100×100 = 10,000 points, the curve doesn't change very much. For instance, for a 20×20 grid the early warning becomes noticeable 20 years later than when computed for the

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7

Figure 3 shows color maps representing different configurations when the lake moves to‐ wards a catastrophic shift from clear to turbid water. Figure 3(A) to (C)correspond, respec‐ tively, to pictures of the lake at t = 250 yrs (when still *F=H*), at t = 437 yrs (30 years before the peak in σ<sup>s</sup> ) and at t = 467 yrs (just at this peak of σs). For *t* = 467 typically there occur patches

**Figure 3.** Maps of P density (values of *P*(*x,y;t*) at each lattice cell) for a 100x100 grid. (A) *t*=250, well before the transi‐

Finally let us have a look to *σ <sup>t</sup>* and compare it versus *σ <sup>s</sup>*. Figure 4 shows both variances. The temporal variance exhibits a temporal delay of around 40 years leaving much less margin for eventual remedial management actions. This delay can be easily understood since *σ <sup>t</sup>*

entire lattice.

covering a wide scale of sizes (Figure 3(C)).

tion,. (B) *t*=437. (C): *t*=467, where σ*s* has it maximum value.

**Figure 4.** σ*s* (solid line) vs. σ *<sup>t</sup>* (dashed line).

employs data in times where the fluctuations are still small.

**Figure 1.** blue) and σs (green) vs. time (in years). The inset is a zoom showing in detail the time region around the catastrophic transition, which shows clearly the delay between both quantities.

**Figure 2.** σ*s* computed for different lattice sizes: L=5, 10, 20, 100 & 400. The larger the lattice the earliest the signal.

Concerning the practical use of *σ <sup>s</sup>* as an early warning, Figure 2 shows that the time at which the signal becomes appreciable depends on the number of points of the grid (or lat‐ tice size *L*). As expected, there is a trade-off between anticipation and cost: the larger the number of points involved in the statistics, the sooner will be this early signal, until, for grids above 100×100 = 10,000 points, the curve doesn't change very much. For instance, for a 20×20 grid the early warning becomes noticeable 20 years later than when computed for the entire lattice.

Figure 3 shows color maps representing different configurations when the lake moves to‐ wards a catastrophic shift from clear to turbid water. Figure 3(A) to (C)correspond, respec‐ tively, to pictures of the lake at t = 250 yrs (when still *F=H*), at t = 437 yrs (30 years before the peak in σ<sup>s</sup> ) and at t = 467 yrs (just at this peak of σs). For *t* = 467 typically there occur patches covering a wide scale of sizes (Figure 3(C)).

**Figure 3.** Maps of P density (values of *P*(*x,y;t*) at each lattice cell) for a 100x100 grid. (A) *t*=250, well before the transi‐ tion,. (B) *t*=437. (C): *t*=467, where σ*s* has it maximum value.

Finally let us have a look to *σ <sup>t</sup>* and compare it versus *σ <sup>s</sup>*. Figure 4 shows both variances. The temporal variance exhibits a temporal delay of around 40 years leaving much less margin for eventual remedial management actions. This delay can be easily understood since *σ <sup>t</sup>* employs data in times where the fluctuations are still small.

**Figure 4.** σ*s* (solid line) vs. σ *<sup>t</sup>* (dashed line).

Figure 1). Therefore *σ <sup>s</sup>* provides an early warning of the coming shift that varies between 40

**Figure 1.** blue) and σs (green) vs. time (in years). The inset is a zoom showing in detail the time region around the

**Figure 2.** σ*s* computed for different lattice sizes: L=5, 10, 20, 100 & 400. The larger the lattice the earliest the signal.

Concerning the practical use of *σ <sup>s</sup>* as an early warning, Figure 2 shows that the time at which the signal becomes appreciable depends on the number of points of the grid (or lat‐ tice size *L*). As expected, there is a trade-off between anticipation and cost: the larger the

catastrophic transition, which shows clearly the delay between both quantities.

and 60 years.

6 Emerging Applications of Cellular Automata

#### **2.4. Discussion and some Remarks**

We have proposed a CA that describes the evolution of the P concentration, which domi‐ nates the eutrophication process in lakes, with its parameters calibrated for Lake Mendota. We have considered different possible early warnings for eutrophication shifts in lakes. In the case of *σ <sup>s</sup>*, by measuring samples of *P* on a grid of points on the lake surface, it was found that a grid containing few points might be sufficient for the purpose of extracting an appropriate signal, and that a significant growth in *σ <sup>s</sup>* could serve as an early warning of an imminent transition. The spatial variance appears to have an advantage over the temporal one, as *σ <sup>t</sup>* is delayed with respect to *σ <sup>s</sup>*.

It is worth to remark that the quantitative details of our conclusions depend on the choice of parameter values employed in our model. We have verified that the qualitative behavior of our results do not depend strongly on those values. Furthermore our model as presented in this work is schematic in the sense that the quality of the lake's water is dependent of a sin‐ gle parameter, the amount of Phosphorous in solution. In real cases other environmental factors might be playing an important role in the evolution, and the model and its predic‐ tions could be much more complicated. Nevertheless it appears that our main conclusions should hold in the more realistic case: spatial variance of critical quantities is an earlier sig‐ nal than the temporal variance, and its associated cluster structure of the patterns formed in the eutrophication process could be the fastest detectable warning that a catastrophic change

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

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9

**3.** *Barro colorado island* **cellular automaton: species competition in**

The model combines features from both niche-based models of interspecific competition [20]

The CA considered in this section is based on the Lotka–Volterra competition equations for *N* sp species [20]: *dns/*d*t* =*g <sup>s</sup> n <sup>s</sup>*(1 *− <sup>r</sup> α sr n <sup>r</sup>*), where *s* and *r* run from 1 to *N*sp, *n <sup>s</sup>* is the density of species *s*; *g <sup>s</sup>* its maximum per capita growth rate, and the coefficients *α sr* represent the competitive interaction of species *r* over species *s*. Estimating the interaction strengths among species is, however, far from trivial because of the large number of potential interac‐ tions in diverse natural communities, feedback effects and non-linearities in interactionstrength functions [22]. In fact the Lotka*-Volterra Competition Model* (LVCM) include *N* sp×(*N* sp-1) competition coefficients *α sr*. In the case of a tropical forest, where there are typically some hundreds of different coexisting species, estimating the corresponding several thou‐ sand coefficients becomes definitely an impossible task. Thus community ecologists have re‐ sorted to different approaches to include these diverse interaction strengths into their dynamics. Following the commonly used MacArthur and Levins' approach [23,20], we con‐ sider the simplest one-dimensional finite niche wherein the resource utilization function of

each species *s* can be represented by a normal distribution *P <sup>s</sup>*(*x*) = exp[*−*(*x − μ <sup>s</sup>*)

mean *μ <sup>s</sup>* and a standard deviation *σ <sup>s</sup>* that measures the niche width. Then for each pair of species *s* and *r*, the strength of its competition is determined by their niche overlap, i.e. the overlapping between *P <sup>s</sup>*(*x*) and *P <sup>r</sup>*(*x*) (see Figure 5). We call this model the *Niche Lotka-Vol‐ terra Competition Model* (NLVCM). Assuming a finite normalized niche (*x ∈* [0, 1]), the com‐

<sup>2</sup> */*2*σ <sup>s</sup>* 2 ], with

**3.1. Niche Lotka-Volterra Competition Cellular Automaton**

and from the neutral theory of biodiversity [21].

petition coefficients *α sr* can thus be computed as:

is about to occur.

**physical and niche spaces**

When studying the origin of the rise in *σ <sup>s</sup>*, we found that it is connected to the appearance of spatial patterns, in the form of clusters of clear and turbid water. The spatial patchiness un‐ der routine conditions is quite different from spatial patchiness near a regime shift. This is because in the last case in some places the clear water state is realized while in others the turbid water state occurs. In mathematical terms, close to the regime shift there are two com‐ peting attractors or alternative states (clear and turbid water) one occurs in a given set of cells and the other in the set of remaining cells. This explains the spatial fluctuations in the concentration of P. Under routine conditions, either before or after the catastrophic shift, on‐ ly one attractor remains.

We then conclude that the visualization of the onset of those patches, for example by aerial or satellite imaging of the lake surface, may be an effective way of anticipating an eutrophi‐ cation transition.

However a note of caution is required, in thermally stratified real lakes two main layers can be distinguished. The hypolimnion is the dense, bottom layer of water in thermally-strati‐ fied lakes, which is isolated from surface wind-mixing. The epilimnion is top-most layer, oc‐ curring above the deeper hypolimnion and being exposed at the surface, it typically becomes turbulently mixed as a result of surface wind-mixing. On the one hand, the flow of phosphorus in recycling occurs first from sediment to and also from decomposition in the hypolimnion. Dissolved P in the hypolimnion tends to be rather well-mixed and homogene‐ ous. Then vertical mixing of hypolimnetic and epilimnetic water occurs from time to time (roughly 10-15 day return time [18]. Horizontal dispersion of dissolved P is fast after a verti‐ cal mixing event, within a day or so, smoothing over the patchiness structure in the epilimn‐ ion. As a consequence patches could never survive during a year, which is the unit of time in our model. On the other hand, some lakes show important spatial differences in terms of total phosphorus related to the spatial distribution of submerged plants or cyanobacteria blooms. The biomass tends to accumulate in several zones, by the hydrodynamic and wind actions, and determine strong differences of algal biomass as well as total phosphorous [19]. In any case, since the spatial model we are analyzing is two-dimensional and there is no depth (the hypolimnion and the epilimnion are collapsed to a single layer), the above proc‐ esses are not included. Indeed it seems that the patches produced by the spatial model, in‐ stead of long lasting structures, should be interpreted as frequent popping in and out clusters. This dynamic patchy pattern changes many times during the course of one year.

It is worth to remark that the quantitative details of our conclusions depend on the choice of parameter values employed in our model. We have verified that the qualitative behavior of our results do not depend strongly on those values. Furthermore our model as presented in this work is schematic in the sense that the quality of the lake's water is dependent of a sin‐ gle parameter, the amount of Phosphorous in solution. In real cases other environmental factors might be playing an important role in the evolution, and the model and its predic‐ tions could be much more complicated. Nevertheless it appears that our main conclusions should hold in the more realistic case: spatial variance of critical quantities is an earlier sig‐ nal than the temporal variance, and its associated cluster structure of the patterns formed in the eutrophication process could be the fastest detectable warning that a catastrophic change is about to occur.

### **3.** *Barro colorado island* **cellular automaton: species competition in physical and niche spaces**

#### **3.1. Niche Lotka-Volterra Competition Cellular Automaton**

**2.4. Discussion and some Remarks**

8 Emerging Applications of Cellular Automata

is delayed with respect to *σ <sup>s</sup>*.

one, as *σ <sup>t</sup>*

ly one attractor remains.

cation transition.

We have proposed a CA that describes the evolution of the P concentration, which domi‐ nates the eutrophication process in lakes, with its parameters calibrated for Lake Mendota. We have considered different possible early warnings for eutrophication shifts in lakes. In the case of *σ <sup>s</sup>*, by measuring samples of *P* on a grid of points on the lake surface, it was found that a grid containing few points might be sufficient for the purpose of extracting an appropriate signal, and that a significant growth in *σ <sup>s</sup>* could serve as an early warning of an imminent transition. The spatial variance appears to have an advantage over the temporal

When studying the origin of the rise in *σ <sup>s</sup>*, we found that it is connected to the appearance of spatial patterns, in the form of clusters of clear and turbid water. The spatial patchiness un‐ der routine conditions is quite different from spatial patchiness near a regime shift. This is because in the last case in some places the clear water state is realized while in others the turbid water state occurs. In mathematical terms, close to the regime shift there are two com‐ peting attractors or alternative states (clear and turbid water) one occurs in a given set of cells and the other in the set of remaining cells. This explains the spatial fluctuations in the concentration of P. Under routine conditions, either before or after the catastrophic shift, on‐

We then conclude that the visualization of the onset of those patches, for example by aerial or satellite imaging of the lake surface, may be an effective way of anticipating an eutrophi‐

However a note of caution is required, in thermally stratified real lakes two main layers can be distinguished. The hypolimnion is the dense, bottom layer of water in thermally-strati‐ fied lakes, which is isolated from surface wind-mixing. The epilimnion is top-most layer, oc‐ curring above the deeper hypolimnion and being exposed at the surface, it typically becomes turbulently mixed as a result of surface wind-mixing. On the one hand, the flow of phosphorus in recycling occurs first from sediment to and also from decomposition in the hypolimnion. Dissolved P in the hypolimnion tends to be rather well-mixed and homogene‐ ous. Then vertical mixing of hypolimnetic and epilimnetic water occurs from time to time (roughly 10-15 day return time [18]. Horizontal dispersion of dissolved P is fast after a verti‐ cal mixing event, within a day or so, smoothing over the patchiness structure in the epilimn‐ ion. As a consequence patches could never survive during a year, which is the unit of time in our model. On the other hand, some lakes show important spatial differences in terms of total phosphorus related to the spatial distribution of submerged plants or cyanobacteria blooms. The biomass tends to accumulate in several zones, by the hydrodynamic and wind actions, and determine strong differences of algal biomass as well as total phosphorous [19]. In any case, since the spatial model we are analyzing is two-dimensional and there is no depth (the hypolimnion and the epilimnion are collapsed to a single layer), the above proc‐ esses are not included. Indeed it seems that the patches produced by the spatial model, in‐ stead of long lasting structures, should be interpreted as frequent popping in and out clusters. This dynamic patchy pattern changes many times during the course of one year.

The model combines features from both niche-based models of interspecific competition [20] and from the neutral theory of biodiversity [21].

The CA considered in this section is based on the Lotka–Volterra competition equations for *N* sp species [20]: *dns/*d*t* =*g <sup>s</sup> n <sup>s</sup>*(1 *− <sup>r</sup> α sr n <sup>r</sup>*), where *s* and *r* run from 1 to *N*sp, *n <sup>s</sup>* is the density of species *s*; *g <sup>s</sup>* its maximum per capita growth rate, and the coefficients *α sr* represent the competitive interaction of species *r* over species *s*. Estimating the interaction strengths among species is, however, far from trivial because of the large number of potential interac‐ tions in diverse natural communities, feedback effects and non-linearities in interactionstrength functions [22]. In fact the Lotka*-Volterra Competition Model* (LVCM) include *N* sp×(*N* sp-1) competition coefficients *α sr*. In the case of a tropical forest, where there are typically some hundreds of different coexisting species, estimating the corresponding several thou‐ sand coefficients becomes definitely an impossible task. Thus community ecologists have re‐ sorted to different approaches to include these diverse interaction strengths into their dynamics. Following the commonly used MacArthur and Levins' approach [23,20], we con‐ sider the simplest one-dimensional finite niche wherein the resource utilization function of each species *s* can be represented by a normal distribution *P <sup>s</sup>*(*x*) = exp[*−*(*x − μ <sup>s</sup>*) <sup>2</sup> */*2*σ <sup>s</sup>* 2 ], with mean *μ <sup>s</sup>* and a standard deviation *σ <sup>s</sup>* that measures the niche width. Then for each pair of species *s* and *r*, the strength of its competition is determined by their niche overlap, i.e. the overlapping between *P <sup>s</sup>*(*x*) and *P <sup>r</sup>*(*x*) (see Figure 5). We call this model the *Niche Lotka-Vol‐ terra Competition Model* (NLVCM). Assuming a finite normalized niche (*x ∈* [0, 1]), the com‐ petition coefficients *α sr* can thus be computed as:

*.*

$$\alpha sr = \frac{\int\_{\mathbb{P}\_s} \mathbb{P}\_s(\mathbf{x}) P\_r(\mathbf{x}) d\mathbf{x}}{(1 \; \bigg|\; 2) \left[ \int\_{\mathbb{P}\_s} \mathbb{P}\_s^2(\mathbf{x}) d\mathbf{x} + \oint\_{\mathbb{P}} \mathbb{P}\_r^2(\mathbf{x}) d\mathbf{x} \right]} \tag{10}$$

The model is defined as follows:

of initially coexisting species.

\* Landscape. We work on a *L*× *L* = *N* square lattice with periodic boundary conditions to avoid border effects. Each lattice cell (i,j) is always occupied by one individual belonging to a certain species specified by its niche position *μ <sup>s</sup>*(i,j), s =1,2,...,*N* sp, where *N* sp is the number

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\* Initial conditions. Because the initial spatial distribution of individuals at BCI is of course unknowable, we run the model from 100 different initial spatial configurations of complete‐

\* Dynamics and update rule. The dynamics is asynchronous. At each time step *t* a focal cell is chosen at random. The LVC dynamics is implemented by using the "copy the fittest" rule i.e. by replacing each focal individual by the individual having the maximum fitness in the Moore neighborhood (the focal individual and its eight surrounding neighbors), which

In this subsection we will contrast the model against field data from a well-studied ecologi‐ cal community, the tropical forest at Barro Colorado Island (BCI), Panama [25,26,27]. Popu‐ lation structure and spatial patterns have been particularly important themes in the ecology of tropical trees [21]. The long-term research program coordinated by the Smithsonian Cen‐ ter for Tropical Forest Science [28] includes inventory data collected for the first time in 1982 and then every 5 years from 1985 to 2005, with measurements of all trees with diameter at

We set the initial number of species *N*sp = 320. This is the maximum number of species found for BCI in the six censuses for 1982 and 1985, although this number steadily declined afterward from census to census until in 2005 the number of species recorded was 283 [29]. The number of individuals was kept fixed as *N* = 250,000 (*L* = 500) that is close to the maxi‐ mum number of individuals observed in the set of BCI censuses. Therefore, the only free model parameters to be fitted are *σ*, controlling the intensity of interspecies competition, and

This second parameter is needed because we are interested in transient states rather than in the steady state. The reason for this is simple: the set of censuses for tropical forests collected by the Center for Tropical Forest Science reveal that they are non-stationary systems: species richness decreases from one census to the next in *all* tropical forests and in many cases at a considerable pace [28]. Just to cite some examples: at BCI (Panama), roughly 3% of the num‐ ber of species have been disappearing every 5 years; for Bukit Timah (Singapore), this per‐ centage varies between 3% and 8% between consecutive censuses; for Edoro (Congo) 12% of the species disappeared between the 1994 census and the 2000 census, etc. The dynamics of

ly random assignation of species (i.e. corresponding to the maximum entropy).

eventually can be itself and thus no change occurs at the focal node.

**3.2. Comparison with empirical data from tropical forests**

breast height (dbh) *>* 1 cm for a plot of 50 ha in BCI [29].

*3.3. Fitting parameters and biodiversity indices*

the number of steps of simulation *t*.

We will compute in addition of RSA and SAR, aggregation properties.

**Figure 5.** Competition and niche overlap: the gray area common to the normal distributions for species *s* and *r,* cen‐ tered respectively at μ *<sup>s</sup>* = 0.4 and μ *<sup>r</sup>* = 0.6, is equal to the value of the competition coefficient between them α *sr .*

On the other hand, inspired by neutral theory and to keep things as simple as possible, we assume two important simplifications. Firstly, that all species have the same maximum growth rate i.e. *g <sup>s</sup>* = 1 and the same niche width i.e. *σ <sup>s</sup>* = *σ*. The robustness of this simplifica‐ tion was recently analyzed for the NLVCM [24]. Therefore, each species' growth rate de‐ pends both on its position in the niche axis and on the niche positions of the individuals of other species that make its competitive neighborhood at each time step. The (intrinsic) func‐ tional equivalence between species is consistent with the chosen normalization for the *α rs* in (Fig. 5) that ensures that the matrix *α* is symmetric and allows it to be expressed as:

$$\alpha\_{rs} = e^{-(\left(\mu\_r - \mu\_s\right)/2\sigma)^2} \frac{\text{erf}(\left(2-\mu\_r - \mu\_s\right)/2\sigma) + \text{erf}(\left(\mu\_r + \mu\_s\right)/2\sigma)}{\text{erf}(\left(1-\mu\_r\right)/\sigma) + \text{erf}(\left(\mu\_r\right)/\sigma) + \text{erf}(\left(1-\mu\_s\right)/\sigma)\text{erf}(\left(\mu\_s\right)/\sigma)}\tag{11}$$

where 'erf' denotes the standard error rate function. Secondly, we consider that the com‐ munity has a constant, finite size *N* [21] such that local reproductive and mortality events are tightly synchronized (see the used update rule below); this is equivalent to the 'zero-sum assumption' of Hubbell's neutral model of biodiversity [21].

The model is defined as follows:

*αsr* =

10 Emerging Applications of Cellular Automata

*.*

*αrs* =*e*

−((*μr*−*μs*)72*σ*)

assumption' of Hubbell's neutral model of biodiversity [21].

*∫* 0

1 *Ps* 2 (*x*)*dx* +*∫* 0

(1 / 2)(*∫* 0 *Ps*(*x*)*Pr*(*x*)*dx*

**Figure 5.** Competition and niche overlap: the gray area common to the normal distributions for species *s* and *r,* cen‐ tered respectively at μ *<sup>s</sup>* = 0.4 and μ *<sup>r</sup>* = 0.6, is equal to the value of the competition coefficient between them α *sr .*

On the other hand, inspired by neutral theory and to keep things as simple as possible, we assume two important simplifications. Firstly, that all species have the same maximum growth rate i.e. *g <sup>s</sup>* = 1 and the same niche width i.e. *σ <sup>s</sup>* = *σ*. The robustness of this simplifica‐ tion was recently analyzed for the NLVCM [24]. Therefore, each species' growth rate de‐ pends both on its position in the niche axis and on the niche positions of the individuals of other species that make its competitive neighborhood at each time step. The (intrinsic) func‐ tional equivalence between species is consistent with the chosen normalization for the *α rs* in

(Fig. 5) that ensures that the matrix *α* is symmetric and allows it to be expressed as:

<sup>2</sup> erf((2−*μr* −*μs*) / 2*σ*) + erf((*μr* + *μs*) / 2*σ*)

where 'erf' denotes the standard error rate function. Secondly, we consider that the com‐ munity has a constant, finite size *N* [21] such that local reproductive and mortality events are tightly synchronized (see the used update rule below); this is equivalent to the 'zero-sum

erf((1−*μr*) / *<sup>σ</sup>*) + erf((*μr*) / *<sup>σ</sup>*) + erf((1−*μs*) / *<sup>σ</sup>*)erf((*μs*) / *<sup>σ</sup>*) (11)

1 *Pr* 2

(*x*)*dx*) (10)

1

\* Landscape. We work on a *L*× *L* = *N* square lattice with periodic boundary conditions to avoid border effects. Each lattice cell (i,j) is always occupied by one individual belonging to a certain species specified by its niche position *μ <sup>s</sup>*(i,j), s =1,2,...,*N* sp, where *N* sp is the number of initially coexisting species.

\* Initial conditions. Because the initial spatial distribution of individuals at BCI is of course unknowable, we run the model from 100 different initial spatial configurations of complete‐ ly random assignation of species (i.e. corresponding to the maximum entropy).

\* Dynamics and update rule. The dynamics is asynchronous. At each time step *t* a focal cell is chosen at random. The LVC dynamics is implemented by using the "copy the fittest" rule i.e. by replacing each focal individual by the individual having the maximum fitness in the Moore neighborhood (the focal individual and its eight surrounding neighbors), which eventually can be itself and thus no change occurs at the focal node.

We will compute in addition of RSA and SAR, aggregation properties.

#### **3.2. Comparison with empirical data from tropical forests**

In this subsection we will contrast the model against field data from a well-studied ecologi‐ cal community, the tropical forest at Barro Colorado Island (BCI), Panama [25,26,27]. Popu‐ lation structure and spatial patterns have been particularly important themes in the ecology of tropical trees [21]. The long-term research program coordinated by the Smithsonian Cen‐ ter for Tropical Forest Science [28] includes inventory data collected for the first time in 1982 and then every 5 years from 1985 to 2005, with measurements of all trees with diameter at breast height (dbh) *>* 1 cm for a plot of 50 ha in BCI [29].

#### *3.3. Fitting parameters and biodiversity indices*

We set the initial number of species *N*sp = 320. This is the maximum number of species found for BCI in the six censuses for 1982 and 1985, although this number steadily declined afterward from census to census until in 2005 the number of species recorded was 283 [29]. The number of individuals was kept fixed as *N* = 250,000 (*L* = 500) that is close to the maxi‐ mum number of individuals observed in the set of BCI censuses. Therefore, the only free model parameters to be fitted are *σ*, controlling the intensity of interspecies competition, and the number of steps of simulation *t*.

This second parameter is needed because we are interested in transient states rather than in the steady state. The reason for this is simple: the set of censuses for tropical forests collected by the Center for Tropical Forest Science reveal that they are non-stationary systems: species richness decreases from one census to the next in *all* tropical forests and in many cases at a considerable pace [28]. Just to cite some examples: at BCI (Panama), roughly 3% of the num‐ ber of species have been disappearing every 5 years; for Bukit Timah (Singapore), this per‐ centage varies between 3% and 8% between consecutive censuses; for Edoro (Congo) 12% of the species disappeared between the 1994 census and the 2000 census, etc. The dynamics of the RSA between censuses in each forest are also far from being steady. It is noteworthy that most analyses of the BCI (and of other tropical forests) largely involve fitting separately sev‐ eral metrics of community structure and organization (e.g. SAR, RSA, indices of diversity) for each census rather than considering them as a dynamic sequence whose attributes needs be fitted together [25], [30],[31].

Moreover, the BCI CA reaches a steady state with four species after a long transient of the order of 3 billion time steps (data not shown). This a known result for MF [32] that was ana‐ lyzed in detail in [24][33]. It is of course impossible to decide whether this steady state could be a reasonable result in a more or less distant future, and it is indeed absurd to run any ecological model for such a long time horizon without including the effect of speciation at this very long temporal scale. Rather the BCI CA aims to describe the transient correspond‐ ing to an entire set of censuses (rather than viewing them as separate snapshots as is gener‐ ally done), which requires specifying the simulation time *t* as a parameter.

The decrease (although moderate) of Shannon entropy provides another way to verify the non-equilibrium nature of the BCI forest and this negative entropy production corresponds to the loss of species and the changes in community structure over time (see below).

We searched for the combination of *σ* and the number of time steps that produce the best agreement between the theoretical and the empirically observed biodiversity for the entire set of six BCI censuses. In addition to RSA we used two well known and often used indices of diversity, the Shannon entropy *S* and the Simpson index 1- *D* [34], to describe community structure for each census. Normalized versions of these indices are given by

$$\mathcal{S} = -\frac{1}{\text{InN}\_{sp}} \sum\_{s=1}^{N\_{sp}} \langle f\_s \ln f\_s \rangle\_{\prime} \tag{12}$$

empirical one as *σ* approaches 0.09. The Simpson diversity index 1 *− D*1 = 0*.*949 observed for

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

http://dx.doi.org/10.5772/52068

13

Second, for *σ* = 0*.*09, we look for the best fit to the entire sequence of six BCI censuses. We found that the optimal fit occurs for *t* <sup>1</sup> = 37 million of time steps (Mts) for the first 1982 cen‐ sus and then there is a correspondence of 1 year *←→* 250 000 time steps. Therefore, the cen‐ sus years 1982, 1985, 1990, 1995, 2000 and 2005 correspond, respectively, to *t* = 37, 37.75, 39, 40.25, 41.5 and 42.75 Mts. The Barro Colorado cellular automaton (BCI CA) is then defined

Figure 6 shows the close agreement between the *S* and 1 *− D* calculated for BCI and the pre‐ dicted average values obtained for 100 simulations each starting from different initial condi‐ tions, for the sequence of six censuses. This figure also shows the MF results, i.e. numerical integration of LVC equations, indicating that this MF cannot simultaneously and accurately

**Figure 6.** *S* and 1 − D: the six BCI censuses and averages produced by BCI CA with error bars corresponding to 1 stand‐

In addition, the RSA curves for the MF approximation showed a poorer agreement with the

As in most biological communities, most trees species on BCI have few individuals: only five species *Hybanthus prunifolius*, *Faramea occidentalis*, *Trichilia tuberculata*, *Desmopsis pana‐ mensis* and *Alseis blackiana* individually had more than 5% of the total community size.

Figure 7 compares the predicted and observed dominance–diversity curves for the BCI 1995 and 2005 censuses. In both cases the predicted curve slightly underestimates the abundance (notice the logarithmic vertical scale) of rare species, i.e. those representing a RSA smaller than 0.05% which is a percentage well below 0.35% and 0.31%, the average values of RSA at BCI for 1995 and 2005, respectively (horizontal lines in figure 7). It turns out that for values

of *σ* departing from 0.09 this agreement found for common species becomes worse.

the first census served to narrow even further the range of values of *σ* around 0.09.

by *L* = 500, *N*sp = 320, *σ* = 0*.*09 and *t* 1 = 37 Mts.

predict the values of both biodiversity indices.

ard deviation (SD) and those of MF (dashed lines) for σ = 0*.*09 (see text).

empirical ones for each census (not shown).

*3.4. Dominance–diversity curves*

and

$$\mathbf{1} \cdot \mathbf{D} = \mathbf{1} \cdot \sum\_{s=1}^{N\_{\text{eff}}} f\_s \, ^{2} \mathbf{1} \tag{13}$$

where *fs* is the relative abundance of species *s*. For both indices, the greater the value, the larger the sample diversity.

The procedure to find the optimal *σ* and *t* is as follows:

First, for a given value of *σ*, each simulation is run until the entropy *S* becomes equal to the entropy corresponding to the BCI population distribution for the first census (1982), i.e. *S*1 = 0*.*694. It turns out that for most values of *σ* except for a narrow set of values centered around *σ* = 0*.*09, the RSA are clearly different from those of BCI in 1982. A useful representation in ecology to perform such comparison between theoretical and empirical RSA is provided by the commonly used *dominance–diversity curves*. These curves are obtained by ranking the species according to their population abundance and plotting the RSA (%) versus species rank. Specifically, we found that the theoretical dominance–diversity curve converges to the empirical one as *σ* approaches 0.09. The Simpson diversity index 1 *− D*1 = 0*.*949 observed for the first census served to narrow even further the range of values of *σ* around 0.09.

Second, for *σ* = 0*.*09, we look for the best fit to the entire sequence of six BCI censuses. We found that the optimal fit occurs for *t* <sup>1</sup> = 37 million of time steps (Mts) for the first 1982 cen‐ sus and then there is a correspondence of 1 year *←→* 250 000 time steps. Therefore, the cen‐ sus years 1982, 1985, 1990, 1995, 2000 and 2005 correspond, respectively, to *t* = 37, 37.75, 39, 40.25, 41.5 and 42.75 Mts. The Barro Colorado cellular automaton (BCI CA) is then defined by *L* = 500, *N*sp = 320, *σ* = 0*.*09 and *t* 1 = 37 Mts.

Figure 6 shows the close agreement between the *S* and 1 *− D* calculated for BCI and the pre‐ dicted average values obtained for 100 simulations each starting from different initial condi‐ tions, for the sequence of six censuses. This figure also shows the MF results, i.e. numerical integration of LVC equations, indicating that this MF cannot simultaneously and accurately predict the values of both biodiversity indices.

**Figure 6.** *S* and 1 − D: the six BCI censuses and averages produced by BCI CA with error bars corresponding to 1 stand‐ ard deviation (SD) and those of MF (dashed lines) for σ = 0*.*09 (see text).

In addition, the RSA curves for the MF approximation showed a poorer agreement with the empirical ones for each census (not shown).

#### *3.4. Dominance–diversity curves*

the RSA between censuses in each forest are also far from being steady. It is noteworthy that most analyses of the BCI (and of other tropical forests) largely involve fitting separately sev‐ eral metrics of community structure and organization (e.g. SAR, RSA, indices of diversity) for each census rather than considering them as a dynamic sequence whose attributes needs

Moreover, the BCI CA reaches a steady state with four species after a long transient of the order of 3 billion time steps (data not shown). This a known result for MF [32] that was ana‐ lyzed in detail in [24][33]. It is of course impossible to decide whether this steady state could be a reasonable result in a more or less distant future, and it is indeed absurd to run any ecological model for such a long time horizon without including the effect of speciation at this very long temporal scale. Rather the BCI CA aims to describe the transient correspond‐ ing to an entire set of censuses (rather than viewing them as separate snapshots as is gener‐

The decrease (although moderate) of Shannon entropy provides another way to verify the non-equilibrium nature of the BCI forest and this negative entropy production corresponds

We searched for the combination of *σ* and the number of time steps that produce the best agreement between the theoretical and the empirically observed biodiversity for the entire set of six BCI censuses. In addition to RSA we used two well known and often used indices of diversity, the Shannon entropy *S* and the Simpson index 1- *D* [34], to describe community

( *f <sup>s</sup>*ln *f <sup>s</sup>*), (12)

´2, (13)

to the loss of species and the changes in community structure over time (see below).

ally done), which requires specifying the simulation time *t* as a parameter.

structure for each census. Normalized versions of these indices are given by

1 - *D* =1- ∑

*s*=1 *Nsp f s*

where *fs* is the relative abundance of species *s*. For both indices, the greater the value, the

First, for a given value of *σ*, each simulation is run until the entropy *S* becomes equal to the entropy corresponding to the BCI population distribution for the first census (1982), i.e. *S*1 = 0*.*694. It turns out that for most values of *σ* except for a narrow set of values centered around *σ* = 0*.*09, the RSA are clearly different from those of BCI in 1982. A useful representation in ecology to perform such comparison between theoretical and empirical RSA is provided by the commonly used *dominance–diversity curves*. These curves are obtained by ranking the species according to their population abundance and plotting the RSA (%) versus species rank. Specifically, we found that the theoretical dominance–diversity curve converges to the

*<sup>S</sup>* = - <sup>1</sup> ln*Nsp* ∑ *s*=1 *Nsp*

be fitted together [25], [30],[31].

12 Emerging Applications of Cellular Automata

and

larger the sample diversity.

The procedure to find the optimal *σ* and *t* is as follows:

As in most biological communities, most trees species on BCI have few individuals: only five species *Hybanthus prunifolius*, *Faramea occidentalis*, *Trichilia tuberculata*, *Desmopsis pana‐ mensis* and *Alseis blackiana* individually had more than 5% of the total community size.

Figure 7 compares the predicted and observed dominance–diversity curves for the BCI 1995 and 2005 censuses. In both cases the predicted curve slightly underestimates the abundance (notice the logarithmic vertical scale) of rare species, i.e. those representing a RSA smaller than 0.05% which is a percentage well below 0.35% and 0.31%, the average values of RSA at BCI for 1995 and 2005, respectively (horizontal lines in figure 7). It turns out that for values of *σ* departing from 0.09 this agreement found for common species becomes worse.

ing species with similar niches even when space is introduced only implicitly in a MF model [35]. Furthermore, space is known to be important in the formation and maintenance of sta‐ ble vegetation patterns [36,37]. Thus, the local competitive interactions might actually rein‐

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

http://dx.doi.org/10.5772/52068

15

**Figure 8.** Theoretical and empirical distributions of abundant species: A) A patch 100*×*100 produced by the BCI CA showing the spatial distribution for three species with larger values of μ*s* (i.e. located at the right end of the niche axis).B) A comparable patch, extracted from the entire 500m*×*1000m BCI plot as recorded in the 1982 census, showing the locations of three species: *Faramea occidentalis* (green), *Trichilia tuberculata* (red) and *Alseis blackiana* (blue).

There are many options to quantify whether there is spatial segregation among these domi‐ nant species. One of them is to partition a square patch *L <sup>P</sup>* of side multiple of 3 (e.g. *L <sup>P</sup>* = 99) into 3×3 Moore neighborhoods and measure the fraction *f <sup>c</sup>* of these (*L <sup>P</sup>*/3)2 in which there is coexistence of species (at least two of the three species are present). Then by repeating the same procedure for exactly the same number of individuals belonging to each of these three species but *randomly* distributed, and calculating the fraction *fr <sup>c</sup>*, the segregation index γ can

When γ < 1 (>1) there is (there is not) spatial segregation among the species. We found γ = 0.41, confirming then that these three dominant species, whose niches are tightly packed at

The SAR (average number of species in a sampled area) constitute one of the main metrics in spatial community ecology [14], [21]. To generate SAR, we proceed as in [34] by dividing the entire area into non-overlapping squares and rectangles and counting the number of species present in quadrats of 20 *×* 20, 25 *×* 20, 25 *×* 25, 50 *×* 25, 50 *×* 50, 100*×*50, 100*×*100, 250*×*100, 250*×*250 and 500*×*500. This procedure yields the mean number of species in each size quad‐ rat that make up the SAR for each year. Figure 9 shows that after 39 Mts, the model produ‐ ces a SAR that fits quite well to the one obtained from the 1990 BCI census [34] for large area plots (above 1 ha). However, as the area of plots decreases, the agreement with data wor‐

the right end of the niche axis, are actually spatially segregated.

*γ* ≡ *f <sup>c</sup>* / *f rc* (14)

be constructed as:

force the *emergent neutrality* found in non-spatial competition models [38].

**Figure 7.** Dominance–diversity curves: BCI 1995 and 2005 censuses (*×*) and BCI CA predictions with error bars corre‐ sponding to 1 SD (black).The dot-dashed horizontal line corresponds to the average BCI RSA (%).

The most abundant species that yield the BCI CA are those at both ends of the niche axis, i.e. *μ <sup>s</sup>* 0 or 1. The explanation is quite simple: species at the ends of the niche axis are exposed to less competition than those which depart from the ends and experiment competition from the two sides instead of from only one.

#### *3.5. Spatial patterns*

A prediction of BCI CA model is that local interactions in physical space introduce an in‐ teresting fact: individuals of different species close in niche space become spatially segregat‐ ed. This is shown in Figure 8-A which corresponds to a 100 × 100 patch of the entire lattice depicting the spatial distribution of the three most abundant species at the right end of the niche axis (i.e. all with *μ <sup>s</sup>* close to one) after 40 Mts. In order to contrast this against empirical data, Figure 8-B reproduces the spatial distribution found in 1982 for three spe‐ cies of quite similar trees with comparable population sizes, *Faramea occidentalis*, *Trichilia tu‐ berculata* and *Alseis blackiana,* in a 141 m × 141 m 1 patch of the BCI 50 ha plot [25,26,27]. Notice that these three species also show spatial segregation, although the empirical segre‐ gation seems to be lower than the theoretical one. A simple explanation for this is that the BCI CA is little too simplistic, only local replacement of species is taking into account. In‐ deed we have checked that by introducing a global migration parameter *m,* modeling dis‐ persal of seeds by winds or birds, the agreement between theoretical and empirical segregation patterns considerably improves.

The message is then that, while individuals of these three species could potentially have strong competitive interactions because of similarities of their niches, their spatial segrega‐ tion attenuates the strength of their interspecific competitive interactions whenever these are local. Spatial segregation is known to be a mechanism that allows the coexistence of compet‐

<sup>1</sup> Since we use a sqaure 500 ( 500 lattice CA to represent the rectangular 500 m ( 1000m BCI plot, it turns that the lattice spacing corresponds to (2 ( 1.41 m.

ing species with similar niches even when space is introduced only implicitly in a MF model [35]. Furthermore, space is known to be important in the formation and maintenance of sta‐ ble vegetation patterns [36,37]. Thus, the local competitive interactions might actually rein‐ force the *emergent neutrality* found in non-spatial competition models [38].

**Figure 7.** Dominance–diversity curves: BCI 1995 and 2005 censuses (*×*) and BCI CA predictions with error bars corre‐

The most abundant species that yield the BCI CA are those at both ends of the niche axis, i.e. *μ <sup>s</sup>* 0 or 1. The explanation is quite simple: species at the ends of the niche axis are exposed to less competition than those which depart from the ends and experiment competition from

A prediction of BCI CA model is that local interactions in physical space introduce an in‐ teresting fact: individuals of different species close in niche space become spatially segregat‐ ed. This is shown in Figure 8-A which corresponds to a 100 × 100 patch of the entire lattice depicting the spatial distribution of the three most abundant species at the right end of the niche axis (i.e. all with *μ <sup>s</sup>* close to one) after 40 Mts. In order to contrast this against empirical data, Figure 8-B reproduces the spatial distribution found in 1982 for three spe‐ cies of quite similar trees with comparable population sizes, *Faramea occidentalis*, *Trichilia tu‐ berculata* and *Alseis blackiana,* in a 141 m × 141 m 1 patch of the BCI 50 ha plot [25,26,27]. Notice that these three species also show spatial segregation, although the empirical segre‐ gation seems to be lower than the theoretical one. A simple explanation for this is that the BCI CA is little too simplistic, only local replacement of species is taking into account. In‐ deed we have checked that by introducing a global migration parameter *m,* modeling dis‐ persal of seeds by winds or birds, the agreement between theoretical and empirical segregation

The message is then that, while individuals of these three species could potentially have strong competitive interactions because of similarities of their niches, their spatial segrega‐ tion attenuates the strength of their interspecific competitive interactions whenever these are local. Spatial segregation is known to be a mechanism that allows the coexistence of compet‐

1 Since we use a sqaure 500 ( 500 lattice CA to represent the rectangular 500 m ( 1000m BCI plot, it turns that the

sponding to 1 SD (black).The dot-dashed horizontal line corresponds to the average BCI RSA (%).

the two sides instead of from only one.

14 Emerging Applications of Cellular Automata

patterns considerably improves.

lattice spacing corresponds to (2 ( 1.41 m.

*3.5. Spatial patterns*

**Figure 8.** Theoretical and empirical distributions of abundant species: A) A patch 100*×*100 produced by the BCI CA showing the spatial distribution for three species with larger values of μ*s* (i.e. located at the right end of the niche axis).B) A comparable patch, extracted from the entire 500m*×*1000m BCI plot as recorded in the 1982 census, showing the locations of three species: *Faramea occidentalis* (green), *Trichilia tuberculata* (red) and *Alseis blackiana* (blue).

There are many options to quantify whether there is spatial segregation among these domi‐ nant species. One of them is to partition a square patch *L <sup>P</sup>* of side multiple of 3 (e.g. *L <sup>P</sup>* = 99) into 3×3 Moore neighborhoods and measure the fraction *f <sup>c</sup>* of these (*L <sup>P</sup>*/3)2 in which there is coexistence of species (at least two of the three species are present). Then by repeating the same procedure for exactly the same number of individuals belonging to each of these three species but *randomly* distributed, and calculating the fraction *fr <sup>c</sup>*, the segregation index γ can be constructed as:

$$
\gamma = f\_c \mid f \; r\_c \tag{14}
$$

When γ < 1 (>1) there is (there is not) spatial segregation among the species. We found γ = 0.41, confirming then that these three dominant species, whose niches are tightly packed at the right end of the niche axis, are actually spatially segregated.

The SAR (average number of species in a sampled area) constitute one of the main metrics in spatial community ecology [14], [21]. To generate SAR, we proceed as in [34] by dividing the entire area into non-overlapping squares and rectangles and counting the number of species present in quadrats of 20 *×* 20, 25 *×* 20, 25 *×* 25, 50 *×* 25, 50 *×* 50, 100*×*50, 100*×*100, 250*×*100, 250*×*250 and 500*×*500. This procedure yields the mean number of species in each size quad‐ rat that make up the SAR for each year. Figure 9 shows that after 39 Mts, the model produ‐ ces a SAR that fits quite well to the one obtained from the 1990 BCI census [34] for large area plots (above 1 ha). However, as the area of plots decreases, the agreement with data wor‐ sens. This is expected, because we know that our model underestimates the abundance of the scarcest species as shown in the dominance-diversity curves (figure 7) whose presence becomes more and more significant as the plot sizes become smaller than in larger plots that typically contain more individuals of many different species. Another useful spatial metric is provided by species–individuals curves, which are obtained exactly as the SAR but using the mean number of individuals in quadrats of each area instead of sampled area in the hor‐ izontal axis. Again, there is a qualitatively good agreement between model predictions and field data (not shown here).

munity biodiversity, respectively. Both are fundamental scientific questions that go beyond ecology and require a major interdisciplinary effort and would have a significant impact on ecosystem management and conservation. The two CA represent minimal spatially explicit ecological models, that could be called "physicist's models" since they include the minimal

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

http://dx.doi.org/10.5772/52068

17

First, let us consider some guidelines for future work concerning the problem of eutrophica‐ tion of lakes and the more general problem of the development of early warning signals of catastrophic shifts. The eutrophication is a result of a combination of natural processes and human impacts [39]. CA like Mendota could serve to disentangle the effect of these natural processes, like climate change [40], from anthropic effects. This could be done using paleo‐ limnological analysis to estimate and connect historic input and run-off rates with tempera‐ ture and humidity indices [41]. Another interesting extension of this CA could be from 2D to 3D, to take into account the depth, in order to describe the flow of phosphorus in recycling from sediment and also from decomposition in the hypolimnion. Going beyond lakes, the analysis of spatially explicit models is relevant, for example, to understand phenomena like clumping and spatial segregation in plant communities [42]. It was shown that vegetation patches, which have been extensively studied for arid lands, can be approached as a pattern formation phenomenon [43]-[44]. It has been hypothesized that this vegetation patchiness could be used as a signature of imminent catastrophic shifts between alternative states [36]. Evidences that the patch-size distribution of vegetation follows a power law were later found in arid Mediterranean ecosystems [37]. This implies that vegetation patches were present over a wide range of size scales, thus displaying scale invariance. It was also found that with increasing grazing pressure, the field data revealed deviations from power laws. Hence, the authors proposed that this power law behavior may be a warning signal for the onset of desertification. These spatial early warnings complement temporal ones like the

Second, in the case of tropical forests, these are extremely diverse ecosystems -many of which contain in 50 ha sample plots more tree species as occur in all of US and Canada com‐ bined- for which a huge volumes of data (covering 47 permanent plots in 21 Countries with 4.5 million trees of 8,500 species [28]) has been accumulated throughout the last thirty years. This mega diversity, together with the abundance of available data, makes tropical forests a paradigm for research on the interdisciplinary field of complex systems dynamics. The BCI CA parameters could be easily calibrated for modelling other tropical forests different from BCI and contrast with empiric data. Future natural extensions of this CA that are worth con‐ sidering are the inclusion of a migration rate, due to animals or wind propagating seeds be‐ yond local neighborhoods, and noise. These two factors would add a dose of stochasticity, making the model more realistic. Considering to other realms, beyond ecology, the type of competition of BCI CA could be applied to understand other natural as well as non-natural communities. For example in the case of social and economic systems [45], communities formed by different kinds of interacting firms [46] -for example stores, banks, restaurants, etc- wherein territorial competition between heterogeneous individuals (i.e. occupying dif‐ ferent niche positions) occurring in distinct local neighborhoods is a key factor controlling

number of adjustable parameters.

variance of time series.

**Figure 9.** SAR: model for σ = 0*.*09 with error bars corresponding to 2 SD and data from BCI 1990 census (*×*).

#### **3.6. On the importance of the locality of the competition and possible extensions to other realms.**

The BCI CA is a model of local competition that jointly considers organisms in both physical and niche space. It must be viewed as a minimalistic model of local competition, with basi‐ cally only two parameters: *σ*, the species' niche width controlling the intensity of the inter‐ specific competition that drives the spatiotemporal dynamics, and the simulation time *t* (since it is focused on transients rather than in the equilibrium state). This 'microscopic' model of local competition can predict with reasonable accuracy the dynamic sequence of patterns of community structure, species-packing and the spatial distribution of forest spe‐ cies. Interestingly, the model shows that when the well-mixing assumption underlying the MF approximation breaks down, species that are clumped in niche space appear spatially segregated. Including space is well known to allow the long-term coexistence of strongly competing species and to permit the formation of stable patterns [5],[35],[37].

#### **4. Conclusions**

Each one of the two CA presented here address an open problem in ecology, namely how to anticipate catastrophic shifts in ecosystems and understanding the forces that shape com‐ munity biodiversity, respectively. Both are fundamental scientific questions that go beyond ecology and require a major interdisciplinary effort and would have a significant impact on ecosystem management and conservation. The two CA represent minimal spatially explicit ecological models, that could be called "physicist's models" since they include the minimal number of adjustable parameters.

sens. This is expected, because we know that our model underestimates the abundance of the scarcest species as shown in the dominance-diversity curves (figure 7) whose presence becomes more and more significant as the plot sizes become smaller than in larger plots that typically contain more individuals of many different species. Another useful spatial metric is provided by species–individuals curves, which are obtained exactly as the SAR but using the mean number of individuals in quadrats of each area instead of sampled area in the hor‐ izontal axis. Again, there is a qualitatively good agreement between model predictions and

**Figure 9.** SAR: model for σ = 0*.*09 with error bars corresponding to 2 SD and data from BCI 1990 census (*×*).

competing species and to permit the formation of stable patterns [5],[35],[37].

**3.6. On the importance of the locality of the competition and possible extensions to other**

The BCI CA is a model of local competition that jointly considers organisms in both physical and niche space. It must be viewed as a minimalistic model of local competition, with basi‐ cally only two parameters: *σ*, the species' niche width controlling the intensity of the inter‐ specific competition that drives the spatiotemporal dynamics, and the simulation time *t* (since it is focused on transients rather than in the equilibrium state). This 'microscopic' model of local competition can predict with reasonable accuracy the dynamic sequence of patterns of community structure, species-packing and the spatial distribution of forest spe‐ cies. Interestingly, the model shows that when the well-mixing assumption underlying the MF approximation breaks down, species that are clumped in niche space appear spatially segregated. Including space is well known to allow the long-term coexistence of strongly

Each one of the two CA presented here address an open problem in ecology, namely how to anticipate catastrophic shifts in ecosystems and understanding the forces that shape com‐

field data (not shown here).

16 Emerging Applications of Cellular Automata

**realms.**

**4. Conclusions**

First, let us consider some guidelines for future work concerning the problem of eutrophica‐ tion of lakes and the more general problem of the development of early warning signals of catastrophic shifts. The eutrophication is a result of a combination of natural processes and human impacts [39]. CA like Mendota could serve to disentangle the effect of these natural processes, like climate change [40], from anthropic effects. This could be done using paleo‐ limnological analysis to estimate and connect historic input and run-off rates with tempera‐ ture and humidity indices [41]. Another interesting extension of this CA could be from 2D to 3D, to take into account the depth, in order to describe the flow of phosphorus in recycling from sediment and also from decomposition in the hypolimnion. Going beyond lakes, the analysis of spatially explicit models is relevant, for example, to understand phenomena like clumping and spatial segregation in plant communities [42]. It was shown that vegetation patches, which have been extensively studied for arid lands, can be approached as a pattern formation phenomenon [43]-[44]. It has been hypothesized that this vegetation patchiness could be used as a signature of imminent catastrophic shifts between alternative states [36]. Evidences that the patch-size distribution of vegetation follows a power law were later found in arid Mediterranean ecosystems [37]. This implies that vegetation patches were present over a wide range of size scales, thus displaying scale invariance. It was also found that with increasing grazing pressure, the field data revealed deviations from power laws. Hence, the authors proposed that this power law behavior may be a warning signal for the onset of desertification. These spatial early warnings complement temporal ones like the variance of time series.

Second, in the case of tropical forests, these are extremely diverse ecosystems -many of which contain in 50 ha sample plots more tree species as occur in all of US and Canada com‐ bined- for which a huge volumes of data (covering 47 permanent plots in 21 Countries with 4.5 million trees of 8,500 species [28]) has been accumulated throughout the last thirty years. This mega diversity, together with the abundance of available data, makes tropical forests a paradigm for research on the interdisciplinary field of complex systems dynamics. The BCI CA parameters could be easily calibrated for modelling other tropical forests different from BCI and contrast with empiric data. Future natural extensions of this CA that are worth con‐ sidering are the inclusion of a migration rate, due to animals or wind propagating seeds be‐ yond local neighborhoods, and noise. These two factors would add a dose of stochasticity, making the model more realistic. Considering to other realms, beyond ecology, the type of competition of BCI CA could be applied to understand other natural as well as non-natural communities. For example in the case of social and economic systems [45], communities formed by different kinds of interacting firms [46] -for example stores, banks, restaurants, etc- wherein territorial competition between heterogeneous individuals (i.e. occupying dif‐ ferent niche positions) occurring in distinct local neighborhoods is a key factor controlling their dynamics. Of particular interest would be study the role of space in the winner-takesall markets [47], in which in principle slight differences in performance of the firms can lead to enormous differences in reward. Long familiar in sports and entertainment, this payoff pattern has increasingly permeated law, finance, fashion, publishing, and other fields [48].

[9] Fernández, A., & Fort, H. (2009). Catastrophic phase transitions and early warnings in a spatial ecological model. *Jour. Stat. Mech.*, 09014, http://iopscience.iop.org/

Two Cellular Automata Designed for Ecological Problems: Mendota CA and Barro Colorado Island CA

http://dx.doi.org/10.5772/52068

19

[10] Donangelo, R., Fort, H., Dakos, V., Scheffer, M. E. H., & van Nes, E. (2010). Early warnings of catastrophic shifts in ecosystems: Comparison between spatial and tem‐

[11] Preston, F.W. (1948). The Commonness, And Rarity, of Species. *Ecology*, 29, 254-283.

[12] Gaston, K. J., & Blackburn, T. M. (2000). *Pattern and Process in Macroecology.*, Oxford,

[14] Harte, J., et al. (2005). A theory of Spatial-Abundance and Species-Abundance Distri‐ butions in Ecological Communities at Multiple Spatial Scales. *Ecol. Monogr.*, 75-179.

[15] Begon, M., Townsend, C., & Harper, J. (2006). *Ecology: From Individuals to Ecosystems*,

[16] Carpenter, S. (2005). Eutrophication of aquatic ecosystems: Bistability and soil phos‐

[17] Gilmore, R. (1981). *Catastrophe Theory for Scientists and Engineers.*, New York, Dover.

[20] May, R.M. (1974). *Stability and Complexity in Model Ecosystems.*, Princeton, NJ, Prince‐

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[24] Fort, H., Scheffer, M., & van Nes, E. (2010). Biodiversity patterns from an individualbased competition model on niche and physical spaces. *Jour. Stat. Mech.*, 05005,

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[26] Hubbell, S. P., Condit, R., & Foster, R. B. (2005). Barro Colorado Forest Census Plot

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ton University Press.

#### **Acknowledgements**

I thank financial support from ANII (SNI), Uruguay. I am indebted to Steven Carpenter and Néstor Mazzeo for discussion on the material presented in section 2 and to Pablo Inchausti since the material of section 3 is based on a recent paper we wrote [49].

#### **Author details**

H. Fort\*

Complex Systems and Statistical Physics Group Instituto de Física, Facultad de Ciencias, Universidad de la República, Iguá 4225, 11400 Montevideo, Uruguay

#### **References**


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I thank financial support from ANII (SNI), Uruguay. I am indebted to Steven Carpenter and Néstor Mazzeo for discussion on the material presented in section 2 and to Pablo Inchausti

Complex Systems and Statistical Physics Group Instituto de Física, Facultad de Ciencias,

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**Chapter 2**

**Validating Spatial Patterns of Urban Growth from a**

The dynamics of urban growth are the direct consequence of the actions of individuals, and public and private organisations, which act to change the urban landscape simultaneously over space and time. Since previous urban form has a strong influence on the present, a prime concern of urban planners, spatial scientists and government authorities is to under‐ stand how a city has grown in the past in order to predict the growth of the city in the fu‐ ture. This requires flexible tools that allow planners to examine the impacts and potential consequences of applying different development policies, strategies and future plans [1]. However, traditional linear, static and top-down models are unable to adequately capture the processes underlying urban change. The non-linearity of spatial and temporal relation‐ ships and irregular, uncoordinated and uncontrolled local decision-making gives rise to seemingly coordinated global patterns that define the size and shape of cities in familiar ways [2-7]. Cities are now increasingly recognized as complex systems and display many of the characteristic traits of complexity, i.e. non-linearity, self-organization and emergence. Cellular Automata (CA) offer a modeling framework and a set of techniques for modelling the dynamic processes and outcomes of self-organizing systems [8-13]. Since the late 1980s they have demonstrated significant potential benefits for urban modelling through their simplicity, flexibility and transparency [8, 14-17]. CA are capable of generating complex pat‐ terns in aggregate form by using relatively simple local transition rules, i.e. by recursive de‐ velopment decisions being made at individual cells or sites [2, 15, 18]. However, cities are also influenced by global factors representing government polices (such as broader social, economic and technological factors). This has led to a number of hybrid-type urban growth models, which take into consideration local, regional and global factors [19-22]. When inte‐ grated with other technologies such as GIS and remote sensing, the potential of CA for geo‐

> © 2013 Al-Ahmadi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Al-Ahmadi et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Cellular Automata Model**

http://dx.doi.org/10.5772/51708

**1. Introduction**

Khalid Al-Ahmadi, Linda See and Alison Heppenstall

Additional information is available at the end of the chapter

## **Validating Spatial Patterns of Urban Growth from a Cellular Automata Model**

Khalid Al-Ahmadi, Linda See and Alison Heppenstall

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51708

#### **1. Introduction**

The dynamics of urban growth are the direct consequence of the actions of individuals, and public and private organisations, which act to change the urban landscape simultaneously over space and time. Since previous urban form has a strong influence on the present, a prime concern of urban planners, spatial scientists and government authorities is to under‐ stand how a city has grown in the past in order to predict the growth of the city in the fu‐ ture. This requires flexible tools that allow planners to examine the impacts and potential consequences of applying different development policies, strategies and future plans [1]. However, traditional linear, static and top-down models are unable to adequately capture the processes underlying urban change. The non-linearity of spatial and temporal relation‐ ships and irregular, uncoordinated and uncontrolled local decision-making gives rise to seemingly coordinated global patterns that define the size and shape of cities in familiar ways [2-7]. Cities are now increasingly recognized as complex systems and display many of the characteristic traits of complexity, i.e. non-linearity, self-organization and emergence. Cellular Automata (CA) offer a modeling framework and a set of techniques for modelling the dynamic processes and outcomes of self-organizing systems [8-13]. Since the late 1980s they have demonstrated significant potential benefits for urban modelling through their simplicity, flexibility and transparency [8, 14-17]. CA are capable of generating complex pat‐ terns in aggregate form by using relatively simple local transition rules, i.e. by recursive de‐ velopment decisions being made at individual cells or sites [2, 15, 18]. However, cities are also influenced by global factors representing government polices (such as broader social, economic and technological factors). This has led to a number of hybrid-type urban growth models, which take into consideration local, regional and global factors [19-22]. When inte‐ grated with other technologies such as GIS and remote sensing, the potential of CA for geo‐

© 2013 Al-Ahmadi et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Al-Ahmadi et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

spatial, temporal and sectoral studies increases significantly through the ability of CA to utilise physical, environmental, social and economic data in their simulations [23]. For ex‐ ample, remote sensing and GIS can be integrated with CA for providing detailed land use information as well as information on other characteristics of cities to produce realistic simu‐ lations of urban change [24].

inspection, accuracy and spatial statistics, metrics for spatial pattern and district structure detection as well as spatial multi-resolution validation. Results of these methods are given followed by a discussion of the usefulness of the different validation approaches in relation

**2. The Fuzzy Cellular Urban Growth Model (FCUGM) for the City of**

The Fuzzy Cellular Automata Urban Growth Model (FCUGM) is driven by the following

is greater than or equal to a transition threshold parameter, *λ*, which is determined through

and a stochastic disturbance factor. The development suitability is, in turn, a function of four

*<sup>t</sup>* , *TCF ij*

The four driving forces of urban growth (TSF, UAAF, TCF and PRF) are themselves func‐

*<sup>t</sup>* , *AMRij*

*<sup>t</sup>* , *AECSESij*

*<sup>t</sup>* , *EAij*

where the TSF is a function of Accessibility to Local Roads (ALR), Accessibility to Main Roads (AMR) and Accessibility to Major Roads (AMJR); the UAAF is determined by a combination of Urban Density (UD), Accessibility to Town Centres (ATC) and Accessibili‐ ty to Employment Centres and Socio-Economic Services (AECSES); the TCF is a function of

*<sup>t</sup>* <sup>=</sup> *<sup>f</sup>* (*Gij t* , *Aij*

*<sup>t</sup>* <sup>=</sup> *<sup>f</sup>* (*PAij*

*<sup>t</sup>* , *PPRF ij*

*<sup>t</sup>* , *AMJRij*

*<sup>t</sup>* , *ATCij*

*t*

*t*

*<sup>t</sup>* , *UAAF ij*

the calibration process. The *DP* is a function of the development suitability (*DSij*

*<sup>t</sup>*+1 =Urban, Otherwise=Non-Urban (1)

Validating Spatial Patterns of Urban Growth from a Cellular Automata Model

) , urban agglomeration and attractiveness

) and a factor that encompasses planning policies

*t*

http://dx.doi.org/10.5772/51708

25

*<sup>t</sup>* ) (2)

*<sup>t</sup>* ) (3)

*<sup>t</sup>* ) (4)

*<sup>t</sup>* ) (5)

*<sup>t</sup>* ) (6)

) of a cell

*<sup>t</sup>*+1 , is created at time *t*+1 if the cell's development possibility (*DP*)

to the assessment of the FCUGM.

simple rule of development:

where a new urban cell, *Sij*

and regulations (*PPRFij*

If *DPij*

driving forces, i.e. transportation (*TSFij*

) , topographical constraints (*TSFij*

*t* ) :

*DSij*

*<sup>t</sup>* <sup>=</sup> *<sup>f</sup>* (*TSF ij*

tions of a series of fuzzy input variables expressed as follows:

*<sup>t</sup>* <sup>=</sup> *<sup>f</sup>* (*ALRij*

*<sup>t</sup>* <sup>=</sup> *<sup>f</sup>* (*UDij*

*TCF ij*

*PPRF ij*

*TSF ij*

*UAAF ij*

*<sup>t</sup>* <sup>≥</sup>*<sup>λ</sup>* Then *Sij*

**Riyadh**

(*UAAFij t*

A current challenge facing CA urban growth models is the lack of rigorous calibration pro‐ cedures [21, 25-27]. Progress in the evolution of algorithms, particularly from artificial intel‐ ligence (AI), has, however, created many new options for calibrating these complex models. For example, [28] suggested that heuristic-based searches using AI would be an effective ap‐ proach for optimising spatial problems, since they offer many advantages for model calibra‐ tion compared to traditional methods. An example of an urban growth CA model calibrated using AI was developed in [29-31]. They presented an urban planning tool for the city of Riyadh, Saudi Arabia, which is one of the world's major cities undergoing rapid develop‐ ment. At the core of the system is a Fuzzy Cellular Urban Growth Model (FCUGM), which is capable of simulating and predicting the complexities of urban growth. This model was shown to be capable of replicating the trends and characteristics of an urban environment during three periods: 1987-1997, 1997-2005 and 1987-2005.

Along with calibration, one of the most significant aspects of any model is to verify, validate and assess its performance. This is normally undertaken by verifying the model's output against the real-world system through evaluation of goodness-of-fit tests. Validation can be defined as 'a demonstration that a model within its domain of applicability possesses a satis‐ factory range of accuracy consistent with the intended application of the model' [32]. In terms of urban CA models, the validation process refers to the approach by which the per‐ formance of the model is assessed by comparing the simulated map (one generated by the model) with the observed map (based on ground truth). The observed map should be accu‐ rate and shape the benchmark for comparison. A good performing urban CA model gener‐ ates outcomes that capture the basic features of urban forms between simulated and observed spatial patterns [1]. Researchers have utilised a combination of different methods for validating CA models. For example, in [33], thirty-three urban CA models were re‐ viewed and compared using a number of different criteria including the types of validation method employed. In some cases no validation method was used since the models were largely hypothetical or idealized, while in other models, a range of different methods were employed including one or a combination of the following approaches: visual comparison, confusion matrices [21], statistical measures [18], a fractal index and analysis [8, 21, 34], landscape metrics [35], spatial statistics, for example, Moran's I index [8, 25] and structural measurements such as the Lee-Sallee index [25]. It it clear from the review [33], however, that there is no consensus on how CA models of urban growth should be validated and re‐ search in this area has not progressed that much [26-27].

The focus of this chapter is on the techniques used to validate the performance of the FCUGM model; however these approaches are applicable to urban CA models more gener‐ ally. A brief overview of the fuzzy cellular urban growth model (FCUGM) for the city of Riyadh is first provided. We then present seven different validation metrics including visual inspection, accuracy and spatial statistics, metrics for spatial pattern and district structure detection as well as spatial multi-resolution validation. Results of these methods are given followed by a discussion of the usefulness of the different validation approaches in relation to the assessment of the FCUGM.

spatial, temporal and sectoral studies increases significantly through the ability of CA to utilise physical, environmental, social and economic data in their simulations [23]. For ex‐ ample, remote sensing and GIS can be integrated with CA for providing detailed land use information as well as information on other characteristics of cities to produce realistic simu‐

A current challenge facing CA urban growth models is the lack of rigorous calibration pro‐ cedures [21, 25-27]. Progress in the evolution of algorithms, particularly from artificial intel‐ ligence (AI), has, however, created many new options for calibrating these complex models. For example, [28] suggested that heuristic-based searches using AI would be an effective ap‐ proach for optimising spatial problems, since they offer many advantages for model calibra‐ tion compared to traditional methods. An example of an urban growth CA model calibrated using AI was developed in [29-31]. They presented an urban planning tool for the city of Riyadh, Saudi Arabia, which is one of the world's major cities undergoing rapid develop‐ ment. At the core of the system is a Fuzzy Cellular Urban Growth Model (FCUGM), which is capable of simulating and predicting the complexities of urban growth. This model was shown to be capable of replicating the trends and characteristics of an urban environment

Along with calibration, one of the most significant aspects of any model is to verify, validate and assess its performance. This is normally undertaken by verifying the model's output against the real-world system through evaluation of goodness-of-fit tests. Validation can be defined as 'a demonstration that a model within its domain of applicability possesses a satis‐ factory range of accuracy consistent with the intended application of the model' [32]. In terms of urban CA models, the validation process refers to the approach by which the per‐ formance of the model is assessed by comparing the simulated map (one generated by the model) with the observed map (based on ground truth). The observed map should be accu‐ rate and shape the benchmark for comparison. A good performing urban CA model gener‐ ates outcomes that capture the basic features of urban forms between simulated and observed spatial patterns [1]. Researchers have utilised a combination of different methods for validating CA models. For example, in [33], thirty-three urban CA models were re‐ viewed and compared using a number of different criteria including the types of validation method employed. In some cases no validation method was used since the models were largely hypothetical or idealized, while in other models, a range of different methods were employed including one or a combination of the following approaches: visual comparison, confusion matrices [21], statistical measures [18], a fractal index and analysis [8, 21, 34], landscape metrics [35], spatial statistics, for example, Moran's I index [8, 25] and structural measurements such as the Lee-Sallee index [25]. It it clear from the review [33], however, that there is no consensus on how CA models of urban growth should be validated and re‐

The focus of this chapter is on the techniques used to validate the performance of the FCUGM model; however these approaches are applicable to urban CA models more gener‐ ally. A brief overview of the fuzzy cellular urban growth model (FCUGM) for the city of Riyadh is first provided. We then present seven different validation metrics including visual

lations of urban change [24].

24 Emerging Applications of Cellular Automata

during three periods: 1987-1997, 1997-2005 and 1987-2005.

search in this area has not progressed that much [26-27].

### **2. The Fuzzy Cellular Urban Growth Model (FCUGM) for the City of Riyadh**

The Fuzzy Cellular Automata Urban Growth Model (FCUGM) is driven by the following simple rule of development:

$$\text{If } DP\_{\vec{i}\vec{j}} \ge \lambda \text{ Then } \mathbb{S}\_{\vec{i}\vec{j}}^{t+1} = \text{Urban}, \text{ Otherwise= Non-Urban} \tag{1}$$

where a new urban cell, *Sij <sup>t</sup>*+1 , is created at time *t*+1 if the cell's development possibility (*DP*) is greater than or equal to a transition threshold parameter, *λ*, which is determined through the calibration process. The *DP* is a function of the development suitability (*DSij t* ) of a cell and a stochastic disturbance factor. The development suitability is, in turn, a function of four driving forces, i.e. transportation (*TSFij t* ) , urban agglomeration and attractiveness (*UAAFij t* ) , topographical constraints (*TSFij t* ) and a factor that encompasses planning policies and regulations (*PPRFij t* ) :

$$\text{DS}^t\_{\vec{\text{ij}}} = f\left(\text{TSF}^t\_{\vec{\text{ij}}\prime} \mid \text{UAA}\text{F}^t\_{\vec{\text{ij}}\prime} \mid \text{TCF}^t\_{\vec{\text{ij}}\prime} \mid \text{PPRF}^t\_{\vec{\text{ij}}\prime}\right) \tag{2}$$

The four driving forces of urban growth (TSF, UAAF, TCF and PRF) are themselves func‐ tions of a series of fuzzy input variables expressed as follows:

$$TSF^{t}\_{\stackrel{\scriptstyle \!\!/}{\!\!/}} = f\left(ALR^{t}\_{\stackrel{\scriptstyle \!\!/}{\!\!/}}\\_AMR^{t}\_{\stackrel{\scriptstyle \!\!/}{\!\!/}}\\_AMJ\mathcal{R}^{t}\_{\stackrel{\scriptstyle \!\!/}{\!\!/}}\right) \tag{3}$$

$$\text{ULAAF}^{t}\_{\vec{\text{ij}}} = f\left(\text{UD}^{t}\_{\vec{\text{ij}}\prime} \mid A \text{ECSES}^{t}\_{\vec{\text{ij}}\prime} \mid A \text{TC}^{t}\_{\vec{\text{ij}}\prime}\right) \tag{4}$$

$$T\mathbb{C}F\_{i\bar{j}}^t = f\left(\mathbb{G}\_{i\bar{j}}^t, \ A\_{i\bar{j}}^t\right) \tag{5}$$

$$\text{APPRF}^{t}\_{\,\,\,ij} = f\left(\text{PA}^{t}\_{\,\,ij\prime} \to \text{A}^{t}\_{\,\,ij}\right) \tag{6}$$

where the TSF is a function of Accessibility to Local Roads (ALR), Accessibility to Main Roads (AMR) and Accessibility to Major Roads (AMJR); the UAAF is determined by a combination of Urban Density (UD), Accessibility to Town Centres (ATC) and Accessibili‐ ty to Employment Centres and Socio-Economic Services (AECSES); the TCF is a function of Gradient (G) and Altitude (A); and the PPRF takes Planned Areas (PA) and Excluded Areas (EA) into account. These drivers of urban growth are integrated via a fuzzy rule base, where the membership functions and the rules are determined through calibration. A fuzzy infer‐ ence engine is used to process the fuzzy rules and produce a fuzzy development suitabili‐ ty score at each cell. These fuzzy values are then defuzzified and used in combination with the stochastic disturbance factor and the transition threshold to determine whether a giv‐ en cell becomes an area of further urban development. The full details of the model are provided in [29, 31].

and UGB I+II (1987–2005) using the calibrated weights and parameters derived from the GA. Figures 1 to 3 show the simulations for the three time periods respectively for the three top performing simulations, i.e. M1-S4, M2-S4 and M3-S1. The new urban developments that are simulated by the model are shown in red while blue cells indicate those areas that have al‐ ready been developed. For UGB I (1987-1997), simulation M1-S4 shows more compact urban patterns compared with the other two simulations (M2-S4 and M3-S1), where the latter show more urban development across the peripheral areas, in particular for M3-S1. This might be attributed to the high weight assigned to the urban density variable for M1-S4 and to the form of the distance decay effect captured through the membership functions. How‐ ever, the morphology of the simulated urban spatial structure that is located to the north and north east shows quite some dispersed and scattered development. Generally, develop‐ ment sites are more linked in order to provide necessary infrastructure and service facilities. However, dispersed development is one of the characteristics of Riyadh's urban pattern. Typically, urban sprawl is produced by the three simulations regardless of the overall mac‐ roscopic pattern. This sprawl might be attributed to a lack of implementation of a policy to limit urban growth, which the government introduced to prevent chaotic development. In addition, this sprawl mimics the non-continuous or leap-frog pattern of urban growth char‐

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Figures 1 to 3 also show that the direction of growth is generally radial, where urban growth takes place around most of the already developed lands. In particular, most of the growth is to the south west and to the east of the city, while only moderate growth is simulated in the top south eastern part. Growth also rarely occurs to the west of the city. The pattern of growth might be a result of the government's free grant program. Most of the lands in these two areas were granted by the government to households with low incomes. Another rea‐ son may be the lower price of this land compared with the high price of land located to the north of the city. Moreover, moderate growth in the south east of the city could be due to the concentration of heavy industry in this part of the city and to the low urban environ‐ mental quality due to proximity to industrial zones and the oil refinery. It can also be seen that there is almost no urban growth simulated to the west of the city, where areas are either steep or located at higher altitudes, indicating that topographical constraint factors have confined growth in such areas. Topographical characteristics have also constrained growth in the south western part of the city, where the steep areas located between the two big ur‐

In UGB II (1997-2005), the simulated urban pattern contrasts with that shown in UGB I (1987-1997) where the pattern showed compact development around those areas already de‐ veloped, and dispersed in the outskirts of the city and peripheral areas. During this second period (UGB II), the simulated pattern followed an in-filling strategy, where most of the de‐ velopment took place within already developed lands and no development occurred be‐ yond the boundary of the developed areas. This can be seen where small simulated clusters (shown in red) are located within the existing urban areas (shown in blue). This is also an expected finding, since during this historical period, the planning authority in Riyadh strict‐ ly applied a policy to limit urban growth to avoid further urban sprawl that characterised

acteristic of this period.

ban clusters are simulated as non-urban.

To calibrate the model, a stratified random sample consisting of 60% urban and 40% nonurban cells was utilised in combination with a genetic algorithm (GA) where a single objec‐ tive function consisting of the mean squared error and the root mean squared error was employed. The use of these two measures together was designed to penalise model instan‐ ces in which the parameters fell outside of an acceptable range. Nine different model instan‐ ces were developed, which are listed in Table 1. These nine instances were based on different complexities of fuzzy rule (the modes) and different drivers (the scenarios). Mode 1 included fuzzy rules with only a single driver, e.g. transportation or topography while modes 2 and 3 had multiple drivers connected by the AND operator in the fuzzy rules. Sce‐ narios considered different combinations of drivers in order to determine how well the dif‐ ferent drivers were able to explain the observed urban development on their own and in combination. Thus M3-S1 is the most complex of the FCUGM instances. The top three per‐ forming models were M1-S4, M2-S4 and M3-S1, which clearly indicates that all the drivers are important in explaining urban growth in the city of Riyadh. These three simulations will be the focus of the validation process in this chapter.


**Table 1.** Modes and scenarios of the FCUGM.

Once calibrated, the FCUGM was used to simulate the Urban Growth Boundary (UGB) in the city of Riyadh for the following three periods: UGB I (1987–1997), UGB II (1997–2005) and UGB I+II (1987–2005) using the calibrated weights and parameters derived from the GA. Figures 1 to 3 show the simulations for the three time periods respectively for the three top performing simulations, i.e. M1-S4, M2-S4 and M3-S1. The new urban developments that are simulated by the model are shown in red while blue cells indicate those areas that have al‐ ready been developed. For UGB I (1987-1997), simulation M1-S4 shows more compact urban patterns compared with the other two simulations (M2-S4 and M3-S1), where the latter show more urban development across the peripheral areas, in particular for M3-S1. This might be attributed to the high weight assigned to the urban density variable for M1-S4 and to the form of the distance decay effect captured through the membership functions. How‐ ever, the morphology of the simulated urban spatial structure that is located to the north and north east shows quite some dispersed and scattered development. Generally, develop‐ ment sites are more linked in order to provide necessary infrastructure and service facilities. However, dispersed development is one of the characteristics of Riyadh's urban pattern. Typically, urban sprawl is produced by the three simulations regardless of the overall mac‐ roscopic pattern. This sprawl might be attributed to a lack of implementation of a policy to limit urban growth, which the government introduced to prevent chaotic development. In addition, this sprawl mimics the non-continuous or leap-frog pattern of urban growth char‐ acteristic of this period.

Gradient (G) and Altitude (A); and the PPRF takes Planned Areas (PA) and Excluded Areas (EA) into account. These drivers of urban growth are integrated via a fuzzy rule base, where the membership functions and the rules are determined through calibration. A fuzzy infer‐ ence engine is used to process the fuzzy rules and produce a fuzzy development suitabili‐ ty score at each cell. These fuzzy values are then defuzzified and used in combination with the stochastic disturbance factor and the transition threshold to determine whether a giv‐ en cell becomes an area of further urban development. The full details of the model are

To calibrate the model, a stratified random sample consisting of 60% urban and 40% nonurban cells was utilised in combination with a genetic algorithm (GA) where a single objec‐ tive function consisting of the mean squared error and the root mean squared error was employed. The use of these two measures together was designed to penalise model instan‐ ces in which the parameters fell outside of an acceptable range. Nine different model instan‐ ces were developed, which are listed in Table 1. These nine instances were based on different complexities of fuzzy rule (the modes) and different drivers (the scenarios). Mode 1 included fuzzy rules with only a single driver, e.g. transportation or topography while modes 2 and 3 had multiple drivers connected by the AND operator in the fuzzy rules. Sce‐ narios considered different combinations of drivers in order to determine how well the dif‐ ferent drivers were able to explain the observed urban development on their own and in combination. Thus M3-S1 is the most complex of the FCUGM instances. The top three per‐ forming models were M1-S4, M2-S4 and M3-S1, which clearly indicates that all the drivers are important in explaining urban growth in the city of Riyadh. These three simulations will

provided in [29, 31].

26 Emerging Applications of Cellular Automata

be the focus of the validation process in this chapter.

Mode 1 – Scenario 2 M1-S2 Urban density-attractiveness

Mode 2 – Scenario 1 M1-S1 Transportation and topography

Mode 1 – Scenario 4 M1-S4 Transportation, urban density-attractiveness and topography

Mode 2 – Scenario 4 M2-S4 Transportation, urban density-attractiveness and topography Mode 3 – Scenario 1 M3-S1 Transportation, urban density-attractiveness and topography

Once calibrated, the FCUGM was used to simulate the Urban Growth Boundary (UGB) in the city of Riyadh for the following three periods: UGB I (1987–1997), UGB II (1997–2005)

Mode 2 – Scenario 2 M2-S2 Transportation and urban density-attractiveness Mode 2 – Scenario 3 M2-S3 Topography and urban density-attractiveness

**Mode/Scenario Acronym Name of Simulation** Mode 1 – Scenario 1 M1-S1 Transportation

Mode 1 – Scenario 3 M1-S3 Topography

**Table 1.** Modes and scenarios of the FCUGM.

Figures 1 to 3 also show that the direction of growth is generally radial, where urban growth takes place around most of the already developed lands. In particular, most of the growth is to the south west and to the east of the city, while only moderate growth is simulated in the top south eastern part. Growth also rarely occurs to the west of the city. The pattern of growth might be a result of the government's free grant program. Most of the lands in these two areas were granted by the government to households with low incomes. Another rea‐ son may be the lower price of this land compared with the high price of land located to the north of the city. Moreover, moderate growth in the south east of the city could be due to the concentration of heavy industry in this part of the city and to the low urban environ‐ mental quality due to proximity to industrial zones and the oil refinery. It can also be seen that there is almost no urban growth simulated to the west of the city, where areas are either steep or located at higher altitudes, indicating that topographical constraint factors have confined growth in such areas. Topographical characteristics have also constrained growth in the south western part of the city, where the steep areas located between the two big ur‐ ban clusters are simulated as non-urban.

In UGB II (1997-2005), the simulated urban pattern contrasts with that shown in UGB I (1987-1997) where the pattern showed compact development around those areas already de‐ veloped, and dispersed in the outskirts of the city and peripheral areas. During this second period (UGB II), the simulated pattern followed an in-filling strategy, where most of the de‐ velopment took place within already developed lands and no development occurred be‐ yond the boundary of the developed areas. This can be seen where small simulated clusters (shown in red) are located within the existing urban areas (shown in blue). This is also an expected finding, since during this historical period, the planning authority in Riyadh strict‐ ly applied a policy to limit urban growth to avoid further urban sprawl that characterised the period UGB I. As a result of this policy, most of the development occurred on vacant land with the greatest development possibility occurring within existing developed areas. This particular pattern was simulated by all three model instances.

**Figure 2.** Model simulations from the FCUGM for the period 1997 – 2005 for the three scenarios: (a) M1-S4; (b) M2-

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In contrast to the two individual periods (UGB I and UGB II), the simulated urban growth over the combined period UGB I +II shows a more consistent pattern in terms of trend and direction of growth. This is not surprising since the simulation is for a period of 18 years, where more macroscopic urban growth patterns can be identified. The three model simula‐ tions produced a broadly similar direction of urban growth where the highest growth took place to the east of the city followed by a moderate growth to the south west and south east and a low growth to the north and south for the reasons noted above. However, there is a notable variation between the three scenarios in terms of urban morphological pattern. M1- S4 produced highly compact urban patterns while M2-S4 and M3-S1 both generated more

S4; and (c) M3-S1.

**Figure 1.** Model simulations from the FCUGM for the period 1987 – 1997 for the three scenarios: (a) M1-S4; (b) M2- S4; and (c) M3-S1.

Validating Spatial Patterns of Urban Growth from a Cellular Automata Model http://dx.doi.org/10.5772/51708 29

the period UGB I. As a result of this policy, most of the development occurred on vacant land with the greatest development possibility occurring within existing developed areas.

**Figure 1.** Model simulations from the FCUGM for the period 1987 – 1997 for the three scenarios: (a) M1-S4; (b) M2-

S4; and (c) M3-S1.

This particular pattern was simulated by all three model instances.

28 Emerging Applications of Cellular Automata

**Figure 2.** Model simulations from the FCUGM for the period 1997 – 2005 for the three scenarios: (a) M1-S4; (b) M2- S4; and (c) M3-S1.

In contrast to the two individual periods (UGB I and UGB II), the simulated urban growth over the combined period UGB I +II shows a more consistent pattern in terms of trend and direction of growth. This is not surprising since the simulation is for a period of 18 years, where more macroscopic urban growth patterns can be identified. The three model simula‐ tions produced a broadly similar direction of urban growth where the highest growth took place to the east of the city followed by a moderate growth to the south west and south east and a low growth to the north and south for the reasons noted above. However, there is a notable variation between the three scenarios in terms of urban morphological pattern. M1- S4 produced highly compact urban patterns while M2-S4 and M3-S1 both generated more dispersed patterns. The patterns produced by M3-S1 contained less noise (i.e. unrealistic scattered urban lands) compared to M2-S4, which can be clearly viewed in the north eastern part of the city. However, the non-uniform dispersal of lands, as shown in these simulations, is one of the characteristics of Riyadh's historical pattern of urban growth.

**3. Methods of CA Validation**

global method would not adequately capture [40].

Urban.

Seven different methods are described in this chapter; these approaches have all been used to validate the FCUGM model for the city of Riyadh. These include: visual validation; meas‐ ures of accuracy; urban cell correspondence; the Lee-Sallee index; a spatial pattern measure; a spatial district measure; and multi-resolution validation. The first method, or visual vali‐ dation, compares the observed results and simulated images by overlaying one image on top of the other and comparing the patterns qualitatively. Such an approach has been used in a number of studies to compare the overall spatial distribution and urban patterns of ob‐ served and simulated images, see e.g. [25, 36-39]. Visual comparison by itself may be prone to bias as it is based on the judgment of the researcher or planner. For this reason, more ob‐ jective methods are required such as those described below. However, visual examination is still an essential part of the validation process since the human brain is particularly good at recognising spatial patterns (and highlighting missing ones), which a more automated or

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One of the most common methods for assessing the performance of urban CA models quan‐ titatively is through the calculation of an error or confusion matrix. This approach has been widely used by several authors to compare simulated results against the actual ones for ur‐ ban CA models [21, 25, 38-39, 41]. The error matrix is a square array, where the rows and columns represent the number of categories whose classification accuracies are being as‐ sessed. Typically, the columns represent the observed data and the rows indicate the simu‐ lated data. Table 2 shows the error matrix for evaluating the FCUGM where the cells that are categorized in agreement with their observed data are located along the major diagonal of the matrix from the upper left to the lower right. These include urban areas that were si‐ mulated and are also observed, i.e. the true urban areas (TU) and areas that are not urban in both the observed and simulated data (true not urban or TNU). The cells off the diagonal represent errors that are underestimated (FNU or false non-urban) or overestimated (FU or

> **Observed Image Urban Non-Urban Overall**

Urban TU FNU TU+FNU

**Table 2.** Error matrix of the FCUGM. **TU** = True Urban, **FU** = False Urban, **TNU** = True Non-Urban and **FNU** = False Non-

From this error matrix, the accuracy can be calculated, which assesses the overall perform‐ ance of the model by calculating the proportion of the total number of simulated cells that

Overall TU+FU FNU+TNU TU+FU+TNU+FNU

false urban) in the simulated image when compared to the observed image.

**Simulated Image** Non-Urban FU TNU FU+TNU

Overall the model outputs verify that the model is replicating the main processes and driv‐ ers as would be expected given knowledge of policies and city structure in the past. In the following sections, more formal methods of model validation are considered.

**Figure 3.** Model simulations from the FCUGM for the period 1987 – 2005 for the three scenarios: (a) M1-S4; (b) M2- S4; and (c) M3-S1.

#### **3. Methods of CA Validation**

dispersed patterns. The patterns produced by M3-S1 contained less noise (i.e. unrealistic scattered urban lands) compared to M2-S4, which can be clearly viewed in the north eastern part of the city. However, the non-uniform dispersal of lands, as shown in these simulations,

Overall the model outputs verify that the model is replicating the main processes and driv‐ ers as would be expected given knowledge of policies and city structure in the past. In the

**Figure 3.** Model simulations from the FCUGM for the period 1987 – 2005 for the three scenarios: (a) M1-S4; (b) M2-

S4; and (c) M3-S1.

is one of the characteristics of Riyadh's historical pattern of urban growth.

30 Emerging Applications of Cellular Automata

following sections, more formal methods of model validation are considered.

Seven different methods are described in this chapter; these approaches have all been used to validate the FCUGM model for the city of Riyadh. These include: visual validation; meas‐ ures of accuracy; urban cell correspondence; the Lee-Sallee index; a spatial pattern measure; a spatial district measure; and multi-resolution validation. The first method, or visual vali‐ dation, compares the observed results and simulated images by overlaying one image on top of the other and comparing the patterns qualitatively. Such an approach has been used in a number of studies to compare the overall spatial distribution and urban patterns of ob‐ served and simulated images, see e.g. [25, 36-39]. Visual comparison by itself may be prone to bias as it is based on the judgment of the researcher or planner. For this reason, more ob‐ jective methods are required such as those described below. However, visual examination is still an essential part of the validation process since the human brain is particularly good at recognising spatial patterns (and highlighting missing ones), which a more automated or global method would not adequately capture [40].

One of the most common methods for assessing the performance of urban CA models quan‐ titatively is through the calculation of an error or confusion matrix. This approach has been widely used by several authors to compare simulated results against the actual ones for ur‐ ban CA models [21, 25, 38-39, 41]. The error matrix is a square array, where the rows and columns represent the number of categories whose classification accuracies are being as‐ sessed. Typically, the columns represent the observed data and the rows indicate the simu‐ lated data. Table 2 shows the error matrix for evaluating the FCUGM where the cells that are categorized in agreement with their observed data are located along the major diagonal of the matrix from the upper left to the lower right. These include urban areas that were si‐ mulated and are also observed, i.e. the true urban areas (TU) and areas that are not urban in both the observed and simulated data (true not urban or TNU). The cells off the diagonal represent errors that are underestimated (FNU or false non-urban) or overestimated (FU or false urban) in the simulated image when compared to the observed image.


**Table 2.** Error matrix of the FCUGM. **TU** = True Urban, **FU** = False Urban, **TNU** = True Non-Urban and **FNU** = False Non-Urban.

From this error matrix, the accuracy can be calculated, which assesses the overall perform‐ ance of the model by calculating the proportion of the total number of simulated cells that match the corresponding ones in the observed image using Equation 7. In addition the percen‐ tages of agreement and disagreement can be calculated as expressed in Equations 8 and 9:

$$\text{Accuracy } \begin{pmatrix} \% \\ \end{pmatrix} = \begin{pmatrix} TU + TNU \\ \end{pmatrix} \text{ / } \begin{pmatrix} TU + FU + FNU + TNU \\ \end{pmatrix} \text{ \tag{7}}$$

tern as the observed one. Finally, the total number of correct cells is summed and compared against the results generated by the cell-by-cell analysis. In practice, a pre-designed kernel matrix is moved across the whole study area which simultaneously compares the number of neighbours for each cell both in the simulated and observed images. When the number of neighbours (cells) is the same for this particular cell, a value of 1 is assigned to the output

where Ω S*ij* is the number of simulated urban cells *ij* within a neighbourhood Ω; and Ω O*ij* is the number of observed urban cells *ij* within a neighbourhood Ω. To calculate this meas‐ ure, a special kernel matrix is designed as a neighbourhood measure to mimic the common urban block shape in Riyadh. The general urban pattern can be characterised as a grid-iron pattern. The most common shape and size of urban blocks in the contemporary and future districts of Riyadh are rectangular shapes of 180m length and 60m width. In the FCUGM (with a cell size of 20m), this is equivalent to 9 cells in length and 3 cells in width. Thus, a neighbourhood with a rectangular shape of 180m in length and 60m in width is used to vali‐ date the performance of the model in terms of spatial pattern. The SPM compares the num‐ ber of developed land cells within this neighbourhood shape and size in both the simulated

A measure that captures the spatial district structure (SDS) is also used to validate the struc‐ tural similarity between the simulated and observed urban growth in terms of urban neigh‐ bourhood (Figure 4a). It would also be possible to assess this in terms of urban subneighbourhood (Figure 4b) and urban block (Figure 4c), where the boundaries of these zones are shown for the city of Riyadh in Figure 4. Figure 5 shows what these structures look like when zooming into a section of the city. In this chapter, only the spatial structure

**Figure 4.** The boundaries of three spatial structures in the city of Riyadh: (a) urban neighbourhoods; (b) urban sub-

IF S O ; then SPM 1; otherwise SPM 0 åW = åW *ij ij ij* = *ij* = (12)

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image. This can be expressed mathematically as shown below:

and observed images.

of the urban neighbourhood is examined.

neighbourhoods; and (c) urban blocks.

$$\text{Agreement } \begin{pmatrix} \% \end{pmatrix} = \begin{pmatrix} \begin{pmatrix} TU+TNU \end{pmatrix} \end{pmatrix} \text{ } \begin{pmatrix} TU+FU+FNU+TNU \end{pmatrix} \text{)\*100} \tag{8}$$

$$\text{Disagreement } \text{(\%)} \quad = \left( \left( FU + FNU \right) \right.\\ \text{ / } \left( TU + FU + FNU + TNU \right) \text{[\*100]} \tag{9}$$

However, if the study area includes a large number of non-urban cells and a small number of urban cells, the accuracy measure might overstate the model performance due to the high number of non-urban simulated cells that match the non-urban observed ones (i.e. true nonurban (TNU) in Table 2). Such a situation renders it difficult to differentiate between the true performances of different simulations as they might yield similarly high values of accuracy. A validation measure which overcomes this problem is the urban cell correspondence (UCC), since it considers only the True Urban and False Urban cells from the error matrix, as outlined in Equation 10:

$$\text{LCCC} = \frac{TU}{(TU \text{ \* } FU)} \tag{10}$$

Another problem with the error matrix is that it is not able to assess and estimate the form and shape of patterns because it is based on independent comparisons between pairs of cells. Once such measure that does take shape into account and which has been used fre‐ quently for assessing the urban shape produced by CA models is the Lee-Sallee Index (LSI) [18, 38-39, 42-43]. The LSI is calculated as the ratio of the intersection between the observed and simulated urban areas against the union of these areas in the two images as follows:

$$LSI = \sum \{ \mathbf{S}\_{\vec{\eta}\vec{\eta}} \cap \mathbf{O}\_{\vec{\eta}} \} / \sum \{ \mathbf{S}\_{\vec{\eta}\vec{\eta}} \cup \mathbf{O}\_{\vec{\eta}} \} \tag{11}$$

where *Sij* is a simulated urban cell *ij* and *Oij* is an observed urban cell *ij* .

Another validation measure that considers shape is the spatial pattern measure (SPM). Most cell-by-cell based analyses like those described above ignore the underlying presence of neighbourhoods. In the case of the SPM, a cell is regarded as erroneous if the category in the observed map differs from the category in the simulated map, irrespective of whether the category is found in the neighbouring cell or nowhere near the cell. In this sense, the SPM evaluates the performance based on the agreement within a neighbourhood. If a simulated cell and its corresponding observed urban cell have the same number of adjacent urban neighbours within a predefined neighbourhood, then the cell in question gets a value of 1, indicating that this simulated cell and its neighbours have the same simulated spatial pat‐ tern as the observed one. Finally, the total number of correct cells is summed and compared against the results generated by the cell-by-cell analysis. In practice, a pre-designed kernel matrix is moved across the whole study area which simultaneously compares the number of neighbours for each cell both in the simulated and observed images. When the number of neighbours (cells) is the same for this particular cell, a value of 1 is assigned to the output image. This can be expressed mathematically as shown below:

match the corresponding ones in the observed image using Equation 7. In addition the percen‐ tages of agreement and disagreement can be calculated as expressed in Equations 8 and 9:

Accuracy % ( ) = + ++ + (*TU TNU TU FU FNU TNU* ) / ( ) (7)

Agreement % ( ) = + ++ + ((*TU TNU TU FU FNU TNU* ) / ( ))\*100 (8)

Disagreement % ( ) = + ++ + ((*FU FNU TU FU FNU TNU* ) / ( ))\*100 (9)

(*TU* <sup>+</sup> *FU* ) (10)

) (11)

However, if the study area includes a large number of non-urban cells and a small number of urban cells, the accuracy measure might overstate the model performance due to the high number of non-urban simulated cells that match the non-urban observed ones (i.e. true nonurban (TNU) in Table 2). Such a situation renders it difficult to differentiate between the true performances of different simulations as they might yield similarly high values of accuracy. A validation measure which overcomes this problem is the urban cell correspondence (UCC), since it considers only the True Urban and False Urban cells from the error matrix, as

*UCC* <sup>=</sup> *TU*

*LSI* = ∑ (*Sij* ∩*Oij*

is a simulated urban cell *ij* and *Oij*

Another problem with the error matrix is that it is not able to assess and estimate the form and shape of patterns because it is based on independent comparisons between pairs of cells. Once such measure that does take shape into account and which has been used fre‐ quently for assessing the urban shape produced by CA models is the Lee-Sallee Index (LSI) [18, 38-39, 42-43]. The LSI is calculated as the ratio of the intersection between the observed and simulated urban areas against the union of these areas in the two images as follows:

)/ ∑ (*Sij* ∪*Oij*

Another validation measure that considers shape is the spatial pattern measure (SPM). Most cell-by-cell based analyses like those described above ignore the underlying presence of neighbourhoods. In the case of the SPM, a cell is regarded as erroneous if the category in the observed map differs from the category in the simulated map, irrespective of whether the category is found in the neighbouring cell or nowhere near the cell. In this sense, the SPM evaluates the performance based on the agreement within a neighbourhood. If a simulated cell and its corresponding observed urban cell have the same number of adjacent urban neighbours within a predefined neighbourhood, then the cell in question gets a value of 1, indicating that this simulated cell and its neighbours have the same simulated spatial pat‐

is an observed urban cell *ij* .

outlined in Equation 10:

32 Emerging Applications of Cellular Automata

where *Sij*

$$\text{IF } \Sigma \Omega \text{ Sij} = \begin{array}{c} \Sigma \Omega \text{ Oij} \end{array} \text{ then } \text{SPM}ij = \begin{array}{c} 1; \text{ otherwise } \text{SPM}ij = 0 \end{array} \tag{12}$$

where Ω S*ij* is the number of simulated urban cells *ij* within a neighbourhood Ω; and Ω O*ij* is the number of observed urban cells *ij* within a neighbourhood Ω. To calculate this meas‐ ure, a special kernel matrix is designed as a neighbourhood measure to mimic the common urban block shape in Riyadh. The general urban pattern can be characterised as a grid-iron pattern. The most common shape and size of urban blocks in the contemporary and future districts of Riyadh are rectangular shapes of 180m length and 60m width. In the FCUGM (with a cell size of 20m), this is equivalent to 9 cells in length and 3 cells in width. Thus, a neighbourhood with a rectangular shape of 180m in length and 60m in width is used to vali‐ date the performance of the model in terms of spatial pattern. The SPM compares the num‐ ber of developed land cells within this neighbourhood shape and size in both the simulated and observed images.

A measure that captures the spatial district structure (SDS) is also used to validate the struc‐ tural similarity between the simulated and observed urban growth in terms of urban neigh‐ bourhood (Figure 4a). It would also be possible to assess this in terms of urban subneighbourhood (Figure 4b) and urban block (Figure 4c), where the boundaries of these zones are shown for the city of Riyadh in Figure 4. Figure 5 shows what these structures look like when zooming into a section of the city. In this chapter, only the spatial structure of the urban neighbourhood is examined.

**Figure 4.** The boundaries of three spatial structures in the city of Riyadh: (a) urban neighbourhoods; (b) urban subneighbourhoods; and (c) urban blocks.

The final validation method considers the effect of spatial scale or resolution on the model results. The effects of scale have been considered in previous studies of urban growth mod‐ elling by [40, 44-45]. For example, in [40], a multiple-resolution comparison was conducted between the reference and modelled images by demonstrating a pixel aggregation proce‐ dure by which four adjacent pixels were averaged at increasingly coarser levels of resolu‐ tion. To investigate the influence of spatial resolution on the outputs from the FCUGM model, a similar multiple-resolution validation experiment was conducted to that of [40]. The model output from simulation M3-S1 over the period UGB I+II and the observed image for the corresponding period were aggregated from higher to lower levels of spatial resolu‐ tion whereby four neighbouring pixels were averaged at each coarser resolution. Thus, cells at the next level up had twice the width and height of the previous cell size. The initial cell size was 20 m and the experiments were conducted for 40, 80, 160, 320 and 640 m pixel sizes.

show the same comparison but for UGB II (1997-2005) and UGB I+II (1987-2005) respective‐

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The first two classes indicate that the simulation is correct while the latter two are incorrect.

**i.** starting urban (i.e. already developed lands before the year of the simulation); and

For the period UGB I (1987 – 1997), the urban development for the three scenarios in most areas of the city such as north, north east or south west is relatively well estimated (as shown in red). However, areas located at the immediate edges of boundaries of urbanised areas are overestimated (as shown in yellow). This is not surprising because those cells are adjacent to urban land and nearby to attractions, which are more likely to be urban than non-urban. Although the model was able to simulate the pattern or distribution of the devel‐ oped land of the city reasonably well, it is clear that the FCUGM was not able to reproduce all of the actual urban development that took place, for example, at the extreme south east‐ ern edge of the city (right-bottom corner of Figure 6, coloured in black), which resulted in an underestimation of these lands. Additionally, some small clusters at the extreme edge of the mid-east, west and south west of the city are also underestimated. This underestimation could be attributed to the fact that these areas are widely scattered from one another and from the boundaries of the other urbanised areas, and they are located at some distance from most attractions (e.g. the town centre, developed lands and other services), which in turn were assigned a low possibility of being developed. Thus these areas would have been simulated as non-urban. Another possible explanation is misclassification of the satellite im‐ ages during the image processing procedure. However, this underestimation is reasonably small, indicating that the model was able to capture the majority of chaotic and fragmented

In contrast to UGB I (1987 – 1997), during the period UGB II (1997 – 2005) (as shown in Figure 7), the correctly estimated urban areas are hard to distinguish and detect, because most of the developments are in small urbanised clusters located within the boundaries of already devel‐ oped areas. Moreover, this particular period was characterised by significant levels of 'leap‐ frog' development, which might explain why most of the areas in the maps are coloured blue (starting urban) and the urban match that is coloured in red is marginal and scarcely to be seen. However, M1-S4 and M3-S1 seem to have estimated the urban development reasona‐ bly well. It is very hard to detect any urban matching in the M2-S4 simulation, which might suggest that the topographical constraints factor can be considered as a significant influence

ly. In the comparison of the images, four main categories were mapped:

**i.** non-urban match (non-urban in observation and simulation);

**iii.** underestimated (urban in observation but non-urban in simulation); and

**iv.** overestimated (non-urban in the observation but urban in the simulation).

**ii.** urban match (urban in observation and simulation);

Two other classes have been added to facilitate the comparison:

**ii.** agricultural areas.

development that occurred during this period.

**Figure 5.** a) A section of the city of Riyadh with delineations for (b) urban neighbourhood; (c) urban sub-neighbour‐ hood; and (d) an urban block.

#### **4. Results**

This section provides the results from the application of the seven validation methods as de‐ scribed in section 3.

#### **4.1. Visual Validation of Urban Growth Patterns**

The simulated images were overlaid on the observed patterns of development for each of the three time periods and for the three model simulations. Figure 6 shows the overlaid out‐ puts for the period UGB I (1987-1997) for M1-S4, M2-S4 and M3-S1 while Figures 7 and 8 show the same comparison but for UGB II (1997-2005) and UGB I+II (1987-2005) respective‐ ly. In the comparison of the images, four main categories were mapped:


The first two classes indicate that the simulation is correct while the latter two are incorrect. Two other classes have been added to facilitate the comparison:


The final validation method considers the effect of spatial scale or resolution on the model results. The effects of scale have been considered in previous studies of urban growth mod‐ elling by [40, 44-45]. For example, in [40], a multiple-resolution comparison was conducted between the reference and modelled images by demonstrating a pixel aggregation proce‐ dure by which four adjacent pixels were averaged at increasingly coarser levels of resolu‐ tion. To investigate the influence of spatial resolution on the outputs from the FCUGM model, a similar multiple-resolution validation experiment was conducted to that of [40]. The model output from simulation M3-S1 over the period UGB I+II and the observed image for the corresponding period were aggregated from higher to lower levels of spatial resolu‐ tion whereby four neighbouring pixels were averaged at each coarser resolution. Thus, cells at the next level up had twice the width and height of the previous cell size. The initial cell size was 20 m and the experiments were conducted for 40, 80, 160, 320 and 640 m pixel sizes.

**Figure 5.** a) A section of the city of Riyadh with delineations for (b) urban neighbourhood; (c) urban sub-neighbour‐

This section provides the results from the application of the seven validation methods as de‐

The simulated images were overlaid on the observed patterns of development for each of the three time periods and for the three model simulations. Figure 6 shows the overlaid out‐ puts for the period UGB I (1987-1997) for M1-S4, M2-S4 and M3-S1 while Figures 7 and 8

hood; and (d) an urban block.

34 Emerging Applications of Cellular Automata

**4. Results**

scribed in section 3.

**4.1. Visual Validation of Urban Growth Patterns**

For the period UGB I (1987 – 1997), the urban development for the three scenarios in most areas of the city such as north, north east or south west is relatively well estimated (as shown in red). However, areas located at the immediate edges of boundaries of urbanised areas are overestimated (as shown in yellow). This is not surprising because those cells are adjacent to urban land and nearby to attractions, which are more likely to be urban than non-urban. Although the model was able to simulate the pattern or distribution of the devel‐ oped land of the city reasonably well, it is clear that the FCUGM was not able to reproduce all of the actual urban development that took place, for example, at the extreme south east‐ ern edge of the city (right-bottom corner of Figure 6, coloured in black), which resulted in an underestimation of these lands. Additionally, some small clusters at the extreme edge of the mid-east, west and south west of the city are also underestimated. This underestimation could be attributed to the fact that these areas are widely scattered from one another and from the boundaries of the other urbanised areas, and they are located at some distance from most attractions (e.g. the town centre, developed lands and other services), which in turn were assigned a low possibility of being developed. Thus these areas would have been simulated as non-urban. Another possible explanation is misclassification of the satellite im‐ ages during the image processing procedure. However, this underestimation is reasonably small, indicating that the model was able to capture the majority of chaotic and fragmented development that occurred during this period.

In contrast to UGB I (1987 – 1997), during the period UGB II (1997 – 2005) (as shown in Figure 7), the correctly estimated urban areas are hard to distinguish and detect, because most of the developments are in small urbanised clusters located within the boundaries of already devel‐ oped areas. Moreover, this particular period was characterised by significant levels of 'leap‐ frog' development, which might explain why most of the areas in the maps are coloured blue (starting urban) and the urban match that is coloured in red is marginal and scarcely to be seen. However, M1-S4 and M3-S1 seem to have estimated the urban development reasona‐ bly well. It is very hard to detect any urban matching in the M2-S4 simulation, which might suggest that the topographical constraints factor can be considered as a significant influence during this period. This is corroborated by the fact that each of the simulations that includ‐ ed this factor (such as M1-S4 and M3-S1) produced better results than M2-S4.

S1 UGB I+II. However, these high values of accuracy are mainly achieved through the high matching of non-urban cells, of which there are a very large number in this test area (i.e. it ranges between 3,250,000 and 3,450,000). This implies the need to use a measure that allows for better discrimination between the different simulations, i.e. the UCC, which considers only the matching of the urban cells. Note that the accuracy drops across all simulations, re‐ sulting in 0.053 (lowest) and 0.743 (highest) for M2-S4 during UGB II and M3-S1 during UGB I+II, respectively. The UCC measure reveals that the FCUGM simulated the urban growth more accurately over the period UGB I+II (ranging between 0.635 – 0.743) followed by UGB I (0.500 – 0.525), while over the period UGB II, the model produced the poorest re‐

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**Figure 6.** Comparison of the simulated (FCUGM) versus observed cells for the period UGB I (1987 – 1997) for: (a) M1-

sults (0.053 – 0.376).

S4; (b) M2-S4; and (c) M3-S1.

With respect to UGB I+II as shown in Figure 8, the main result of this analysis is that there is a good visual similarity between the maps, and the simulation results resemble the real city. It can be noted that urban development is largely estimated by the three simulations where M3-S1 has estimated most of the urban development followed by M2-S4 and M1-S4. How‐ ever, some clusters of land cells are underestimated, mainly in the peripheral areas, showing a different shape in comparison with the actual city. This particular area (coloured in black) is under-estimated highly, moderately and slightly by M1-S4, M2-S4 and M3-S1 respective‐ ly. The locations of these cells are very difficult to model since they are located in a highly non-linear and chaotic pattern (e.g. they are far from already developed lands, distant from attractions and services, etc.). Furthermore, it can be noted that M3-S1 was capable of repro‐ ducing such complex features to a large extent. This can be attributed to that fact that in this particular simulation, the three urban growth driving forces (TSF, UAAF and TCF) are em‐ bedded in each single fuzzy rule, while in M2-S4 and M1-S4 only two and one of these fac‐ tors are embedded, respectively. This explains why M3-S1 performed well over all periods. It can also be noted that few cells are overestimated compared with the other two periods (i.e. UGB I, UGB II).

Thus overall, the visual analysis of the simulated images shows that they compare well with the patterns that actually occurred in Riyadh during these periods for most of the three simulations, which is a positive reflection of the model's ability to simulate urban growth in the past.

#### **4.2. Accuracy Assessment and Spatial Statistical Measures**

An accuracy assessment is another commonly used validation method where the first step is to calculate the error matrix as shown in Table 3 for the three simulations over the three time periods. It can be seen from Table 3-A that the observed urban development during the peri‐ od UGB I (1987-1997) was about 261,000 cells. The FCUGM simulated around 265,000, 303,000 and 269,000 urban cells in M1-S4, M2-S4 and M3-S4, respectively. Amongst those si‐ mulated cells, about 135,000, 152,000 and 142,000 cells were correctly simulated and match‐ ed the observed image. However, 129,000, 151,000 and 128,000 were overestimated, and approximately 125,000, 109,000 and 119,000 were underestimated. For the period UGB II (1997-2005), the results were less good as shown in Table 3-B. The simulated urban cells that were generated by the simulations M1-S4, M2-S4 and M3-S4 were only 82,000, 10,000 and 108,000 compared with 237,000 observed ones. In contrast, the simulated results for both pe‐ riods together UGB I+II (1987-2005) (Table 3-C) showed an improvement and resulted in a higher correspondence of urban cells compared with the two preceding periods (UGB I and UGB II). The urban cells that were correctly matched reached 222,000, 343,000 and 355,000 compared with 464,000 urban observed cells.

From the error matrix, the accuracy measures and the UCC were calculated as shown in Ta‐ ble 4. The LSI is also provided. The results show that the overall accuracy of all simulations is quite high, ranging between 0.890 for simulation M2-S4 UGB II and 0.937 for scenario M3S1 UGB I+II. However, these high values of accuracy are mainly achieved through the high matching of non-urban cells, of which there are a very large number in this test area (i.e. it ranges between 3,250,000 and 3,450,000). This implies the need to use a measure that allows for better discrimination between the different simulations, i.e. the UCC, which considers only the matching of the urban cells. Note that the accuracy drops across all simulations, re‐ sulting in 0.053 (lowest) and 0.743 (highest) for M2-S4 during UGB II and M3-S1 during UGB I+II, respectively. The UCC measure reveals that the FCUGM simulated the urban growth more accurately over the period UGB I+II (ranging between 0.635 – 0.743) followed by UGB I (0.500 – 0.525), while over the period UGB II, the model produced the poorest re‐ sults (0.053 – 0.376).

during this period. This is corroborated by the fact that each of the simulations that includ‐

With respect to UGB I+II as shown in Figure 8, the main result of this analysis is that there is a good visual similarity between the maps, and the simulation results resemble the real city. It can be noted that urban development is largely estimated by the three simulations where M3-S1 has estimated most of the urban development followed by M2-S4 and M1-S4. How‐ ever, some clusters of land cells are underestimated, mainly in the peripheral areas, showing a different shape in comparison with the actual city. This particular area (coloured in black) is under-estimated highly, moderately and slightly by M1-S4, M2-S4 and M3-S1 respective‐ ly. The locations of these cells are very difficult to model since they are located in a highly non-linear and chaotic pattern (e.g. they are far from already developed lands, distant from attractions and services, etc.). Furthermore, it can be noted that M3-S1 was capable of repro‐ ducing such complex features to a large extent. This can be attributed to that fact that in this particular simulation, the three urban growth driving forces (TSF, UAAF and TCF) are em‐ bedded in each single fuzzy rule, while in M2-S4 and M1-S4 only two and one of these fac‐ tors are embedded, respectively. This explains why M3-S1 performed well over all periods. It can also be noted that few cells are overestimated compared with the other two periods

Thus overall, the visual analysis of the simulated images shows that they compare well with the patterns that actually occurred in Riyadh during these periods for most of the three simulations, which is a positive reflection of the model's ability to simulate urban growth in

An accuracy assessment is another commonly used validation method where the first step is to calculate the error matrix as shown in Table 3 for the three simulations over the three time periods. It can be seen from Table 3-A that the observed urban development during the peri‐ od UGB I (1987-1997) was about 261,000 cells. The FCUGM simulated around 265,000, 303,000 and 269,000 urban cells in M1-S4, M2-S4 and M3-S4, respectively. Amongst those si‐ mulated cells, about 135,000, 152,000 and 142,000 cells were correctly simulated and match‐ ed the observed image. However, 129,000, 151,000 and 128,000 were overestimated, and approximately 125,000, 109,000 and 119,000 were underestimated. For the period UGB II (1997-2005), the results were less good as shown in Table 3-B. The simulated urban cells that were generated by the simulations M1-S4, M2-S4 and M3-S4 were only 82,000, 10,000 and 108,000 compared with 237,000 observed ones. In contrast, the simulated results for both pe‐ riods together UGB I+II (1987-2005) (Table 3-C) showed an improvement and resulted in a higher correspondence of urban cells compared with the two preceding periods (UGB I and UGB II). The urban cells that were correctly matched reached 222,000, 343,000 and 355,000

From the error matrix, the accuracy measures and the UCC were calculated as shown in Ta‐ ble 4. The LSI is also provided. The results show that the overall accuracy of all simulations is quite high, ranging between 0.890 for simulation M2-S4 UGB II and 0.937 for scenario M3-

**4.2. Accuracy Assessment and Spatial Statistical Measures**

compared with 464,000 urban observed cells.

ed this factor (such as M1-S4 and M3-S1) produced better results than M2-S4.

(i.e. UGB I, UGB II).

36 Emerging Applications of Cellular Automata

the past.

**Figure 6.** Comparison of the simulated (FCUGM) versus observed cells for the period UGB I (1987 – 1997) for: (a) M1- S4; (b) M2-S4; and (c) M3-S1.

than M3-S1. The M1-S4 simulation produced moderately accurate results with UCC values of

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**Figure 8.** Comparison of the simulated (FCUGM) versus observed cells for the period UGB I+II (1987 – 2005) for: (a)

With respect to the agreement between the shape of the simulated and observed images in the form of the LSI, Clark and Gaydos (1998) reported that the practical accuracy of the LSI is only around 0.3 while Cheng and Masser (2004) reported values of 0.383 for their model simulations. However, LSI values of greater than 0.35 were achieved in six out of nine simu‐ lations from the FCUGM, indicating a better performance than other CA urban growth models. It can also be noted that the LSI and UCC are highly related to one another where a correlation coefficient of 0.969 was obtained between the two measures for all simulations. Thus, simulations with high UCC are more likely to achieve high LSI, indicating a consisten‐

M1-S4; (b) M2-S4; and (c) M3-S1.

0.635, 0.511 and 0.373 for the three periods UGB I+II, UGB I and UGB II, respectively.

**Figure 7.** Comparison of the simulated (FCUGM) versus observed cells for the period UGB II (1997 – 2005) for: (a) M1- S4; (b) M2-S4; and (c) M3-S1.

The M3-S1 simulation over all periods yielded the most accurate results compared to the other two simulations, achieving UCC accuracies of 0.743, 0.525 and 0.376 for the periods UGB I +II, UGB I and UGB II, respectively. In contrast, M2-S4 produced the poorest performance across the two periods UGB I and UGB II with a UCC accuracy of 0.500 and 0.053, respective‐ ly, while over the period UGB I+II, this simulation performed better than M1-S4 but worse than M3-S1. The M1-S4 simulation produced moderately accurate results with UCC values of 0.635, 0.511 and 0.373 for the three periods UGB I+II, UGB I and UGB II, respectively.

**Figure 8.** Comparison of the simulated (FCUGM) versus observed cells for the period UGB I+II (1987 – 2005) for: (a) M1-S4; (b) M2-S4; and (c) M3-S1.

With respect to the agreement between the shape of the simulated and observed images in the form of the LSI, Clark and Gaydos (1998) reported that the practical accuracy of the LSI is only around 0.3 while Cheng and Masser (2004) reported values of 0.383 for their model simulations. However, LSI values of greater than 0.35 were achieved in six out of nine simu‐ lations from the FCUGM, indicating a better performance than other CA urban growth models. It can also be noted that the LSI and UCC are highly related to one another where a correlation coefficient of 0.969 was obtained between the two measures for all simulations. Thus, simulations with high UCC are more likely to achieve high LSI, indicating a consisten‐

**Figure 7.** Comparison of the simulated (FCUGM) versus observed cells for the period UGB II (1997 – 2005) for: (a) M1-

The M3-S1 simulation over all periods yielded the most accurate results compared to the other two simulations, achieving UCC accuracies of 0.743, 0.525 and 0.376 for the periods UGB I +II, UGB I and UGB II, respectively. In contrast, M2-S4 produced the poorest performance across the two periods UGB I and UGB II with a UCC accuracy of 0.500 and 0.053, respective‐ ly, while over the period UGB I+II, this simulation performed better than M1-S4 but worse

S4; (b) M2-S4; and (c) M3-S1.

38 Emerging Applications of Cellular Automata

cy in performance and stability between the spatial shape measure and the cell-by-cell accu‐ racy measure.

Observed Urban Non-Urban Overall

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Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

**(%) Accuracy UCC LSI**

Urban 108,066 178,113 286,179

Overall 237,566 3,463,421 3,700,987

Urban 222,175 127,517 349,692

Overall 464,167 3,236,820 3,700,987

Urban 342,960 124,755 467,715

Overall 464,167 3,236,820 3,700,987

Urban 355,315 122,649 477,964

Overall 464,167 3,236,820 3,700,987

**M3-S1** Simulated Non-Urban 129,500 3,285,308 3,414,808

**M1-S4** Simulated Non-Urban 241,992 3,109,303 3,351,295

**M2-S4** Simulated Non-Urban 121,207 3,112,065 3,233,272

**M3-S1** Simulated Non-Urban 108,852 3,114,171 3,223,023

**Table 3.** The error matrices for the three FCUGM simulations over the period: (a) UGB I (1987 – 1997); (b) UGB II (1997

**Disagreement**

M1-S4 UGB I 93.1 6.9 0.931 0.511 0.347 M2-S4 UGB I 92.9 7.1 0.929 0.500 0.367 M3-S1 UGB I 93.3 6.7 0.933 0.525 0.364 M1-S4 UGB II 91.6 8.4 0.916 0.373 0.259 M2-S4 UGB II 89.0 11.0 0.890 0.053 0.024 M3-S1 UGB II 92.1 7.9 0.921 0.376 0.218 M1-S4 UGB I+II 90.0 10.0 0.900 0.635 0.375 M2-S4 UGB I+II 93.3 6.7 0.933 0.733 0.582 M3-S1 UGB I+II 93.7 6.3 0.937 0.743 0.605

**Table 4.** Statistical performance of the FCUGM for the three different simulations over the three periods of growth

– 2005); and (c) UGB I+II (1987 – 2005).

**Simulation Agreement**

**(%)**

UGB I (1987 – 1977), UGB II (1997 – 2005) and UGB I+II (1987 – 2005).

**(c): UGB I+II** Observed

The accuracy of the LSI produced by the FCUGM was relatively good with the majority above 0.35. During the period UGB I+II, the model was reasonably good at capturing the shape of the simulated urban areas with values ranging between 0.375 and 0.602. In contrast, the poorest LSI was produced for the period UGB II with values falling to between 0.024 and 0.259. The LSI during the period UGB I was acceptable, ranging between 0.347 and 0.364. The simulation M3-S1 during the period UGB I+II generated the highest shape matching, whilst simulation M2-S4 over the period UGB II showed the poorest performance.


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cy in performance and stability between the spatial shape measure and the cell-by-cell accu‐

The accuracy of the LSI produced by the FCUGM was relatively good with the majority above 0.35. During the period UGB I+II, the model was reasonably good at capturing the shape of the simulated urban areas with values ranging between 0.375 and 0.602. In contrast, the poorest LSI was produced for the period UGB II with values falling to between 0.024 and 0.259. The LSI during the period UGB I was acceptable, ranging between 0.347 and 0.364. The simulation M3-S1 during the period UGB I+II generated the highest shape matching,

Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Urban 135,934 129,948 265,882

Overall 261,339 3,439,648 3,700,987

Urban 151,902 151,773 303,675

Overall 261,339 3,439,648 3,700,987

Urban 141,904 128,060 269,964

Overall 261,339 3,439,648 3,700,987

Urban 81,630 135,234 216,864

Overall 237,566 3,463,421 3,700,987

Urban 10,099 178,113 188,212

Overall 237,566 3,463,421 3,700,987

whilst simulation M2-S4 over the period UGB II showed the poorest performance.

**M1-S4** Simulated Non-Urban 125,405 3,309,700 3,435,105

**M2-S4** Simulated Non-Urban 109,437 3,287,875 3,397,312

**M3-S1** Simulated Non-Urban 119,435 3,311,588 3,431,023

**M1-S4** Simulated Non-Urban 155,936 3,328,187 3,484,123

**M2-S4** Simulated Non-Urban 227,467 3,285,308 3,512,775

**(b): UGB II** Observed

**(a): UGB I** Observed

racy measure.

40 Emerging Applications of Cellular Automata

**Table 3.** The error matrices for the three FCUGM simulations over the period: (a) UGB I (1987 – 1997); (b) UGB II (1997 – 2005); and (c) UGB I+II (1987 – 2005).


**Table 4.** Statistical performance of the FCUGM for the three different simulations over the three periods of growth UGB I (1987 – 1977), UGB II (1997 – 2005) and UGB I+II (1987 – 2005).

#### **4.3. Spatial Pattern Measure (SPM)**

Tables 5 and 6 show the error matrix and statistical indices for the spatial pattern measure for the three FCUGM simulations over the three time periods. These measures include the percentage of agreement, disagreement, accuracy, UCC and LSI when considering the un‐ derlying neighbourhood (Equation 12). It can be seen from Tables 5 and 6 that the perform‐ ance of the FCUGM taking the spatial pattern of neighbourhoods into account shows relatively positive results. For example, the percentage of agreement across all simulations lies between 88% and 94%. The UCC indicates a very high degree of matching between the simulated and observed urban lands with the UCC accuracy as high as 0.765 generated by simulation M3-S1 over the period UGB I+II, and a value of 0.542 generated by simulation M1-S4 and M2-S4 over the period UGB II. The least satisfactory performance was generated by M2-S4 during the period UGB II. In terms of the shape index, this measure shows fairly consistent results similar to the accuracy and the UCC. Those results with high values of UCC and accuracy also generated high shape agreements similar to the findings in section 4.2 where the underlying neighbourhood was not taken into account.

**(a) UGB I** Observed

**M1-S4** Simulated Non-Urban 119,435 3,600,558 3,719,993

**M2-S4** Simulated Non-Urban 119,437 3,611,426 3,730,863

**M3-S1** Simulated Non-Urban 125,405 3,630,959 3,756,364

**M1-S4** Simulated Non-Urban 155,936 3,328,187 3,484,123

**M2-S4** Simulated Non-Urban 237,464 3,285,308 3,522,772

**M3-S1** Simulated Non-Urban 129,500 3,285,308 3,414,808

**M1-S4** Simulated Non-Urban 241,992 3,393,273 3,635,265

**(c) UGB I+II** Observed

**(b) UGB II** Observed

Urban Non-Urban Overall

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Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Urban Non-Urban Overall

Urban 141,904 128,060 269,964

Overall 261,339 3,728,618 3,989,957

Urban 141,902 151,773 293,675

Overall 261,339 3,763,199 4,024,538

Urban 135,934 129,948 265,882

Overall 261,339 3,760,907 4,022,246

Urban 81,630 135,234 216,864

Overall 237,566 3,463,421 3,700,987

Urban 99 178,113 178,212

Overall 237,563 3,463,421 3,700,984

Urban 108,066 178,113 286,179

Overall 237,566 3,463,421 3,700,987

Urban 222,175 127,517 349,692

The performance of the different simulations based on both cell-by-cell and spatial pattern methods of validation are provided in Figure 9. If the validation measures from applying the SPM method produced better results than the cell-by-cell ones, this would be under‐ standable since it is extremely difficult to simulate and predict the precise location of urban lands due to the complexity of the urban system. However, the results in Figure 9 indicate a high degree of consistency and stability in the model. From this it can be inferred that the FCUGM has simulated urban growth based on both local and neighbourhood configura‐ tions to a large extent.

**Figure 9.** Comparison of the FCUGM performance between the cell-by-cell measures and spatial pattern measures for the different simulations and time periods UGB I (1987 – 1977), UGB II (1997 – 2005) and UGB I+II (1987 – 2005)

#### Validating Spatial Patterns of Urban Growth from a Cellular Automata Model http://dx.doi.org/10.5772/51708 43


**4.3. Spatial Pattern Measure (SPM)**

42 Emerging Applications of Cellular Automata

tions to a large extent.

Tables 5 and 6 show the error matrix and statistical indices for the spatial pattern measure for the three FCUGM simulations over the three time periods. These measures include the percentage of agreement, disagreement, accuracy, UCC and LSI when considering the un‐ derlying neighbourhood (Equation 12). It can be seen from Tables 5 and 6 that the perform‐ ance of the FCUGM taking the spatial pattern of neighbourhoods into account shows relatively positive results. For example, the percentage of agreement across all simulations lies between 88% and 94%. The UCC indicates a very high degree of matching between the simulated and observed urban lands with the UCC accuracy as high as 0.765 generated by simulation M3-S1 over the period UGB I+II, and a value of 0.542 generated by simulation M1-S4 and M2-S4 over the period UGB II. The least satisfactory performance was generated by M2-S4 during the period UGB II. In terms of the shape index, this measure shows fairly consistent results similar to the accuracy and the UCC. Those results with high values of UCC and accuracy also generated high shape agreements similar to the findings in section

The performance of the different simulations based on both cell-by-cell and spatial pattern methods of validation are provided in Figure 9. If the validation measures from applying the SPM method produced better results than the cell-by-cell ones, this would be under‐ standable since it is extremely difficult to simulate and predict the precise location of urban lands due to the complexity of the urban system. However, the results in Figure 9 indicate a high degree of consistency and stability in the model. From this it can be inferred that the FCUGM has simulated urban growth based on both local and neighbourhood configura‐

**Figure 9.** Comparison of the FCUGM performance between the cell-by-cell measures and spatial pattern measures for the different simulations and time periods UGB I (1987 – 1977), UGB II (1997 – 2005) and UGB I+II (1987 – 2005)

4.2 where the underlying neighbourhood was not taken into account.


With respect to the simulations, the model results generated from simulations M2-S4 and M3-S1 show good matching with the observed data, while M1-S4 moderately underestimat‐ ed some actual developed areas. During the period UGB II, simulation M3-S1 performed better than the other two simulations (M1-S4 and M3-S1), while over the period UGB I, the

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**Figure 10.** The development profiles by the districts in Riyadh for (a) UGB I (1987 – 1977); (b) UGB II (1997 – 2005);

and (c) UGB I+II (1987 – 2005).

three simulations produced similar moderate levels of urban matching.

**Table 5.** The error matrix for the FCUGM using the spatial pattern measure for the period: (a) UGB I (1987 – 1997); (b) UGB II (1997 – 2005); and (c) UGB I+II (1987 – 2005).


**Table 6.** Statistical performance of the spatial pattern measure for three FCUGM simulations and three time periods: UGB I (1987 – 1977), UGB II (1997 – 2005) and UGB I+II (1987 – 2005).

#### **4.4. Spatial District Structural Measure**

Figure 10 presents the results of the spatial structure indicator, which plots the number of developed areas in each district against the observed ones for the city of Riyadh over the three periods UGB I, II and I+II producing a profile of development by district ID. During the periods UGB I+II and UGB I, the simulation results generated similar patterns to the ob‐ served while, in contrast, the simulation results over the period UGB II underestimated large areas for districts with IDs between 47-50, 75-90 and 110-150.

With respect to the simulations, the model results generated from simulations M2-S4 and M3-S1 show good matching with the observed data, while M1-S4 moderately underestimat‐ ed some actual developed areas. During the period UGB II, simulation M3-S1 performed better than the other two simulations (M1-S4 and M3-S1), while over the period UGB I, the three simulations produced similar moderate levels of urban matching.

Overall 464,167 3,520,790 3,984,957

Urban 342,960 124,775 467,735

Overall 464,167 3,525,790 3,989,957

Urban 355,315 122,649 477,964

Overall 464,167 3,525,779 3,989,946

**M2-S4** Simulated Non-Urban 121,207 3,401,015 3,522,222

**M3-S1** Simulated Non-Urban 108,852 3,403,130 3,511,982

**Table 5.** The error matrix for the FCUGM using the spatial pattern measure for the period: (a) UGB I (1987 – 1997); (b)

**Disagreement**

M1-S4 UGB I 93.7 6.3 0.937 0.542 0.364 M2-S4 UGB I 93.2 6.8 0.932 0.542 0.343 M3-S1 UGB I 93.6 6.4 0.936 0.520 0.347 M1-S4 UGB II 92.1 7.9 0.921 0.343 0.218 M2-S4 UGB II 88.7 11.3 0.887 0.004 0.002 M3-S1 UGB II 91.6 8.4 0.916 0.454 0.259 M1-S4 UGB I+II 90.2 9.8 0.902 0.478 0.375 M2-S4 UGB I+II 93.8 6.2 0.938 0.738 0.582 M3-S1 UGB I+II 94.1 5.9 0.941 0.765 0.605

**Table 6.** Statistical performance of the spatial pattern measure for three FCUGM simulations and three time periods:

Figure 10 presents the results of the spatial structure indicator, which plots the number of developed areas in each district against the observed ones for the city of Riyadh over the three periods UGB I, II and I+II producing a profile of development by district ID. During the periods UGB I+II and UGB I, the simulation results generated similar patterns to the ob‐ served while, in contrast, the simulation results over the period UGB II underestimated

UGB II (1997 – 2005); and (c) UGB I+II (1987 – 2005).

44 Emerging Applications of Cellular Automata

**Simulation Agreement**

**(%)**

UGB I (1987 – 1977), UGB II (1997 – 2005) and UGB I+II (1987 – 2005).

large areas for districts with IDs between 47-50, 75-90 and 110-150.

**4.4. Spatial District Structural Measure**

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

**(%) Accuracy UCC LSI**

**Figure 10.** The development profiles by the districts in Riyadh for (a) UGB I (1987 – 1977); (b) UGB II (1997 – 2005); and (c) UGB I+II (1987 – 2005).

#### **4.5. Spatial Multi-Resolution Validation**

Table 7 shows the error matrix for the FCUGM for simulation M3-S1 over the period UGB I +II (1987 – 2005) at the original and five increasingly coarser spatial resolutions while Table 8 presents the statistical indicators derived from the error matrix.

**Cell Size (m) Agreement (%) Disagreement (%) Accuracy UCC LSI** 93.744 6.256 0.937 0.743 0.605 94.454 5.455 0.944 0.841 0.705 97.467 2.543 0.974 0.981 0.959 94.499 6.251 0.944 0.841 0.707 94.637 5.363 0.946 0.856 0.711 94.367 5.633 0.943 0.847 0.706

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**Table 8.** Statistical performance of the FCUGM for simulation M3-S1 over the period UGB I+II (1987 – 2005) at the

The results show that there are improvements to all the measures reported in Table 8 as the cell size increases from 20 m to a higher resolution. However, some of these improvements are very small and they remain relatively stable as the resolution continues to increase. For example, the accuracy at a 20m resolution is 93.7%, while accuracies at higher resolutions are all around 94%. The only exception is at 80 m where the performance according to all measures is the highest. In terms of urban cell matching, the lowest performance (0.841) was found at 40 and 160 m while moderate UCC accuracies (0.856 and 0.847) were found at spa‐ tial resolutions of 320 and 640 m respectively. Thus, the UCC does not appear to improve very much with a coarser spatial resolution and is likewise quite stable at higher resolutions. With respect to matching the shape between the output of the model and the actual urban image, the LSI also indicates similar values at the higher resolutions with the exceptional performance at a resolution of 80 m. Overall the results suggest that the simulated urban im‐ ages produced by the FCUGM are not that sensitive to spatial resolution, which indicates that a significant feature of the model is its stability and consistency of accuracy over vari‐

Simulating the main processes and drivers of urban growth is a challenging area; research‐ ers are increasingly turning to individual-based models to handle the complexity of these systems. To have any confidence in the outputs of these models, rigorous calibration and validation tests need to be applied. Within this chapter, a series of different measures were used to validate the FCUGM, a complex CA model, for the city of Riyadh. While no one val‐ idation method was found to 'outperform' the others, there was great benefit in using a combination of several approaches. Three different simulations of the FCUGM applied to three different time periods of urban growth were considered. It is clear from the results that the characteristics and patterns of urban development over a particular time period have a large influence on the performance of the model and the resulting accuracy of a given simu‐ lation. For example, over UGB II, urban development has mainly followed a pattern of infill‐ ing of urban growth, i.e. the non-urban areas surrounded by urban areas were converted to

original and five coarser spatial resolutions

**5. Discussion and Conclusions**

ous cell sizes.


**Table 7.** The error matrix of the FCUGM for simulation M3-S1 over the period UGB I+II (1987 – 2005) at the original and five coarser spatial resolutions.


**Table 8.** Statistical performance of the FCUGM for simulation M3-S1 over the period UGB I+II (1987 – 2005) at the original and five coarser spatial resolutions

The results show that there are improvements to all the measures reported in Table 8 as the cell size increases from 20 m to a higher resolution. However, some of these improvements are very small and they remain relatively stable as the resolution continues to increase. For example, the accuracy at a 20m resolution is 93.7%, while accuracies at higher resolutions are all around 94%. The only exception is at 80 m where the performance according to all measures is the highest. In terms of urban cell matching, the lowest performance (0.841) was found at 40 and 160 m while moderate UCC accuracies (0.856 and 0.847) were found at spa‐ tial resolutions of 320 and 640 m respectively. Thus, the UCC does not appear to improve very much with a coarser spatial resolution and is likewise quite stable at higher resolutions. With respect to matching the shape between the output of the model and the actual urban image, the LSI also indicates similar values at the higher resolutions with the exceptional performance at a resolution of 80 m. Overall the results suggest that the simulated urban im‐ ages produced by the FCUGM are not that sensitive to spatial resolution, which indicates that a significant feature of the model is its stability and consistency of accuracy over vari‐ ous cell sizes.

#### **5. Discussion and Conclusions**

**4.5. Spatial Multi-Resolution Validation**

46 Emerging Applications of Cellular Automata

presents the statistical indicators derived from the error matrix.

**Cell Size** Observed

Table 7 shows the error matrix for the FCUGM for simulation M3-S1 over the period UGB I +II (1987 – 2005) at the original and five increasingly coarser spatial resolutions while Table 8

**20 (Original)** Simulated Non-Urban 108,852 3,114,171 3,223,023

**40** Simulated Non-Urban 27,264 890,363 917,627

**80** Simulated Non-Urban 6,738 222,551 229,289

**160** Simulated Non-Urban 1,721 55,577 57,298

**320** Simulated Non-Urban 380 13,933 14,313

**640** Simulated Non-Urban 104 3,459 3,563

and five coarser spatial resolutions.

**Table 7.** The error matrix of the FCUGM for simulation M3-S1 over the period UGB I+II (1987 – 2005) at the original

Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Observed Urban Non-Urban Overall

Urban 355,315 122,649 477,964

Overall 464,167 3,236,820 3,700,987

Urban 144,635 33,206 177,841

Overall 171,899 923,569 1,095,468

Urban 361,127 8,430 369,557

Overall 367,865 230,981 598,846

Urban 9,120 2,045 11,165

Overall 10,841 57,622 68,463

Urban 2,269 538 2,807

Overall 2,649 14,471 17,120

Urban 579 137 716

Overall 683 3,596 4,279

Simulating the main processes and drivers of urban growth is a challenging area; research‐ ers are increasingly turning to individual-based models to handle the complexity of these systems. To have any confidence in the outputs of these models, rigorous calibration and validation tests need to be applied. Within this chapter, a series of different measures were used to validate the FCUGM, a complex CA model, for the city of Riyadh. While no one val‐ idation method was found to 'outperform' the others, there was great benefit in using a combination of several approaches. Three different simulations of the FCUGM applied to three different time periods of urban growth were considered. It is clear from the results that the characteristics and patterns of urban development over a particular time period have a large influence on the performance of the model and the resulting accuracy of a given simu‐ lation. For example, over UGB II, urban development has mainly followed a pattern of infill‐ ing of urban growth, i.e. the non-urban areas surrounded by urban areas were converted to urban, while very limited development took place on the margins or fringe areas of the city. This type of development exhibits a highly non-linear pattern, where the new potential de‐ veloped land occurs in very small clusters that are surrounded by very large urban clusters. Consequently, the simulation results over this period were the least satisfactory when com‐ pared with the other two time periods. It is worth noting that this pattern was generated as a result of applying the urban growth limit regulations (as advocated by the planning local authority of Riyadh) to prevent urban sprawl. The urban growth pattern over the periods UGB I and UGB I+II can be characterised by a pattern of edge-expansion, where the newly developed urban areas spread out from the fringes or margins of existing urban patches. This feature was modeled in a satisfactory manner during these two periods of growth.

identical fractal dimensions. Thus, this measure tells us very little about how similar the two maps may be in terms of local structures. Although the approach reflects how much space is filled correctly across a range of scales, it does not seem to be valid when dealing with nonurban situations [1]. However, other approaches involving comparison with null models re‐ quire further investigation [40]. What remains clear from this study and the current state of validation approaches in the CA urban modelling literature is that there is no one best meth‐ od or set of approaches available for validating CA urban growth models. Many different methods are available and the best approach appears to be validation using multiple meas‐ ures. Ultimately, these measures must be linked to confidence in the model performance and the ability to simulate future growth especially when they move from an academic and

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49

experimental environment to real world applications by planners.

\*Address all correspondence to: a.j.heppenstall@leeds.ac.uk

4 School of Geography, University of Leeds, Leeds, UK

[2] Batty, M. (1995). New Ways of Looking at Cities. *Nature*, 574.

plexity. Amsterdam Gordon and Breach Science.

[3] Portugali, J. (2000). Self-Organization and the City. Berlin: Springer-Verlag.

tems Analysis (IIASA), Laxenburg, Austria

, Linda See2,3 and Alison Heppenstall4\*

1 King Abdulaziz City for Science and Technology (KACST), Riyadh, Saudi Arabia

2 Ecosystems Services and Management Programme, International Institute of Applied Sys‐

[1] White, R., & Engelen, G. (2000). High Resolution Integrated Modelling of the Spatial Dynamics of Urban and Regional Systems. *Computers, Environment and Urban Sys‐*

[4] Allen, PM. (1997). Cities and Regions as Self-organizing Systems: Models of Com‐

[5] Batty, M., & Longley, P. (1994). Fractal Cities: A Geometry of Form and Function.

3 Centre for Applied Spatial Analysis (CASA), University College London, London, UK

**Author details**

Khalid Al-Ahmadi1

**References**

*tems*, 24, 383-440.

London Academic Press.

Similarly, the characteristics of the simulation are another factor that can have a significant impact on the results, which was clearly supported by consistency across the different vali‐ dation measures when examining the three simulations, i.e. simulation M3-S1 produced the best spatial simulation over all of the periods followed by M1-S4 and M2-S4. It is worth not‐ ing that the three urban growth factors, i.e. transportation, urban density and attractiveness, and topographical constraints, were part of all three simulations M1-S4, M2-S4 and M3-S1. However, the difference between these model instances has to do with the form of the fuzzy rules and how many factors are combined in each rule. M1-S4 embeds only one factor, M2- S4 embeds two factors and M3-S1 combines all three factors in each fuzzy rule. Embedding all factors into the fuzzy rules and combining these via the AND operator appears to have produced the best performing model. However, M1-S4, with only one factor per fuzzy rule, generally outperformed M2-S4 with two factors in each rule but containing all three factors in the model with more rules needed to capture all the possible pairs of factors. Perhaps re‐ stricting the model to rules with only two factors produced a model that was actually more complex than the simple M1-S4 and even the M3-S1 simulation, but less able to capture ur‐ ban growth as adequately.

Overall there was consistency between the measures regarding which model instance per‐ formed better and for which growth periods. The visual inspection provided an overall qualitative assessment that would not have been possible using any of the quantitative measures and is therefore always recommended as a method of model validation. The accu‐ racy measures are very sensitive to the number of non-urban cells and should mostly likely not be used or reported in conjunction with the UCC, which took only urban cells into ac‐ count. This measure provides a much better assessment of model performance. The meas‐ ures that took shape or underlying neighbourhood into account are also valuable. In this case, they provided a consistent message regarding model performance but they could help to identify models that are good global predictors but are not spatially or locally very good. Finally the analysis at multiple resolutions provides a good indication of model stability across spatial scales and should be implemented as a minimum measure of validation as ad‐ vocated in [40].

While the validation techniques used in this work provided a comprehensive assessment of the model outputs, there are other techniques available, e.g. fractal dimensional analysis [34, 46]. However, this approach has limitations, e.g. two maps that seem different may have identical fractal dimensions. Thus, this measure tells us very little about how similar the two maps may be in terms of local structures. Although the approach reflects how much space is filled correctly across a range of scales, it does not seem to be valid when dealing with nonurban situations [1]. However, other approaches involving comparison with null models re‐ quire further investigation [40]. What remains clear from this study and the current state of validation approaches in the CA urban modelling literature is that there is no one best meth‐ od or set of approaches available for validating CA urban growth models. Many different methods are available and the best approach appears to be validation using multiple meas‐ ures. Ultimately, these measures must be linked to confidence in the model performance and the ability to simulate future growth especially when they move from an academic and experimental environment to real world applications by planners.

#### **Author details**

urban, while very limited development took place on the margins or fringe areas of the city. This type of development exhibits a highly non-linear pattern, where the new potential de‐ veloped land occurs in very small clusters that are surrounded by very large urban clusters. Consequently, the simulation results over this period were the least satisfactory when com‐ pared with the other two time periods. It is worth noting that this pattern was generated as a result of applying the urban growth limit regulations (as advocated by the planning local authority of Riyadh) to prevent urban sprawl. The urban growth pattern over the periods UGB I and UGB I+II can be characterised by a pattern of edge-expansion, where the newly developed urban areas spread out from the fringes or margins of existing urban patches. This feature was modeled in a satisfactory manner during these two periods of growth.

Similarly, the characteristics of the simulation are another factor that can have a significant impact on the results, which was clearly supported by consistency across the different vali‐ dation measures when examining the three simulations, i.e. simulation M3-S1 produced the best spatial simulation over all of the periods followed by M1-S4 and M2-S4. It is worth not‐ ing that the three urban growth factors, i.e. transportation, urban density and attractiveness, and topographical constraints, were part of all three simulations M1-S4, M2-S4 and M3-S1. However, the difference between these model instances has to do with the form of the fuzzy rules and how many factors are combined in each rule. M1-S4 embeds only one factor, M2- S4 embeds two factors and M3-S1 combines all three factors in each fuzzy rule. Embedding all factors into the fuzzy rules and combining these via the AND operator appears to have produced the best performing model. However, M1-S4, with only one factor per fuzzy rule, generally outperformed M2-S4 with two factors in each rule but containing all three factors in the model with more rules needed to capture all the possible pairs of factors. Perhaps re‐ stricting the model to rules with only two factors produced a model that was actually more complex than the simple M1-S4 and even the M3-S1 simulation, but less able to capture ur‐

Overall there was consistency between the measures regarding which model instance per‐ formed better and for which growth periods. The visual inspection provided an overall qualitative assessment that would not have been possible using any of the quantitative measures and is therefore always recommended as a method of model validation. The accu‐ racy measures are very sensitive to the number of non-urban cells and should mostly likely not be used or reported in conjunction with the UCC, which took only urban cells into ac‐ count. This measure provides a much better assessment of model performance. The meas‐ ures that took shape or underlying neighbourhood into account are also valuable. In this case, they provided a consistent message regarding model performance but they could help to identify models that are good global predictors but are not spatially or locally very good. Finally the analysis at multiple resolutions provides a good indication of model stability across spatial scales and should be implemented as a minimum measure of validation as ad‐

While the validation techniques used in this work provided a comprehensive assessment of the model outputs, there are other techniques available, e.g. fractal dimensional analysis [34, 46]. However, this approach has limitations, e.g. two maps that seem different may have

ban growth as adequately.

48 Emerging Applications of Cellular Automata

vocated in [40].

Khalid Al-Ahmadi1 , Linda See2,3 and Alison Heppenstall4\*

\*Address all correspondence to: a.j.heppenstall@leeds.ac.uk

1 King Abdulaziz City for Science and Technology (KACST), Riyadh, Saudi Arabia

2 Ecosystems Services and Management Programme, International Institute of Applied Sys‐ tems Analysis (IIASA), Laxenburg, Austria

3 Centre for Applied Spatial Analysis (CASA), University College London, London, UK

4 School of Geography, University of Leeds, Leeds, UK

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**Chapter 3**

**Cellular Automata for Pattern Recognition**

Cellular Automata (CA) are spatiotemporal discrete systems (Neumann, 1966) that can mod‐ el dynamic complex systems. A variety of problem domains have been reported to date in suc‐ cessful CA applications. In this regard, digital image processing is one of those as reported by Wongthanavasu et. al. (Wongthanavasu et al., 2003; 2004; 2007) and Rosin (Rosin, 2006).

Generalized Multiple Attractor CA (GMACA) is introduced for elementary pattern recogni‐ tion (Ganguly et al., 2002; Maji et al., 2003; 2008). It is a promising pattern classifier using a simple local network of Elementary Cellular Automata (ECA) (Wolfram, 1994), called attrac‐ tor basin that is a reverse tree-graph. GMACA utilizes a reverse engineering technique and genetic algorithm in ordering the CA rules. This leads to a major drawback of computational complexity, as well as recognition performance. There are reports in successful applications of GMACA in error correcting problem with only one bit noise. It shows the promising re‐ sults for the restricted one bit noise, but becomes combinatorial explosion in complexity, us‐

Due to the drawbacks of complexity and recognition performance stated previously, the bi‐ nary CA-based classifier, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artificial point (2C2-GMACA), is presented. In this regard, a pattern recogni‐ tion of error correcting capability is implemented comprehensively in comparison with GMACA. Following this, the basis on CA for pattern recognition and GMACA's configura‐ tion are presented. Then, the 2C2-GMACA model and its performance evaluation in com‐

> © 2013 Wongthanavasu and Ponkaew; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

© 2013 Wongthanavasu and Ponkaew; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

parison with GMACA are provided. Finally, conclusions and discussions are given.

ing associative memory, when a number of bit noises increases.

properly cited.

Sartra Wongthanavasu and Jetsada Ponkaew

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52364

**1. Introduction**


## **Cellular Automata for Pattern Recognition**

Sartra Wongthanavasu and Jetsada Ponkaew

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52364

#### **1. Introduction**

[33] Santé, I., Garcia, A. M., Miranda, D., & Crecente, R. (2010). Cellular Automata Mod‐ els for the Simulation of Real-world Urban Processes: A Review and Analysis. *Land‐*

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[35] Soares-Filho, B., Coutinho-Cerqueira, G., & Lopes-Pennachin, C. (2002). DINAMICA-Stochastic Cellular Automata Model Designed to Simulate the Landscape Dynamics

[36] Clarke, K. C., Hoppen, S., & Gaydos, L. (1997). A Self-modifying Cellular Automaton Model of Historical Urbanization in the San Francisco Bay Area. *Environment and*

[37] Ward, D. P., Murray, AT, & Phinn, S. R. (2000). A Stochastically Constrained Cellular Model of Urban Growth. *Computers, Environment and Urban Systems.*, 24, 539-558. [38] Barredo, J. I., Demicheli, L., Lavalle, C., Kasanko, M., & Mc Cormick, N. (2004). Mod‐ elling Future Urban Scenarios in Developing Countries: An Application Case Study

[39] Cheng, J., & Masser, I. (2004). Understanding Spatial and Temporal Process of Urban Growth: Cellular Automata Modeling. *Environment and Planning B.*, 31, 167-194. [40] Pontius, R., Huffaker, D., & Denman, K. (2004). Useful Techniques of Validation for Spatially Explicit Land-change Model. *Ecological Modeling.*, 179(4), 445-461.

[41] Wu, F., & Webster, C. J. (1998). Simulation of Land Development through the Inte‐ gration of Cellular Automata and Multi-criteria Evaluation. *Environment and Planning*

[42] Lee, D., & Sallee, T. (1974). Theoretical Patterns of Farm Shape and Central Place Lo‐

[43] Jantz, C., & Goetz, S. (2005). Analysis of Scale Dependencies in an Urban Land-use Change Model. *International Journal of Geographical Information Science.*, 19, 271-241.

[44] Kok, K., & Veldkamp, A. (2001). Evaluating Impact of Spatial Scales on Land Use Pattern Analysis in Central America. *Agricultural, Ecosystems and Environment.*, 85,

[45] Kok, K., Farrow, A., Veldkamp, A., & Verburg, P. (2001). A Method and Application of Multi-scale Validation in Spatial and Land Use Models. *Agricultural, Ecosystems*

[46] Frankhauser, P., & Sadler, R. (1991). Fractal Analysis of Agglomerations. In: Natural Structures: Principles, Strategies, and Models in Architecture and Nature. M. Hilliges

in Lagos, Nigeria. *Environment and Planning B.*, 31(1), 65-84.

cation. *Journal of Regional Science*, 14(3), 423-430.

(Ed.), 57-65, Stuttgart: University of Stuttgart.

in an Amazonian Colonization Frontier. *Ecological Modelling.*, 154, 217-235.

*scape and Urban Planning*, 96(2), 108-122.

*ning A*, 25(8), 1175-1199.

52 Emerging Applications of Cellular Automata

*Planning B.*, 24(2), 247-261.

*B.*, 25, 103-126.

205-221.

*and Environment.*, 85, 223-238.

Cellular Automata (CA) are spatiotemporal discrete systems (Neumann, 1966) that can mod‐ el dynamic complex systems. A variety of problem domains have been reported to date in suc‐ cessful CA applications. In this regard, digital image processing is one of those as reported by Wongthanavasu et. al. (Wongthanavasu et al., 2003; 2004; 2007) and Rosin (Rosin, 2006).

Generalized Multiple Attractor CA (GMACA) is introduced for elementary pattern recogni‐ tion (Ganguly et al., 2002; Maji et al., 2003; 2008). It is a promising pattern classifier using a simple local network of Elementary Cellular Automata (ECA) (Wolfram, 1994), called attrac‐ tor basin that is a reverse tree-graph. GMACA utilizes a reverse engineering technique and genetic algorithm in ordering the CA rules. This leads to a major drawback of computational complexity, as well as recognition performance. There are reports in successful applications of GMACA in error correcting problem with only one bit noise. It shows the promising re‐ sults for the restricted one bit noise, but becomes combinatorial explosion in complexity, us‐ ing associative memory, when a number of bit noises increases.

Due to the drawbacks of complexity and recognition performance stated previously, the bi‐ nary CA-based classifier, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artificial point (2C2-GMACA), is presented. In this regard, a pattern recogni‐ tion of error correcting capability is implemented comprehensively in comparison with GMACA. Following this, the basis on CA for pattern recognition and GMACA's configura‐ tion are presented. Then, the 2C2-GMACA model and its performance evaluation in com‐ parison with GMACA are provided. Finally, conclusions and discussions are given.

properly cited.

© 2013 Wongthanavasu and Ponkaew; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is © 2013 Wongthanavasu and Ponkaew; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### **2. Cellular Automata for Pattern Recognition**

Elementary Cellular Automata (ECA) (Wolfram, 1994) is generally utilized as a basis on pat‐ tern recognition. It is the simplest class of one dimension (1d) CA with *n* cells, 2 states and 3 neighbors. A state is changed in discrete time and space ( *Si <sup>t</sup>* <sup>→</sup>*Si <sup>t</sup>*+1 ; where *Si <sup>t</sup>* is the present state and *Si <sup>t</sup>*+1 is the next state for the *i th* cell) by considering it nearest neighbor ( *Si*-1 *<sup>t</sup>* , *Si t* , *Si*+1 *<sup>t</sup>* ) of the present state.

*Si*

*ji*

mal.

lowing:

present state ( *S <sup>t</sup>*

*<sup>t</sup>*+1 is the next state of the *i*

is the 3 neighbouring values ( *Si*-1

*th* cell.

( )

*t*

classifiers based on the evolution of the ECA is defined as following

*t*

**3. Generalized Multiple Attractor Cellular Automata**

by Hamming distance (*r*) defined as follows:

<sup>+</sup> <sup>ì</sup>

,

*M S*

=

*t Si t Si*+1

1 11 1 01 1

+ ++ +

*t tt t*

= ¼

*S SS S*

( , ,, ) ( 0, , 1, , 1 , )

= ¼ -

<sup>1</sup> ( ) , ,

<sup>ï</sup> <sup>Ï</sup> <sup>=</sup> <sup>í</sup>

*t M S if S Y <sup>S</sup>*

ïî

*<sup>t</sup>* ) of the present state at the *i*

*n*


The next state (*S <sup>t</sup>*+1) for *n*-cell ECA calculated is also defined by the rule matrix *M* as fol‐

( ) ( ) ( )

*M j M j Mn j*

*n*


01 1

Suppose a system designed with a rule matrix (*M*) comprises a set of solutions *Y*= { *yi* <sup>|</sup> *yi* <sup>∈</sup>{0,1}*<sup>n</sup>*} and an input *x*;*<sup>x</sup>* <sup>∈</sup>{0,1}*<sup>n</sup>* , where *i* =*1, 2…, N*. Consequently, the pattern

,

*S and stop otherwise*

For an input *x* , it must be identified a solution from *Y* using the equation (3). Firstly, the

matrix *M* until it reaches some solution ( *S <sup>t</sup>* ∈*Y* ). The structure for pattern classification us‐ ing ECA can be represented by a simple local network called attractor basin. It consists of a cyclic and non-cyclic states. The cyclic state contains a pivotal point which is a solution to clas‐ sification problem, while the transient states (all possible inputs) are contained in the non-cy‐ clic states. The attractor cycle lengths (height) in the GMACA (Oliveira, et al., 2006; Sipper, 1996) are greater than or equal to one, while Multiple Attractor Cellular Automata (MACA) (Das, et al., 2008; Maji, et al., 2003; Sipper, 1996) is limited to one. In Fig. 1(b), two attractor ba‐ sins of 4-bit pattern of MACA with null boundary condition are designed with a rule vector <60, 150, 60, 240>. The target solution patterns are 0000 and 1011, respectively. The rule vec‐ tor is ordered by the evolution of heuristic search using simulated annealing algorithm.

This section gives the detailed configuration of GMACA and its application in ECC. Sup‐ pose *an n*-bit pattern is sent in a communication system. Let *X* be the sender's pattern and *Y* be the receiver's pattern. Thus, the number of different bits between *X* and *Y* is determined

) will be set to *x* . Then, the next state ( *S <sup>t</sup>*+1 ) will be generated using the rule

*t t*

*th* cell decoded in deci‐

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364

(2)

55

(3)

**Figure 1.** Elementary Cellular Automata (ECA) and Generalized Multiple Attractor Cellular Automata (GMACA).

For *n*-cell ECA, the next state function ( *Si <sup>t</sup>* <sup>→</sup>*Si <sup>t</sup>*+1) can be represented by a rule matrix (*M*) with size |*n*x8| and the nearest neighbour configuration ( *Si*-1 *<sup>t</sup>* , *Si t* , *Si*+1 *<sup>t</sup>* ) of the present state. Suppose a*n n*-cell ECA ( *S*<sup>0</sup> *t S*1 *t S*2 *<sup>t</sup>* …*Sn*-1 *<sup>t</sup>* ) at time 't' is changed in discrete time by a rule vector <*R*0, *R*1, …, *Rn*-1 > . A truth table is a simplified form of the rule vector as illustrated in Fig. 1(a). It comprises the possible 3 neighbor values of *Si*-1 *t Si t Si*+1 *<sup>t</sup>* from 000 to 111, and the next states for the rule *R <sup>i</sup>* ; where *i*=0, 1, 2…, n-1. Each rule is represented in binary num‐ bers (*b7 b6 b5 b4 b3b2b1b0*). If the binary numbers are decoded into decimal, it must equal to the number *Ri* such as '01011010' for the rule-90. Simultaneously, A rule matrix (M) can also be represented the rule vector.

Let *M*(*i,j*) be an element of the matrix at the *i th* (*i=0,1,2,...,n-1*) row and the *j th* (*j=0,1,2,...,7*) col‐ umn. The *M*(*i,j*) is contained *bj* of the rule-*Ri* . For example, *M*(2,3) is *b*3 of the rule *R2* (the rule-90) that is '1'. Consequently, the next state ( *Si <sup>t</sup>*+1 ) for the *i th* cell is represented by the *M*(*i,j*) as the following:

$$S\_{\ell}^{\prime t+1} = M\left(t, j\_{\ell}\right) \tag{1}$$

where;

*Si <sup>t</sup>*+1 is the next state of the *i th* cell.

**2. Cellular Automata for Pattern Recognition**

neighbors. A state is changed in discrete time and space ( *Si*

*<sup>t</sup>*+1 is the next state for the *i*

*<sup>t</sup>* ) of the present state.

For *n*-cell ECA, the next state function ( *Si*

Let *M*(*i,j*) be an element of the matrix at the *i*

rule-90) that is '1'. Consequently, the next state ( *Si*

state. Suppose a*n n*-cell ECA ( *S*<sup>0</sup>

the next states for the rule *R <sup>i</sup>*

represented the rule vector.

*M*(*i,j*) as the following:

umn. The *M*(*i,j*) is contained *bj*

number *Ri*

where;

with size |*n*x8| and the nearest neighbour configuration ( *Si*-1

*t S*1 *t S*2 *<sup>t</sup>* …*Sn*-1

in Fig. 1(a). It comprises the possible 3 neighbor values of *Si*-1

state and *Si*

54 Emerging Applications of Cellular Automata

( *Si*-1 *<sup>t</sup>* , *Si t* , *Si*+1

Elementary Cellular Automata (ECA) (Wolfram, 1994) is generally utilized as a basis on pat‐ tern recognition. It is the simplest class of one dimension (1d) CA with *n* cells, 2 states and 3

**Figure 1.** Elementary Cellular Automata (ECA) and Generalized Multiple Attractor Cellular Automata (GMACA).

*<sup>t</sup>* <sup>→</sup>*Si*

vector <*R*0, *R*1, …, *Rn*-1 > . A truth table is a simplified form of the rule vector as illustrated

bers (*b7 b6 b5 b4 b3b2b1b0*). If the binary numbers are decoded into decimal, it must equal to the

of the rule-*Ri*

( ) <sup>1</sup> , *<sup>t</sup>*

such as '01011010' for the rule-90. Simultaneously, A rule matrix (M) can also be

*<sup>t</sup>* <sup>→</sup>*Si*

*<sup>t</sup>*+1 ; where *Si*

*th* cell) by considering it nearest neighbor

*<sup>t</sup>*+1) can be represented by a rule matrix (*M*)

*<sup>t</sup>* ) of the present

*<sup>t</sup>* from 000 to 111, and

*th* (*j=0,1,2,...,7*) col‐

*<sup>t</sup>* , *Si t* , *Si*+1

*t Si t Si*+1

; where *i*=0, 1, 2…, n-1. Each rule is represented in binary num‐

*th* (*i=0,1,2,...,n-1*) row and the *j*

*<sup>t</sup>* ) at time 't' is changed in discrete time by a rule

. For example, *M*(2,3) is *b*3 of the rule *R2* (the

*i i S Mij* <sup>+</sup> <sup>=</sup> (1)

*<sup>t</sup>*+1 ) for the *i th* cell is represented by the

*<sup>t</sup>* is the present

*ji* is the 3 neighbouring values ( *Si*-1 *t Si t Si*+1 *<sup>t</sup>* ) of the present state at the *i th* cell decoded in deci‐ mal.

The next state (*S <sup>t</sup>*+1) for *n*-cell ECA calculated is also defined by the rule matrix *M* as fol‐ lowing:

$$\begin{aligned} \boldsymbol{S}^{t+1} &= (\boldsymbol{S}\_0^{t+1}, \boldsymbol{S}\_1^{t+1}, \dots, \boldsymbol{S}\_{n-1}^{t+1}) \\ \boldsymbol{S}^t &= (\boldsymbol{M}\left(\boldsymbol{0}, \boldsymbol{j}\_0\right), \boldsymbol{M}\left(\boldsymbol{1}, \boldsymbol{j}\_1\right), \dots, \boldsymbol{M}\left(\boldsymbol{n} - \boldsymbol{1}, \boldsymbol{j}\_{n-1}\right)) \\ \boldsymbol{S}^t &= \left(\boldsymbol{M}, \boldsymbol{S}^t\right) \end{aligned} \tag{2}$$

Suppose a system designed with a rule matrix (*M*) comprises a set of solutions *Y*= { *yi* <sup>|</sup> *yi* <sup>∈</sup>{0,1}*<sup>n</sup>*} and an input *x*;*<sup>x</sup>* <sup>∈</sup>{0,1}*<sup>n</sup>* , where *i* =*1, 2…, N*. Consequently, the pattern classifiers based on the evolution of the ECA is defined as following

$$S^{-t+1} = \begin{cases} (M, S^t), \text{if } S^t \not\in Y\\ S^t \text{ and stop, otherwise} \end{cases} \tag{3}$$

For an input *x* , it must be identified a solution from *Y* using the equation (3). Firstly, the present state ( *S <sup>t</sup>* ) will be set to *x* . Then, the next state ( *S <sup>t</sup>*+1 ) will be generated using the rule matrix *M* until it reaches some solution ( *S <sup>t</sup>* ∈*Y* ). The structure for pattern classification us‐ ing ECA can be represented by a simple local network called attractor basin. It consists of a cyclic and non-cyclic states. The cyclic state contains a pivotal point which is a solution to clas‐ sification problem, while the transient states (all possible inputs) are contained in the non-cy‐ clic states. The attractor cycle lengths (height) in the GMACA (Oliveira, et al., 2006; Sipper, 1996) are greater than or equal to one, while Multiple Attractor Cellular Automata (MACA) (Das, et al., 2008; Maji, et al., 2003; Sipper, 1996) is limited to one. In Fig. 1(b), two attractor ba‐ sins of 4-bit pattern of MACA with null boundary condition are designed with a rule vector <60, 150, 60, 240>. The target solution patterns are 0000 and 1011, respectively. The rule vec‐ tor is ordered by the evolution of heuristic search using simulated annealing algorithm.

#### **3. Generalized Multiple Attractor Cellular Automata**

This section gives the detailed configuration of GMACA and its application in ECC. Sup‐ pose *an n*-bit pattern is sent in a communication system. Let *X* be the sender's pattern and *Y* be the receiver's pattern. Thus, the number of different bits between *X* and *Y* is determined by Hamming distance (*r*) defined as follows:

$$r = \sum\_{\iota=0}^{n-1} |\mathbf{x}\_{\iota} - \mathbf{y}\_{\iota}| \tag{4}$$

tractor basin equals *pAll*. Then original messages are randomly mapped into pivotal points while its possible errors are also randomly mapped into transient states at the same attract basin. Finally, the search heuristics, such as simulated annealing (SA) and genetic algorithm (GA) (Holland, 1992; Shuai, et al., 2007; Jie, et al., 2002) have been taken to explore the opti‐ mal structure. The search heuristics then iteratively changes directions and height of the at‐

As reported in Ganguly, et al., 2002, Maji, et al., 2003 and Maji, et al., 2008, the GMACA pro‐ vides the best performance of pattern recognition if it is trained with the *rmax* having a value of 1. Although percentage of recognition in testing is high when deals with the *rmax* equals 1,

Due to the drawbacks of recognition performance resulting from the increasing *rmax* and search space complexity in rule ordering, the proposed method, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artificial point (2C2-GMACA) (Pon‐ kaew, et al., 2011; Ponkaew, et al., 2011), is introduced. The 2C2-GMACA is designed based on two class classifier architecture basis. In this regard, two classes are taken to process at a time and a solution is binary answer +1 or -1, which is a pointer to the class label of solution. There are two kinds of attractor basins: a positive attractor basin that returns the +1 as the

, *yi*

the positive and negative attractor basins, respectively. Given *x* ∈{0,1}*<sup>n</sup>* as an input, the x must be assigned a class label which is a solution to the pattern recognition. The 2C2-GMA‐

equation (2) to the next state ( *S <sup>t</sup>*+1 ). Next, the binary decision function will take *S <sup>t</sup>*+1 and

1 1


0 0

*i i f S A sgn S A S A*

= =

*ii ii*

( 1) 1 1

*th* cell.

*th* cell.

( ,) ( . .) *n n t tt*

artificial point (*A)* as parameters as the equation (7) to assign the class.

*th* class label and *i*=*1,2,…N.* Let *L* + and *L* -

), where *xi* <sup>∈</sup>{0,1}*<sup>n</sup>* is the *<sup>i</sup>*

) to *x* . Then, the *S <sup>t</sup>* will be evolved with the

<sup>=</sup> å å- (7)

*th* cell.

*th* pattern

be a class label of

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364 57

it sharply decreases the recognition performance when the *rmax* is greater than 1.

tractor basins until the satisfied rule vector is acquired.

**4. Proposed 2C2-GMACA Model**

result and a negative attractor basin, otherwise.

Suppose a system consists of patterns (*xi*

} is the *i*

CA begins with setting the present state ( *S <sup>t</sup>*

, *L* -

*sgn*(\_ ) denotes the sign function.

*<sup>t</sup>*+1 represents the next state for the *i*

represents the artificial point for the *i*

*t*+1 represents a bit complement of the next state for the *i*

, *and yi* <sup>∈</sup>{*<sup>L</sup>* <sup>+</sup>

where

*Si*

*Ai*

*S*¯ *i*

where *X* = *x*0*x*1… *xn*-1; *xi* ∈{0,1} and *Y* = *y*0*y*1… *yn*-1; *yi* ∈{0,1} *.*

The number of possible error patterns (*pr*) for a given *r* of *n*-bit communication can be ex‐ pressed as follow:

$$p\_r = \binom{n}{r} \tag{5}$$

Then, the number of all possible error patterns (*pAll*) for a given *rmax*, where *rmax* ∈(0, *n*) is the maximum permissible noise, is given by:

$$p\_{All} = \sum\_{r=0}^{r\_{\text{max}}} \binom{n}{r} \tag{6}$$

The maximum permissible noise (*rmax*) is the highest value of *r* allowed to occur in the com‐ munication system. The Hamming distance model of a message (pattern) and it errors are also represented by an attractor basin—that is, the messages is a pivotal point while the er‐ rors are transient states. Thus, the error correcting codes can be solved by the Generalized Multiple Attractor Cellular Automata (GMACA).

**Figure 2.** Two-Class Classifier GMACA with artificial point (2C2-GMACA): <232,212,178,142 >.

Suppose a communication system comprises *k* original messages of *n*-bit data and the maxi‐ mum permissible noise *rmax*. If error messages are corrected using the GMACA, thus a satis‐ fied rule vector is required. The rule vector is a result of a reverse engineering technique. Firstly, *k* attractor basins are randomly constructed with the number of nodes for each at‐ tractor basin equals *pAll*. Then original messages are randomly mapped into pivotal points while its possible errors are also randomly mapped into transient states at the same attract basin. Finally, the search heuristics, such as simulated annealing (SA) and genetic algorithm (GA) (Holland, 1992; Shuai, et al., 2007; Jie, et al., 2002) have been taken to explore the opti‐ mal structure. The search heuristics then iteratively changes directions and height of the at‐ tractor basins until the satisfied rule vector is acquired.

As reported in Ganguly, et al., 2002, Maji, et al., 2003 and Maji, et al., 2008, the GMACA pro‐ vides the best performance of pattern recognition if it is trained with the *rmax* having a value of 1. Although percentage of recognition in testing is high when deals with the *rmax* equals 1, it sharply decreases the recognition performance when the *rmax* is greater than 1.

#### **4. Proposed 2C2-GMACA Model**

Due to the drawbacks of recognition performance resulting from the increasing *rmax* and search space complexity in rule ordering, the proposed method, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artificial point (2C2-GMACA) (Pon‐ kaew, et al., 2011; Ponkaew, et al., 2011), is introduced. The 2C2-GMACA is designed based on two class classifier architecture basis. In this regard, two classes are taken to process at a time and a solution is binary answer +1 or -1, which is a pointer to the class label of solution. There are two kinds of attractor basins: a positive attractor basin that returns the +1 as the result and a negative attractor basin, otherwise.

Suppose a system consists of patterns (*xi* , *yi* ), where *xi* <sup>∈</sup>{0,1}*<sup>n</sup>* is the *<sup>i</sup> th* pattern , *and yi* <sup>∈</sup>{*<sup>L</sup>* <sup>+</sup> , *L* - } is the *i th* class label and *i*=*1,2,…N.* Let *L* + and *L* be a class label of the positive and negative attractor basins, respectively. Given *x* ∈{0,1}*<sup>n</sup>* as an input, the x must be assigned a class label which is a solution to the pattern recognition. The 2C2-GMA‐ CA begins with setting the present state ( *S <sup>t</sup>* ) to *x* . Then, the *S <sup>t</sup>* will be evolved with the equation (2) to the next state ( *S <sup>t</sup>*+1 ). Next, the binary decision function will take *S <sup>t</sup>*+1 and artificial point (*A)* as parameters as the equation (7) to assign the class.

$$f\left(S^{\left(t+1\right)}, A\right) = \text{sgn}(\sum\_{l=0}^{n-1} S\_l^{t+1} \mathcal{A}\_l - \sum\_{l=0}^{n-1} \overline{S}\_l^{t+1} \mathcal{A}\_l) \tag{7}$$

where

1

*n r xy* -

0 *i i*

=

*r*

*p*

*All*

*p*

**Figure 2.** Two-Class Classifier GMACA with artificial point (2C2-GMACA): <232,212,178,142 >.

Suppose a communication system comprises *k* original messages of *n*-bit data and the maxi‐ mum permissible noise *rmax*. If error messages are corrected using the GMACA, thus a satis‐ fied rule vector is required. The rule vector is a result of a reverse engineering technique. Firstly, *k* attractor basins are randomly constructed with the number of nodes for each at‐

where *X* = *x*0*x*1… *xn*-1; *xi* ∈{0,1} and *Y* = *y*0*y*1… *yn*-1; *yi* ∈{0,1} *.*

pressed as follow:

56 Emerging Applications of Cellular Automata

maximum permissible noise, is given by:

Multiple Attractor Cellular Automata (GMACA).

*i*

The number of possible error patterns (*pr*) for a given *r* of *n*-bit communication can be ex‐

*n*

*r* æ ö <sup>=</sup> ç ÷

Then, the number of all possible error patterns (*pAll*) for a given *rmax*, where *rmax* ∈(0, *n*) is the

*n*

max 0

*r* <sup>=</sup> æ ö <sup>=</sup> ç ÷

The maximum permissible noise (*rmax*) is the highest value of *r* allowed to occur in the com‐ munication system. The Hamming distance model of a message (pattern) and it errors are also represented by an attractor basin—that is, the messages is a pivotal point while the er‐ rors are transient states. Thus, the error correcting codes can be solved by the Generalized

*r*

*r*

= - å (4)

è ø (5)

è ø <sup>å</sup> (6)

*sgn*(\_ ) denotes the sign function.

*Si <sup>t</sup>*+1 represents the next state for the *i th* cell.

*Ai* represents the artificial point for the *i th* cell.

*S*¯ *i t*+1 represents a bit complement of the next state for the *i th* cell. (.) denotes "AND" logical operator.

Finally, the *x* is considered to be a member of the positive attractor basin and returns *L+* if  *f* (*S <sup>t</sup>*+1, *A* ) = + 1 , and returns *L* - , otherwise.

*Example 1*: Consider two attractor basins of *4*-bit recognizer of 2C2-GMACA with periodic boundary condition given in Fig. 2, they are designed by a rule vector <232,212,178,142> rep‐ resenting in a matrix *M*, and an artificial point (*A*) of '0001'. Suppose a class label of the posi‐ tive (*L+* ) and the negative attractor basins ( *L* - ) are '1101' and '0010', respectively. For an input *x'* ='1100', firstly the present state ( *S <sup>t</sup>* ;*t* =0) is set to *x'* and then evolved with the giv‐ en rule vector to the next state (*S <sup>t</sup>*+1;*t* +1=1) by the equation (2), resulting

$$\mathcal{S}^1 \equiv \{ \mathcal{S}^0\_{0'} \: \mathcal{S}^0\_{1'} \: \mathcal{S}^0\_{2'} \: \mathcal{S}^0\_3 \} \tag{8} \\ \text{=} \{ M\langle 0, j\_0 \rangle\_{\boldsymbol{\varbeta}} \: M\langle 1, j\_1 \rangle\_{\boldsymbol{\upbeta}} \: M\langle 2, j\_2 \rangle\_{\boldsymbol{\upbeta}} \: M\langle 3, j\_3 \rangle \} \tag{8}$$

where *ji* is the 3 neighbour values ( *Si*-1 *t Si t Si*+1 *<sup>t</sup>* ) for the *i th* cell decoded in decimal. That is, *j*0= (011)2=3, *j1*= (110)2=6, *j*2= (100)2=4 and *j*3= (001)2=1. Thus, the above equation is replaced with the *ji* in decimal as following

$$\mathbf{H} = \begin{pmatrix} M\begin{pmatrix} 0,3 \end{pmatrix}, M\begin{pmatrix} 1,6 \end{pmatrix}, M\begin{pmatrix} 2,4 \end{pmatrix}, M\begin{pmatrix} 3,1 \end{pmatrix} \end{pmatrix} = \mathbf{1111} \tag{9}$$

According to a binary classifier, 2C2-GMACA conducts multiclass classification by DDAG (Decision Directed Acyclic Graph), One-versus-All, One-versus-One, etc., for example. However, this paper focuses on DDAG approach [28]. Suppose that a set of three patterns {y1, y2, y3}, where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1, 2, 3, is constructed using the DDAG scheme. Thus, total number of binary classifier is ( 3∙2/2) = 3. That is, (1 vs 3), (1 vs 2) and (2 vs 3) and the num‐ ber of levels is *log*23 = 2. A root node is (1 vs 3) contained in the *0th*-level. Then, (1 vs 2) and (2 vs 3) are contained in the *1st*-level. Finally, the solutions {3, 2, 1} are labeled in the leaf no‐ des of the *2nd*-level. In order to assign a class label for an unknown input *x* ∈{0,1}*<sup>n</sup>* , it is first evaluated at the root node. The node is exited through the left edge if the binary decision function is -1. On the other hand, it is exited via the right edge if the binary decision function is +1. The *x* is evaluated until it reaches final level. At this point, a leaf node connecting to

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364 59

A majority voting rule is utilized to synthesize a rule vector for two attractor basins. It is one time step process which is different from a reverse engineering technique (Maji, et al., 2003; Maji, et al., 2008) using in GMACA. Reverse engineering technique continues reconstructing attractor basins randomly until arriving at the rule vector with the lowest collision. In this regard, 2C2-GMACA's time complexity for ordering the rule is simply O(1). However, it must search for an optimal artificial point which applies evolutionary heuristic search. The

the edge of the binary decision function is assigned as the solution.

2C2-GMACA synthesis scheme comprises three phases as follows.

**Figure 3.** GMACA synthesize scheme under the majority voting rule.

**4.2. Design of Rule Vector**

Finally, the binary decision function will process the *S <sup>t</sup>*+1 , which equals "1111" using the artificial point *A*=0001 as co-parameters resulting in the following

$$f\left(\mathbf{S}^{t+1},\ A\_{\cdot}\right) = \operatorname{sgn}\left(\sum\_{i=0}^{n-1} S\_i^{t+1}, A\_{\cdot} \cdot \sum\_{i=0}^{n-1} S\_i^{t} \cdot A\_{\cdot}\right) = \operatorname{sgn}\left(\{1.0+1.0+1.0+1.1\} \cdot \begin{pmatrix} \cdot & \cdot & \cdot & \cdot\\ 1.0+1.0+1.0+1.1\end{pmatrix}\right) = +1 \quad \text{(10)}$$

The function returns 1 meaning that the input *x'* is a member of positive attractor basin and then the label '1101' is assigned as the solution.

#### **4.1. 2C2-GMACA with Associative and Nonassociative Memories**

Given a set of patterns *<sup>Y</sup>* ={*y*1, *<sup>y</sup>*2…, *yk* } represents original messages; where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1,2…,*k*. 2C2-GMACA takes two patterns { *yi* , *y <sup>j</sup>* }: *yi* ≠ *y <sup>j</sup>* and *yi* , *y <sup>j</sup>* ∈*Y* to process at a time. For associative memory learning, all possible transient states of the *yi* and *yj* are gener‐ ated using the equation (6) with the maximum permissible noise (*rmax*), while all transient states are randomly generated *r* ∈ 0, *rmax* ] for non-associative memory. Then, all states of *yi* and *yj* are mapped into the leaf nodes of the positive and negative attractor basins, respec‐ tively. After two attractor basins are completely constructed, it will be synthesized by a ma‐ jority voting technique to arrive at the rule vector. In other word, the rule vector is determined in only one time step which is different from GMACA in that it is iteratively de‐ termined through the evolution of heuristic search. In this regard, complexity is the main drawback excluding recognition performance.

According to a binary classifier, 2C2-GMACA conducts multiclass classification by DDAG (Decision Directed Acyclic Graph), One-versus-All, One-versus-One, etc., for example. However, this paper focuses on DDAG approach [28]. Suppose that a set of three patterns {y1, y2, y3}, where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1, 2, 3, is constructed using the DDAG scheme. Thus, total number of binary classifier is ( 3∙2/2) = 3. That is, (1 vs 3), (1 vs 2) and (2 vs 3) and the num‐ ber of levels is *log*23 = 2. A root node is (1 vs 3) contained in the *0th*-level. Then, (1 vs 2) and (2 vs 3) are contained in the *1st*-level. Finally, the solutions {3, 2, 1} are labeled in the leaf no‐ des of the *2nd*-level. In order to assign a class label for an unknown input *x* ∈{0,1}*<sup>n</sup>* , it is first evaluated at the root node. The node is exited through the left edge if the binary decision function is -1. On the other hand, it is exited via the right edge if the binary decision function is +1. The *x* is evaluated until it reaches final level. At this point, a leaf node connecting to the edge of the binary decision function is assigned as the solution.

#### **4.2. Design of Rule Vector**

(.) denotes "AND" logical operator.

58 Emerging Applications of Cellular Automata

 *f* (*S <sup>t</sup>*+1, *A* ) = + 1 , and returns *L* -

*<sup>S</sup>* <sup>1</sup> =(*S*<sup>0</sup> 0 , *S*<sup>1</sup> 0 , *S*<sup>2</sup> 0 , *S*<sup>3</sup> 0

in decimal as following

*i*=0 *n*-1 *Si*

) and the negative attractor basins ( *L* -

input *x'* ='1100', firstly the present state ( *S <sup>t</sup>*

is the 3 neighbour values ( *Si*-1

*<sup>t</sup>*+1.*Ai* - <sup>∑</sup> *i*=0 *n*-1 *S i t*+1 .*Ai*

then the label '1101' is assigned as the solution.

*i*=1,2…,*k*. 2C2-GMACA takes two patterns { *yi*

drawback excluding recognition performance.

tive (*L+*

where *ji*

*f* (*S <sup>t</sup>*+1, *A* ) =*sgn*(∑

and *yj*

the *ji*

Finally, the *x* is considered to be a member of the positive attractor basin and returns *L+*

*Example 1*: Consider two attractor basins of *4*-bit recognizer of 2C2-GMACA with periodic boundary condition given in Fig. 2, they are designed by a rule vector <232,212,178,142> rep‐ resenting in a matrix *M*, and an artificial point (*A*) of '0001'. Suppose a class label of the posi‐

0), *M* (1, *j*

(011)2=3, *j1*= (110)2=6, *j*2= (100)2=4 and *j*3= (001)2=1. Thus, the above equation is replaced with

Finally, the binary decision function will process the *S <sup>t</sup>*+1 , which equals "1111" using the

The function returns 1 meaning that the input *x'* is a member of positive attractor basin and

Given a set of patterns *<sup>Y</sup>* ={*y*1, *<sup>y</sup>*2…, *yk* } represents original messages; where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and

ated using the equation (6) with the maximum permissible noise (*rmax*), while all transient states are randomly generated *r* ∈ 0, *rmax* ] for non-associative memory. Then, all states of *yi*

, *y <sup>j</sup>*

 are mapped into the leaf nodes of the positive and negative attractor basins, respec‐ tively. After two attractor basins are completely constructed, it will be synthesized by a ma‐ jority voting technique to arrive at the rule vector. In other word, the rule vector is determined in only one time step which is different from GMACA in that it is iteratively de‐ termined through the evolution of heuristic search. In this regard, complexity is the main

}: *yi* ≠ *y <sup>j</sup>*

) <sup>=</sup>*sgn*((1.0 + 1.0 + 1.0 + 1.1) - (1

*<sup>t</sup>* ) for the *i*

1), *M* (2, *j*

=(*M* (0,3), *M* (1,6), *M* (2,4), *M* (3,1))=1111 (9)


and *yi*

, otherwise.

en rule vector to the next state (*S <sup>t</sup>*+1;*t* +1=1) by the equation (2), resulting

) =(*M* (0, *j*

artificial point *A*=0001 as co-parameters resulting in the following

**4.1. 2C2-GMACA with Associative and Nonassociative Memories**

time. For associative memory learning, all possible transient states of the *yi*

*t Si t Si*+1 if

) are '1101' and '0010', respectively. For an

*th* cell decoded in decimal. That is, *j*0=

3)) (8)

.1)) =+1 (10)

, *y <sup>j</sup>* ∈*Y* to process at a

are gener‐

and *yj*

;*t* =0) is set to *x'* and then evolved with the giv‐

2), *M* (3, *j*

A majority voting rule is utilized to synthesize a rule vector for two attractor basins. It is one time step process which is different from a reverse engineering technique (Maji, et al., 2003; Maji, et al., 2008) using in GMACA. Reverse engineering technique continues reconstructing attractor basins randomly until arriving at the rule vector with the lowest collision. In this regard, 2C2-GMACA's time complexity for ordering the rule is simply O(1). However, it must search for an optimal artificial point which applies evolutionary heuristic search. The 2C2-GMACA synthesis scheme comprises three phases as follows.

**Figure 3.** GMACA synthesize scheme under the majority voting rule.

*Phase I*--- Two attractor basins, namely, positive and negative attractor basins, are generated. In this phase, two patterns { *yl* , *ym* }, where *yl* ≠ *ym* and *yl* , *ym*∈*Y* are chosen from a set of learnt patterns to process according to the multiclass classification scheme [28]. Suppose *yl* is assigned to a class label of *L <sup>+</sup>* . Thus, the *ym* is a class label of *L -* . Then, transient states of the *yl* and *ym* are generated into the leaf nodes of the positive and negative attractor basins, respectively.

**4.3. Design of Artificial Point**

artificial point as follows:

**5. Performance Evaluation**

**5.1. Reduction of Search Space**

depends only on a parameter *n.*

the transient states are constructed to be attractor basins.

An artificial point (A) takes a major role in the binary decision function. It interprets the next state ( *S <sup>t</sup>*+1 ) in features space to be a pointer identifying the class label of solution. In this respect, Genetic Algorithm (GA) (Holland, 1992; Buhmann, et al., 1989) is implemented to determine the optimal artificial point. A chromosome with n genes in GA represents an n-bit

Selection is done by using a random pairing approach and a traditional single point cross‐ over is also performed by random at the same point of the *n* element array of the selected two parents. Mutation makes a small change in the bits in the list of a chromosome with a small percentage. The fitness function is calculated as a cost for each chromosome. It is creat‐ ed from a true positive (*TP*) and a false positive (*FP*) of the confusion matrix (Simon, et al.,

The search space complexity for rule ordering of the 2C2-GMACA is the all possible pat‐

This section reports performance evaluation of the proposed method in comparison with GMACA on a set of measured matrices consisting of search space and classification com‐ plexities, recognition percentage, evolution time for rule ordering, and effects of the number of pivotal point, permissible noises, p-parameter, pattern size on error correcting problem.

Given a set of learnt patterns *<sup>Y</sup>* ={*y*1, *<sup>y</sup>*2…, *yk* } , where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1,2…,*k*, is original messages. The 2C2-GMACA and GMACA based associative memory learning will generate all transient states using the equation (6) with the maximum permissible noise (*r max*). Then,

*Theorem 1:* In training phase, a search space complexity of the GMACA ( *SGMACA* ) depends on parameters of bit patterns (*n*)*,* the maximum permissible noise (*r max*) and the maximum permissible height (*h max*), while the search space complexity of 2C2-GMACA ( *S*2*C*2-*GMACA* )

2010) calculated by the below equation (8). The fitness function is given as following

<sup>1</sup> *TP Fitness*

terns of the artificial point, *000…000 to 111….111*, which is *2 <sup>n</sup>*, i.e. O(*2 <sup>n</sup>*).

012 1 *<sup>n</sup> chromosome b b b b* = ¼ - é ù ë û (11)

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364 61

*TP FP* = - <sup>+</sup> (12)

*Example 1*: Fig. 3(a) represents two attractor basins based on associative memory learning of 4 bit patterns with *rmax*=1. Suppose *Y*={1101, 0010} is a set of learnt patterns. The 2C2-GMA‐ CA takes two patterns {*y <sup>1</sup>*=1101, *y2*=0010} to process according to the multiclass classification algorithm. Let a class label of the positive (*L* +) and negative (*L* - ) attractor basins be '1101' and '0010', respectively. Then, two sets of noisy patterns with *rmax*=1 are generated resulting in {1101, 0101, 1001, 1111, 1100} and {0010, 1010, 0110, 0000, 0011}, respectively. Then, all pat‐ terns are mapped into leaf nodes of attractor basins corresponding with its label as shown in Fig. 3(a).

*Phase II*--- Let *M* <sup>+</sup> and *M* be matrices with size |nx8|, and *M* +(*i*, *j*) and *M* - (*i*, *j*),where i=0,1,2,...,n-1 and j=0,1,2,...,7, be an element of the matrices *M* <sup>+</sup>  *and M* - , respectively. The *M* +(*i*, *j*) represents numbers of nodes from the positive attractor basin where the 3 neigh‐ bors, ( *Si*-1 *t Si t Si*+1 *<sup>t</sup>* ), for the ith cell is decoded in decimal satisfying the jth column. The negative attractor basin considers the *M* - (*i*, *j*) under the similar condition with the positive one.

*Example 2*: As shown in Fig. 3(b), two matrices *M* <sup>+</sup> *and M* are constructed with size |4x8|, each element of which is represented the numbers of nodes from corresponding attractor ba‐ sin. For example, *M* +(1, 1) represents an element of matrix *M* <sup>+</sup> at the *1* st row and the *1* st column; it is a total number of leaf nodes from the positive attractor basin where 3 neighbors ( *S*<sup>0</sup> *t S*1 *t S*2 *t* ) of the *1* st cell decoded in decimal equal to *1*, i.e. *j*=1=0012=( *Si*-1 *t Si t Si*+1 *<sup>t</sup>* )2 where *i*=1.

*Phase III*--- Rule matrix *M* is determined. The matrix *M* with size |nx8| is the simplified form of the rule vector (*RV*), while an element *M* (*i*, *j*) represents the next state for the *i th* cell, where the 3 neighbor (*Si*-1 *<sup>t</sup> Si t Si*+1 *<sup>t</sup>* ) of the cell decoded in decimal equal to *j*. The *M* is de‐ signed by comparing between *M* +(*i*, *j*) and *M* - (*i*, *j*), where *i*=0,1,2,...,*n*-1 and *j*=0,1,2,...,7, due to the following conditions:

1) if *M* +(*i*, *j*) >*M* - (*i*, *j*) then *M* (*i*, *j*)=1

2) if *M* +(*i*, *j*)≤*M* - (*i*, *j*) then *M* (*i*, *j*)=0

Fig. 3(c) shows that a rule vector <232, 212, 178, 142> is obtained by the majority voting tech‐ nique. The rule vector (matrix rule) is utilized to evolve the given pattern in one time step to the pattern at the next time step which becomes one of parameters of the binary decision function.

#### **4.3. Design of Artificial Point**

*Phase I*--- Two attractor basins, namely, positive and negative attractor basins, are generated.

learnt patterns to process according to the multiclass classification scheme [28]. Suppose *yl*

*Example 1*: Fig. 3(a) represents two attractor basins based on associative memory learning of 4 bit patterns with *rmax*=1. Suppose *Y*={1101, 0010} is a set of learnt patterns. The 2C2-GMA‐ CA takes two patterns {*y <sup>1</sup>*=1101, *y2*=0010} to process according to the multiclass classification

and '0010', respectively. Then, two sets of noisy patterns with *rmax*=1 are generated resulting in {1101, 0101, 1001, 1111, 1100} and {0010, 1010, 0110, 0000, 0011}, respectively. Then, all pat‐ terns are mapped into leaf nodes of attractor basins corresponding with its label as shown in

*M* +(*i*, *j*) represents numbers of nodes from the positive attractor basin where the 3 neigh‐

each element of which is represented the numbers of nodes from corresponding attractor ba‐

column; it is a total number of leaf nodes from the positive attractor basin where 3 neighbors

*Phase III*--- Rule matrix *M* is determined. The matrix *M* with size |nx8| is the simplified form of the rule vector (*RV*), while an element *M* (*i*, *j*) represents the next state for the *i th* cell,

Fig. 3(c) shows that a rule vector <232, 212, 178, 142> is obtained by the majority voting tech‐ nique. The rule vector (matrix rule) is utilized to evolve the given pattern in one time step to the pattern at the next time step which becomes one of parameters of the binary decision

) of the *1* st cell decoded in decimal equal to *1*, i.e. *j*=1=0012=( *Si*-1

be matrices with size |nx8|, and *M* +(*i*, *j*) and *M* -

*<sup>t</sup>* ), for the ith cell is decoded in decimal satisfying the jth column. The negative

(*i*, *j*) under the similar condition with the positive one.

*<sup>t</sup>* ) of the cell decoded in decimal equal to *j*. The *M* is de‐

algorithm. Let a class label of the positive (*L* +) and negative (*L* -

i=0,1,2,...,n-1 and j=0,1,2,...,7, be an element of the matrices *M* <sup>+</sup>

*Example 2*: As shown in Fig. 3(b), two matrices *M* <sup>+</sup> *and M* -

*<sup>t</sup> Si t Si*+1

(*i*, *j*) then *M* (*i*, *j*)=1

(*i*, *j*) then *M* (*i*, *j*)=0

signed by comparing between *M* +(*i*, *j*) and *M* -

sin. For example, *M* +(1, 1) represents an element of matrix *M* <sup>+</sup>

. Thus, the *ym* is a class label of *L -*

and *ym* are generated into the leaf nodes of the positive and negative attractor basins,

, *ym*∈*Y* are chosen from a set of

. Then, transient states of

) attractor basins be '1101'

 *and M* -

(*i*, *j*),where

, respectively. The

are constructed with size |4x8|,

*t Si t Si*+1

(*i*, *j*), where *i*=0,1,2,...,*n*-1 and *j*=0,1,2,...,7,

at the *1* st row and the *1* st

*<sup>t</sup>* )2 where *i*=1.

, *ym* }, where *yl* ≠ *ym* and *yl*

In this phase, two patterns { *yl*

60 Emerging Applications of Cellular Automata

is assigned to a class label of *L <sup>+</sup>*

the *yl*

Fig. 3(a).

bors, ( *Si*-1

( *S*<sup>0</sup> *t S*1 *t S*2 *t*

*Phase II*--- Let *M* <sup>+</sup>

*t Si t Si*+1

attractor basin considers the *M* -

where the 3 neighbor (*Si*-1

1) if *M* +(*i*, *j*) >*M* -

2) if *M* +(*i*, *j*)≤*M* -

function.

due to the following conditions:

and *M* -

respectively.

An artificial point (A) takes a major role in the binary decision function. It interprets the next state ( *S <sup>t</sup>*+1 ) in features space to be a pointer identifying the class label of solution. In this respect, Genetic Algorithm (GA) (Holland, 1992; Buhmann, et al., 1989) is implemented to determine the optimal artificial point. A chromosome with n genes in GA represents an n-bit artificial point as follows:

$$
clrow{osome} = \begin{bmatrix} b\_0 \ b\_1 \ b\_2 \ \dots \ b\_{n-1} \end{bmatrix} \tag{11}$$

Selection is done by using a random pairing approach and a traditional single point cross‐ over is also performed by random at the same point of the *n* element array of the selected two parents. Mutation makes a small change in the bits in the list of a chromosome with a small percentage. The fitness function is calculated as a cost for each chromosome. It is creat‐ ed from a true positive (*TP*) and a false positive (*FP*) of the confusion matrix (Simon, et al., 2010) calculated by the below equation (8). The fitness function is given as following

$$Fitness = l - \frac{TP}{TP + FP} \tag{12}$$

The search space complexity for rule ordering of the 2C2-GMACA is the all possible pat‐ terns of the artificial point, *000…000 to 111….111*, which is *2 <sup>n</sup>*, i.e. O(*2 <sup>n</sup>*).

#### **5. Performance Evaluation**

This section reports performance evaluation of the proposed method in comparison with GMACA on a set of measured matrices consisting of search space and classification com‐ plexities, recognition percentage, evolution time for rule ordering, and effects of the number of pivotal point, permissible noises, p-parameter, pattern size on error correcting problem.

#### **5.1. Reduction of Search Space**

Given a set of learnt patterns *<sup>Y</sup>* ={*y*1, *<sup>y</sup>*2…, *yk* } , where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1,2…,*k*, is original messages. The 2C2-GMACA and GMACA based associative memory learning will generate all transient states using the equation (6) with the maximum permissible noise (*r max*). Then, the transient states are constructed to be attractor basins.

*Theorem 1:* In training phase, a search space complexity of the GMACA ( *SGMACA* ) depends on parameters of bit patterns (*n*)*,* the maximum permissible noise (*r max*) and the maximum permissible height (*h max*), while the search space complexity of 2C2-GMACA ( *S*2*C*2-*GMACA* ) depends only on a parameter *n.*

*Proof:* From the set *Y* ={*y*1, *y*2…, *yk* } , GMACA constructs *k* attractor basins randomly until a satisfied rule vector is acquired. Thus, the search space of the GMACA (*S GMACA*) is all pos‐ sible patterns of *k* attractor basins defined by

$$\mathbf{L}S\_{GMACA} = \mathbf{G}^k \tag{13}$$

When comparing the search space complexity between GMACA and 2C2-GMACA, we found that GMACA can only be implemented if it is considered at the *hmax=2* and *rmax* =*1,* while 2C2-GMACA can be implemented whatsoever with the exact solution through heuris‐ tic search. This corresponds to the reports in Maji, et al., 2003 and Maji, et al., 2008, the GMACA provides the best performance of pattern recognition when it is trained with the *rmax*=1 and *hmax=2*. However, the percentage of recognition in testing is also high if the Ham‐ ming distance of patterns is less than or equal to 1 and it is decreased sharply when the

*Theorem 2:* In worst case scenario of learning based on associative memory model, the classi‐

*Proof:* In general, time spent in classifying *n* nodes of GMACA depends on an arrangement of nodes in attractor basins. At worst, the attractor basin is a linear tree. Thus, time for classi‐ fying *n* nodes is the summation of the number of traversal paths from each node to a pivotal point. For example, the number of traversal paths of a pivotal point is 0 while the *nth*-node is (*n-*1). This can be solved by arithmetic series ( *Sn* ). Given the common different *d* is 1 and an

( )

( )

As being designed the height of attractor basis of 2C2-GMACA is limited to 1, the time of

Pattern classifiers based on an associative memory is independent from the number of pat‐ terns to be learnt, because all possible distorted patterns are generated into learning system. Suppose a set of pivotal points *<sup>Y</sup>* ={*y*1, *<sup>y</sup>*2…, *yk* }, where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1, 2…, *k,* is origi‐

*l*, *m*=1, 2…,*k*, to process at a time using the DDAG scheme. Thus, the number of classifiers of the 2C2-GMACA is *k* ∙(*k* - 1) / 2, while GMACA takes all pivotal points to process at once.

This section reports recognition rate and evolution time for rule ordering between 2C2- GMACA and GMACA based on associative memory. Table 1 presents the recognition rate at different sizes of bit patterns (*n*) and the number of attractor basins (*k*). It generates pat‐

, *ym* }, where *yl*

[2(0) 1 (1)]

[2 1 ]

initial term (*a <sup>1</sup>*) is 0, the equation in determining the summation is given as follows.

1

*<sup>n</sup> S and*

= +-

2

*<sup>n</sup> <sup>n</sup>*

= +-

= -Î

2

**5.3. Performance Analysis of 2C2-GMACA on Associative Memory**

nal messages. 2C2-GMACA takes two pivotal points { *yl*

*n*

<sup>2</sup> <sup>2</sup>

( ) () 2 2

*n n O n*

), while 2C2-GMACA is O(*n*).

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364 63

(19)

, *ym*∈*Y* , *yl* ≠ *ym* and

Hamming distance is greater than 1.

classifying *n* nodes is *n* , *ie*. *O*(*n* ).

*5.3.1. Recognition and Evolution Time*

**5.2. Reduction of Classification Complexity**

fication complexity of *n*-bit pattern for GMACA is O(*n2*

where *G* is the number of learnt patterns in each attractor basin previously defined by Cay‐ ley 's formula (Maji, et al., 2003) as follows:

$$G = p^{p-2} \tag{14}$$

where *p* is the number of possible transient states calculated from (6). Therefore, the above equation is defined following

$$S\_{GMACA} = \mathbf{I} \sum\_{r=0}^{r\_{\text{max}}} \binom{n}{r}^{\sum\_{r=0}^{\text{max}} \binom{n}{r} - 2} \mathbf{J}^k \in O(n!) \tag{15}$$

It shows that search space complexity of GMACA is factorial growth *O*(*n* ! ), which depends on parameters *n* and *rmax*. In real world application, it must face a severe search space in which the search heuristics cannot reach the optimal solution if *n* or *rmax* is considered at a high num‐ ber. In this regard, GMACA tries to examine the optimal values of the *rmax* and *hmax*. GMACA shows that the search space complexity can be reduced to O(*nn*) if the *rmax*=1 as shown following

$$S\_{GMACA} = (n+l)^{kn-k} \in O(n^n) \tag{16}$$

The search space complexity in Maji, et al., 2003 and Maji, et al., 2008 is examined under the *hmax=2* and the *rmax*=*1* as described below.

$$\mathbf{S}\_{\text{GMACA}} = \mathfrak{n}^k \in O\left(\mathfrak{n}^c\right); c > 1 \tag{17}$$

For the proposed 2C2-GMACA, the search space is the number of possible patterns (*G*) of artificial point: *000…000 to 111….111*—that is; 2n. Due to DDAG approach for multiclass classification algorithm, the machine consists of *k*(*k-1*)*/2* binary classifier. Thus, the search space complexity of the 2C2-GMACA (*S2C2-GMACA*) is:

$$\begin{aligned} S\_{2C2-GMAC4} &= \frac{k(k-l)}{2} G\\ \cong k^2(\mathfrak{L}^\mu) \in O(\mathfrak{L}^\mu) \end{aligned} \tag{18}$$

When comparing the search space complexity between GMACA and 2C2-GMACA, we found that GMACA can only be implemented if it is considered at the *hmax=2* and *rmax* =*1,* while 2C2-GMACA can be implemented whatsoever with the exact solution through heuris‐ tic search. This corresponds to the reports in Maji, et al., 2003 and Maji, et al., 2008, the GMACA provides the best performance of pattern recognition when it is trained with the *rmax*=1 and *hmax=2*. However, the percentage of recognition in testing is also high if the Ham‐ ming distance of patterns is less than or equal to 1 and it is decreased sharply when the Hamming distance is greater than 1.

#### **5.2. Reduction of Classification Complexity**

*Proof:* From the set *Y* ={*y*1, *y*2…, *yk* } , GMACA constructs *k* attractor basins randomly until a satisfied rule vector is acquired. Thus, the search space of the GMACA (*S GMACA*) is all pos‐

*k*

where *G* is the number of learnt patterns in each attractor basin previously defined by Cay‐

where *p* is the number of possible transient states calculated from (6). Therefore, the above

( ) max

] ( !)[

It shows that search space complexity of GMACA is factorial growth *O*(*n* ! ), which depends on parameters *n* and *rmax*. In real world application, it must face a severe search space in which the search heuristics cannot reach the optimal solution if *n* or *rmax* is considered at a high num‐ ber. In this regard, GMACA tries to examine the optimal values of the *rmax* and *hmax*. GMACA shows that the search space complexity can be reduced to O(*nn*) if the *rmax*=1 as shown following

( 1) ( ) *kn k n*

The search space complexity in Maji, et al., 2003 and Maji, et al., 2008 is examined under the

( ); 1 *k c*

For the proposed 2C2-GMACA, the search space is the number of possible patterns (*G*) of artificial point: *000…000 to 111….111*—that is; 2n. Due to DDAG approach for multiclass classification algorithm, the machine consists of *k*(*k-1*)*/2* binary classifier. Thus, the search

> ( 1) 2



0 2

å

*r r <sup>n</sup> <sup>r</sup> <sup>r</sup> <sup>k</sup>*

*<sup>n</sup> <sup>S</sup> <sup>r</sup> O n* <sup>=</sup>

æ ö ç <sup>è</sup> <sup>÷</sup> Î= ø

max

*GMACA*

0

=

*r*

*GMACA S G*= (13)

*<sup>p</sup>* <sup>2</sup> *G p* - = (14)

å (15)

*GMACA S n On* - =+ Î (16)

*GMACA S n On c* =Î > (17)

(18)

sible patterns of *k* attractor basins defined by

62 Emerging Applications of Cellular Automata

ley 's formula (Maji, et al., 2003) as follows:

*hmax=2* and the *rmax*=*1* as described below.

space complexity of the 2C2-GMACA (*S2C2-GMACA*) is:

2 2 2

*C GMACA*

*k O* -

@ Î

(2 ) (2 )

*n n k k S G*

equation is defined following

*Theorem 2:* In worst case scenario of learning based on associative memory model, the classi‐ fication complexity of *n*-bit pattern for GMACA is O(*n2* ), while 2C2-GMACA is O(*n*).

*Proof:* In general, time spent in classifying *n* nodes of GMACA depends on an arrangement of nodes in attractor basins. At worst, the attractor basin is a linear tree. Thus, time for classi‐ fying *n* nodes is the summation of the number of traversal paths from each node to a pivotal point. For example, the number of traversal paths of a pivotal point is 0 while the *nth*-node is (*n-*1). This can be solved by arithmetic series ( *Sn* ). Given the common different *d* is 1 and an initial term (*a <sup>1</sup>*) is 0, the equation in determining the summation is given as follows.

$$S\_n = \frac{n}{2} \{2a\_1 + (n-1)d\}$$

$$S = \frac{n}{2} [2(0) + (n-1)(1)]\tag{19}$$

$$S = (\frac{n^2}{2} - \frac{n}{2}) \in O(n^2)$$

As being designed the height of attractor basis of 2C2-GMACA is limited to 1, the time of classifying *n* nodes is *n* , *ie*. *O*(*n* ).

#### **5.3. Performance Analysis of 2C2-GMACA on Associative Memory**

Pattern classifiers based on an associative memory is independent from the number of pat‐ terns to be learnt, because all possible distorted patterns are generated into learning system. Suppose a set of pivotal points *<sup>Y</sup>* ={*y*1, *<sup>y</sup>*2…, *yk* }, where *yi* <sup>∈</sup>{0,1}*<sup>n</sup>* and *i*=1, 2…, *k,* is origi‐ nal messages. 2C2-GMACA takes two pivotal points { *yl* , *ym* }, where *yl* , *ym*∈*Y* , *yl* ≠ *ym* and *l*, *m*=1, 2…,*k*, to process at a time using the DDAG scheme. Thus, the number of classifiers of the 2C2-GMACA is *k* ∙(*k* - 1) / 2, while GMACA takes all pivotal points to process at once.

#### *5.3.1. Recognition and Evolution Time*

This section reports recognition rate and evolution time for rule ordering between 2C2- GMACA and GMACA based on associative memory. Table 1 presents the recognition rate at different sizes of bit patterns (*n*) and the number of attractor basins (*k*). It generates pat‐ terns with maximum permissible noise in training phase (*rmax*) and testing with different sizes of noise *r; r* ∈(1, *rmax*) . Table 2 presents the evolution time in second for the genetic algorithm in determining the well-fitting attractor basins and artificial point with different values of *n* and *k*. The results show that 2C2-GMACA is superior to GMACA both recogni‐ tion performance and times spent in rule ordering. This corresponds the previous mention that search space is the major problem of GMACA for ordering the rules when deals with high number of *rmax*.

The effects of the number of transient states (*p* ; *p* ∈*I* <sup>+</sup>

examined and shown in Fig. 7. During the training phase, the number of bit pattern (*n*) is set to 100, while the maximum permissible noise (*rmax*) is set nearly to 3 / 4∙*n* ≈ 75. Then, the per‐ centage of recognition is observed at different numbers of *p*---that is 2000, 4000 and 10000. The results show that the average percentage of recognition is highest if it is trained with the

n=50 and rmax=3 **Figure 4.** The effect of *k*-parameter on the percentage of recognition of 2C2-GMACA based on associative memory.

k=15 and rmax=2 **Figure 5.** The effect of *n*-parameter on the percentage of recognition of 2C2-GMACA based on associative memory.

n=100, k=2 and p=2000

**Figure 6.** The effect of *rmax* parameter on the percentage of recognition of 2C2-GMACA based on non-associative

memory.

highest number of *p*. However, it is memory consumptions as already mentioned.

) for two attractor basins (*k*=2) are

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364 65

#### *5.3.2. Effects of Number of Pivotal Points and Pattern Size*

A pivotal point in 2C2-GMACA represents an original message in communication systems. Fig. 4 shows the effects of the number of pivotal points (*k*) in the recognition performance of the proposed 2C2-GMACA based on associative memory learning at a particular *rmax* and bit pattern. It shows that if is trained by *rmax*= 3 the recognition rate is almost 100% when the number of bit noises (*r*) is not greater than 5 no matter of the number of classes (*k*), and de‐ clined sharply when the number of bit noises increases. The less the number of classes, the better the recognition performance. Fig. 5 shows the effects of the number of bit pattern in recognition performance of the 2C2-GMACA based on associative memory learning by fix‐ ing *rmax* and the number of classes (*k*). In this regard, when the number of bit noises in testing increases, the recognition of different number of bit patterns decreases in distinguishable manner. The more the number of bit patterns, the less the recognition performance.

#### **5.4. Performance Analysis of 2C2-GMACA on Non-Associative Memory**

The memory capacity becomes a serious problem of pattern classifier based on an associa‐ tive memory learning if the classfier deal with the high values of *n, rmax* and *k.* It generates a large number of transient states. In ordet to solve this problem, the 2C2-GMACA based on non-associtive memory is presented. The transient states will be generated by randomly choosing bit noise *<sup>r</sup>* <sup>∈</sup>(0, *rmax*) , the number of which is limited into some number *p*; *<sup>p</sup>* <sup>∈</sup>*<sup>I</sup>* <sup>+</sup> .

#### *5.4.1. Effects of Maximum Permissible Noise and P-Parameter*

In order to examine the effects of the maximum permissible noise *rmax* on the error correcting problem of 2C2-GMACA based non-associative memory, two pivotal points are randomly generated and then the number of transient states is limited to some number *p*; *p* ∈*I* <sup>+</sup> . Thus, the transient states are randomly generated from the equation (6) using *r* ∈(0, *rmax*) until the number of states equals to *p*. This method is called uniform distribution learning. Fig. 6 shows the effects of the *rmax* at 1/4∙*n* , 2/4∙*n* and 3/4∙*n* ; where *n*=100 and *n* is bits pattern. The number of pivotal points (*k*) and transient states (*p*) is fixed to 2 and 2000, respectively. Results are plotted in the inverted bell curve. It shows that the 2C2-GMACA has the lowest capability in range of *r* ∈(0 , 1/2∙*n* ) if it is trained by the *rmax* ≈3/4∙*n* , which opposed to the *r max*= 1 / 2∙*n* . However, overall average percentage of the *rmax* ≈3/4∙*n* is the highest value.

The effects of the number of transient states (*p* ; *p* ∈*I* <sup>+</sup> ) for two attractor basins (*k*=2) are examined and shown in Fig. 7. During the training phase, the number of bit pattern (*n*) is set to 100, while the maximum permissible noise (*rmax*) is set nearly to 3 / 4∙*n* ≈ 75. Then, the per‐ centage of recognition is observed at different numbers of *p*---that is 2000, 4000 and 10000. The results show that the average percentage of recognition is highest if it is trained with the highest number of *p*. However, it is memory consumptions as already mentioned.

terns with maximum permissible noise in training phase (*rmax*) and testing with different sizes of noise *r; r* ∈(1, *rmax*) . Table 2 presents the evolution time in second for the genetic algorithm in determining the well-fitting attractor basins and artificial point with different values of *n* and *k*. The results show that 2C2-GMACA is superior to GMACA both recogni‐ tion performance and times spent in rule ordering. This corresponds the previous mention that search space is the major problem of GMACA for ordering the rules when deals with

A pivotal point in 2C2-GMACA represents an original message in communication systems. Fig. 4 shows the effects of the number of pivotal points (*k*) in the recognition performance of the proposed 2C2-GMACA based on associative memory learning at a particular *rmax* and bit pattern. It shows that if is trained by *rmax*= 3 the recognition rate is almost 100% when the number of bit noises (*r*) is not greater than 5 no matter of the number of classes (*k*), and de‐ clined sharply when the number of bit noises increases. The less the number of classes, the better the recognition performance. Fig. 5 shows the effects of the number of bit pattern in recognition performance of the 2C2-GMACA based on associative memory learning by fix‐ ing *rmax* and the number of classes (*k*). In this regard, when the number of bit noises in testing increases, the recognition of different number of bit patterns decreases in distinguishable

manner. The more the number of bit patterns, the less the recognition performance.

The memory capacity becomes a serious problem of pattern classifier based on an associa‐ tive memory learning if the classfier deal with the high values of *n, rmax* and *k.* It generates a large number of transient states. In ordet to solve this problem, the 2C2-GMACA based on non-associtive memory is presented. The transient states will be generated by randomly choosing bit noise *<sup>r</sup>* <sup>∈</sup>(0, *rmax*) , the number of which is limited into some number *p*; *<sup>p</sup>* <sup>∈</sup>*<sup>I</sup>* <sup>+</sup>

In order to examine the effects of the maximum permissible noise *rmax* on the error correcting problem of 2C2-GMACA based non-associative memory, two pivotal points are randomly generated and then the number of transient states is limited to some number *p*; *p* ∈*I* <sup>+</sup>

Thus, the transient states are randomly generated from the equation (6) using *r* ∈(0, *rmax*) until the number of states equals to *p*. This method is called uniform distribution learning. Fig. 6 shows the effects of the *rmax* at 1/4∙*n* , 2/4∙*n* and 3/4∙*n* ; where *n*=100 and *n* is bits pattern. The number of pivotal points (*k*) and transient states (*p*) is fixed to 2 and 2000, respectively. Results are plotted in the inverted bell curve. It shows that the 2C2-GMACA has the lowest capability in range of *r* ∈(0 , 1/2∙*n* ) if it is trained by the *rmax* ≈3/4∙*n* , which opposed to the *r max*= 1 / 2∙*n* . However, overall average percentage of the *rmax*

.

.

**5.4. Performance Analysis of 2C2-GMACA on Non-Associative Memory**

*5.4.1. Effects of Maximum Permissible Noise and P-Parameter*

≈3/4∙*n* is the highest value.

high number of *rmax*.

64 Emerging Applications of Cellular Automata

*5.3.2. Effects of Number of Pivotal Points and Pattern Size*

**Figure 4.** The effect of *k*-parameter on the percentage of recognition of 2C2-GMACA based on associative memory.

**Figure 5.** The effect of *n*-parameter on the percentage of recognition of 2C2-GMACA based on associative memory.

**Figure 6.** The effect of *rmax* parameter on the percentage of recognition of 2C2-GMACA based on non-associative memory.

**Author details**

**References**

Press.

Sartra Wongthanavasu1\* and Jetsada Ponkaew2

\*Address all correspondence to: wongsar@kku.ac.th

1 Machine Learning and Intelligent Systems (MLIS) Laboratory, Department of Computer

Cellular Automata for Pattern Recognition http://dx.doi.org/10.5772/52364 67

2 Cellular Automata and Knowledge Engineering (CAKE) Laboratory, Department of Com‐

[1] Neumann, J.V. (1966). Theory of Self-Reproducing Automata. University of Illinois

[2] Wongthanavasu, S., & Sadananda, R. (2003). A CA-based edge operator and its perform‐ ance evaluation. *Journal of Visual Communication and Image Representation*, 14, 2, 83-96. [3] Wongthanavasu, S., & Lursinsap, C. (2004). A 3D CA-based edge operator for 3D im‐ ages. *in Image Processing*, ICIP'04. 2004 International Conference on, 2004, 235-238. [4] Wongthanavasu, S., & Tangvoraphonkchai, V. (2007). Cellular Automata-Based Al‐ gorithm and its Application in Medical Image Processing. *Image Processing*, ICIP

[5] Rosin, P.L. (2006). Training cellular automata for image processing. *IEEE Transactions*

[6] Wolfram, S. (1994). Cellular automata and complexity: collected papers. Addison-

[7] Delgado, O.G., Encinas, L.H., White, S.H., Rey, A.M.d. and Sanchez, G.R.,. (2005). Characterization of the reversibility of Wolfram cellular automata with rule number

[8] Das, S., & Chowdhury, D. R. (2008). An Efficient n x n Boolean Mapping Using Addi‐ tive Cellular Automata. *Proceedings of the 8th international conference on Cellular Autom‐*

[9] Maji, P., Shaw, C., Ganguly, N., Sikdar, B.K., & Chaudhuri, P.P. (2003). Theory and application of cellular automata for pattern classification. *Fundam. Inf.*, 58(3-4),

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321-354.

puter Science, Faculty of Science, Khon Kaen University, Khon Kaen, Thailand

**Figure 7.** The effect of *p*-parameter on the percentage of recognition of 2C2-GMACA based on non-associative mem‐ ory.

#### **6. Conclusions and Discussions**

This chapter presents a non-uniform cellular automata-based algorithm with binary classifi‐ er, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artifi‐ cial point (2C2-GMACA), for pattern recognition. The 2C2-GMACA is built around the simple structure of evolving non-uniform cellular automata called attractor basin, and clas‐ sify the patterns on the basis of two-class classifier architecture similar to support vector ma‐ chines. To reduce computational time complexity in ordering the rules, 2C2-GMACA is limited the height of attractor basin to 1, while GMACA can have its height to n, where n is a number of bit pattern. Genetic algorithm is utilized to determine the CA's best rules for classification. In this regard, GMACA designs one chromosome consists of k-genes, where k is a number of classes (target patterns) to be classified. This leads to abundant state spaces and combinatorial explosion in computation, especially when a number of bit noises in‐ creases. For the design of 2C2-GMACA, a chromosome represents an artificial point which is consists of n-bit pattern. Consequently, the state space is minimal and feasible in compu‐ tation in general pattern recognition problem. The 2C2-GMACA reduces search space for or‐ dering a rule vector from GMACA which is O(*nn*) to O(*1*)+O(*2n*). In addition, multiple errors correcting problem is empirically experimented in comparison between the proposed meth‐ od and GMACA based on associative and non-associative memories for performance evalu‐ ation. The results show that the proposed method provides the 99.98% recognition rate superior to GMACA which reports 72.50% when used associative memory, and 95.00% and 64.30% when used non-associative memory, respectively. For computational times in order‐ ing the rules through genetic algorithm, the proposed method provides 7 to 14 times faster than GMACA. These results suggests the extension of 2C2-GMACA to other pattern recog‐ nition tasks. In this respect, we are improving and extending the 2C2-GMACA to cope with complicated patterns in which state of the art methods, SVM, ANN, etc., for example, poorly report the classification performance, and hope to report our findings soon.

#### **Author details**

Sartra Wongthanavasu1\* and Jetsada Ponkaew2

\*Address all correspondence to: wongsar@kku.ac.th

1 Machine Learning and Intelligent Systems (MLIS) Laboratory, Department of Computer Science, Faculty of Science, Khon Kaen University, Khon Kaen, Thailand

2 Cellular Automata and Knowledge Engineering (CAKE) Laboratory, Department of Com‐ puter Science, Faculty of Science, Khon Kaen University, Khon Kaen, Thailand

#### **References**

n=100, k=2 and rmax=75

**Figure 7.** The effect of *p*-parameter on the percentage of recognition of 2C2-GMACA based on non-associative mem‐

This chapter presents a non-uniform cellular automata-based algorithm with binary classifi‐ er, called Two-class Classifier Generalized Multiple Attractor Cellular Automata with artifi‐ cial point (2C2-GMACA), for pattern recognition. The 2C2-GMACA is built around the simple structure of evolving non-uniform cellular automata called attractor basin, and clas‐ sify the patterns on the basis of two-class classifier architecture similar to support vector ma‐ chines. To reduce computational time complexity in ordering the rules, 2C2-GMACA is limited the height of attractor basin to 1, while GMACA can have its height to n, where n is a number of bit pattern. Genetic algorithm is utilized to determine the CA's best rules for classification. In this regard, GMACA designs one chromosome consists of k-genes, where k is a number of classes (target patterns) to be classified. This leads to abundant state spaces and combinatorial explosion in computation, especially when a number of bit noises in‐ creases. For the design of 2C2-GMACA, a chromosome represents an artificial point which is consists of n-bit pattern. Consequently, the state space is minimal and feasible in compu‐ tation in general pattern recognition problem. The 2C2-GMACA reduces search space for or‐ dering a rule vector from GMACA which is O(*nn*) to O(*1*)+O(*2n*). In addition, multiple errors correcting problem is empirically experimented in comparison between the proposed meth‐ od and GMACA based on associative and non-associative memories for performance evalu‐ ation. The results show that the proposed method provides the 99.98% recognition rate superior to GMACA which reports 72.50% when used associative memory, and 95.00% and 64.30% when used non-associative memory, respectively. For computational times in order‐ ing the rules through genetic algorithm, the proposed method provides 7 to 14 times faster than GMACA. These results suggests the extension of 2C2-GMACA to other pattern recog‐ nition tasks. In this respect, we are improving and extending the 2C2-GMACA to cope with complicated patterns in which state of the art methods, SVM, ANN, etc., for example, poorly

report the classification performance, and hope to report our findings soon.

ory.

**6. Conclusions and Discussions**

66 Emerging Applications of Cellular Automata


[11] Ganguly, N., Maji, P., Sikdar, B.K., & Chaudhuri, P.P. (2002). Generalized Multiple Attractor Cellular Automata (GMACA) For Associative Memory. *International Journal of Pattern Recognition and Artificial Intelligence, Special Issue: Computational Intelligence for Pattern Recognition*, 16(7), 781-795.

**Chapter 4**

**Using Cellular Automata and Global Sensitivity**

Advait A. Apte, Stephen S. Fong and Ryan S. Senger

Additional information is available at the end of the chapter

**L-Arabinose Operon**

http://dx.doi.org/10.5772/52085

**1. Introduction**

**Analysis to Study the Regulatory Network of the**

The field of computational biologyhas grown significantly in recent years, allowing re‐ searchers to investigatecomplex biological systems *in silico.* In this chapter, a new method of combining cellular automata (CA) with global sensitivity analysis (GSA) is introduced. This method has the potential to determine which mechanisms of a regulated biological network contribute to characteristics such as stability and responsiveness. For dynamic models of bi‐ ochemical reactions and networks, determining the correct values ofthe kineticparameters that govern the system is often problematic[12, 13, 19, 29, 30, 33, 36, 46, 47, 49, 50, 55, 57]. This isoften due to the difficulty ofobtaining accurate experimental measurements ofvital ki‐ neticconstants[6, 10, 17, 21, 32, 37]. This is currently the case for biochemical reactions occur‐ ring in complex environments *in vivo* that cannot be approximated through more simple experiments*in vitro*.Complex gene regulatory networks are a prime example.Modelling these systems using aCA approach allows researchers to easily change kineticparameter val‐ ues (individuallyand in combination) to study their effects on system function[4, 7, 8, 25, 38, 51]. Since the overall system function (i.e., the output molecule or action) is something that can be measured easily, CA modelling provides a method for determining "difficult-tomeasure" parameters using "easy-to-measure" observations of the system being stud‐ ied.However, the real challenge after conducting a CA study with combinatorial parameter variation is interpreting the results. It has been found that GSAis an extremely useful tool for identifying the parameters that most significantly affect overall model performance[9, 11, 16, 24, 28, 31, 44, 45, 56]. In this chapter,a new approach that combinesCA and GSA is ap‐ plied for analysing regulatory mechanisms of the L-arabinose (*ara*) operon. In particular, the influence of the negative autoregulatory (NAR) action of the transcription factor AraC on

> © 2013 Apte et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Apte et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.


## **Using Cellular Automata and Global Sensitivity Analysis to Study the Regulatory Network of the L-Arabinose Operon**

Advait A. Apte, Stephen S. Fong and Ryan S. Senger

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52085

#### **1. Introduction**

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354-359.

*and Humans, IEEE Transactions on*, 33(4), 466-480.

The field of computational biologyhas grown significantly in recent years, allowing re‐ searchers to investigatecomplex biological systems *in silico.* In this chapter, a new method of combining cellular automata (CA) with global sensitivity analysis (GSA) is introduced. This method has the potential to determine which mechanisms of a regulated biological network contribute to characteristics such as stability and responsiveness. For dynamic models of bi‐ ochemical reactions and networks, determining the correct values ofthe kineticparameters that govern the system is often problematic[12, 13, 19, 29, 30, 33, 36, 46, 47, 49, 50, 55, 57]. This isoften due to the difficulty ofobtaining accurate experimental measurements ofvital ki‐ neticconstants[6, 10, 17, 21, 32, 37]. This is currently the case for biochemical reactions occur‐ ring in complex environments *in vivo* that cannot be approximated through more simple experiments*in vitro*.Complex gene regulatory networks are a prime example.Modelling these systems using aCA approach allows researchers to easily change kineticparameter val‐ ues (individuallyand in combination) to study their effects on system function[4, 7, 8, 25, 38, 51]. Since the overall system function (i.e., the output molecule or action) is something that can be measured easily, CA modelling provides a method for determining "difficult-tomeasure" parameters using "easy-to-measure" observations of the system being stud‐ ied.However, the real challenge after conducting a CA study with combinatorial parameter variation is interpreting the results. It has been found that GSAis an extremely useful tool for identifying the parameters that most significantly affect overall model performance[9, 11, 16, 24, 28, 31, 44, 45, 56]. In this chapter,a new approach that combinesCA and GSA is ap‐ plied for analysing regulatory mechanisms of the L-arabinose (*ara*) operon. In particular, the influence of the negative autoregulatory (NAR) action of the transcription factor AraC on

© 2013 Apte et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Apte et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

system dynamics and stability is calculated. The purpose of this chapter is to provide in‐ struction through a detailed example of how to apply CA and GSA simultaneously to analy‐ sea biological network that must be studied *in vivo*. An in-depth explanation of GSA and the regulatory elements of the *ara* operon are provided in the Introduction. Detailed descrip‐ tions of CA model building and system parameters as well as a comprehensive GSA tutorial are presented in the Materials and Methods section.Computational experiments illuminat‐ ing the influence of the NAR mechanism on the studied regulatory network are presented and discussed throughout the rest of thechapter.

usage of the *ara* operon isone of the best-studied gene regulation systems and is well-charac‐ terized[5, 15, 22, 23, 34, 42, 52, 59]. The entire arabinose system consists of the following:

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71

Ultimately, the *ara* operon is responsible for the conversion of arabinose to D-xylulose-5 phosphate, which then enters the pentose phosphate pathway. The AraC TF regulates the *ara* operon [1, 14, 18, 20, 26, 52, 54, 59].Recent studies have shown that *araC*and the *ara* oper‐ on share a common regulatory protein,cAMP Receptor Protein (CRP),which is activated by cAMP. AraC both activates and represses the *ara* operonusing a DNA looping mechanism. As a negative regulator, AraC isde-activated by L-arabinose, allowing transcription of the

In this research, the overall influence of NAR on the dynamics of a regulated biological net‐ work was studied by applying a unique combination of CA and GSA. Here, the expression

The CA approach was used to simulate this network given altered kinetic rate constants and initial concentrations. Then, the GSA approach was applied to determine which of these pa‐

**2.** absence of NAR, the difference in GSA results give clues to the influence of the NAR

In the case studied in this research, NAR was found to equally distribute model sensitivity across all input parameters. This dramatically increases stability and responsivenessof the regulatory network. The approach presented in this chapter of combining CA and GSA can be applied to virtually any biologicalnetwork using the methods presented in this chapter.

The *ara* operon model was constructed using NetLogo simulation environment [53]. To per‐ form a CA simulation, individual (agents) (i.e., interacting molecules) were allowed to move among (cells) (i.e., spatial locations) inside the simulation environment and undergobio‐

rameters most directly influence *araBAD* expression. When applied to the

mechanism in regulating the system dynamics.

*ara* operon. AraC represses its own promoter through a NAR motif[1, 14, 26, 43, 58].

**1.** the system specific transcription factor (TF) *araC*,

**1.3. Goals of the Modelling Effort**

of *araBAD* was calculated in the

**2.** absence of NAR by AraC.

**1.** presence and

**1.** presence and

**2. Methods**

**2.1. Model Construction**

**2.** the arabinose transporters (*araE, araFGH,* and *araJ*), and

**3.** the *ara* operon containing the arabinose catalytic enzymes, *araBAD*.

#### **1.1. Global sensitivity analysis**

GSA uses Monte Carlo simulations to calculate the outputs of a model over the entire range of all input parameters [39, 40]. This variance-based method calculates the contribution of each input parameter to the total variability of the model output. In other words, GSA is used to determine which inputs most significantly influence the output. This provides the investigator one or multiple targets that can be manipulated to effectively engineer the sys‐ tem. GSA differs significantly from the traditionally used method of partial gradient-based sensitivity analysis (SA). With traditional SA, the change in a model output is calculated by allowing only one parameter to vary, while keeping others constant. This variance in the model output is likely to change if all other parameters are held constant at different values. The GSA approach takes this into account and enables the consideration of multiple param‐ eters simultaneously over the entire range of each parameter. To consider the model output variance caused by only a single parameter is a "first-order" analysis. Two parameters may be considered simultaneously to develop a measure of their interactions in a "second-order" analysis. Or, a single parameter can be considered with is interactions with all other parame‐ ters in the model. This is called the "total effect index" [39, 40]. Thus, another significant ad‐ vantage of GSA over SA is that GSA accounts for the influence and interactions between input parameters over the entire input space. A simplified tutorial of GSA has been publish‐ ed for a deterministic ammonia emissions model [35]. The GSA methods are also presented in detail in this chapter for the *ara* operon model system.

#### **1.2. The L-arabinose operon**

Transcriptional regulation networks are largely made up of recurring regulatory patterns called network motifs. These network motifs have been shown to carry out many signal transduction functions[2, 3, 27, 48]. One of the most abundantly found network motifs is thenegative autoregulation (NAR) motif. In an NAR motif, a transcription factor (TF) nega‐ tively regulates the promoter of its own gene or operon. This has been found to


The L-arabinose system is an example of an NAR network.L-arabinose,which is a five-car‐ bon sugar foundin plant cell walls, is used as a carbon source by many organisms. The *ara* operon contains genes encoding enzymes leading to L-arabinose catabolism. The selective usage of the *ara* operon isone of the best-studied gene regulation systems and is well-charac‐ terized[5, 15, 22, 23, 34, 42, 52, 59]. The entire arabinose system consists of the following:


Ultimately, the *ara* operon is responsible for the conversion of arabinose to D-xylulose-5 phosphate, which then enters the pentose phosphate pathway. The AraC TF regulates the *ara* operon [1, 14, 18, 20, 26, 52, 54, 59].Recent studies have shown that *araC*and the *ara* oper‐ on share a common regulatory protein,cAMP Receptor Protein (CRP),which is activated by cAMP. AraC both activates and represses the *ara* operonusing a DNA looping mechanism. As a negative regulator, AraC isde-activated by L-arabinose, allowing transcription of the *ara* operon. AraC represses its own promoter through a NAR motif[1, 14, 26, 43, 58].

#### **1.3. Goals of the Modelling Effort**

In this research, the overall influence of NAR on the dynamics of a regulated biological net‐ work was studied by applying a unique combination of CA and GSA. Here, the expression of *araBAD* was calculated in the

**1.** presence and

system dynamics and stability is calculated. The purpose of this chapter is to provide in‐ struction through a detailed example of how to apply CA and GSA simultaneously to analy‐ sea biological network that must be studied *in vivo*. An in-depth explanation of GSA and the regulatory elements of the *ara* operon are provided in the Introduction. Detailed descrip‐ tions of CA model building and system parameters as well as a comprehensive GSA tutorial are presented in the Materials and Methods section.Computational experiments illuminat‐ ing the influence of the NAR mechanism on the studied regulatory network are presented

GSA uses Monte Carlo simulations to calculate the outputs of a model over the entire range of all input parameters [39, 40]. This variance-based method calculates the contribution of each input parameter to the total variability of the model output. In other words, GSA is used to determine which inputs most significantly influence the output. This provides the investigator one or multiple targets that can be manipulated to effectively engineer the sys‐ tem. GSA differs significantly from the traditionally used method of partial gradient-based sensitivity analysis (SA). With traditional SA, the change in a model output is calculated by allowing only one parameter to vary, while keeping others constant. This variance in the model output is likely to change if all other parameters are held constant at different values. The GSA approach takes this into account and enables the consideration of multiple param‐ eters simultaneously over the entire range of each parameter. To consider the model output variance caused by only a single parameter is a "first-order" analysis. Two parameters may be considered simultaneously to develop a measure of their interactions in a "second-order" analysis. Or, a single parameter can be considered with is interactions with all other parame‐ ters in the model. This is called the "total effect index" [39, 40]. Thus, another significant ad‐ vantage of GSA over SA is that GSA accounts for the influence and interactions between input parameters over the entire input space. A simplified tutorial of GSA has been publish‐ ed for a deterministic ammonia emissions model [35]. The GSA methods are also presented

Transcriptional regulation networks are largely made up of recurring regulatory patterns called network motifs. These network motifs have been shown to carry out many signal transduction functions[2, 3, 27, 48]. One of the most abundantly found network motifs is thenegative autoregulation (NAR) motif. In an NAR motif, a transcription factor (TF) nega‐

tively regulates the promoter of its own gene or operon. This has been found to

**2.** increase the stability of the gene product concentration response to noise [26, 41].

The L-arabinose system is an example of an NAR network.L-arabinose,which is a five-car‐ bon sugar foundin plant cell walls, is used as a carbon source by many organisms. The *ara* operon contains genes encoding enzymes leading to L-arabinose catabolism. The selective

and discussed throughout the rest of thechapter.

in detail in this chapter for the *ara* operon model system.

**1.** dramatically increase response acceleration and

**1.1. Global sensitivity analysis**

70 Emerging Applications of Cellular Automata

**1.2. The L-arabinose operon**

**2.** absence of NAR by AraC.

The CA approach was used to simulate this network given altered kinetic rate constants and initial concentrations. Then, the GSA approach was applied to determine which of these pa‐ rameters most directly influence *araBAD* expression. When applied to the


In the case studied in this research, NAR was found to equally distribute model sensitivity across all input parameters. This dramatically increases stability and responsivenessof the regulatory network. The approach presented in this chapter of combining CA and GSA can be applied to virtually any biologicalnetwork using the methods presented in this chapter.

#### **2. Methods**

#### **2.1. Model Construction**

The *ara* operon model was constructed using NetLogo simulation environment [53]. To per‐ form a CA simulation, individual (agents) (i.e., interacting molecules) were allowed to move among (cells) (i.e., spatial locations) inside the simulation environment and undergobio‐ chemical reactions with other agents in their Von Neumann neighbourhood. In all simula‐ tions, a two-dimensional16 x 16 matrix of cells was used as the simulation environment, and 10 time steps were executed. Whether a reaction occurs between interacting agents is gov‐ erned by probability. The agents of the *ara* operon model are:

**2.2. System parameters**

**Parameter**

**Table 1.** The *M1* matrix of GSA.

tail below.

A mathematical model was created to simulate the dynamics of *ara* operon activation in the presence and absence of NAR by AraC. CA was applied by allowing critical parameters to vary over thousands of simulations of the system. GSA was then applied to the results to determine which system parameters most influence *ara* operon activation.The following pa‐ rameters were taken into consideration while building and simulating the model. The upper and lower bounds of the parameters and a description of their functions are described in de‐

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73

**Simulation 1 2 … 1000**

*conc\_cAMP* 105 188 … 32 *conc\_arabinose* 159 179 … 244 *rate\_CRP\_activate* 0.019 0.103 … 0.500 *rate\_araC\_activate* 0.418 0.391 … 0.406 *rate\_araBAD\_activate* 0.577 0.326 … 0.445 *rate\_araC\_autoreg* 0 0 … 0 **Output (***Y***1) 9 3 ... 296**

**1.** *conc\_arabinose*: The initial concentration of L-arabinose was allowed to vary between 1 and 250 cells. Upon binding,L-arabinoseactivates the TF and autoregulatorAraC.Un‐

**2.** *conc\_cAMP*: Initial concentration of the second messenger cAMP was allowed to vary between 1 to 250cells. cAMP binds to CRP causing its activation. The cAMP-CRP posi‐

**3.** *rate\_CRP\_activate*: This parameter controls the probability of CRP activation by cAMP. This parameter was varied between 0 and 1 for the simulations described in this chap‐

**4.** *rate\_araC\_activate*: This parameter controls the probability of AraC activation by L-arabi‐ nose and was allowed to vary between 0 and 1.This probabilityultimately controlsthe

**5.** *rate\_araBAD\_activate*: This parameter controls the probability of *araBAD* activation by

**6.** *rate\_araC\_autoreg*: This parameter controls the NARbyAraC protein and was allowed to‐ vary between 0 and 1.Thisrepresents the rate at which AraC supresses its promoter.

ter. This probability parameter represents the rate of activation of CRP.

activity of the *araC*gene and transcription of the *ara* operon.

bound AraC inhibits transcription of the *ara* operon.

tively regulates transcription of the *ara* operon.

CRP and was allowed tovary between 0 and 1.


Characteristics and reaction rules for individual agents were predefined at model initializa‐ tion in NetLogo. Basal expression levels for CRP, AraC, and AraBAD were set at 10, 0, and 0 cells respectively.The number of agents occupying cells represents concentration in agent based modelling. For example, CRP was present in 10 cells of the simulation environment upon initializing the simulation. Specifics of the varied model parameters are discussed in detail in the next section. These included


Monte Carlo methods were used to select 2000 values of each parameter to perform CA sim‐ ulations. This was followed by 1000 independent iterations of the model to perform GSA. The CA simulation records activation of the *ara* operon measured as number of AraBAD agents present in cells at the end of the simulation. Simulations are also often run to record the number of model "events" required to reach a specified concentration of an agent of in‐ terest (e.g., AraBAD). Independent simulations use different values of the varied parame‐ ters, resulting in different values of the targeted agent. Two common approaches use


The first approach was used in this study. Two different scenarios for NAR byAraC regula‐ tion were simulated in this research:


These simulations seek to understand the influence of the NAR mechanism on overall ri‐ gidity and robustness of the *ara* operon regulatory network.

#### **2.2. System parameters**

chemical reactions with other agents in their Von Neumann neighbourhood. In all simula‐ tions, a two-dimensional16 x 16 matrix of cells was used as the simulation environment, and 10 time steps were executed. Whether a reaction occurs between interacting agents is gov‐

Characteristics and reaction rules for individual agents were predefined at model initializa‐ tion in NetLogo. Basal expression levels for CRP, AraC, and AraBAD were set at 10, 0, and 0 cells respectively.The number of agents occupying cells represents concentration in agent based modelling. For example, CRP was present in 10 cells of the simulation environment upon initializing the simulation. Specifics of the varied model parameters are discussed in

Monte Carlo methods were used to select 2000 values of each parameter to perform CA sim‐ ulations. This was followed by 1000 independent iterations of the model to perform GSA. The CA simulation records activation of the *ara* operon measured as number of AraBAD agents present in cells at the end of the simulation. Simulations are also often run to record the number of model "events" required to reach a specified concentration of an agent of in‐ terest (e.g., AraBAD). Independent simulations use different values of the varied parame‐

**2.** a different number of model events required to reach a specified concentration of the

The first approach was used in this study. Two different scenarios for NAR byAraC regula‐

These simulations seek to understand the influence of the NAR mechanism on overall ri‐

ters, resulting in different values of the targeted agent. Two common approaches use

**1.** AraC is not allowed to negatively autoregulate its own promoter and

erned by probability. The agents of the *ara* operon model are:

**5.** AraBAD (representing gene products of the *ara* operon).

**1.** L-arabinose,

**3.** AraC (the TF regulator),

72 Emerging Applications of Cellular Automata

detail in the next section. These included

**1.** the concentrations of L-arabinose and cAMP,

**2.** the probabilities of biochemical reactions, and

**1.** a set number of model events to derive a target agent or

gidity and robustness of the *ara* operon regulatory network.

**2.** cAMP,

**4.** CRP, and

**3.** NAR by ArgC.

target agent.

tion were simulated in this research:

**2.** NAR by AraC is allowed.

A mathematical model was created to simulate the dynamics of *ara* operon activation in the presence and absence of NAR by AraC. CA was applied by allowing critical parameters to vary over thousands of simulations of the system. GSA was then applied to the results to determine which system parameters most influence *ara* operon activation.The following pa‐ rameters were taken into consideration while building and simulating the model. The upper and lower bounds of the parameters and a description of their functions are described in de‐ tail below.


**Table 1.** The *M1* matrix of GSA.


#### **2.3. Global sensitivity analysis**

GSA was performed on the *ara* operon activation model described in this chapter. The fol‐ lowing step-by-step tutorial is given to combine CA with GSA.

The procedure starts with the derivation of the *M1* and *M2* matrices shown in Tables 1 and 2, respectively. To build each table, 1000 CA simulations were run given random values of the system parameters. For each simulation, the model output (activated *araBAD*or ex‐ pressed AraBAD) was calculated and recorded. The estimated unconditional means ( *E* ^ *<sup>Y</sup>* ) and estimated unconditional variances ( *V* ^ *<sup>Y</sup>* ) of the model outputs were calculated for both matrices according to the following, where *N* is the number of simulations (i.e., 1000 for this study).

$$\begin{aligned} \stackrel{\circ}{E}\_{Y\_1} &= \frac{1}{N} \sum\_{i=1}^{N} Y\_1^{(i)} \\ \stackrel{\circ}{E}\_{Y\_2} &= \frac{1}{N} \sum\_{i=1}^{N} Y\_2^{(i)} \end{aligned} \tag{1}$$

Next, the total effect index was calculated by creation of the *R* matrix for each parameter. The first-order index describes the influence of a single parameter on the model output di‐ rectly. The total effect index takes into account all interactions of a parameter with all other

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75

**Simulation 1 2 … 1000**

*conc\_cAMP* 190 189 … 227 *conc\_arabinose* 67 212 … 208 *rate\_CRP\_activate* 0.887 0.281 … 0.671 *rate\_araC\_activate* 0.638 0.413 … 0.693 *rate\_araBAD\_activate* 0.530 0.620 … 0.891 *rate\_araC\_autoreg* 0 0 … 0 **Output (***Y***2) 608 93 ... 3043**

**Simulation 1 2 … 1000**

*conc\_cAMP (M2)* 190 189 … 227 *conc\_arabinose (M1)* 159 179 … 244 *rate\_CRP\_activate (M1)* 0.019 0.103 … 0.500 *rate\_araC\_activate (M1)* 0.418 0.391 … 0.406 *rate\_araBAD\_activate (M1)* 0.577 0.326 … 0.445 *rate\_araC\_autoreg (M1)* 0 0 … 0 **Output (***YP***) 12 196 ... 6**

The *R* matrix is shown in Table 4 for the *conc\_cAMP*parameter example. To build the *R* ma‐ trix, the parameter values from *M1* for *conc\_cAMP*were used along with parameter values from *M2* for all other parameters. An additional 1000 CA simulations are required to calcu‐ late the model outputs for the *R* matrix. The calculation of the total effect index for rcAMP

> 1 1

*R R i U YY N* <sup>=</sup>

() () 1 1

<sup>=</sup> - å (5)

*<sup>N</sup> i i*

parameters when determining the effect on model output.

**Parameter**

**Table 2.** The *M2* matrix of GSA.

**Parameter**

**Table 3.** The *P* matrix of GSA.

( *ST* (*conc*\_*cAMP*) ) was calculated as follows.

$$\begin{aligned} \hat{\boldsymbol{V}}\_{\boldsymbol{Y}\_{1}} &= \frac{1}{N-1} \sum\_{l=1}^{N} \left( \boldsymbol{Y}\_{1}^{(l)} \right)^{2} - \left( \stackrel{\circ}{\boldsymbol{E}}\_{\boldsymbol{Y}\_{1}} \right)^{2} \\ \hat{\boldsymbol{V}}\_{\boldsymbol{Y}\_{2}} &= \frac{1}{N-1} \sum\_{l=1}^{N} \left( \boldsymbol{Y}\_{2}^{(l)} \right)^{2} - \left( \stackrel{\circ}{\boldsymbol{E}}\_{\boldsymbol{Y}\_{1}} \right)^{2} \end{aligned} \tag{2}$$

Next, the *P* matrix was created for the calculation of the first-order sensitivity index for each model parameter. To illustrate this example, the model parameter *conc\_cAMP*was used. The *P* matrix for this case is shown in Table 3.

The *P* matrix consists of the *conc\_cAMP*parameter values from matrix *M2*, and all other pa‐ rameters are assigned their values from *M1*. Then model outputs were calculated for the *P* matrix using these new inputs. Thus, 1000 more simulations are required for each parameter a first-order sensitivity index is desired. The first-order sensitivity index ( *Sconc*\_*cAMP* ) was calculated by the following.

$$U\_{\rho} = \frac{1}{N-1} \sum\_{l=1}^{N} Y\_{l}^{(l)} Y\_{p}^{(l)} \tag{3}$$

$$\mathcal{S}\_{\text{cowc\\_c4MP}} = 1 - \left(\frac{U\_p - \stackrel{\wedge}{E\_{Y1}}\stackrel{\wedge}{E\_{Y2}}}{\stackrel{\wedge}{V}\_{Y1}}\right) \tag{4}$$

Next, the total effect index was calculated by creation of the *R* matrix for each parameter. The first-order index describes the influence of a single parameter on the model output di‐ rectly. The total effect index takes into account all interactions of a parameter with all other parameters when determining the effect on model output.


**Table 2.** The *M2* matrix of GSA.

**2.3. Global sensitivity analysis**

74 Emerging Applications of Cellular Automata

study).

and estimated unconditional variances ( *V*

*P* matrix for this case is shown in Table 3.

calculated by the following.

GSA was performed on the *ara* operon activation model described in this chapter. The fol‐

The procedure starts with the derivation of the *M1* and *M2* matrices shown in Tables 1 and 2, respectively. To build each table, 1000 CA simulations were run given random values of the system parameters. For each simulation, the model output (activated *araBAD*or ex‐ pressed AraBAD) was calculated and recorded. The estimated unconditional means ( *E*

matrices according to the following, where *N* is the number of simulations (i.e., 1000 for this

( )

1 1

( )

2 2

2 2

2 1

^

1

*E Y N*

=

=

*<sup>N</sup> <sup>i</sup> <sup>Y</sup> i <sup>N</sup> <sup>i</sup> <sup>Y</sup> i*

=

å

=

( ) ( )

1 1

æ ö <sup>=</sup> - ç ÷ - è ø

<sup>1</sup> <sup>1</sup>

*<sup>N</sup> <sup>i</sup> <sup>Y</sup> <sup>Y</sup> i*

*V YE*

å

Ù Ù = Ù Ù =

<sup>2</sup> <sup>2</sup>

*<sup>N</sup> <sup>i</sup> <sup>Y</sup> <sup>Y</sup> i*

*V YE*

1 1

\_

*S*

*conc cAMP*

*P P i U YY N* <sup>=</sup>

å

( ) ( )

Next, the *P* matrix was created for the calculation of the first-order sensitivity index for each model parameter. To illustrate this example, the model parameter *conc\_cAMP*was used. The

The *P* matrix consists of the *conc\_cAMP*parameter values from matrix *M2*, and all other pa‐ rameters are assigned their values from *M1*. Then model outputs were calculated for the *P* matrix using these new inputs. Thus, 1000 more simulations are required for each parameter a first-order sensitivity index is desired. The first-order sensitivity index ( *Sconc*\_*cAMP* ) was

> () () 1 1

> > 1 2

Ù Ù

1

*Y*

è ø

<sup>=</sup> - å (3)

*<sup>N</sup> i i*

1 *p YY*

*U EE*

*V*

Ù æ ö - ç ÷ = - ç ÷

2 1

æ ö <sup>=</sup> - ç ÷ - è ø

å

1

*E Y N*

1

Ù

Ù

*N*

*N*

2

^ *<sup>Y</sup>* )

(1)

(2)

(4)

*<sup>Y</sup>* ) of the model outputs were calculated for both

lowing step-by-step tutorial is given to combine CA with GSA.


**Table 3.** The *P* matrix of GSA.

The *R* matrix is shown in Table 4 for the *conc\_cAMP*parameter example. To build the *R* ma‐ trix, the parameter values from *M1* for *conc\_cAMP*were used along with parameter values from *M2* for all other parameters. An additional 1000 CA simulations are required to calcu‐ late the model outputs for the *R* matrix. The calculation of the total effect index for rcAMP ( *ST* (*conc*\_*cAMP*) ) was calculated as follows.

$$U\_{\kappa} = \frac{1}{N-1} \sum\_{\ell=1}^{N} Y\_{\mathfrak{l}}^{(\ell)} Y\_{\kappa}^{(\ell)} \tag{5}$$

$$\hat{S}\_{T(\text{conc\\_c}^\circ \text{AMP})} = 1 - \left(\frac{U\_R - \stackrel{\text{\\_}^2}{E\_{Y1}}}{\stackrel{\text{\\_}}{V}\_{Y1}}\right) \tag{6}$$

significant because the *ara* operon is known to require the presence of L-arabinose to be ac‐

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77

**Calculation Value**

(estimated unconditional mean of *M1*) 256.78

(estimated unconditional mean of *M2*) 228.49

(estimated unconditional variance of *M2*) N/A

*UP* 4210.45 *Sconc*\_*cAMP*(first-order sensitivity index of conc\_cAMP) 0.18

*UR* 83451.88

**Figure 1.** First-order indices calculated by GSA for the case without NAR by AraC.

(total effect index of conc\_cAMP) 0.94

The first-order indices for each parameter for the *ara* operonmodel with NAR by AraC are reported in Fig. 3. By activating the NAR role of AraC, the first-order indices show very close index values for all parameters (~16.5%). In other words, the NAR reduced the exces‐ sive influence of CRP activation over *araBAD*activation. The total effect indices are shown in Fig. 4. A pattern similar to that revealed by first-order indices was obtained. All total effect indices were also similar for all parameters (~16.5%).Adding the NAR by AraCto the regula‐ tion network dramatically increased the influence of L-arabinose concentration on *ara‐*

(estimated unconditional variance of *M1*) 309411.56

tive in the cell.

*E* ^ *<sup>Y</sup>* 11

*E* ^ *<sup>Y</sup>* 12

*V* ^ *Y*1

*V* ^ *Y*2

*ST* (*conc*\_*cAMP*)

*BAD*activation.

**Table 5.** GSA calculations for the *rcAMP* parameter

#### **3. Results**

The CA modeling of the *ara* operon was performed for two cases


GSA was applied to both cases in order to determine how the NAR mechanism impacts overall system dynamics. To simulate the model without NAR, the parameter *rate\_araC\_au‐ toreg* was held constant at 0. The results of the GSA calculations derived from Eqs. 1-6 and the values in Tables 1-4 are given in Table 5. This case was simulated without NAR by AraC.


**Table 4.** The *R* matrix of GSA.

The first-order sensitivity indices for each system parameter for the *ara* operonmodel with‐ out NAR by AraCare reported in Fig. 1. These values are reported as a percentage of the summation of all first-order index values. When interactions between single parameters are not taken into consideration, the probability of CRP activation was found to be the single most important parameter significantly influencing the *araBAD*activation(29.58%).All other parameters show similar influence (~17%). The total effect indices for the *ara* operon model without NAR by AraC is shown in Fig. 2. When all the interactions between all parameters were considered, probability of CRP activation (*rate\_CRP\_activate* parameter) was shown to have most influence (29.47%) over *araBAD*activation. The more noticeable result is the small contribution from the initial concentration of L-arabinose (*conc\_arabinose* parameter). This is significant because the *ara* operon is known to require the presence of L-arabinose to be ac‐ tive in the cell.


**Table 5.** GSA calculations for the *rcAMP* parameter

( )

*U E <sup>S</sup>*

\_

*T conc cAMP*

The CA modeling of the *ara* operon was performed for two cases

**1.** without NAR (i.e., negative autoregulation) by AraC and

**2.** with NAR by AraC (as is observed experimentally).

**3. Results**

76 Emerging Applications of Cellular Automata

**Parameter**

**Table 4.** The *R* matrix of GSA.

2 1

(6)

Ù

1

*Y*

è ø

*V*

Ù æ ö ç ÷ - = - ç ÷

1 *<sup>Y</sup> <sup>R</sup>*

GSA was applied to both cases in order to determine how the NAR mechanism impacts overall system dynamics. To simulate the model without NAR, the parameter *rate\_araC\_au‐ toreg* was held constant at 0. The results of the GSA calculations derived from Eqs. 1-6 and the values in Tables 1-4 are given in Table 5. This case was simulated without NAR by AraC.

**Simulation 1 2 … 1000**

*conc\_cAMP (M1)* 105 188 … 32 *conc\_arabinose (M2)* 67 212 … 208 *rate\_CRP\_activate (M2)* 0.887 0.281 … 0.671 *rate\_araC\_activate (M2)* 0.638 0.413 … 0.693 **rate\_araBAD\_activate (M2)** 0.530 0.620 … 0.891 *rate\_araC\_autoreg (M2)* 0 0 … 0 **Output (***YR***) 288 519 ... 324**

The first-order sensitivity indices for each system parameter for the *ara* operonmodel with‐ out NAR by AraCare reported in Fig. 1. These values are reported as a percentage of the summation of all first-order index values. When interactions between single parameters are not taken into consideration, the probability of CRP activation was found to be the single most important parameter significantly influencing the *araBAD*activation(29.58%).All other parameters show similar influence (~17%). The total effect indices for the *ara* operon model without NAR by AraC is shown in Fig. 2. When all the interactions between all parameters were considered, probability of CRP activation (*rate\_CRP\_activate* parameter) was shown to have most influence (29.47%) over *araBAD*activation. The more noticeable result is the small contribution from the initial concentration of L-arabinose (*conc\_arabinose* parameter). This is

**Figure 1.** First-order indices calculated by GSA for the case without NAR by AraC.

The first-order indices for each parameter for the *ara* operonmodel with NAR by AraC are reported in Fig. 3. By activating the NAR role of AraC, the first-order indices show very close index values for all parameters (~16.5%). In other words, the NAR reduced the exces‐ sive influence of CRP activation over *araBAD*activation. The total effect indices are shown in Fig. 4. A pattern similar to that revealed by first-order indices was obtained. All total effect indices were also similar for all parameters (~16.5%).Adding the NAR by AraCto the regula‐ tion network dramatically increased the influence of L-arabinose concentration on *ara‐ BAD*activation.

**1.** increasing network stability and

logical and environmental) of the system.

**2.** increasing the response of the network to L-arabinose concentrations.

**Figure 4.** Total effect indices calculated by GSA for the case with NAR by AraC.

gene regulatory networks.

, Stephen S. Fong2

**Author details**

Advait A. Apte1

ty, Richmond, VA, USA

Equal distribution of variation among all parameters suggests that the NAR mechanism in‐ creases network robustness,providing protection against random perturbations (both bio‐

Using Cellular Automata and Global Sensitivity Analysis to Study the Regulatory Network of the L-Arabinose Operon

http://dx.doi.org/10.5772/52085

79

GSA has shown that parameter sensitivity indices can provide useful insight in interpreting the results of CA simulations.Thus, the combination of CA and GSA provides a valuable tool for the identification of source of output variability. In the case of the *ara*operon, all model parameters showed to contribute equally to the variance of *araBAD*activationlevel. While all the systemparameters are important and can significantly influence *araBAD*activa‐ tion, those parameters with higher first-order sensitivities can have profound effects on reg‐ ulation of the *ara* operon if NAR function by AraC is lost. This scenario demonstrates the potential of CA and GSA for identifying targets for manipulating highly interconnected

and Ryan S. Senger1\*

2 Department of Chemical and Life Science Engineering, Virginia Commonwealth Universi‐

1 Department of Biological Systems Engineering, Virginia Tech, Blacksburg, VA, USA

**Figure 2.** Total effect indices calculated by GSA for the case without NAR by AraC.

**Figure 3.** First-order indices calculated by GSA for the case with NAR by AraC.

#### **4. Discussion**

In this study, a unique combination of CA and GSA were used to study the parameters that influence the dynamics of the *ara* operon regulatory network. The results of the GSA study revealed the degree to which individual parameters affect the output of a biological mod‐ el.GSA was used to explorethe influence of NAR on the regulatory network by calculating the impact of parameter variance on model output.Comparing first-order and total effect sensitivity indices with and without NAR by AraC elucidates the roles NAR plays in the sig‐ naling network. These include


Equal distribution of variation among all parameters suggests that the NAR mechanism in‐ creases network robustness,providing protection against random perturbations (both bio‐ logical and environmental) of the system.

**Figure 4.** Total effect indices calculated by GSA for the case with NAR by AraC.

GSA has shown that parameter sensitivity indices can provide useful insight in interpreting the results of CA simulations.Thus, the combination of CA and GSA provides a valuable tool for the identification of source of output variability. In the case of the *ara*operon, all model parameters showed to contribute equally to the variance of *araBAD*activationlevel. While all the systemparameters are important and can significantly influence *araBAD*activa‐ tion, those parameters with higher first-order sensitivities can have profound effects on reg‐ ulation of the *ara* operon if NAR function by AraC is lost. This scenario demonstrates the potential of CA and GSA for identifying targets for manipulating highly interconnected gene regulatory networks.

#### **Author details**

**Figure 2.** Total effect indices calculated by GSA for the case without NAR by AraC.

78 Emerging Applications of Cellular Automata

**Figure 3.** First-order indices calculated by GSA for the case with NAR by AraC.

In this study, a unique combination of CA and GSA were used to study the parameters that influence the dynamics of the *ara* operon regulatory network. The results of the GSA study revealed the degree to which individual parameters affect the output of a biological mod‐ el.GSA was used to explorethe influence of NAR on the regulatory network by calculating the impact of parameter variance on model output.Comparing first-order and total effect sensitivity indices with and without NAR by AraC elucidates the roles NAR plays in the sig‐

**4. Discussion**

naling network. These include

Advait A. Apte1 , Stephen S. Fong2 and Ryan S. Senger1\*

1 Department of Biological Systems Engineering, Virginia Tech, Blacksburg, VA, USA

2 Department of Chemical and Life Science Engineering, Virginia Commonwealth Universi‐ ty, Richmond, VA, USA

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**Chapter 5**

**Cellular Learning Automata and Its Applications**

Cellular Automata are mathematical models for systems consisting of large number of sim‐ ple identical components with local interactions. Cellular Automata is a non-linear dynami‐ cal system in which space and time are discrete. It is called cellular because it is made up of cells like points in a lattice or like squares of checker boards, and it is called automata be‐

Informally, a d-dimensional Cellular Automata consists of an infinite d-dimensional lattice of identical cells. Each cell can assume a state from a finite set of states. The cells update their states synchronously on discrete steps according to a local rule [4]. The new state of each cell depends on the previous states of a set of cells, including the cell itself, and consti‐ tutes its neighbourhood. The state of all cells in the lattice is described by a configuration. A

Cellular Automata provided a potential solution and is probably the most popular techni‐ que to model the dynamics of many processes, since they can predict complex global space

However, Cellular Automata is usually associated to bi-dimensional matrixes of rectangular identical cells that are not the most adequate to model and tessellate a real world geograph‐

Regular grids, or more particularly: rectangular grids are the standard grid structure that is used in previous Cellular Automata studies. Broadly, a regular grid assumes that the struc‐ ture of the cell grid and the number of neighbours are homogenous for every location in the cellular space. This assumption seems highly implausible as an empirical description of the geographical or social space that underlies the processes typically studied in Cellular Au‐

> © 2013 Navid and Aghababa; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Navid and Aghababa; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

configuration can be described as the state of the whole lattice [11].

pattern dynamic evolution using a set of simple local rules.

Amir Hosein Fathy Navid and

Additional information is available at the end of the chapter

Amir Bagheri Aghababa

http://dx.doi.org/10.5772/52953

cause it follows a simple rule.

**1. Introduction**

ic area.


## **Cellular Learning Automata and Its Applications**

Amir Hosein Fathy Navid and Amir Bagheri Aghababa

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52953

#### **1. Introduction**

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e4560.

84 Emerging Applications of Cellular Automata

Cellular Automata are mathematical models for systems consisting of large number of sim‐ ple identical components with local interactions. Cellular Automata is a non-linear dynami‐ cal system in which space and time are discrete. It is called cellular because it is made up of cells like points in a lattice or like squares of checker boards, and it is called automata be‐ cause it follows a simple rule.

Informally, a d-dimensional Cellular Automata consists of an infinite d-dimensional lattice of identical cells. Each cell can assume a state from a finite set of states. The cells update their states synchronously on discrete steps according to a local rule [4]. The new state of each cell depends on the previous states of a set of cells, including the cell itself, and consti‐ tutes its neighbourhood. The state of all cells in the lattice is described by a configuration. A configuration can be described as the state of the whole lattice [11].

Cellular Automata provided a potential solution and is probably the most popular techni‐ que to model the dynamics of many processes, since they can predict complex global space pattern dynamic evolution using a set of simple local rules.

However, Cellular Automata is usually associated to bi-dimensional matrixes of rectangular identical cells that are not the most adequate to model and tessellate a real world geograph‐ ic area.

Regular grids, or more particularly: rectangular grids are the standard grid structure that is used in previous Cellular Automata studies. Broadly, a regular grid assumes that the struc‐ ture of the cell grid and the number of neighbours are homogenous for every location in the cellular space. This assumption seems highly implausible as an empirical description of the geographical or social space that underlies the processes typically studied in Cellular Au‐

tomata modelling, like opinion formation or neighbourhood segregation. However, to our knowledge there are virtually no insights into how regular vs. irregular grid structures af‐ fect cellular dynamics [14].

*A STV* ~( , , ) (1)

Cellular Learning Automata and Its Applications

http://dx.doi.org/10.5772/52953

87

Figure1 shows the regular cellular Automata with regular cells.

Voronoi region is defined as:

Cellular Automata is shown.

**Figure 1.** Regular Cellular Automata

The Voronoi spatial model is a tessellation of space that is constructed by decomposing the entire space into a set of Voronoi regions around each spatial object. By definition, points in the Voronoi region of a spatial object are closest to the spatial object than to any other spatial object [5]. The generations of Voronoi regions can be considered as 'expanding' spatial ob‐ jects at a unique rate until these areas meet each other. The mathematical expression of the

In this equation, the Voronoi region of spatial object *pi*, *V*(*pi*), is the region defined by the set of locations *p* in space where the distance from *p* to the spatial object *pi*, *d*(*p, pi*), is less than or equal to the distance from *p* to any other spatial object *pj*. In figure 2, the Voronoi based

Voronoi region boundaries are convex polygons. Points along a common boundary between Voronoi regions are equidistant to the corresponding spatial objects. Objects which share a

In this section, Irregular Cellular Automata has been defined in context but for a better un‐ derstanding, we have also explained Voronoi diagrams concept and modelling irregular

common boundary are neighbors to each other in the Voronoi spatial model [5, 12].

grid structures using a Voronoi diagram further in this section.

( ) { | ( , ) ( , ), , 1... } *V p pdpp dpp j ij n i ij* = £ ¹= (2)

Cellular Automata extensions using Voronoi spatial models have been previously proposed to overcome this problem. In these approaches one uses convex cells with different sizes and shapes that can provide a much more adequate terrain partition.

A different problem lies in the fact that, on regular Cellular Automata, each cell has a finite set of possible states, and transition between states is a crisp function of present cell state and neighbour cells state. Crisp data modelling and crisp transition mechanisms have known limitations when one trying to model and simulate real-world processes where un‐ certainty and imprecision is present and cannot simply be ignored [28].

The most prominent reason is that Cellular Automata can be seen as multi-agent system based on locality with overlapping interaction structures. In this perspective, Cellular Au‐ tomata is attractive as a modelling framework that may provide a better understanding of micro/macro relations.

We will give some background specific to the study of cellular automata, and then back‐ ground from other fields that are necessary for the work here [17].

We will then conclude with some potential questions that merit future investigation, and where appropriate we will discuss potential consequences of such questions.

#### **2. Irregular Cellular Automata**

Practically all social science applications of cellular modelling use a regular grid as the un‐ derlying network structure. More in particular, the standard grid structure used is a rectan‐ gular regular grid. Other regular grids could be hexagonal or triangular structures. In general, we denote grids as regular where all inner cells (i.e. cells that are not at the border of the grid) have the same number of neighbours, whatever our neighbourhood definition may be - von Neumann neighbourhood or a Moore neighbourhood of a given size. On a regular torus, this definition generalises even to border cells [7].

Regular Cellular Automata has cells with identical shape and size. Since geographic features in nature are usually not distributed uniformly, regular spatial tessellation obviously limits modelling and simulation potential of regular Cellular Automata. In order to overcome this limitation, several authors have extended the Cellular Automata model to irregular cells. The most successful approaches use the Voronoi spatial model [10].

A Cellular Automata is a system composed by several identical automata, physically organ‐ ized as a 2 dimensional array of rectangular cells, where each cell is considered an autom‐ aton, *A*, with a set of rules, *T*, which gets its inputs from its own state and from neighboring cells states *V*:

$$A \sim (S, T, V) \tag{1}$$

Figure1 shows the regular cellular Automata with regular cells.

tomata modelling, like opinion formation or neighbourhood segregation. However, to our knowledge there are virtually no insights into how regular vs. irregular grid structures af‐

Cellular Automata extensions using Voronoi spatial models have been previously proposed to overcome this problem. In these approaches one uses convex cells with different sizes and

A different problem lies in the fact that, on regular Cellular Automata, each cell has a finite set of possible states, and transition between states is a crisp function of present cell state and neighbour cells state. Crisp data modelling and crisp transition mechanisms have known limitations when one trying to model and simulate real-world processes where un‐

The most prominent reason is that Cellular Automata can be seen as multi-agent system based on locality with overlapping interaction structures. In this perspective, Cellular Au‐ tomata is attractive as a modelling framework that may provide a better understanding of

We will give some background specific to the study of cellular automata, and then back‐

We will then conclude with some potential questions that merit future investigation, and

Practically all social science applications of cellular modelling use a regular grid as the un‐ derlying network structure. More in particular, the standard grid structure used is a rectan‐ gular regular grid. Other regular grids could be hexagonal or triangular structures. In general, we denote grids as regular where all inner cells (i.e. cells that are not at the border of the grid) have the same number of neighbours, whatever our neighbourhood definition may be - von Neumann neighbourhood or a Moore neighbourhood of a given size. On a

Regular Cellular Automata has cells with identical shape and size. Since geographic features in nature are usually not distributed uniformly, regular spatial tessellation obviously limits modelling and simulation potential of regular Cellular Automata. In order to overcome this limitation, several authors have extended the Cellular Automata model to irregular cells.

A Cellular Automata is a system composed by several identical automata, physically organ‐ ized as a 2 dimensional array of rectangular cells, where each cell is considered an autom‐ aton, *A*, with a set of rules, *T*, which gets its inputs from its own state and from neighboring

shapes that can provide a much more adequate terrain partition.

certainty and imprecision is present and cannot simply be ignored [28].

ground from other fields that are necessary for the work here [17].

regular torus, this definition generalises even to border cells [7].

The most successful approaches use the Voronoi spatial model [10].

where appropriate we will discuss potential consequences of such questions.

fect cellular dynamics [14].

86 Emerging Applications of Cellular Automata

micro/macro relations.

cells states *V*:

**2. Irregular Cellular Automata**

The Voronoi spatial model is a tessellation of space that is constructed by decomposing the entire space into a set of Voronoi regions around each spatial object. By definition, points in the Voronoi region of a spatial object are closest to the spatial object than to any other spatial object [5]. The generations of Voronoi regions can be considered as 'expanding' spatial ob‐ jects at a unique rate until these areas meet each other. The mathematical expression of the Voronoi region is defined as:

$$V(p\_\perp) = \{ p \mid d(p, p\_\perp) \le d(p, p\_\perp), j \ne i, j = 1 \ldots n \} \tag{2}$$

In this equation, the Voronoi region of spatial object *pi*, *V*(*pi*), is the region defined by the set of locations *p* in space where the distance from *p* to the spatial object *pi*, *d*(*p, pi*), is less than or equal to the distance from *p* to any other spatial object *pj*. In figure 2, the Voronoi based Cellular Automata is shown.

Voronoi region boundaries are convex polygons. Points along a common boundary between Voronoi regions are equidistant to the corresponding spatial objects. Objects which share a common boundary are neighbors to each other in the Voronoi spatial model [5, 12].

In this section, Irregular Cellular Automata has been defined in context but for a better un‐ derstanding, we have also explained Voronoi diagrams concept and modelling irregular grid structures using a Voronoi diagram further in this section.

**Figure 1.** Regular Cellular Automata

**Figure 2.** Voronoi Based Cellular Automata

#### **2.1. Voronoi Diagrams**

We begin with a description of elementary, though important, properties of the Voronoi dia‐ gram that will suggest some feelings for this structure. We also introduce notation used throughout this paper.

In this section, we introduce concepts of Voronoi Diagrams and describe necessary steps to create them. A naive approach to construct a Voronoi diagram is to determine the region for each point using Euclidean distance [16]. For points *p= (x <sup>p</sup> , y <sup>p</sup> )* and *q= (x <sup>q</sup> , y <sup>q</sup> )* in the plane, equation 3 denote their Euclidean distance.

$$d(p,q) = \sqrt{\left(\mathbf{x}\_p - \mathbf{x}\_q\right)^2 + \left(\mathbf{y}\_p - \mathbf{y}\_q\right)^2} \tag{3}$$

Given a set of *S* points *p <sup>1</sup> , p <sup>2</sup> ,..., p <sup>n</sup>* in the plane, a Voronoi diagram divides the plane into *n*

The points *p <sup>1</sup> ,..., p <sup>n</sup>* are called Voronoi sites. The Voronoi diagram for two sites *p <sup>i</sup>*

Such diagrams would consist of two unbounded Voronoi regions, denoted by *V(p <sup>i</sup> )* and *V(p <sup>j</sup> )*, in equation 4. In general, a Voronoi region *V(p <sup>i</sup> )* is defined as the intersection of *n − 1* half-planes formed by taking the perpendicular bisector of the segment for all where *i* ≠ *j* .

In this notation, *H(p <sup>i</sup> p <sup>j</sup> )* refers to the half-plane formed by taking the perpendicular bisec‐

convex region bounded by a set of connected line segments. These line segments form the boundaries of Voronoi regions and are called Voronoi edges. The endpoints of these edges

The points on Voronoi edges of Voronoi diagram are in equal distance of Voronoi sites *p <sup>i</sup>*

. You can see an example of Voronoi diagram in Figure 5.

in figure 4. We know that the intersection of any number of half-planes forms a

can be easily constructed by drawing the perpendicular bisector of line segment *pi*¯*<sup>q</sup> <sup>j</sup>*

, then the Euclidian distance from *p <sup>i</sup>*

is any other point in *S*.

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to *q*, where *p <sup>j</sup>*

1 2 ( ) ( ) ( ) ... ( ) *V p H pp H pp H pp i i* = Ç ÇÇ *<sup>i</sup> i n* (4)

to *q* will

89

and *p <sup>j</sup>*

.

Voronoi regions with the following properties:

**•** If a point *q* ∉*S* lies in the same region as *p <sup>i</sup>*

be shorter than the Euclidian distance from *p <sup>j</sup>*

lies in exactly one region.

**•** Each point *p <sup>i</sup>*

tor of *p <sup>i</sup> p <sup>j</sup>*

and *p <sup>j</sup>*

are called Voronoi vertices [8, 12].

**Figure 4.** Create half-plane by perpendicular bisector

By *<sup>p</sup>*¯*<sup>q</sup>* , we denote the line segment from *p* to *q*. To draw Voronoi diagram, we use perpen‐ dicular bisectors of point set on the 2D space as it is shown in figure 3.

**Figure 3.** Divided plane with perpendicular bisector of two points

Given a set of *S* points *p <sup>1</sup> , p <sup>2</sup> ,..., p <sup>n</sup>* in the plane, a Voronoi diagram divides the plane into *n* Voronoi regions with the following properties:

**•** Each point *p <sup>i</sup>* lies in exactly one region.

**Figure 2.** Voronoi Based Cellular Automata

88 Emerging Applications of Cellular Automata

equation 3 denote their Euclidean distance.

**Figure 3.** Divided plane with perpendicular bisector of two points

We begin with a description of elementary, though important, properties of the Voronoi dia‐ gram that will suggest some feelings for this structure. We also introduce notation used

In this section, we introduce concepts of Voronoi Diagrams and describe necessary steps to create them. A naive approach to construct a Voronoi diagram is to determine the region for each point using Euclidean distance [16]. For points *p= (x <sup>p</sup> , y <sup>p</sup> )* and *q= (x <sup>q</sup> , y <sup>q</sup> )* in the plane,

By *<sup>p</sup>*¯*<sup>q</sup>* , we denote the line segment from *p* to *q*. To draw Voronoi diagram, we use perpen‐

dicular bisectors of point set on the 2D space as it is shown in figure 3.

2 2 (,) ( ) ( ) *pq p q d pq x x y y* = - +- (3)

**2.1. Voronoi Diagrams**

throughout this paper.

**•** If a point *q* ∉*S* lies in the same region as *p <sup>i</sup>* , then the Euclidian distance from *p <sup>i</sup>* to *q* will be shorter than the Euclidian distance from *p <sup>j</sup>* to *q*, where *p <sup>j</sup>* is any other point in *S*.

The points *p <sup>1</sup> ,..., p <sup>n</sup>* are called Voronoi sites. The Voronoi diagram for two sites *p <sup>i</sup>* and *p <sup>j</sup>* can be easily constructed by drawing the perpendicular bisector of line segment *pi*¯*<sup>q</sup> <sup>j</sup>* .

Such diagrams would consist of two unbounded Voronoi regions, denoted by *V(p <sup>i</sup> )* and *V(p <sup>j</sup> )*, in equation 4. In general, a Voronoi region *V(p <sup>i</sup> )* is defined as the intersection of *n − 1* half-planes formed by taking the perpendicular bisector of the segment for all where *i* ≠ *j* .

$$W(p\_i) = H(p\_i p\_1) \cap H(p\_i p\_2) \cap \dots \cap H(p\_i p\_n) \tag{4}$$

In this notation, *H(p <sup>i</sup> p <sup>j</sup> )* refers to the half-plane formed by taking the perpendicular bisec‐ tor of *p <sup>i</sup> p <sup>j</sup>* in figure 4. We know that the intersection of any number of half-planes forms a convex region bounded by a set of connected line segments. These line segments form the boundaries of Voronoi regions and are called Voronoi edges. The endpoints of these edges are called Voronoi vertices [8, 12].

**Figure 4.** Create half-plane by perpendicular bisector

The points on Voronoi edges of Voronoi diagram are in equal distance of Voronoi sites *p <sup>i</sup>* and *p <sup>j</sup>* . You can see an example of Voronoi diagram in Figure 5.

**Figure 5.** Voronoi diagrams of collection points on plane

Equation (5) shows the Voronoi region of p with respect to *S*, for *p*, *q* ∈*S* .

$$W(p, S) = \bigcap\_{q \in S, q \star p} H(p, q) \tag{5}$$

Given a triangle Δ*abc*, the perpendicular bisector of each edge will intersect at a common point *q* called the circumcenter. The circumcenter is equi-distant from points *a*, *b*, *c* and these points all lie on a circle with *q* as its center. This circle is called the circumcircle for triangle Δ*abc* [16].

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91

**Figure 6.** The circumcircle with circumcenter *q*

**•** *a, b, c* would be Voronoi sites

**•** *q* would be a Voronoi vertex

If a circumcircle is empty in its interior then in a Voronoi diagram:

**•** The perpendicular bisectors of Δ*abc* would be Voronoi edges.

**Figure 7.** Voronoi Diagrams with (a) 10 random points (b) 10 simultaneous points

Figure 7 shows, on the left, the Voronoi regions corresponding to 10 randomly selected points in a square; the density function is constant. The dots are the Voronoi generators and

Finally, the Voronoi diagram of *S* is defined by equation 6.

$$Voronoi(S) = \bigcup\_{p,q \in S, p \neq q} \overline{V(p,S)} \cap \overline{V(q,S)}\tag{6}$$

By definition, each Voronoi region is the intersection of *n – 1* open half-planes containing the site *p*.

#### **2.2. Properties of Voronoi Diagrams**


Given a triangle Δ*abc*, the perpendicular bisector of each edge will intersect at a common point *q* called the circumcenter. The circumcenter is equi-distant from points *a*, *b*, *c* and these points all lie on a circle with *q* as its center. This circle is called the circumcircle for triangle Δ*abc* [16].

**Figure 6.** The circumcircle with circumcenter *q*

If a circumcircle is empty in its interior then in a Voronoi diagram:

**•** *a, b, c* would be Voronoi sites

**Figure 5.** Voronoi diagrams of collection points on plane

90 Emerging Applications of Cellular Automata

**2.2. Properties of Voronoi Diagrams**

*<sup>j</sup> )* will share a common edge.

vex hull of *S*.

**•** The number of Voronoi vertices is at most *2n − 5*.

**•** The number of Voronoi edges is at most *3n − 6*.

site *p*.

Equation (5) shows the Voronoi region of p with respect to *S*, for *p*, *q* ∈*S* .

Finally, the Voronoi diagram of *S* is defined by equation 6.

, (,) (,) *q Sq p V pS H pq* Î ¹

() (, ) (, )

By definition, each Voronoi region is the intersection of *n – 1* open half-planes containing the

**•** If site *p <sup>i</sup> ∈ S* is the nearest neighbor of site *p <sup>j</sup> ∈ S*, then the Voronoi regions *V(p <sup>i</sup> )* and *V(p*

**•** Region *V(p)* is unbounded iff *p* is an extreme point of *S*. That is, *p* will be part of the con‐

, ,

**•** Each Voronoi vertex is the common intersection point of exactly three edges.

*pq S p q Voronoi S V p S V q S* Î ¹

<sup>=</sup> I (5)

<sup>=</sup> U <sup>Ç</sup> (6)


**Figure 7.** Voronoi Diagrams with (a) 10 random points (b) 10 simultaneous points

Figure 7 shows, on the left, the Voronoi regions corresponding to 10 randomly selected points in a square; the density function is constant. The dots are the Voronoi generators and the circles are the centroids of the corresponding Voronoi regions. Note that the generators and the centroids do not coincide. On the right, the 10 dots are simultaneously the genera‐ tors for the Voronoi tessellation and the centroids of the Voronoi regions.

Figure 8 shows a decisive feature of irregular grids: even the cells inside the grid have differ‐ ent numbers of next neighbors. The figure shows locations of three different cells with 6, 8 and 12 next neighbors in an irregular grid. We define a next neighbor cell here as a cell that has a common border (not just a common edge) with the focal cell. Notice that this defini‐ tion implies a von Neumann neighborhood on a rectangular grid. More in general, it has been found in simulation analyses that in a Voronoi graph, the number of neighbor cells

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The idea of irregular cellular automata was suggested in mid 80s, but due to the computa‐ tionally intensive operations required to search irregular neighborhood, it has been paid less attention to since then. In an informal way, Irregular Cellular Automata is a configuration of points in the space with no prior restriction. Each point has a number of other points as its

In this section, we present the learning automata concept, cellular learning automata and ir‐

Learning Automaton is a simple entity which operates in an unknown random environ‐ ment. In a simple form, the automaton has a finite set of actions to choose from, and at each stage its choice (action) depends upon its action probability vector. For each action chosen by the automaton, the environment gives a reinforcement signal with fixed unknown proba‐ bility distribution. The automaton then updates its action probability vector depending on

Learning Automata is an abstract model which randomly selects one action out of its finite set of actions and performs it on a random environment. Environment, then evaluates the selected action and responses to the automata with a reinforcement signal. Based on the se‐ lected action and received signal, the automata updates its internal state and selects its next

Environment can be defined by the triple *E={α, β, c}* where *α={α <sup>1</sup> , α <sup>2</sup> …, α <sup>r</sup> }* represents a finite input set, *β={β <sup>1</sup> , β <sup>2</sup> , …, β <sup>r</sup> }* represents the output set, and *c={c <sup>1</sup> , c <sup>2</sup> , …, c <sup>r</sup> }* is a set of

ing automata are classified into fixed structure stochastic, and variable structure stochastic

. Learn‐

the reinforcement signal at that stage, and evolves to some final desired behavior [1].

action. Figure 9 depicts the relationship between an automata and its environment.

penalty probabilities where each element *c <sup>i</sup>* of *c* corresponds to one input action *α <sup>i</sup>*

[17, 18]. In the following, we consider only variable structure automata.

varies between 3 and 14 [10].s

neighbors such as figure 8. [8].

**3. Learning Automata Concepts**

regular cellular learning automata.

**3.1. Learning Automata**

#### **2.3. Constructing Voronoi Diagrams**

#### *2.3.1. Naive Approach*

A naive approach to construct a Voronoi diagram is to determine the region for each site one at a time. Since each region is the intersection of *n− 1* half-planes, we can use an *O*(*n log n*) half-plane intersection algorithm to determine this region. Repeating for all *n* points, we have an *O*(*n*2 *log n*) algorithm.

#### *2.3.2. Divide and Conquer*

To construct a Voronoi diagram using the divide and conquer method, first partition the set of points *S* into two sets *L* and *R* based on x-coordinates. Next, construct the Voronoi dia‐ grams for the left and right subset *V(L)*and *V(R)*. Finally, merge the two diagrams to pro‐ duce *V(S)*. If the merge step can be carried out in linear time, then the construction of *V(S)* can be accomplished in *O*(*n log n*) time [16].

#### **2.4. Irregular Grids in a Cellular Automaton**

To model irregular grid structures, we use a Voronoi diagram. The crosses in Voronoi dia‐ gram are the generators of the grid. The edges of the resulting polygons are points with equal distance to their neighboring generators [10].

**Figure 8.** Neighborhoods of three different cells in an irregular field. Focal cells are gray and neighbor cells are red.

Figure 8 shows a decisive feature of irregular grids: even the cells inside the grid have differ‐ ent numbers of next neighbors. The figure shows locations of three different cells with 6, 8 and 12 next neighbors in an irregular grid. We define a next neighbor cell here as a cell that has a common border (not just a common edge) with the focal cell. Notice that this defini‐ tion implies a von Neumann neighborhood on a rectangular grid. More in general, it has been found in simulation analyses that in a Voronoi graph, the number of neighbor cells varies between 3 and 14 [10].s

The idea of irregular cellular automata was suggested in mid 80s, but due to the computa‐ tionally intensive operations required to search irregular neighborhood, it has been paid less attention to since then. In an informal way, Irregular Cellular Automata is a configuration of points in the space with no prior restriction. Each point has a number of other points as its neighbors such as figure 8. [8].

#### **3. Learning Automata Concepts**

In this section, we present the learning automata concept, cellular learning automata and ir‐ regular cellular learning automata.

#### **3.1. Learning Automata**

the circles are the centroids of the corresponding Voronoi regions. Note that the generators and the centroids do not coincide. On the right, the 10 dots are simultaneously the genera‐

A naive approach to construct a Voronoi diagram is to determine the region for each site one at a time. Since each region is the intersection of *n− 1* half-planes, we can use an *O*(*n log n*) half-plane intersection algorithm to determine this region. Repeating for all *n* points, we

To construct a Voronoi diagram using the divide and conquer method, first partition the set of points *S* into two sets *L* and *R* based on x-coordinates. Next, construct the Voronoi dia‐ grams for the left and right subset *V(L)*and *V(R)*. Finally, merge the two diagrams to pro‐ duce *V(S)*. If the merge step can be carried out in linear time, then the construction of *V(S)*

To model irregular grid structures, we use a Voronoi diagram. The crosses in Voronoi dia‐ gram are the generators of the grid. The edges of the resulting polygons are points with

**Figure 8.** Neighborhoods of three different cells in an irregular field. Focal cells are gray and neighbor cells are red.

tors for the Voronoi tessellation and the centroids of the Voronoi regions.

**2.3. Constructing Voronoi Diagrams**

92 Emerging Applications of Cellular Automata

have an *O*(*n*2 *log n*) algorithm.

can be accomplished in *O*(*n log n*) time [16].

**2.4. Irregular Grids in a Cellular Automaton**

equal distance to their neighboring generators [10].

*2.3.2. Divide and Conquer*

*2.3.1. Naive Approach*

Learning Automaton is a simple entity which operates in an unknown random environ‐ ment. In a simple form, the automaton has a finite set of actions to choose from, and at each stage its choice (action) depends upon its action probability vector. For each action chosen by the automaton, the environment gives a reinforcement signal with fixed unknown proba‐ bility distribution. The automaton then updates its action probability vector depending on the reinforcement signal at that stage, and evolves to some final desired behavior [1].

Learning Automata is an abstract model which randomly selects one action out of its finite set of actions and performs it on a random environment. Environment, then evaluates the selected action and responses to the automata with a reinforcement signal. Based on the se‐ lected action and received signal, the automata updates its internal state and selects its next action. Figure 9 depicts the relationship between an automata and its environment.

Environment can be defined by the triple *E={α, β, c}* where *α={α <sup>1</sup> , α <sup>2</sup> …, α <sup>r</sup> }* represents a finite input set, *β={β <sup>1</sup> , β <sup>2</sup> , …, β <sup>r</sup> }* represents the output set, and *c={c <sup>1</sup> , c <sup>2</sup> , …, c <sup>r</sup> }* is a set of penalty probabilities where each element *c <sup>i</sup>* of *c* corresponds to one input action *α <sup>i</sup>* . Learn‐ ing automata are classified into fixed structure stochastic, and variable structure stochastic [17, 18]. In the following, we consider only variable structure automata.

tion of Learning Automatons which can be defined in a form of graph such as tree, mesh, or

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On the other hand, Cellular Automata are mathematical models for systems consisting of large numbers of simple identical components with local interactions. Cellular Automata and Learning Automata are combined to obtain a new model called Cellular Learning Au‐ tomata (CLA). This model is superior to Cellular Automata because of its ability to learn and also is superior to single Learning Automata because it is a collection of Learning Autom‐

Cellular Learning Automata is a mathematical model for dynamical complex systems that consists of large number of simple components. The simple components have learning capa‐ bility and act together to produce complicated behavioral patterns. A Cellular Learning Au‐ tomata is a Cellular Automata in which a Learning Automata will be assigned to its every cell [4]. The learning automaton residing in each cell determines the state of the cell on the basis of its action probability vector. Like Cellular Automata, there is a rule that Cellular Learning Automata operates according to it. The rule of Cellular Learning Automata and the actions selected by the neighboring Learning Automatons of any cell determine the rein‐ forcement signal to the Learning Automata residing in that cell. In Cellular Learning Au‐ tomata, the neighboring Learning Automatons of any cell constitute its local environment. This environment is non-stationary because of the fact that it changes as action probability

The operation of cellular learning automata could be described as follows: At the first step, the internal state of every cell is specified. The state of every cell is determined on the basis of action probability vectors of the learning automata residing in that cell. The initial value of this state may be chosen on the basis of past experience or at random. In the second step, the rule of Cellular Learning Automata determines the reinforcement signal to each learning automaton residing in that cell. Finally, each learning automaton updates its action probabil‐ ity vector on the basis of supplied reinforcement signal and the chosen action. This process continues until the desired result is obtained (figure 10). Formally a d−dimensional Cellular

A d−dimensional cellular learning automata is a structure *A = (Z <sup>d</sup> , Φ, A, N, F)*, here

**3.** *A* is the set of Learning Automatons each of which is assigned to each cell of the Cellu‐

called neighborhood vector where *<sup>m</sup>*¯ repre‐

array, is natural in many applications.

atons which can interact with each other.

vectors of neighboring Learning Automatons vary [7].

is a lattice of d−tuples of integer numbers.

sents the number of neighboring cells and *x*¯*<sup>i</sup>* <sup>∈</sup>*<sup>Z</sup> <sup>d</sup>* .

**4.** *<sup>N</sup>* ={*x*¯ 1, *<sup>x</sup>*¯ 2, ..., *<sup>x</sup>*¯*m*} is a finite subset of *<sup>Z</sup> <sup>d</sup>*

**3.2. Cellular Learning Automata**

Learning Automata is given below.

**2.** *Φ* is a finite set of states.

lar Automata.

**1.** *Zd*

**Figure 9.** Relationship between learning automata and its environment

A variable structure automata is defined by the quadruple *{α, β, p, T}* in which *α={ α <sup>1</sup> , α <sup>2</sup> , …, α <sup>n</sup> }* represents the action set of the automata, *β={ β <sup>1</sup> , β <sup>2</sup> , …, β <sup>r</sup> }* represents the input set, *p={p <sup>1</sup> , p <sup>2</sup> , …, p <sup>r</sup> }* represents the action probability set, and finally *p(n+1)=T[α(n), β(n), p(n)]* represents the learning algorithm. This automaton operates as follows. Based on the ac‐ tion probability set *p*, automaton randomly selects an action *α <sup>i</sup>* , and performs it on the envi‐ ronment. Having received the environment's reinforcement signal, automaton updates its action probability set based on equation (7) for favorable responses, and on equation (8) for unfavorable ones [18].

$$\begin{aligned} p\_i(n+1) &= p\_i(n) + a.(1 - p\_i(n)) \\ p\_\bot(n+1) &= p\_\bot(n) - a.p\_\bot(n) \qquad \forall j \quad j \neq i \end{aligned} \tag{7}$$

$$\begin{aligned} p\_i(n+1) &= (1-b).p\_/(n) \\ p\_/(n+1) &= \frac{b}{r-1} + (1-b).p\_/(n) \qquad \forall j \quad j \neq i \end{aligned} \tag{8}$$

In these two equations, *a* and *b* are reward and penalty parameters, respectively. For *a = b*, learning algorithm is called *L R-P*, for *a << b*, it is called *L RεP*, and for *b=0* it is called *L R-I*. For more information about learning automata the reader may refer to Learning automata that are, by design, "simple agents for doing simple things". The full potential of a Learning Au‐ tomata is realized when multiple automata interact with each other. Interaction may assume different forms such as tree, mesh, array and etc. Depending on the problem that needs to be solved, one of these structures for interaction may be chosen. In most applications, full inter‐ action between all Learning Automatons is not necessary and is not natural. Local interac‐ tion of Learning Automatons which can be defined in a form of graph such as tree, mesh, or array, is natural in many applications.

On the other hand, Cellular Automata are mathematical models for systems consisting of large numbers of simple identical components with local interactions. Cellular Automata and Learning Automata are combined to obtain a new model called Cellular Learning Au‐ tomata (CLA). This model is superior to Cellular Automata because of its ability to learn and also is superior to single Learning Automata because it is a collection of Learning Autom‐ atons which can interact with each other.

#### **3.2. Cellular Learning Automata**

**Figure 9.** Relationship between learning automata and its environment

tion probability set *p*, automaton randomly selects an action *α <sup>i</sup>*

unfavorable ones [18].

94 Emerging Applications of Cellular Automata

A variable structure automata is defined by the quadruple *{α, β, p, T}* in which *α={ α <sup>1</sup> , α <sup>2</sup> , …, α <sup>n</sup> }* represents the action set of the automata, *β={ β <sup>1</sup> , β <sup>2</sup> , …, β <sup>r</sup> }* represents the input set, *p={p <sup>1</sup> , p <sup>2</sup> , …, p <sup>r</sup> }* represents the action probability set, and finally *p(n+1)=T[α(n), β(n), p(n)]* represents the learning algorithm. This automaton operates as follows. Based on the ac‐

ronment. Having received the environment's reinforcement signal, automaton updates its action probability set based on equation (7) for favorable responses, and on equation (8) for

+= - " ¹ (7)

( 1) ( ) .(1 ( )) ( 1) ( ) . ( ) *ii i j jj*

*p n p n ap n j j i*

*p n bp n j j i*

+= +- " ¹ -

In these two equations, *a* and *b* are reward and penalty parameters, respectively. For *a = b*, learning algorithm is called *L R-P*, for *a << b*, it is called *L RεP*, and for *b=0* it is called *L R-I*. For more information about learning automata the reader may refer to Learning automata that are, by design, "simple agents for doing simple things". The full potential of a Learning Au‐ tomata is realized when multiple automata interact with each other. Interaction may assume different forms such as tree, mesh, array and etc. Depending on the problem that needs to be solved, one of these structures for interaction may be chosen. In most applications, full inter‐ action between all Learning Automatons is not necessary and is not natural. Local interac‐

*pn pn a pn*

( 1) (1 ) ( ) <sup>1</sup>

*j j*

+= + -

( 1) (1 ). ( )

*i j*

+=-

*pn bp n b*

*r*

, and performs it on the envi‐

(8)

Cellular Learning Automata is a mathematical model for dynamical complex systems that consists of large number of simple components. The simple components have learning capa‐ bility and act together to produce complicated behavioral patterns. A Cellular Learning Au‐ tomata is a Cellular Automata in which a Learning Automata will be assigned to its every cell [4]. The learning automaton residing in each cell determines the state of the cell on the basis of its action probability vector. Like Cellular Automata, there is a rule that Cellular Learning Automata operates according to it. The rule of Cellular Learning Automata and the actions selected by the neighboring Learning Automatons of any cell determine the rein‐ forcement signal to the Learning Automata residing in that cell. In Cellular Learning Au‐ tomata, the neighboring Learning Automatons of any cell constitute its local environment. This environment is non-stationary because of the fact that it changes as action probability vectors of neighboring Learning Automatons vary [7].

The operation of cellular learning automata could be described as follows: At the first step, the internal state of every cell is specified. The state of every cell is determined on the basis of action probability vectors of the learning automata residing in that cell. The initial value of this state may be chosen on the basis of past experience or at random. In the second step, the rule of Cellular Learning Automata determines the reinforcement signal to each learning automaton residing in that cell. Finally, each learning automaton updates its action probabil‐ ity vector on the basis of supplied reinforcement signal and the chosen action. This process continues until the desired result is obtained (figure 10). Formally a d−dimensional Cellular Learning Automata is given below.

A d−dimensional cellular learning automata is a structure *A = (Z <sup>d</sup> , Φ, A, N, F)*, here


The neighborhood vector determines the relative position of the neighboring cells from any given cell *u* in the lattice *Z <sup>d</sup>* . The neighbors of a particular cell *u* are set of cells {*<sup>u</sup>* <sup>+</sup> *<sup>x</sup>*¯*<sup>i</sup>* <sup>|</sup>*<sup>i</sup>* =1, 2, ..., *<sup>m</sup>*¯} . We assume that there exists a neighborhood function *N*¯(*<sup>u</sup>*) mapping a cell *u* to the set of its neighbors according to equation (9).

$$\overline{N}(\mu) = \langle \mu + \overline{\mathfrak{x}}\_1, \mu + \overline{\mathfrak{x}}\_2, \dots, \mu + \overline{\mathfrak{x}}\_{\mathfrak{m}} \rangle \tag{9}$$

In an informal way, Irregular Cellular Automata is a configuration of points in the space with no prior restriction. The few examples of Irregular Cellular Automata all use Voronoi polygons or the related Delaunay triangulation to divide space and determine the neighbors of each point. Voronoi polygons divide space into regions surrounding objects such that any point in an object's polygon is closer to that object than to any other object, while Delaunay triangulation is a triangulation of the points in a Voronoi diagram where the circumcircle of

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An Irregular Cellular Learning Automata is a combination of Irregular Cellular Automata and Learning Automata (Figure 11). We define Irregular Cellular Learning Automata as an undirected graph in which each vertex represents a cell which is equipped with a learn‐

**Figure 11.** Irregular Cellular Learning Automata, LA means Learning Automata in each neighbor cell.

during evolution of the Irregular Cellular Learning Automata.

An Irregular Cellular Learning Automata is formally defined below.

The Learning Automaton residing in a particular cell determines its state (action) on the ba‐ sis of its action probability vector. Like Cellular Learning Automata, there is a rule that the Irregular Cellular Learning Automata operate according to it. The rule of the Cellular Learn‐ ing Automata and the actions selected by the neighboring Learning Automatons of any par‐ ticular Learning Automata determine the reinforcement signal to the Learning Automata residing in a cell. The neighboring Learning Automatons of any particular Learning Autom‐ ata constitute the local environment of that cell. The local environment of a cell is non-sta‐ tionary because the action probability vectors of the neighboring Learning Automatons vary

each triangle is an empty triangle.

ing automaton.

**Figure 10.** Cellular Learning Automata

A number of applications for Cellular Learning Automata have been developed recently such as rumor diffusion,image processing, modeling of commerce networks, fixed channel assign‐ ment in cellular networks, and VLSI Placement to mention a few (Beigy & Meybodi, 2004).

The Cellular Learning Automata can be classified into two types of *synchronous* and *asynchro‐ nous*. In synchronous Cellular Learning Automata, all cells are synchronized with a global clock and executed at the same time [10]. It is shown that the synchronous Cellular Learning Automata converges to a globally stable state for a class of rules called commutative rules. In some applications such as image processing, a type of Cellular Learning Automata in which the action of each cell in next stage of its evolution not only depends on the local environ‐ ment (actions of its neighbors) but it also depends on the external environments.

#### **3.3. Irregular Cellular Learning Automata**

Irregular Cellular Learning Automata is a generalization of Cellular Learning Automata which removes the restriction of rectangular grid structure in traditional Cellular Learning Automata. This generalization is expected because there are applications which cannot be adequately modeled with rectangular grids [10].

In an informal way, Irregular Cellular Automata is a configuration of points in the space with no prior restriction. The few examples of Irregular Cellular Automata all use Voronoi polygons or the related Delaunay triangulation to divide space and determine the neighbors of each point. Voronoi polygons divide space into regions surrounding objects such that any point in an object's polygon is closer to that object than to any other object, while Delaunay triangulation is a triangulation of the points in a Voronoi diagram where the circumcircle of each triangle is an empty triangle.

The neighborhood vector determines the relative position of the neighboring cells from any

{*<sup>u</sup>* <sup>+</sup> *<sup>x</sup>*¯*<sup>i</sup>* <sup>|</sup>*<sup>i</sup>* =1, 2, ..., *<sup>m</sup>*¯} . We assume that there exists a neighborhood function *N*¯(*<sup>u</sup>*) mapping

A number of applications for Cellular Learning Automata have been developed recently such as rumor diffusion,image processing, modeling of commerce networks, fixed channel assign‐ ment in cellular networks, and VLSI Placement to mention a few (Beigy & Meybodi, 2004).

The Cellular Learning Automata can be classified into two types of *synchronous* and *asynchro‐ nous*. In synchronous Cellular Learning Automata, all cells are synchronized with a global clock and executed at the same time [10]. It is shown that the synchronous Cellular Learning Automata converges to a globally stable state for a class of rules called commutative rules. In some applications such as image processing, a type of Cellular Learning Automata in which the action of each cell in next stage of its evolution not only depends on the local environ‐

Irregular Cellular Learning Automata is a generalization of Cellular Learning Automata which removes the restriction of rectangular grid structure in traditional Cellular Learning Automata. This generalization is expected because there are applications which cannot be

ment (actions of its neighbors) but it also depends on the external environments.

. The neighbors of a particular cell *u* are set of cells

1 2 ( ) { , ,..., } *Nu u x u x u x* =+ + + *<sup>m</sup>* (9)

given cell *u* in the lattice *Z <sup>d</sup>*

96 Emerging Applications of Cellular Automata

**Figure 10.** Cellular Learning Automata

**3.3. Irregular Cellular Learning Automata**

adequately modeled with rectangular grids [10].

a cell *u* to the set of its neighbors according to equation (9).

An Irregular Cellular Learning Automata is a combination of Irregular Cellular Automata and Learning Automata (Figure 11). We define Irregular Cellular Learning Automata as an undirected graph in which each vertex represents a cell which is equipped with a learn‐ ing automaton.

**Figure 11.** Irregular Cellular Learning Automata, LA means Learning Automata in each neighbor cell.

The Learning Automaton residing in a particular cell determines its state (action) on the ba‐ sis of its action probability vector. Like Cellular Learning Automata, there is a rule that the Irregular Cellular Learning Automata operate according to it. The rule of the Cellular Learn‐ ing Automata and the actions selected by the neighboring Learning Automatons of any par‐ ticular Learning Automata determine the reinforcement signal to the Learning Automata residing in a cell. The neighboring Learning Automatons of any particular Learning Autom‐ ata constitute the local environment of that cell. The local environment of a cell is non-sta‐ tionary because the action probability vectors of the neighboring Learning Automatons vary during evolution of the Irregular Cellular Learning Automata.

An Irregular Cellular Learning Automata is formally defined below.

An Irregular Cellular Learning Automata is a structure *A = (G <E, V>, Φ, A, F)*, where


tion the nature of ad hoc networks is dynamically changing. Hence security is hard to achieve due to the dynamic nature of nodes. Routing protocols for WSNs are designed based on the assumption that all participating nodes are completely cooperative. In a closed MANET, all mobile nodes cooperate with each other towards a common destiny, such as emergency search/rescue or military and law enforcement operations. In an open MANET, different mobile nodes with different goals share their resources in order to ensure global connectivity [9, 15]. Lately, significant research efforts have focused on improving the secur‐ ity of ad hoc networks. In WSNs, nodes are both routers and terminals. Due to the lack of a routing infrastructure all the nodes have to cooperate to ensure successful communication. Clearly, cooperation means ensuring correct routing establishment mechanisms, the protec‐ tion of routing information and the security of packet forwarding. One major challenge that was neglected previously is that of making wireless sensor network robust against MAC layer misbehaviors. Significant applications of WSNs include establishing survivable, effi‐ cient, dynamic communication for emergency/rescue operations, disaster relief. Security is a critical problem when implementing WSN. The fast detection of malicious nodes is vital in mobile ad hoc networks, since they rely on the cooperation of nodes for routing and for‐ warding. Also, cooperation of misbehaving nodes can seriously degrade the performance

Cellular Learning Automata and Its Applications

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99

The security difference between wired infrastructure networks and wireless sensor net‐ works motivated researchers to model an intrusion detection system that can handle the new security challenges such as securing routing protocols [21]. We only list here some of

Sterne et al. proposed a dynamic intrusion detection hierarchy that is potentially scalable to large networks with using clustering [24]. This method is similar with Kachirski and Guha, but it can be structured in more than two levels. Thus, nodes on first level are cluster-heads and nodes on the second level are leaf nodes. In this model, every node has the task of moni‐ toring, logging, analyzing, properly responding to intrusions detection if there is enough evidence, and alert or report to cluster-heads. The cluster-heads, in addition, must also per‐

Sumalatha and Reddy proposed an approach for misbehavior detection [25]. Detection sys‐ tem is implemented based on fuzzy logic concept and the DSR has been used as routing pro‐ tocol. Every node implements an instance of the detection system and runs it in two phases. In the initial phase, the detecting system learns about the normal behavior of nodes with re‐ spect to the DSR protocol. Then, the node may leave the protected environment and enters

and jeopardize the functionality of network.

the existent research work that is related to our approach.

**1.** data fusion/integration and data filtering,

**2.** computations of intrusion, and

**3.** security management.

**4.2. Intrusion Detection Protocols**

form:

**•** : *F* F ®*<sup>j</sup>* b is the local rule of the irregular cellular learning automata in each vertex *j* where *<sup>j</sup>* { (, ) } { } *i j* F =F Î +F *ij E* is the set of states of all neighbors of *j,*and *β* is the set of values that the reinforcement signal can take. *β* computes the reinforcement signal for Learning Automata based on the actions selected by the neighboring Learning Autom‐ ata.

Note that in the definition of Irregular Cellular Learning Automata, no explicit definition of neighborhood of each cell is given. This is because neighborhood in Irregular Cellular Learning Automata is implicitly defined in definition of the graph *G*.

In what follows, we consider Irregular Cellular Learning Automata with *n* cells. The learn‐ ing automaton *A <sup>i</sup>* which has a finite action set α*<sup>i</sup>* is associated to cell *i* (for *i=1, 2, …, n*) of the Irregular Cellular Learning Automata. Let the cardinality of α*<sup>i</sup>* be *m <sup>i</sup>* . The state of the Irregu‐ lar Cellular Learning Automata represented by *p=* (*p <sup>1</sup> , p <sup>2</sup> ,..., p <sup>n</sup>*), where *p <sup>i</sup>* =( *p <sup>i</sup>*1, *p <sup>i</sup>*2,..., *p imi* ) is the action probability vector of *A <sup>i</sup>* . The operation of the Irregular Cellular Learning Autom‐ ata takes place as the following iterations. At iteration *k*, each learning automaton chooses an action. Let α*<sup>i</sup>* ∈ α be the action chosen by *A <sup>i</sup>* . Then all learning automata receive a reinforce‐ ment signal. Let *β <sup>i</sup>* ∈*β* be the reinforcement signal received by *A <sup>i</sup>* . This reinforcement signal is produced by the application of local rule *F(Φ <sup>i</sup> )→β*. Finally, each Learning Automata updates its action probability vector on the basis of the supplied reinforcement signal and the chosen action by the cell. This process continues until the desired result is obtained.

There are some applications that apply Irregular Cellular Learning Automata such as Image Processing, Graph Coloring, Social Modeling, Clustering and Sensor network applications like Channel Assignment and Routing. In the following, we introduce a sensor network application.

#### **4. Intrusion Detection in Wireless Sensor Network Using Irregular Cellular Learning Automata**

#### **4.1. Wireless Sensor Networks**

A Wireless Sensor Network contains hundreds or thousands of sensor nodes. Basically, each sensor node comprises sensing, processing, transmission, mobilizer, position finding sys‐ tem, and power units (some of these components are optional like the mobilizer). By defini‐ tion the nature of ad hoc networks is dynamically changing. Hence security is hard to achieve due to the dynamic nature of nodes. Routing protocols for WSNs are designed based on the assumption that all participating nodes are completely cooperative. In a closed MANET, all mobile nodes cooperate with each other towards a common destiny, such as emergency search/rescue or military and law enforcement operations. In an open MANET, different mobile nodes with different goals share their resources in order to ensure global connectivity [9, 15]. Lately, significant research efforts have focused on improving the secur‐ ity of ad hoc networks. In WSNs, nodes are both routers and terminals. Due to the lack of a routing infrastructure all the nodes have to cooperate to ensure successful communication. Clearly, cooperation means ensuring correct routing establishment mechanisms, the protec‐ tion of routing information and the security of packet forwarding. One major challenge that was neglected previously is that of making wireless sensor network robust against MAC layer misbehaviors. Significant applications of WSNs include establishing survivable, effi‐ cient, dynamic communication for emergency/rescue operations, disaster relief. Security is a critical problem when implementing WSN. The fast detection of malicious nodes is vital in mobile ad hoc networks, since they rely on the cooperation of nodes for routing and for‐ warding. Also, cooperation of misbehaving nodes can seriously degrade the performance and jeopardize the functionality of network.

#### **4.2. Intrusion Detection Protocols**

An Irregular Cellular Learning Automata is a structure *A = (G <E, V>, Φ, A, F)*, where

**•** *A* is the set of Learning Automata each of which is assigned to one cell of the Irregular

where *<sup>j</sup>* { (, ) } { } *i j* F =F Î +F *ij E* is the set of states of all neighbors of *j,*and *β* is the set of values that the reinforcement signal can take. *β* computes the reinforcement signal for Learning Automata based on the actions selected by the neighboring Learning Autom‐

Note that in the definition of Irregular Cellular Learning Automata, no explicit definition of neighborhood of each cell is given. This is because neighborhood in Irregular Cellular

In what follows, we consider Irregular Cellular Learning Automata with *n* cells. The learn‐

ata takes place as the following iterations. At iteration *k*, each learning automaton chooses an

is produced by the application of local rule *F(Φ <sup>i</sup> )→β*. Finally, each Learning Automata updates its action probability vector on the basis of the supplied reinforcement signal and the chosen

There are some applications that apply Irregular Cellular Learning Automata such as Image Processing, Graph Coloring, Social Modeling, Clustering and Sensor network applications like Channel Assignment and Routing. In the following, we introduce a sensor network application.

A Wireless Sensor Network contains hundreds or thousands of sensor nodes. Basically, each sensor node comprises sensing, processing, transmission, mobilizer, position finding sys‐ tem, and power units (some of these components are optional like the mobilizer). By defini‐

∈*β* be the reinforcement signal received by *A <sup>i</sup>*

**4. Intrusion Detection in Wireless Sensor Network Using Irregular**

Learning Automata is implicitly defined in definition of the graph *G*.

which has a finite action set α*<sup>i</sup>*

lar Cellular Learning Automata represented by *p=* (*p <sup>1</sup> , p <sup>2</sup> ,..., p <sup>n</sup>*), where *p <sup>i</sup>*

action by the cell. This process continues until the desired result is obtained.

Irregular Cellular Learning Automata. Let the cardinality of α*<sup>i</sup>*

∈ α be the action chosen by *A <sup>i</sup>*

is the local rule of the irregular cellular learning automata in each vertex *j*

is associated to cell *i* (for *i=1, 2, …, n*) of the

. Then all learning automata receive a reinforce‐

. The state of the Irregu‐

. This reinforcement signal

=( *p <sup>i</sup>*1, *p <sup>i</sup>*2,..., *p imi* )

be *m <sup>i</sup>*

. The operation of the Irregular Cellular Learning Autom‐

**•** *G* is an undirected graph, with *V* as the set of vertices and *E* as the set of edges.

**•** *Φ* is a finite set of states.

98 Emerging Applications of Cellular Automata

**•** : *F* F ®*<sup>j</sup>*

ata.

ing automaton *A <sup>i</sup>*

action. Let α*<sup>i</sup>*

ment signal. Let *β <sup>i</sup>*

is the action probability vector of *A <sup>i</sup>*

**Cellular Learning Automata**

**4.1. Wireless Sensor Networks**

Cellular Learning Automata.

b

The security difference between wired infrastructure networks and wireless sensor net‐ works motivated researchers to model an intrusion detection system that can handle the new security challenges such as securing routing protocols [21]. We only list here some of the existent research work that is related to our approach.

Sterne et al. proposed a dynamic intrusion detection hierarchy that is potentially scalable to large networks with using clustering [24]. This method is similar with Kachirski and Guha, but it can be structured in more than two levels. Thus, nodes on first level are cluster-heads and nodes on the second level are leaf nodes. In this model, every node has the task of moni‐ toring, logging, analyzing, properly responding to intrusions detection if there is enough evidence, and alert or report to cluster-heads. The cluster-heads, in addition, must also per‐ form:


Sumalatha and Reddy proposed an approach for misbehavior detection [25]. Detection sys‐ tem is implemented based on fuzzy logic concept and the DSR has been used as routing pro‐ tocol. Every node implements an instance of the detection system and runs it in two phases. In the initial phase, the detecting system learns about the normal behavior of nodes with re‐ spect to the DSR protocol. Then, the node may leave the protected environment and enters the second phase where node finds some of the nodes as malicious and captures each node parameters such as number of route requests, number of route replies and number of up‐ dates at each node in the network. These parameters are used for input of the fuzzy infer‐ ence system and also are fuzzified at the beginning in order to make fuzzy values. To find the crisp value of the calculated trust, trust is assigned to each node in the ad hoc network. This process mainly contains fuzzification, inference by rule base construction and defuzzifi‐ cation processes. The Defuzzification is the process of conversion of fuzzy output set into a single number. In this approach, the authors have used these numbers to detect the mali‐ cious nodes in the network. In one of recent works, the authors suggested learning automa‐ ta-based protocol for intrusion detection (LAID) in wireless sensor networks [27]. LAID functions in a distributed manner and uses the learning automata to optimize the selection of paths in which sampling has to be performed. The system, in essence, tries to identify or approximate the location of the attacker and, thus, it catches the malicious packets sent by the attacker. LAID protocol is not energy-aware and it may not be always practically ideal for resource-constrained networks such as distributed WSN.

**Figure 12.** Network Model

erence to its cluster-heads.

**1.** *Phase 1*

To configure the routing in network, each node constructs its probability vector *{p <sup>1</sup> , p <sup>2</sup> ,…, p <sup>n</sup> }*. Each node sends its *Id* and energy level to its cluster-head and neighbor nodes to form the clusters. Neighbor nodes construct their local routing tables upon receiving this packet. For each received packet, an entry for the node *Id* in the packet is created in routing table,

*EnergyLevel*

Where *p <sup>i</sup>* is the probability of selecting the *i th* neighbor node, *EnergyLevel <sup>i</sup>* is the energy lev‐ el of the *i th* neighbor node and *m* is the total number of neighbor nodes. Indeed each node in the network gets the preference of all nodes that are in the same clusters and sends this pref‐

Systematically, in our protocol, we attach an IDS agent to each mobile node. These IDS agents run independently and monitor local activities to detect abnormal behaviors. We as‐ sume the local IDS agent is tamper resistant. Several software tamper resistance techniques have been proposed that are very hard to crack and suitable for our approach. In this meth‐ od, we have considered two level architecture for each node. The first layer is the internal IDS agent. IDS agent can be divided into the following components: the data collection mod‐ ule (DCM), the data transmission quality (DTQ) module, the cluster aggregation and fusion module (CAFM), and the intrusion response module. A diagram is given in Figure 13.

*i*

*j*

(10)

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101

and initial preference for that node is calculated as follows:

( )

=

*i j and i*

¹

=

1

=

*j*

*m*

å

*EnergyLevel p i*

1, 2,3,...

Further, another learning automata-based intrusion detection protocol (S-LAID) has been proposed [28]. S-LAID functions in a distributed manner with each node functioning inde‐ pendently without any knowledge about the adjacent nodes. S-LAID assumes that the sys‐ tem budget is configured prior to its installation. In this protocol, the authors considered that sampling of a packet consumes energy. In S-LAID, each node continuously samples its interface at a minimum sampling budget. According to S-LAID algorithm if malicious pack‐ ets are found and the detection rate is higher than the penalty threshold, then the sampling rate is increased. The learning functions calculate the sampling rate that should be used dur‐ ing the next instant by the automaton. In order to maintain efficiency and increase lifetime, the authors have bound the value by the sampling rate. They also have used the rate control algorithm to moderate the increase in the sampling rate.

#### **4.3. Irregular Cellular Learning Automata-based Intrusion Detection Protocols**

In this protocol, the entire network is divided into multiple clusters. Nodes are placed into clusters with one cluster-head for each cluster (Figure 12). Each cluster-head node is aware of its cluster information. The authenticity of a node is mostly determined by the nodes that are in same cluster. Each node has an IDS agent for detecting potential abnormalities in packets forwarding process. To reduce the overhead of intrusion detection process, nodes in a cluster will cooperate to select a cluster-head node based on learning automata residing in each node for handling the detection process for the whole cluster. Data packets may tra‐ verse between different clusters. The process of misbehaving nodes detection is performed in 3 sequential phases.


**Figure 12.** Network Model

the second phase where node finds some of the nodes as malicious and captures each node parameters such as number of route requests, number of route replies and number of up‐ dates at each node in the network. These parameters are used for input of the fuzzy infer‐ ence system and also are fuzzified at the beginning in order to make fuzzy values. To find the crisp value of the calculated trust, trust is assigned to each node in the ad hoc network. This process mainly contains fuzzification, inference by rule base construction and defuzzifi‐ cation processes. The Defuzzification is the process of conversion of fuzzy output set into a single number. In this approach, the authors have used these numbers to detect the mali‐ cious nodes in the network. In one of recent works, the authors suggested learning automa‐ ta-based protocol for intrusion detection (LAID) in wireless sensor networks [27]. LAID functions in a distributed manner and uses the learning automata to optimize the selection of paths in which sampling has to be performed. The system, in essence, tries to identify or approximate the location of the attacker and, thus, it catches the malicious packets sent by the attacker. LAID protocol is not energy-aware and it may not be always practically ideal

Further, another learning automata-based intrusion detection protocol (S-LAID) has been proposed [28]. S-LAID functions in a distributed manner with each node functioning inde‐ pendently without any knowledge about the adjacent nodes. S-LAID assumes that the sys‐ tem budget is configured prior to its installation. In this protocol, the authors considered that sampling of a packet consumes energy. In S-LAID, each node continuously samples its interface at a minimum sampling budget. According to S-LAID algorithm if malicious pack‐ ets are found and the detection rate is higher than the penalty threshold, then the sampling rate is increased. The learning functions calculate the sampling rate that should be used dur‐ ing the next instant by the automaton. In order to maintain efficiency and increase lifetime, the authors have bound the value by the sampling rate. They also have used the rate control

for resource-constrained networks such as distributed WSN.

100 Emerging Applications of Cellular Automata

algorithm to moderate the increase in the sampling rate.

in 3 sequential phases.

**4.3. Irregular Cellular Learning Automata-based Intrusion Detection Protocols**

**•** Phase 1: Detection of misbehaving nodes by cluster-head node in same cluster.

**•** Phase 2: Confirmation of misbehaving nodes by neighbor nodes. **•** Phase 3: Reward or penalize misbehaving node by neighbor nodes.

In this protocol, the entire network is divided into multiple clusters. Nodes are placed into clusters with one cluster-head for each cluster (Figure 12). Each cluster-head node is aware of its cluster information. The authenticity of a node is mostly determined by the nodes that are in same cluster. Each node has an IDS agent for detecting potential abnormalities in packets forwarding process. To reduce the overhead of intrusion detection process, nodes in a cluster will cooperate to select a cluster-head node based on learning automata residing in each node for handling the detection process for the whole cluster. Data packets may tra‐ verse between different clusters. The process of misbehaving nodes detection is performed To configure the routing in network, each node constructs its probability vector *{p <sup>1</sup> , p <sup>2</sup> ,…, p <sup>n</sup> }*. Each node sends its *Id* and energy level to its cluster-head and neighbor nodes to form the clusters. Neighbor nodes construct their local routing tables upon receiving this packet. For each received packet, an entry for the node *Id* in the packet is created in routing table, and initial preference for that node is calculated as follows:

$$\begin{aligned} p(i) &= \frac{\textit{EnergyLevel}\_i}{\sum\_{j=1}^{m} \textit{EnergyLevel}\_j} \\ \text{if } i \neq j \\ \textit{and} \\ i &= 1, 2, 3, \dots \end{aligned} \tag{10}$$

Where *p <sup>i</sup>* is the probability of selecting the *i th* neighbor node, *EnergyLevel <sup>i</sup>* is the energy lev‐ el of the *i th* neighbor node and *m* is the total number of neighbor nodes. Indeed each node in the network gets the preference of all nodes that are in the same clusters and sends this pref‐ erence to its cluster-heads.

**1.** *Phase 1*

Systematically, in our protocol, we attach an IDS agent to each mobile node. These IDS agents run independently and monitor local activities to detect abnormal behaviors. We as‐ sume the local IDS agent is tamper resistant. Several software tamper resistance techniques have been proposed that are very hard to crack and suitable for our approach. In this meth‐ od, we have considered two level architecture for each node. The first layer is the internal IDS agent. IDS agent can be divided into the following components: the data collection mod‐ ule (DCM), the data transmission quality (DTQ) module, the cluster aggregation and fusion module (CAFM), and the intrusion response module. A diagram is given in Figure 13.

The second layer is the ICLA. This layer is a combination of the detection engine module and learning automata residing in each node.

( ) ( )

=´ +

1, 2,3,...

l

*i*

In this function, *EnergyLevel <sup>i</sup>*

*i*

*STB*

**2.** *Phase 2*

tion engine.

*Detection Engine Module*

=

has tried to transmit to. This statistic is *STB()*.

*Cluster Aggregation and Fusion Module (CAFM)*

( ) 1 2

stability of the nodal behavior. This quantity is measured as follows:

*i*

å

*i i*

*<sup>i</sup> <sup>i</sup> STB EnergyLevel p i Er IEnergy N SEnergy*

initial energy in each node, and *SEnergy* is the required energy to transmit data. *N* is the number of Data packets or the bucket size. *Er()* is probability of error in the channel. λ<sup>1</sup> and λ2 declare the effect of nodal behavior and node's energy respectively. Also *STB <sup>i</sup> ()* is the

*numberoffcrwardedpackets*

å å (12)

*numberoffcrwardedpackets numberoffrecievedpackets* <sup>=</sup> <sup>+</sup>

Every node measures the number of received acknowledgments from the neighbor nodes it

If a node is inter-cluster node or normal node, it sends the gathered data from the neighbor‐ ing nodes to detection engine on second level. It can send the alarms and reports to its clus‐ ter-head based on voting request. While if the node is a cluster-head node, then the CAFM module receives the alarms and reports from inter-cluster nodes. Also, CAFM module of the cluster-head node allows the voting or prevents it by aggregation and fusion of the received alarms and reports. When the CAFM module of each cluster-head node receives the vote re‐ quest packet, it votes for the suspect node. Voting process is performed base on the results that are calculated by the detection engine of inter-cluster nodes. At the end of the voting process, CAFM module of the cluster-head node sends the number of votes (*V <sup>m</sup>*) to its detec‐

The detection engine identifies the misbehaving nodes according to the received informa‐ tion from CAFM module. The detection engine set a threshold (*τ*) according to equation (13). This threshold is determined based on the behavior and energy level of all nodes that are participating in voting process. In this equation, we use STB and energy level of each

node because these values show the quality of node behavior properly.


is the level of the energy of the *i th* neighbor node. *IEnergy* is the

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(11)

103

 l

#### *Data Collection Module (DCM)*

The functionality of the data collection module is to collect security related data via monitor‐ ing local activities and local behaviors of neighbor nodes. We define misbehaving nodes as those that have aberrations in data exchange patterns. We have used the bucket as a specific count of packets that are transmitted from one node to the other. At the end of every bucket, Data collection modules send the gathered information and statistics to CAFM. This infor‐ mation determines the behavior of the node and its neighbors that are sending and receiving data packets.

#### *Data Transmission Quality (DTQ Function)*

This module has a function to measure the quality of a communication node. In our Method, the DTQ function measures changes in the environment and sends a probability to a higher layer. This probability is calculated as follow:

$$\begin{aligned} p\left(i\right) &= \lambda\_i \times \frac{STB\_i\left(\begin{array}{c} \\ \end{array}\right)}{Er\left(\begin{array}{c} \\ \end{array}\right)} + \lambda\_2 \frac{EnergyLevel\_i}{IEnergy-N \times SEnergy} \\ i &= 1,2,3,... \end{aligned} \tag{11}$$

In this function, *EnergyLevel <sup>i</sup>* is the level of the energy of the *i th* neighbor node. *IEnergy* is the initial energy in each node, and *SEnergy* is the required energy to transmit data. *N* is the number of Data packets or the bucket size. *Er()* is probability of error in the channel. λ<sup>1</sup> and λ2 declare the effect of nodal behavior and node's energy respectively. Also *STB <sup>i</sup> ()* is the stability of the nodal behavior. This quantity is measured as follows:

$$\text{STB}\_{i} = \frac{\sum\_{i} \text{number of ferwardedpackets}}{\sum\_{i} \text{number of ferwardedpackets} + \sum\_{i} \text{number of fireviewedpackets}} \tag{12}$$

Every node measures the number of received acknowledgments from the neighbor nodes it has tried to transmit to. This statistic is *STB()*.

#### **2.** *Phase 2*

The second layer is the ICLA. This layer is a combination of the detection engine module

The functionality of the data collection module is to collect security related data via monitor‐ ing local activities and local behaviors of neighbor nodes. We define misbehaving nodes as those that have aberrations in data exchange patterns. We have used the bucket as a specific count of packets that are transmitted from one node to the other. At the end of every bucket, Data collection modules send the gathered information and statistics to CAFM. This infor‐ mation determines the behavior of the node and its neighbors that are sending and receiving

This module has a function to measure the quality of a communication node. In our Method, the DTQ function measures changes in the environment and sends a probability to a higher

and learning automata residing in each node.

*Data Collection Module (DCM)*

102 Emerging Applications of Cellular Automata

**Figure 13.** Internal Model for the IDS Agent

*Data Transmission Quality (DTQ Function)*

layer. This probability is calculated as follow:

data packets.

#### *Cluster Aggregation and Fusion Module (CAFM)*

If a node is inter-cluster node or normal node, it sends the gathered data from the neighbor‐ ing nodes to detection engine on second level. It can send the alarms and reports to its clus‐ ter-head based on voting request. While if the node is a cluster-head node, then the CAFM module receives the alarms and reports from inter-cluster nodes. Also, CAFM module of the cluster-head node allows the voting or prevents it by aggregation and fusion of the received alarms and reports. When the CAFM module of each cluster-head node receives the vote re‐ quest packet, it votes for the suspect node. Voting process is performed base on the results that are calculated by the detection engine of inter-cluster nodes. At the end of the voting process, CAFM module of the cluster-head node sends the number of votes (*V <sup>m</sup>*) to its detec‐ tion engine.

#### *Detection Engine Module*

The detection engine identifies the misbehaving nodes according to the received informa‐ tion from CAFM module. The detection engine set a threshold (*τ*) according to equation (13). This threshold is determined based on the behavior and energy level of all nodes that are participating in voting process. In this equation, we use STB and energy level of each node because these values show the quality of node behavior properly.

$$\begin{aligned} \tau &= \gamma\_1 \times \frac{STB\_i\left(\begin{array}{c} \\ \end{array}\right)}{\sum\limits\_{j=1}^{M} STB\_j\left(\begin{array}{c} \\ \end{array}\right)} + \gamma\_2 \times \frac{EnergyLevel\_i}{\sum\limits\_{j=1}^{M} EnergyLevel\_j} \\ i &= 1,2,... \\ i &\neq j \end{aligned} \tag{13}$$

them in *STB* table. Second, if the specific node is a cluster-head node, the intrusion response module sends the order of penalization or dismissal of the suspected node to all cluster no‐ des. Therefore, misbehaving nodes won't be permitted to participate in routing process.

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In this section, we have implemented the proposed protocol by MATLAB and Glomosim

The network area size is 2000\*2000 (in m2). The mobility model is the random waypoint model. The minimum speed is 5 m/s, and the maximum speed is 15 m/s. We have used the IEEE 802.11 for distributed wireless sensor networks as the MAC layer protocol. The num‐ ber of nodes varies from 1000 to 3000 nodes. Radio bandwidth is 250000(in bps). Initial ener‐ gy level of each node is 5(mW) and radio transmit power is 10 (in dBm). The size of all data packets is set to 512 bytes. The duration of each simulation is 1800 seconds. The values of γ<sup>1</sup>

**•** Black-hole attack: In this attack, a misbehaving node uses the routing protocol to adver‐ tise itself as having the shortest path to the node whose packets it wants to intercept. The attacker will then receive the traffic which is destined for other nodes, and then it can

**•** Denial of Service: A node prevents itself from receiving and forwarding data packets to

**•** Malicious Flooding: In this attack, the misbehaving node pumps a great deal of useless and garbage packets to the network. In this way, it corrodes the resources of the network

**•** Packet dropping: A node conditionally or randomly drops data packets which are sup‐

The results of simulation in figure 14 show the percentage of detection rate with variation of misbehaving nodes' percentage. In this simulation, the number of nodes is 100 and the num‐ ber of clusters is 10. Obviously, at first, the percentage of detection rate in all attacks has de‐ creased, and afterwards with increase of misbehaving nodes' percentage the detection rate has increased either. The important reason for this behavior is the application of ICLA. In voting process, gathered information of neighbor nodes is increased which have participat‐ ed in voting. In fact, the system learns abnormal behavior by increase of gathered informa‐ tion of learning automata from its environment. Therefore, the misbehaving nodes will be detected accurately. Because of using the energy and behavior factors for detecting the mali‐

simulator, a scalable discrete event simulator developed by UCLA.

In our simulation, we have implemented and used the following attacks:

**4.4. Evaluation the Proposed Protocol**

and γ2 are considered 0.5 in our simulations.

drop or modify the packets.

such as bandwidth and energy.

their destinations.

posed to be forward.

*Simulation Results*

*Simulation settings*

*Simulated Attacks*

In equation (6), *STB <sup>i</sup> ()* is the stability of the nodal behavior which can be calculated by equation (6). *EnergyLevel <sup>i</sup>* is the level of the energy of the *i th* neighbor node. *M* is the total number of nodes that participate in the voting. γ1 and γ2 are numbers between zero and one. According to the information of CAFM module, if the detection engine finds one or more values of *STB* in the table that are less than the threshold (*τ*), then it realizes that there may be one or more misbehaving nodes in its cluster. So it sends a vote request message about the suspect nodes to the CAFM module. In addition, the detection engine module makes a decision in cooperating with ICLA based on the number of vote response messages gathered by CAFM module. According to the results of voting, the node *M* is a well-behaving one and should be rewarded or it is a misbehaving one and should be punished.

#### **3.** *Phase 3*

CAFM module of the cluster-head will gather all the vote responses about suspect node *M*. CAFM module then sends the number of gathered vote response messages (*V <sup>m</sup>*) to the de‐ tection engine module. According to the number of voting for the suspect node's authentici‐ ty, a decision is made as follow:


#### *Intrusion Response Module*

The Intrusion Response Module efficiently penalizes misbehaving nodes based on updated statistics which are created and sent to intrusion response module by learning automata. The intrusion response module performs the following actions according to received statis‐ tics. First, it receives the updated *STB* values (equation 5) from the second layer and saves them in *STB* table. Second, if the specific node is a cluster-head node, the intrusion response module sends the order of penalization or dismissal of the suspected node to all cluster no‐ des. Therefore, misbehaving nodes won't be permitted to participate in routing process.

#### **4.4. Evaluation the Proposed Protocol**

In this section, we have implemented the proposed protocol by MATLAB and Glomosim simulator, a scalable discrete event simulator developed by UCLA.

#### *Simulation settings*

( ) ( ) 1 2

=´ +´

1, 2,...

neighbor nodes of *M* add node *M* to their black lists.

*N* with *a=0.6* according to *L R-P* learning algorithm.

*i i j*

equation (6). *EnergyLevel <sup>i</sup>*

104 Emerging Applications of Cellular Automata

ty, a decision is made as follow:

*Intrusion Response Module*

**3.** *Phase 3*

= ¹

t g

1 1


*j j*

and should be rewarded or it is a misbehaving one and should be punished.

*M M*

*i i*

*STB EnergyLevel*

In equation (6), *STB <sup>i</sup> ()* is the stability of the nodal behavior which can be calculated by

number of nodes that participate in the voting. γ1 and γ2 are numbers between zero and one. According to the information of CAFM module, if the detection engine finds one or more values of *STB* in the table that are less than the threshold (*τ*), then it realizes that there may be one or more misbehaving nodes in its cluster. So it sends a vote request message about the suspect nodes to the CAFM module. In addition, the detection engine module makes a decision in cooperating with ICLA based on the number of vote response messages gathered by CAFM module. According to the results of voting, the node *M* is a well-behaving one

CAFM module of the cluster-head will gather all the vote responses about suspect node *M*. CAFM module then sends the number of gathered vote response messages (*V <sup>m</sup>*) to the de‐ tection engine module. According to the number of voting for the suspect node's authentici‐

**•** If more than 80 percent of the participating nodes in the voting give a positive vote to sus‐ pect node *M*, then this node will be exclude from participating in the routing. Moreover,

**•** If less than 80 percent and more than 50 percent of the participating nodes in the voting give a positive vote to suspect node *M*, then this action will be penalized by learning au‐

**•** If less than 50 percent and more than 30 percent of the participating nodes in the voting give a positive vote to suspect node *M*, then this action will be penalized by learning au‐

**•** If less than 30 percent of the participating nodes in the voting give a positive vote to sus‐ pect node *M*, then this action will be rewarded by learning automata residing in the node

The Intrusion Response Module efficiently penalizes misbehaving nodes based on updated statistics which are created and sent to intrusion response module by learning automata. The intrusion response module performs the following actions according to received statis‐ tics. First, it receives the updated *STB* values (equation 5) from the second layer and saves

tomata residing in the node *N* with *b=0.2* according to *L R-P* learning algorithm.

tomata residing in the node *N* with *b=0.4* according to *L R-P* learning algorithm.

*STB EnergyLevel*

g

*j j*

å å (13)

is the level of the energy of the *i th* neighbor node. *M* is the total

The network area size is 2000\*2000 (in m2). The mobility model is the random waypoint model. The minimum speed is 5 m/s, and the maximum speed is 15 m/s. We have used the IEEE 802.11 for distributed wireless sensor networks as the MAC layer protocol. The num‐ ber of nodes varies from 1000 to 3000 nodes. Radio bandwidth is 250000(in bps). Initial ener‐ gy level of each node is 5(mW) and radio transmit power is 10 (in dBm). The size of all data packets is set to 512 bytes. The duration of each simulation is 1800 seconds. The values of γ<sup>1</sup> and γ2 are considered 0.5 in our simulations.

#### *Simulated Attacks*

In our simulation, we have implemented and used the following attacks:


#### *Simulation Results*

The results of simulation in figure 14 show the percentage of detection rate with variation of misbehaving nodes' percentage. In this simulation, the number of nodes is 100 and the num‐ ber of clusters is 10. Obviously, at first, the percentage of detection rate in all attacks has de‐ creased, and afterwards with increase of misbehaving nodes' percentage the detection rate has increased either. The important reason for this behavior is the application of ICLA. In voting process, gathered information of neighbor nodes is increased which have participat‐ ed in voting. In fact, the system learns abnormal behavior by increase of gathered informa‐ tion of learning automata from its environment. Therefore, the misbehaving nodes will be detected accurately. Because of using the energy and behavior factors for detecting the mali‐ cious nodes in black-hole attack, the results for black-hole attack are detected accurately. Consequently these results are better than that of other attacks with higher population of misbehaving nodes.

In figure 16, we have shown the results of simulation and have discussed the detection rate with variation in number of clusters. In this simulation, detection rate will be decreased with increasing the number of clusters. Because of the constant total number of nodes in the net‐ work and the increase of clusters' number, the number of nodes in each cluster will be de‐ creased. Therefore, the action probability vector in each cluster-head will be decreased, and this causes the detection rate to decrease in each cluster. As it is shown in this figure, after a specific number of clusters (10), the learning rate has increased, and, thus the detection rate

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107

In figure 17, the simulation results of false positive rate in variation with number of clus‐ ters are illustrated. At first, false positive rate increases with increase of clusters' number, because as number of clusters increases, number of nodes inside each clusters decreases. Therefore, the length of action probability vector in cluster-heads' learning automata decreas‐ es, but after specific number of clusters (10) the false positive rate decreases due to increas‐ ing of learning rate increases. This decrease for denial of service attack has been more noticeable than the other attacks because of ICLA application in detecting attacks and no‐

In the next simulation, we have evaluated the effects of mentioned attacks on energy con‐ sumption of network in our method. Figure 18 shows the average energy consumption with variation in number of nodes. For all attacks, energy consumption has increased with in‐ crease of nodes' number. Moreover, average energy consumption for malicious forwarding attack is lower than black-hole attack, and the average energy consumption of black-hole at‐ tack is lower than other attacks. These results were predictable, because the proposed meth‐ od uses the energy level of each node for detecting malicious flooding and black-hole attacks. In addition, the proposed protocol uses each node's behavior for black-hole attack

has increased too. In this state, ICLA performs very well.

**Figure 16.** Detection rate vs. number of cluster

and this causes the energy consumption to increase.

dal behavior.

**Figure 14.** Detection rate vs. percentage of misbehaving nodes

Figure 15 shows the false positive rate with variation in percentage of the misbehaving no‐ des. In this simulation, the number of nodes is 100 and the number of clusters is 10. In this figure, first, the percentage of false positive rate has increased and then the false positive rate has decreased with increasing the percentage of misbehaving nodes. The learning au‐ tomaton in each node gathers the information from the environment and this causes detec‐ tion of misbehaving nodes to be performed properly. So the percentage of false positive rate has decreased after obvious quantity of 40%.

**Figure 15.** False positive rate vs. percentage of misbehaving nodes

In figure 16, we have shown the results of simulation and have discussed the detection rate with variation in number of clusters. In this simulation, detection rate will be decreased with increasing the number of clusters. Because of the constant total number of nodes in the net‐ work and the increase of clusters' number, the number of nodes in each cluster will be de‐ creased. Therefore, the action probability vector in each cluster-head will be decreased, and this causes the detection rate to decrease in each cluster. As it is shown in this figure, after a specific number of clusters (10), the learning rate has increased, and, thus the detection rate has increased too. In this state, ICLA performs very well.

**Figure 16.** Detection rate vs. number of cluster

cious nodes in black-hole attack, the results for black-hole attack are detected accurately. Consequently these results are better than that of other attacks with higher population of

Figure 15 shows the false positive rate with variation in percentage of the misbehaving no‐ des. In this simulation, the number of nodes is 100 and the number of clusters is 10. In this figure, first, the percentage of false positive rate has increased and then the false positive rate has decreased with increasing the percentage of misbehaving nodes. The learning au‐ tomaton in each node gathers the information from the environment and this causes detec‐ tion of misbehaving nodes to be performed properly. So the percentage of false positive rate

misbehaving nodes.

106 Emerging Applications of Cellular Automata

**Figure 14.** Detection rate vs. percentage of misbehaving nodes

has decreased after obvious quantity of 40%.

**Figure 15.** False positive rate vs. percentage of misbehaving nodes

In figure 17, the simulation results of false positive rate in variation with number of clus‐ ters are illustrated. At first, false positive rate increases with increase of clusters' number, because as number of clusters increases, number of nodes inside each clusters decreases. Therefore, the length of action probability vector in cluster-heads' learning automata decreas‐ es, but after specific number of clusters (10) the false positive rate decreases due to increas‐ ing of learning rate increases. This decrease for denial of service attack has been more noticeable than the other attacks because of ICLA application in detecting attacks and no‐ dal behavior.

In the next simulation, we have evaluated the effects of mentioned attacks on energy con‐ sumption of network in our method. Figure 18 shows the average energy consumption with variation in number of nodes. For all attacks, energy consumption has increased with in‐ crease of nodes' number. Moreover, average energy consumption for malicious forwarding attack is lower than black-hole attack, and the average energy consumption of black-hole at‐ tack is lower than other attacks. These results were predictable, because the proposed meth‐ od uses the energy level of each node for detecting malicious flooding and black-hole attacks. In addition, the proposed protocol uses each node's behavior for black-hole attack and this causes the energy consumption to increase.

similar to our model of spatial collective action can be robust with respect to variation in si‐

Cellular Learning Automata and Its Applications

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109

We applied these tools to intrusion detection protocols which is an application from sensor networks. This is a novel approach which used of Irregular Cellular Learning Automata to detect suspect nodes by using and analyzing nodes' behavior during routing process and nodes' energy level. It also implements Irregular Cellular Learning Automata to detect ab‐ normal behaviors. Afterwards, our method starts its voting process in which it decides to reward or penalize suspect node based on learning automata reports. The simulations show that our proposed method not only has a proper detection rate but also is an energy-aware

[1] Abolhasani, S. M., Meybodi, M. R., & Esnaashari, M. (2007). LABER: A Learning Au‐ tomata Based Energy-aware Routing Protocol for Sensor Networks. *IKT conference,*

[2] Al-Karaki, J. N., & Kamal, A. E. Routing techniques in wireless sensor networks: a

[3] Ankit, M., Arpit, M., Deepak, T. J., Venkateswarlu, R., & Janakiram, D. (2006). Tiny‐ LAP: A Scalable learning automata-based energy aware routing protocol for sensor networks. *Communicated to IEEE Wireless and Communications and Networking Confer‐*

[4] Beigy, H., & Meybodi, M. R. (2004). A mathematical framework for cellular learning automata. Advances on Complex Systems Nos. 3-4, September/December, New Jer‐

[5] Carvalho, J. P., Carola, M., & Tomé, A. B. (2002). Using Rule-Based Fuzzy Cognitive Maps to Model Dynamic Cell Behavior in Voronoi Based Cellular Automata. *FCT-*

*Portuguese Foundation for Science and Technology, Lisboa, Portugal*.

multaneous vs.

**Author details**

**References**

*Tehran, Iran.*

*ence, Las Vegas, NV USA*.

sey,., 7, 295-320.

protocol in detecting malicious nodes.

Amir Hosein Fathy Navid1\* and Amir Bagheri Aghababa2

\*Address all correspondence to: Amir.Fathy.n@qiau.ac.ir

2 Islamic Azad University East Tehran Branch, Tehran, Iran

survey. *IEEE Wireless Communications*, 6-28, 11.

1 Islamic Azad University, Hamedan Beranch, Bahar, Hamedan, Iran

**Figure 17.** False positive rate vs. number of clusters

**Figure 18.** Average energy consumption in the network vs. number of nodes

#### **5. Conclusion**

In this Chapter, we have disscussed Cellular Learning Automata which has standard as‐ sumption of a rectangular grid structure. Then we have represented Irregular Cellular Learning Automata which are the models cause to develop tools that allow us use both rec‐ tangular and even irregular grids within one and the same Cellular Automata modeling framework. Of course, we are aware that the regularity of the grid structure is but one of a number of idealizations used in Cellular Automata modeling. For example, it has been dis‐ cussed controversially whether and under what conditions a simple influence dynamics similar to our model of spatial collective action can be robust with respect to variation in si‐ multaneous vs.

We applied these tools to intrusion detection protocols which is an application from sensor networks. This is a novel approach which used of Irregular Cellular Learning Automata to detect suspect nodes by using and analyzing nodes' behavior during routing process and nodes' energy level. It also implements Irregular Cellular Learning Automata to detect ab‐ normal behaviors. Afterwards, our method starts its voting process in which it decides to reward or penalize suspect node based on learning automata reports. The simulations show that our proposed method not only has a proper detection rate but also is an energy-aware protocol in detecting malicious nodes.

#### **Author details**

**Figure 17.** False positive rate vs. number of clusters

108 Emerging Applications of Cellular Automata

**Figure 18.** Average energy consumption in the network vs. number of nodes

In this Chapter, we have disscussed Cellular Learning Automata which has standard as‐ sumption of a rectangular grid structure. Then we have represented Irregular Cellular Learning Automata which are the models cause to develop tools that allow us use both rec‐ tangular and even irregular grids within one and the same Cellular Automata modeling framework. Of course, we are aware that the regularity of the grid structure is but one of a number of idealizations used in Cellular Automata modeling. For example, it has been dis‐ cussed controversially whether and under what conditions a simple influence dynamics

**5. Conclusion**

Amir Hosein Fathy Navid1\* and Amir Bagheri Aghababa2

\*Address all correspondence to: Amir.Fathy.n@qiau.ac.ir

1 Islamic Azad University, Hamedan Beranch, Bahar, Hamedan, Iran

2 Islamic Azad University East Tehran Branch, Tehran, Iran

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**Chapter 6**

**Provisional chapter**

**Interactive Maps on Variant Phase Spaces**

Jeffrey Zheng, Christian Zheng and Tosiyasu Kunii

**1.1. Fundamental models of cellular automata and phase space**

Input *X* →

measured by a unknown function *U*, or expressed as

Input, output and functions are fundamental elements of the wider applications of dynamic systems [3, 5, 21] such applications include: mathematics, probability, physics, statistics,

vectors are linked by an equation where the function may be expressed by *Y* = *f*(*X*) thus:


This is called *a white box model* [3, 15, 28]. Using the white box model, a pair (*X*,*Y*) can be

If there is no explicit expression for a unknown function *U*, a pair of vectors (*X*,*Y*) could be collected for their correspondences on the pair of input-output relationships. Equation *Y* = *U*(*X*) is still satisfied. This is called *a black box model*. i.e. A pair of (*X*,*Y*) can be

> ©2012 Zheng et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Zheng et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2013 Zheng, et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>2</sup> with states, for a given 0-1 function *f* , the pair of 0-1

Given function *<sup>f</sup>* <sup>→</sup> Output *<sup>Y</sup>*. (1)

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Jeffrey Zheng, Christian Zheng and T. L. Kunii

http://dx.doi.org/10.5772/51635

*1.1.1. White and Black Box Models*

classical logic, and cellular automata. For a pair of *N* bit vectors *X*,*Y* ∈ *B<sup>N</sup>*

explicitly calculated by a function *f* .

10.5772/51635

**1. Introduction**

**– From Measurements - Micro Ensembles to Ensemble**

**Matrices on Statistical Mechanics of Particle Models**

**Interactive Maps on Variant Phase Spaces – From**

**Measurements - Micro Ensembles to Ensemble**

**Matrices on Statistical Mechanics of Particle Models**

**Provisional chapter**

#### **Interactive Maps on Variant Phase Spaces – From Measurements - Micro Ensembles to Ensemble Matrices on Statistical Mechanics of Particle Models Interactive Maps on Variant Phase Spaces – From Measurements - Micro Ensembles to Ensemble Matrices on Statistical Mechanics of Particle Models**

Jeffrey Zheng, Christian Zheng and Tosiyasu Kunii Jeffrey Zheng, Christian Zheng and T. L. Kunii

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/51635 10.5772/51635

#### **1. Introduction**

#### **1.1. Fundamental models of cellular automata and phase space**

#### *1.1.1. White and Black Box Models*

Input, output and functions are fundamental elements of the wider applications of dynamic systems [3, 5, 21] such applications include: mathematics, probability, physics, statistics, classical logic, and cellular automata.

For a pair of *N* bit vectors *X*,*Y* ∈ *B<sup>N</sup>* <sup>2</sup> with states, for a given 0-1 function *f* , the pair of 0-1 vectors are linked by an equation where the function may be expressed by *Y* = *f*(*X*) thus:

$$\text{Input } X \to \boxed{\begin{subarray}{c} \text{-} \text{White Box} \\ \text{Given function } f \end{subarray}} \to \text{Output } Y. \tag{1}$$

This is called *a white box model* [3, 15, 28]. Using the white box model, a pair (*X*,*Y*) can be explicitly calculated by a function *f* .

If there is no explicit expression for a unknown function *U*, a pair of vectors (*X*,*Y*) could be collected for their correspondences on the pair of input-output relationships. Equation *Y* = *U*(*X*) is still satisfied. This is called *a black box model*. i.e. A pair of (*X*,*Y*) can be measured by a unknown function *U*, or expressed as

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Zheng et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Zheng, et al., licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Zheng et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative

$$\text{Input } X \to \boxed{\begin{subarray}{c} \text{- Black box} \\ \text{Unknow function } U \end{subarray}} \to \text{Output } Y. \tag{2}$$

(1804-51) recognized that Liouville's works could be used to describe mechanical systems and so placed Liouville's mathematical theorem into a mechanical context. Pücker working in Germany and Cayley and Sylvester in the UK, extended projective geometry beyond the ordinary three dimensions in the 1840s and Grassmann developed an *n* dimensional vector

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 115

Riemannn's work in 1868 developed the geometric properties of multi-dimensional manifolds. This was followed by further developments in the 1870s by E. Betti, F. Klein,

As it was Lagrange who took the first steps, a bottom-up approach is now often described as *a Lagrange expression*. Hamiltonian dynamics is a typical representative under this expression

Robert Boyle (1627-91) developed new physio-mechanical experiments. Boyle's law states that at a constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure using a set of measures characterizing the global properties of a gas. Anders Celsius (1701-1744) proposed a thermometer scale calibrated to the freezing point and boiling point of water. Benjamin Thompson (later known as Count Rumford) (1753-1814) explored cannon barrel-boring experiments and demonstrated the conversion of work into heat via friction in the absence of any additional weight of the object due to such heating being detected. Leonhard Euler (1707-1783) developed a Kinetic Heat Theory based on his description of a calculus of variations to introduce the concept of moving axes in astronomy. Daniel Bernoulli (1700-1782) and Pierre-Simon Laplace (1749-1827) refined Newton's work to represent gas properties through repulsive interactions. Jean Baptiste Joseph Fourier (1768-1830) developed an understanding of the conduction of heat to represent a periodic function as a Fourier series. Poisson (1781-1840) further developed the theories of heat using Fourier series. Thomas Young (1773-1829) expressed the modern formulation of energy, mathematically associated with *mv*2. Sadi Carnot (1796-1832) introduced the concept of ideal gas cycle analysis. William Thomson (later known as Lord Kelvin) (1824-1907) developed a wave theory of heat in homogeneous solid bodies. James Prescott Joule (1818-1889) established the relationship between heat and mechanical work through a series of experiments. John James Waterston (1811-1883) explored a kinetic theory of gases and mean free path. Von Helmholtz (1821-94) further developed the principle of conservation of energy extending Carnot's principle of kinetic energy into a mathematical formulation. Rudolf Clausius (1822-88) explored an expression of the second law for which the only function is the transfer of heat to propose the function *dQ*/*T* to compare heat flows with heat conversions using Carnot's techniques to derive *entropy* and show the two laws of thermodynamics were the equivalent of the older caloric theory. Gustav Robert Kirchhoff (1824-1887) derived from the second law of thermodynamics that objects cannot be distinguished by thermal radiation at a uniform temperature to formulate a black body

Leonhard Euler (1707-1783) provided key methodologies in this direction with a top-down approach known as *a Euler expression*. A Fourier series of a periodic function is a typical representative under this expression founded on *a periodic function composed of a set of simple*

space in 1844.

*1.2.2. Top-down Approaches*

[23, 24, 31, 33].

*harmonic components* [31, 33].

and C. Jordan then more recently by [23, 24, 31, 33].

as it is founded on *a pair of conjugate parameters* [31, 33].

In science and engineering [13, 28, 29], a *black box* is a device, system, or object which can be viewed solely in terms of its input, output and transfer characteristics without any knowledge of its internal works.

From a cellular automata viewpoint, the black box approach is useful in describing a situation where both input and output are in the form of two bit vectors for an unknown function of a digital system.

#### *1.1.2. Characteristic Point and Phase Space*

In mathematics and physics [3, 14, 18, 20, 21, 29], the concept of *a phase space* as introduced by W. Gibbs in 1901 is a space in which all possible states of a system are represented. Here, each possible state of the system corresponds to one unique point in the phase space. For cellular automata, the phase space usually consists of all possible values of pairs of input and output vectors in multiple dimensions.

For either a known function *f* or for an unknown function *U*, when the states of *X*,*Y* reside in the same finite region, it is entirely feasible in principle to undertake an exhaustive procedure to list all pairs of {(*X*,*Y*)}. For a given *N* bit vector *X*, the vector generates a *point* with a unique spatial position to indicate the characteristics of the function and by listing all such possible points, a *phase space* for the function is established.

#### **1.2. Historical review on phase spaces of statistical mechanics**

Top-down and bottom-up are two distinct strategies of intelligent processing and knowledge ordering used in humanistic and scientific theories [4, 13, 15, 25, 28]. In practice, they can be seen as alternative styles of thinking and problem solving. *Top-down* may be taken to mean an approach based on an analysis or decomposition to identify key components within a global target that has been identified for study and from which there may be constructed a hierarchy of local features. *Bottom-up* may be taken to describe a process of synthesis via integration working from local features towards a global target.

#### *1.2.1. Bottom-up Approaches*

Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) gave calculus to the world of mathematics during the decade 1670-1680. This established a systematic methodology for the efficient analysis of local variables in order to reveal global features.

Joseph-Louis Lagrange (1736-1813) took the conservation of energy as the foundation for his system of mechanics where he combined the principle of virtual velocities with the principle of least action. Along these lines, W. R. Hamilton (1805-65) established his approach to dynamics in 1834-1835 with the first description of functions on phase space with pairs of conjugate parameters, together with position and momentum. J. Liouville (1809-82) proposed a theorem on the conservation of volume in phase space in 1838. C. G. Jacobi (1804-51) recognized that Liouville's works could be used to describe mechanical systems and so placed Liouville's mathematical theorem into a mechanical context. Pücker working in Germany and Cayley and Sylvester in the UK, extended projective geometry beyond the ordinary three dimensions in the 1840s and Grassmann developed an *n* dimensional vector space in 1844.

Riemannn's work in 1868 developed the geometric properties of multi-dimensional manifolds. This was followed by further developments in the 1870s by E. Betti, F. Klein, and C. Jordan then more recently by [23, 24, 31, 33].

As it was Lagrange who took the first steps, a bottom-up approach is now often described as *a Lagrange expression*. Hamiltonian dynamics is a typical representative under this expression as it is founded on *a pair of conjugate parameters* [31, 33].

#### *1.2.2. Top-down Approaches*

2 Cellular Automata

Input *X* →

knowledge of its internal works.

*1.1.2. Characteristic Point and Phase Space*

and output vectors in multiple dimensions.

*1.2.1. Bottom-up Approaches*

possible points, a *phase space* for the function is established.

**1.2. Historical review on phase spaces of statistical mechanics**

integration working from local features towards a global target.

a digital system.


In science and engineering [13, 28, 29], a *black box* is a device, system, or object which can be viewed solely in terms of its input, output and transfer characteristics without any

From a cellular automata viewpoint, the black box approach is useful in describing a situation where both input and output are in the form of two bit vectors for an unknown function of

In mathematics and physics [3, 14, 18, 20, 21, 29], the concept of *a phase space* as introduced by W. Gibbs in 1901 is a space in which all possible states of a system are represented. Here, each possible state of the system corresponds to one unique point in the phase space. For cellular automata, the phase space usually consists of all possible values of pairs of input

For either a known function *f* or for an unknown function *U*, when the states of *X*,*Y* reside in the same finite region, it is entirely feasible in principle to undertake an exhaustive procedure to list all pairs of {(*X*,*Y*)}. For a given *N* bit vector *X*, the vector generates a *point* with a unique spatial position to indicate the characteristics of the function and by listing all such

Top-down and bottom-up are two distinct strategies of intelligent processing and knowledge ordering used in humanistic and scientific theories [4, 13, 15, 25, 28]. In practice, they can be seen as alternative styles of thinking and problem solving. *Top-down* may be taken to mean an approach based on an analysis or decomposition to identify key components within a global target that has been identified for study and from which there may be constructed a hierarchy of local features. *Bottom-up* may be taken to describe a process of synthesis via

Isaac Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) gave calculus to the world of mathematics during the decade 1670-1680. This established a systematic methodology for the efficient analysis of local variables in order to reveal global features. Joseph-Louis Lagrange (1736-1813) took the conservation of energy as the foundation for his system of mechanics where he combined the principle of virtual velocities with the principle of least action. Along these lines, W. R. Hamilton (1805-65) established his approach to dynamics in 1834-1835 with the first description of functions on phase space with pairs of conjugate parameters, together with position and momentum. J. Liouville (1809-82) proposed a theorem on the conservation of volume in phase space in 1838. C. G. Jacobi

Unknow function *<sup>U</sup>* <sup>→</sup> Output *<sup>Y</sup>*. (2)

Robert Boyle (1627-91) developed new physio-mechanical experiments. Boyle's law states that at a constant temperature, the volume of a fixed mass of gas is inversely proportional to its pressure using a set of measures characterizing the global properties of a gas. Anders Celsius (1701-1744) proposed a thermometer scale calibrated to the freezing point and boiling point of water. Benjamin Thompson (later known as Count Rumford) (1753-1814) explored cannon barrel-boring experiments and demonstrated the conversion of work into heat via friction in the absence of any additional weight of the object due to such heating being detected. Leonhard Euler (1707-1783) developed a Kinetic Heat Theory based on his description of a calculus of variations to introduce the concept of moving axes in astronomy. Daniel Bernoulli (1700-1782) and Pierre-Simon Laplace (1749-1827) refined Newton's work to represent gas properties through repulsive interactions. Jean Baptiste Joseph Fourier (1768-1830) developed an understanding of the conduction of heat to represent a periodic function as a Fourier series. Poisson (1781-1840) further developed the theories of heat using Fourier series. Thomas Young (1773-1829) expressed the modern formulation of energy, mathematically associated with *mv*2. Sadi Carnot (1796-1832) introduced the concept of ideal gas cycle analysis. William Thomson (later known as Lord Kelvin) (1824-1907) developed a wave theory of heat in homogeneous solid bodies. James Prescott Joule (1818-1889) established the relationship between heat and mechanical work through a series of experiments. John James Waterston (1811-1883) explored a kinetic theory of gases and mean free path. Von Helmholtz (1821-94) further developed the principle of conservation of energy extending Carnot's principle of kinetic energy into a mathematical formulation. Rudolf Clausius (1822-88) explored an expression of the second law for which the only function is the transfer of heat to propose the function *dQ*/*T* to compare heat flows with heat conversions using Carnot's techniques to derive *entropy* and show the two laws of thermodynamics were the equivalent of the older caloric theory. Gustav Robert Kirchhoff (1824-1887) derived from the second law of thermodynamics that objects cannot be distinguished by thermal radiation at a uniform temperature to formulate a black body [23, 24, 31, 33].

Leonhard Euler (1707-1783) provided key methodologies in this direction with a top-down approach known as *a Euler expression*. A Fourier series of a periodic function is a typical representative under this expression founded on *a periodic function composed of a set of simple harmonic components* [31, 33].

#### *1.2.3. Formal Expression of Phase Space in Statistical Mechanics*

Following methodologies established by Hamilton, Lagrange, and Euler, [26] and L. Boltzmann (1844-1906) went on to lay the foundations of statistical mechanics from 1871. They introduced the term *phase* to describe the analogy they saw between the physical trajectories of particles in two dimensional space and Lissajous figures expressed as interactive maps. When two harmonic frequencies exist as rational fractions, period 4 circular patterns occur. However, when the frequency ratio is irrational, the system trajectories visit all points on the plane bounded by the signal amplitude. J. C. Maxwell (1831-79) adopted Boltzmann's expression of phase to describe the state of systems in 1879. William Thomson (Lord Kelvin) was the first to use the word *demon* for Maxwell's Thermodynamics concept in 1874. H. Poincarè (1854-1912) in 1885 took a geometric approach to visualize a saddle point where stable and unstable trajectories intersected in phase space. Various mapping techniques are relevant to such explorations. These include Poincarè sections (maps), fixed-point classifications, and the conservation of phase space as an integrated invariant. Influenced by the work of Maxwell and Boltzmann together with other wider contributions, J. W. Gibbs (1939-1903) proposed his Elementary Principles in Statistical Mechanics in 1902 to describe a phase as represented by a point of 2*n* dimensions.

**Key Maxwell-Boltzmann Darwin-Fowler Gibbs Issue Most probable theory Average Theory Ensemble Theory Assumption** Ergodic Average: Ergodic Average: Equality of PS:

**Phase Space** State State Density functions N combination combination Liouville equation **Cell Unit** Local cell in n particles Complex function Ensemble based n Stirling Approximation non restriction to n non-cell required

**States** with maximal entropy with maximal entropy with maximal entropy

**Prefer** Isolated system Isolated system MCE: isolated system; **System** for { MCE, CE} for { MCE, CE} CE : closed system;

**Model** Black-box White-box Black-box

In the context of the pursuit of an interpretation of quantum mechanics, the state vector or wave function has been widely discussed as a model for describing the individual

The most comprehensive descriptions of an individual physical system are to be found in the various versions of *the Copenhagen interpretation* [8], or in subsequent versions incorporating

An interpretation according to the state vector based not on an individual system but on an ensemble of identically prepared systems is known as *a statistical ensemble interpretation* or

The two alternative strategies of top-down and bottom-up strongly influence the direction of various explorations in the field of quantum mechanism. *The Lagrange expression* emphasizes single particles in a bottom-up strategy. In contrast, *the Euler expression* emphasizes complex

From as early as the turn of the 20th century when Plank started his quantum revolution, various interpretations of quantum behaviors have been explored. Following the Heisenberg matrix approach and the Schrödinger wave function equation, the intellectually absorbing anomalies of quantum mechanics have been linked to intrinsic behaviors associated with

To address the various paradoxes encountered in the development of quantum mechanics during the course of 20th century, a number of different interpretations may be listed [19, 31].

**Table 1.** Key Methods in Statistical Mechanics

particle and wave duality.

components of a system (e.g. an electron).

more briefly just as *a statistical interpretation* [19, 31].

objects treated as ensembles in a top-down strategy.

**Probabilistic interpretation** Max Born 1926 [19]]

**Copenhagen interpretation** N. Bohr and Heisenberg 1927 [8] **Double-solution interpretation** de Broglie 1927, 1953 [9]

minor modifications as in *the hidden variable interpretations* [19, 31].

**Balanced** MPD MPD MPD

**Expression** Lagrange Lagrange Euler **Interaction** No No Yes

TM = PSM TM = PSM EPV in same probability

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 117

not for GCE not for GCE GCE : open system

non function explicit function non function Bottom-up Bottom-up Top-down

From a terminological viewpoint, Gibbs brought us such expressions as *statistical mechanics, micro ensemble, canonical ensemble, and grand ensemble*. He facilitated the establishment of a hierarchy in Statistical Mechanics. However, Gibbs did not use the term *phase space*. The first formal expression of the term *phase space* appeared in the context of ergodic theory in a 1913 publication by A. Rosenthal and M. Plancherel [26, 31, 33].

From 1919 to 1922, Sir Charles Galton Darwin (1887-1962) worked with Sir Ralph Howard Fowler (1889-1944) on statistical mechanics and established the Darwin-Fowler method.

#### *1.2.4. Key Properties in Statistical Mechanics*

[23, 24, 31, 33] noted the usefulness of listing key properties in classical statistical mechanics. A typical comparison is presented below in Table 1.

In general, both Maxwell-Boltzmann and Gibbs follow black-box models without involving explicit local functions. However, the Darwin-Fowler method uses a complex function to describe its unit cell so making it a white box model. Both Maxwell-Boltzmann and Darwin-Fowler schemes use Lagrange expressions to calculate cell unit and to form their fundamentals using a bottom-up strategy. Meanwhile, Gibbs applies Euler expressions for analysis using a top-down strategy without involving explicit cell units.

Table 1 uses abbreviations as follows: TM for Time Measurement, PSM for Phase Space Measurement, PS for Phase Space, EPV for Equal Phase Volume, MPD for Most Probable Distribution, MCE for Micro Canonical Ensemble, CE for Canonical Ensemble, and GCE for Grand Canonical Ensemble.

#### *1.2.5. Common Interpretations of Quantum Mechanics*

Quantum mechanics is a modern legacy with its roots in classical statistical mechanics [11, 12, 17]. Meanwhile, Bose-Einstein, Fermi-Dirac statistics, and Planck's quantum are deeply connected with the statistical mechanics of Boltzmann and Gibbs [17, 19].


**Table 1.** Key Methods in Statistical Mechanics

4 Cellular Automata

*1.2.3. Formal Expression of Phase Space in Statistical Mechanics*

to describe a phase as represented by a point of 2*n* dimensions.

publication by A. Rosenthal and M. Plancherel [26, 31, 33].

*1.2.4. Key Properties in Statistical Mechanics*

Grand Canonical Ensemble.

A typical comparison is presented below in Table 1.

*1.2.5. Common Interpretations of Quantum Mechanics*

Following methodologies established by Hamilton, Lagrange, and Euler, [26] and L. Boltzmann (1844-1906) went on to lay the foundations of statistical mechanics from 1871. They introduced the term *phase* to describe the analogy they saw between the physical trajectories of particles in two dimensional space and Lissajous figures expressed as interactive maps. When two harmonic frequencies exist as rational fractions, period 4 circular patterns occur. However, when the frequency ratio is irrational, the system trajectories visit all points on the plane bounded by the signal amplitude. J. C. Maxwell (1831-79) adopted Boltzmann's expression of phase to describe the state of systems in 1879. William Thomson (Lord Kelvin) was the first to use the word *demon* for Maxwell's Thermodynamics concept in 1874. H. Poincarè (1854-1912) in 1885 took a geometric approach to visualize a saddle point where stable and unstable trajectories intersected in phase space. Various mapping techniques are relevant to such explorations. These include Poincarè sections (maps), fixed-point classifications, and the conservation of phase space as an integrated invariant. Influenced by the work of Maxwell and Boltzmann together with other wider contributions, J. W. Gibbs (1939-1903) proposed his Elementary Principles in Statistical Mechanics in 1902

From a terminological viewpoint, Gibbs brought us such expressions as *statistical mechanics, micro ensemble, canonical ensemble, and grand ensemble*. He facilitated the establishment of a hierarchy in Statistical Mechanics. However, Gibbs did not use the term *phase space*. The first formal expression of the term *phase space* appeared in the context of ergodic theory in a 1913

From 1919 to 1922, Sir Charles Galton Darwin (1887-1962) worked with Sir Ralph Howard Fowler (1889-1944) on statistical mechanics and established the Darwin-Fowler method.

[23, 24, 31, 33] noted the usefulness of listing key properties in classical statistical mechanics.

In general, both Maxwell-Boltzmann and Gibbs follow black-box models without involving explicit local functions. However, the Darwin-Fowler method uses a complex function to describe its unit cell so making it a white box model. Both Maxwell-Boltzmann and Darwin-Fowler schemes use Lagrange expressions to calculate cell unit and to form their fundamentals using a bottom-up strategy. Meanwhile, Gibbs applies Euler expressions for

Table 1 uses abbreviations as follows: TM for Time Measurement, PSM for Phase Space Measurement, PS for Phase Space, EPV for Equal Phase Volume, MPD for Most Probable Distribution, MCE for Micro Canonical Ensemble, CE for Canonical Ensemble, and GCE for

Quantum mechanics is a modern legacy with its roots in classical statistical mechanics [11, 12, 17]. Meanwhile, Bose-Einstein, Fermi-Dirac statistics, and Planck's quantum are deeply

analysis using a top-down strategy without involving explicit cell units.

connected with the statistical mechanics of Boltzmann and Gibbs [17, 19].

In the context of the pursuit of an interpretation of quantum mechanics, the state vector or wave function has been widely discussed as a model for describing the individual components of a system (e.g. an electron).

The most comprehensive descriptions of an individual physical system are to be found in the various versions of *the Copenhagen interpretation* [8], or in subsequent versions incorporating minor modifications as in *the hidden variable interpretations* [19, 31].

An interpretation according to the state vector based not on an individual system but on an ensemble of identically prepared systems is known as *a statistical ensemble interpretation* or more briefly just as *a statistical interpretation* [19, 31].

The two alternative strategies of top-down and bottom-up strongly influence the direction of various explorations in the field of quantum mechanism. *The Lagrange expression* emphasizes single particles in a bottom-up strategy. In contrast, *the Euler expression* emphasizes complex objects treated as ensembles in a top-down strategy.

From as early as the turn of the 20th century when Plank started his quantum revolution, various interpretations of quantum behaviors have been explored. Following the Heisenberg matrix approach and the Schrödinger wave function equation, the intellectually absorbing anomalies of quantum mechanics have been linked to intrinsic behaviors associated with particle and wave duality.

To address the various paradoxes encountered in the development of quantum mechanics during the course of 20th century, a number of different interpretations may be listed [19, 31].

**Probabilistic interpretation** Max Born 1926 [19]] **Copenhagen interpretation** N. Bohr and Heisenberg 1927 [8] **Double-solution interpretation** de Broglie 1927, 1953 [9]

**de Broglie-Bohm Theory** de Broglie 1927, David Bohm 1952 [9] **Standard interpretation** von Neumann 1932 Wigner, Wheeler [32] **Quantum Logic** G. Birkhoff and von Neumann 1936 [19] **Ensemble interpretation** D. Blokhintsev 1949 [6] **Many-world interpretation** H. Everett 1957 [19] **Time-symmetric theory** Y. Aharonov 1964 **Stochastic interpretation** E.Nelson 1966 **Many-minds interpretation** H. Zeh 1970 [34] **Consistent histories** R. Griffiths 1984 **Objective collapse theories** Ghirardi-Rimini-Weber 1986 **Transitional interpretation** J. Cramer 1986 **Rational interpretation** C. Rovelli 1994

In general, the seven key interpretations (Copenhagen, Double-solution, de Broglie-Bohm, Standard, Ensemble, Many-world, and Stochastic) of the first four decades of the 20th century can be separated as follows into the following two general categories [9, 31]:

**The interpreting structure I** includes states, transitions between states, measurement operations and possible information about spatial extension of these elements.

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 119

Applying Einstein's criteria to this set of interpretations, the ensemble interpretation (statistical interpretation) is a minimalist interpretation. It claims to make the fewest assumptions associated with the standard mathematics. The most notable supporter of such

At the 1927 Solvay Congress, Einstein proposed a statistical interpretation in order to avoid conceptual difficulties if the reduction of a wave packet led to the association of wave functions with individual systems. He hoped that someday a complete theory of microphysics would become available to establish a conceptual base as a (preferred)

In 1932, von Neumann established *mathematical foundations for quantum mechanics* as a standard interpretation on Hilbert space to provide a proof rejecting any hidden variable

Influenced by K.V. Nikolskii and V.A. Fock, D. I. Blokhintsev developed a statistical interpretation in the 1940s. He expressed the view that modern quantum mechanics is not a theory of micro-processes but rather a means of studying their properties by the application of statistical ensembles. Menawhile, the approach taken in the publication was borrowed

Landé's 1951 book sought to reconcile the contradictions between the two classical concepts of the particle and the wave by providing something equivalent to the descriptions of physical phenomena in either terms. He emphasized that in diffraction experiments, particles exhibit both maximum and minimum intensities of diffraction through a perfectly normal mechanical process that can be described in terms of a wave explanation. Using transition probability, these experimentally-determinable transition probabilities can be shown to map

Compared to continuous approaches, Heisenberg's matrix offers several advances in handling the case of a single particle. In July 1926, the first question Heisenberg asked Schrödinger was, "Can you use your continuous wave equation to explain black body

Due to the inherent differences between the two strategies it is difficult to find a direct answer to the question under the Copenhagen interpretation, "Is the Schrödinger equation a single

Through statistical interpretation is a minimalist interpretation, it too is not a complete interpretation. During the development of statistical interpretation there were various

Heisenberg questioned as *self contradictory*, Blokhintsev's basic contention that quantum mechanics eliminates the observer and becomes objectively significant due to the fact that the

statistical interpretation was Einstein himself [19, 22, 31].

*1.2.6. Statistical Interpretation of Quantum Mechanics*

alternative to modern quantum mechanics [9, 31].

from classical macro physics [6, 19, 22].

*1.2.7. Main weaknesses in key interpretations*

radiation or quantum effects in photoelectric actions?"

particle description or an equation for a group of particles?" [16, 19, 31].

debates between Blokhintsev and Heisenberg during the 1940s [19, 22].

approach [32].

a matrix [19].


In general, a Lagrange expression is preferred for representing a single quanta while a Euler expression can better describe certain group activities. It is interesting to note that *de Broglie's Double-solution* with a special interpretation can to be involved in both cases [9, 31].

According to Einstein's criteria for quantum mechanics [10], an interpretation of quantum mechanics can be characterized by its treatment of:


Here, an interpretation is taken to mean a correspondence between the elements of the mathematical formalism **M** and the elements of an interpreting structure **I**, where:

**The Mathematical formalism M** consists of the Hilbert space machinery of ket-vectors, self-adjoint operations on the space of ket-vectors, unitary time dependence of the ket-vectors and measurement operations: and ...

**The interpreting structure I** includes states, transitions between states, measurement operations and possible information about spatial extension of these elements.

Applying Einstein's criteria to this set of interpretations, the ensemble interpretation (statistical interpretation) is a minimalist interpretation. It claims to make the fewest assumptions associated with the standard mathematics. The most notable supporter of such statistical interpretation was Einstein himself [19, 22, 31].

#### *1.2.6. Statistical Interpretation of Quantum Mechanics*

6 Cellular Automata

**de Broglie-Bohm Theory** de Broglie 1927, David Bohm 1952 [9] **Standard interpretation** von Neumann 1932 Wigner, Wheeler [32]

**Quantum Logic** G. Birkhoff and von Neumann 1936 [19]

**Objective collapse theories** Ghirardi-Rimini-Weber 1986

In general, the seven key interpretations (Copenhagen, Double-solution, de Broglie-Bohm, Standard, Ensemble, Many-world, and Stochastic) of the first four decades of the 20th century

**1) Lagrange Expression:** comprising the Copenhagen Interpretation (N. Bohr and Heisenberg), the Double-solution (de Broglie), the Standard Interpretation (von Neumann), the Many-world interpretation (H. Everett), and the de Broglie-Bohm Theory

**2) Euler Expression:** comprising the Double-solution (de Broglie), the Ensemble

In general, a Lagrange expression is preferred for representing a single quanta while a Euler expression can better describe certain group activities. It is interesting to note that *de Broglie's*

According to Einstein's criteria for quantum mechanics [10], an interpretation of quantum

Here, an interpretation is taken to mean a correspondence between the elements of the

**The Mathematical formalism M** consists of the Hilbert space machinery of ket-vectors, self-adjoint operations on the space of ket-vectors, unitary time dependence of the

mathematical formalism **M** and the elements of an interpreting structure **I**, where:

interpretation (D. Blokhintsev), and the Stochastic interpretation (E. Nelson)

*Double-solution* with a special interpretation can to be involved in both cases [9, 31].

can be separated as follows into the following two general categories [9, 31]:

**Ensemble interpretation** D. Blokhintsev 1949 [6] **Many-world interpretation** H. Everett 1957 [19]

**Time-symmetric theory** Y. Aharonov 1964 **Stochastic interpretation** E.Nelson 1966 **Many-minds interpretation** H. Zeh 1970 [34]

**Transitional interpretation** J. Cramer 1986 **Rational interpretation** C. Rovelli 1994

**Consistent histories** R. Griffiths 1984

(de Broglie & David Bohm)

• Realism

• Completeness • Local realism • Determinism

mechanics can be characterized by its treatment of:

ket-vectors and measurement operations: and ...

At the 1927 Solvay Congress, Einstein proposed a statistical interpretation in order to avoid conceptual difficulties if the reduction of a wave packet led to the association of wave functions with individual systems. He hoped that someday a complete theory of microphysics would become available to establish a conceptual base as a (preferred) alternative to modern quantum mechanics [9, 31].

In 1932, von Neumann established *mathematical foundations for quantum mechanics* as a standard interpretation on Hilbert space to provide a proof rejecting any hidden variable approach [32].

Influenced by K.V. Nikolskii and V.A. Fock, D. I. Blokhintsev developed a statistical interpretation in the 1940s. He expressed the view that modern quantum mechanics is not a theory of micro-processes but rather a means of studying their properties by the application of statistical ensembles. Menawhile, the approach taken in the publication was borrowed from classical macro physics [6, 19, 22].

Landé's 1951 book sought to reconcile the contradictions between the two classical concepts of the particle and the wave by providing something equivalent to the descriptions of physical phenomena in either terms. He emphasized that in diffraction experiments, particles exhibit both maximum and minimum intensities of diffraction through a perfectly normal mechanical process that can be described in terms of a wave explanation. Using transition probability, these experimentally-determinable transition probabilities can be shown to map a matrix [19].

#### *1.2.7. Main weaknesses in key interpretations*

Compared to continuous approaches, Heisenberg's matrix offers several advances in handling the case of a single particle. In July 1926, the first question Heisenberg asked Schrödinger was, "Can you use your continuous wave equation to explain black body radiation or quantum effects in photoelectric actions?"

Due to the inherent differences between the two strategies it is difficult to find a direct answer to the question under the Copenhagen interpretation, "Is the Schrödinger equation a single particle description or an equation for a group of particles?" [16, 19, 31].

Through statistical interpretation is a minimalist interpretation, it too is not a complete interpretation. During the development of statistical interpretation there were various debates between Blokhintsev and Heisenberg during the 1940s [19, 22].

Heisenberg questioned as *self contradictory*, Blokhintsev's basic contention that quantum mechanics eliminates the observer and becomes objectively significant due to the fact that the wave function does not describe the state of a particle but rather identifies that the particle belongs to a particular ensemble. In this, Heisenberg argued that in order to assign a particle to a particular ensemble, some knowledge of the particle is required on the part of the observer [19], p445.

8. sample results

11. main results 12. conclusions

diagrams.

**2.1. Architecture**

exhaust all possible 22*<sup>n</sup>*

and two parameters in the output group.

*X* a 0-1 vector with *N* elements, *X* ∈ *B<sup>N</sup>*

*SM* a selection on a pair of measurements

*J* a function with *n* variables, *J* ∈ *B*2*<sup>n</sup>*

The three groups of parameters may be listed as follows.

*N* an integer indicates a 0-1 vector with *N* elements *n* an integer indicates *n* variables for a function

∀*J* exhaustive set of all functions of *n* variables with 22*<sup>n</sup>*

*FC* A given configuration for variant logic functions: a 22*n*−<sup>1</sup>

*IP*(*J*, *X*) a set of eight interactive projections related to *ME*(*J*, *X*)

*IM*(*J*) a set of eight interactive maps associated with one *CE*(*J*)

GEM component.

**Input group:**

**Intermediate group:**

9. analysis of visual distributions 10. global symmetry properties

**2. System architecture**

In this section, system architecture and its core components are discussed with the use of

The three components of *a Variant Phase Space System* are *the Creating Micro Ensemble (CME), the Canonical Ensemble (CEIM)* together with *the Interactive Map and the Global Ensemble Matrix (GEM)* as shown in Figure 1. The architecture is shown in Figure 1(a) with the key modules

In the first part of the system, a micro ensemble and its eight projections are created for a given vector and function by the CME component. Next, in order to exhaust all possible 2*<sup>N</sup>* vectors, a CE and eight IMs are established by the CEIM component. Then, in order to

With eight parameters in an input group, there are four parameters in the intermediate group

*ME*(*J*, *X*) a micro ensemble under either multiple or conditional probability measurements

functions, a CE matrix and eight IM matrices are generated by the

elements

× 22*<sup>n</sup>*−<sup>1</sup>

matrix

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 121

of the three core components being shown in Figures 1(b) through 1(d) respectively.

2

*CE*(*J*) a canonical ensemble for an *N* bit vector under an *n* variable function *J*

∀*X* exhaustive set of all states of *N* bit vectors with 2*<sup>N</sup>* elements

2

The main weakness of Blokhintsev's ensemble interpretation is that though its mathematical formula can express wave distributions well it fails to describe particle structure properly. This is a common weakness of similar mathematical constructions based on periodic components of a Fourier series [19, 27, 31].

Similar to difficulties faced by the Schrödinger equation, ensemble construction is suitable for wave representations but is weak in particle description. On the other hand, the Copenhagen interpretation is preferred for a single particle but comes with inherent limitations of expression with respect to wave behaviours which require further reliance on Born's probabilistic interpretation of the wave-function.

#### *1.2.8. Other Challenges on Statistical Mechanics*

Statistical mechanics presents us with several fundemental difficulties [20, 23, 24, 31, 33]:


#### **1.3. Chapter organization**

In this chapter, *variant construction* comprising *variant logic, variant measurement and variant phase space* is explored with a view to addressing the main challenges and difficulties associated with statistical interpretations and statistical mechanics. The focus is on a unified model to illustrate a path leading from local measurements to global matrices on phase space via variant construction.

This chapter is organized into 12 sections addressing the following:


8. sample results

8 Cellular Automata

observer [19], p445.

components of a Fourier series [19, 27, 31].

probabilistic interpretation of the wave-function.

**Analytic apparatus** the construction of asymptotic formulas.

**Logic foundation:** solid logic foundation for statistical mechanics.

This chapter is organized into 12 sections addressing the following:

*1.2.8. Other Challenges on Statistical Mechanics*

replaced by space (phase)-average

continuous systems.

**1.3. Chapter organization**

via variant construction.

2. system architecture 3. creating micro ensemble

6. representation models

1. general introduction (above)

4. canonical ensemble and interactive maps 5. global ensemble and interactive map matrices

7. symbolic representation on selected cases

wave function does not describe the state of a particle but rather identifies that the particle belongs to a particular ensemble. In this, Heisenberg argued that in order to assign a particle to a particular ensemble, some knowledge of the particle is required on the part of the

The main weakness of Blokhintsev's ensemble interpretation is that though its mathematical formula can express wave distributions well it fails to describe particle structure properly. This is a common weakness of similar mathematical constructions based on periodic

Similar to difficulties faced by the Schrödinger equation, ensemble construction is suitable for wave representations but is weak in particle description. On the other hand, the Copenhagen interpretation is preferred for a single particle but comes with inherent limitations of expression with respect to wave behaviours which require further reliance on Born's

Statistical mechanics presents us with several fundemental difficulties [20, 23, 24, 31, 33]:

**Computational Efficiency:** use of modern computing power in tackling complexity.

**Ergodic property:** a time sequence average over a large set of local measurements be

**Discrete via continuous:** relationships between irregular discrete systems and regular

In this chapter, *variant construction* comprising *variant logic, variant measurement and variant phase space* is explored with a view to addressing the main challenges and difficulties associated with statistical interpretations and statistical mechanics. The focus is on a unified model to illustrate a path leading from local measurements to global matrices on phase space


#### **2. System architecture**

In this section, system architecture and its core components are discussed with the use of diagrams.

#### **2.1. Architecture**

The three components of *a Variant Phase Space System* are *the Creating Micro Ensemble (CME), the Canonical Ensemble (CEIM)* together with *the Interactive Map and the Global Ensemble Matrix (GEM)* as shown in Figure 1. The architecture is shown in Figure 1(a) with the key modules of the three core components being shown in Figures 1(b) through 1(d) respectively.

In the first part of the system, a micro ensemble and its eight projections are created for a given vector and function by the CME component. Next, in order to exhaust all possible 2*<sup>N</sup>* vectors, a CE and eight IMs are established by the CEIM component. Then, in order to exhaust all possible 22*<sup>n</sup>* functions, a CE matrix and eight IM matrices are generated by the GEM component.

With eight parameters in an input group, there are four parameters in the intermediate group and two parameters in the output group.

The three groups of parameters may be listed as follows.

#### **Input group:**

*N* an integer indicates a 0-1 vector with *N* elements

*n* an integer indicates *n* variables for a function

*X* a 0-1 vector with *N* elements, *X* ∈ *B<sup>N</sup>* 2

∀*X* exhaustive set of all states of *N* bit vectors with 2*<sup>N</sup>* elements

*J* a function with *n* variables, *J* ∈ *B*2*<sup>n</sup>* 2

∀*J* exhaustive set of all functions of *n* variables with 22*<sup>n</sup>* elements

*SM* a selection on a pair of measurements

*FC* A given configuration for variant logic functions: a 22*n*−<sup>1</sup> × 22*<sup>n</sup>*−<sup>1</sup> matrix

#### **Intermediate group:**

*ME*(*J*, *X*) a micro ensemble under either multiple or conditional probability measurements


$$\begin{array}{l} \{N,n\} \to \begin{array}{l} \{N,n\} \to \begin{array}{l} \text{Creating} \\ X \in B\_{2}^{n} \to \\ \text{Micro} \\ X \to B\_{2}^{n} \to \begin{array}{l} \text{Micro} \\ \text{HNiro} \\ \text{CME} \end{array} \end{array} \to \begin{array}{l} \text{ME(\,J,X)} \to \begin{array}{l} \text{Cianomial} \\ \text{Ensemble} \\ \text{NEsemble} \\ \forall X \to \begin{array}{l} \text{Ensemble} \\ \text{CEIM} \end{array} \end{array} \to \begin{array}{l} \{\text{C2.O}\} \to \begin{array}{l} \text{C2.O}\,\text{M} \\ \text{Micro} \\ \text{CEM} \end{array} \end{array} \to \begin{array}{l} \{\text{C2.O}\} \to \begin{array}{l} \text{Micro} \\ \text{C2.O}\,\text{M} \end{array} \} \end{array}$$

$$\begin{array}{l} \{\text{CE}(\,J),IM(\,J)\} \to \begin{array}{l} \text{Cialob} \\ \text{NEsemble} \\ \text{FC} \to \begin{array}{l} \text{Micro} \\ \text{Micro} \\ \text{GEM} \end{array} \end{array} \to \begin{array}{l} \text{C2.O}\,\text{M} \\ \text{Micro} \\ \text{GEM} \end{array} \} \to \begin{array}{l} \text{C2.O}\,\text{M} \\ \text{Micro} \\ \text{GEM} \end{array} \}$$

$$\text{(a) Archificature}$$

**Input:**

**Output:**

follows:

**Output:**

**Adding Input:**

certain environment.

*& IM Matrices*.

*N* an integer indicating a 0-1 vector with *N* elements *n* an integer indicating *n* variables for a function

2

*ME*(*J*, *X*) a micro ensemble under either multiple or conditional probability measurements

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 123

A point in variant phase space can be determined under a set of conditions. A set of relevant projections can be associated with an interactive environment. The operation of this module transfers each set of input parameters to one micro ensemble signal and its distinct interactive

The CEIM component as shown in Figure 1(c) is composed of two modules: *CE Canonical Ensemble and IP Interactive Projection*. This component inputs three groups of parameters {*ME*(*J*, *X*), *IP*(*J*, *X*), ∀*X*} from the CME component and outputs two sets of canonical ensembles together with its interactive maps {*CE*(*J*), *IM*(*J*)} as distinct distributions under certain environments. One additional input and two output parameters are described as

The CEIM component collects all possible micro ensembles for a given function to form a canonical ensemble on variant phase space. Meanwhile, different interactive maps associated with this CE can be calculated to output as a set of *IM*(*J*) as distinct distributions under

The GIM component as shown in Figure 1(d) is composed of two modules: one for the *SCEIM Set of Cannonical Ensembles together with Interactive Maps, and the other for the CIM CE*

Two outputs {*CE*(*J*), *IM*(*J*)} from CEIM are taken as inputs, while another two parameters

2

*IP*(*J*, *X*) a set of eight interactive projections under the SM condition

*X* a 0-1 vector with *N* elements, *X* ∈ *B<sup>N</sup>*

*SM* a selection on a pair of measurements

projections subject to certain restrictions.

**2.3. CEIM canonical ensemble and interactive map**

∀*X* exhaustive set of all states of *N* bit vectors with 2*<sup>N</sup>* elements

*IM*(*J*) a set of eight interactive maps associated with *CE*(*J*)

**2.4. GIM global ensemble and interactive map matrix**

{∀*J*, *FC*} and two outputs can be described as follows:

*CE*(*J*) a canonical ensemble for an *N* bit vector under an *n* variable function *J*

*J* a function with *n* variables, *J* ∈ *B*2*<sup>n</sup>*

$$\begin{array}{c} \{N, n\} \to \overline{\begin{array}{c} \text{Variant} \\ X \in B\_2^{\mathsf{H}} \to \begin{array}{c} \text{Measure} \\ \text{M} \text{Sameures} \end{array} \to \begin{array}{c} \text{Probability} \\ \text{VM} \end{array} \to \begin{array}{c} \text{Probability} \\ \text{Measurements} \end{array} \to PM(J, X) \end{array}$$

$$\begin{array}{c} PM(J, X) \to \begin{array}{c} \text{Micro} \\ \text{Ensemble} \\ \text{ME} \end{array} \to \begin{array}{c} \text{Intermediate} \\ \text{Projection} \\ \text{IP} \end{array} \to \begin{array}{c} \text{Interactions} \\ \text{Projection} \\ \text{IP} \end{array} \to IP(J, X)$$

(b) CME Creating Micro Ensemble Component

$$\begin{array}{c} \text{ME}(f,X) \to \begin{array}{c} \text{Canonical} \\ \text{IP}(f,X) \to \begin{array}{c} \text{Ensemial} \\ \text{Ensemial} \end{array} \to \begin{array}{c} \text{Interactive} \\ \text{Maps} \\ \text{IM} \end{array} \end{array} \to \begin{array}{c} \text{Interactive} \\ \text{Maps} \\ \text{IM} \end{array} \to \begin{Bmatrix} \text{CE}(f),IM(f) \end{Bmatrix} \to \begin{Bmatrix} \text{ME}(f),IM(f) \end{Bmatrix} \to \begin{Bmatrix} \text{ME}(f),IM(f) \end{Bmatrix}$$

(c) CEIM Canonical Ensemble and Interactive Map Component

$$\begin{array}{c} \{\mathsf{CE}(\mathsf{J}), IM(\mathsf{J})\} \to \begin{array}{c} \text{Sets of } \{\mathsf{CE}(\mathsf{J})\}, \\ \{\mathsf{IM}(\mathsf{J})\} \end{array} \to \begin{array}{c} \text{SCE} \rightarrow \begin{array}{c} \text{CCE} \rightarrow \begin{array}{c} \text{CE} \ \mathsf{k} \ \mathsf{IM} \end{array} \\ \mathsf{SCEIM} \end{array} \to \begin{array}{c} \text{SIM} \rightarrow \begin{array}{c} \text{CE} \ \mathsf{k} \ \mathsf{M} \end{array} \\ \text{FC} \rightarrow \begin{array}{c} \text{Multies} \\ \mathsf{CIM} \end{array} \end{array} \to \begin{array}{c} \text{CEM} \end{array} \end{array}$$

(d) GEM Global Ensemble Matrix Component

**Figure 1.** (a-d) Variant Phase Space System; (a) Architecture; (b) CME Creating Micro Ensemble; (c) CEIM Canonical Ensemble and Interactive Map; (d) GEM Global Ensemble Matrix

#### **Output group:**

*CEM* one CE matrix under FC condition

*IMM* a set of eight IM matrices under FC condition

#### **2.2. CME creating micro ensemble**

The CME component as shown in Figure 1(b) is composed of four modules: *VM Variant Measures, PM Probability Measurements, ME Micro Ensemble and IP Interactive Projection*. Five distinct parameters are shown as input signals {*N*, *n*, *X*, *J*, *SM*} and two groups of vector measurements are performed as a group of output signals {*ME*(*J*, *X*), *IP*(*J*, *X*)} respectively.

The various parameter can be described as follows:

#### **Input:**

10 Cellular Automata

{*N*, *n*} → *X* ∈ *B<sup>N</sup>* <sup>2</sup> <sup>→</sup>

> {*N*, *n*} → *X* ∈ *B<sup>N</sup>* <sup>2</sup> <sup>→</sup>

*J* ∈ *B*2*<sup>n</sup>* <sup>2</sup> <sup>→</sup>

Creating Micro Ensemble CME

→ *ME*(*J*, *X*) → → *IP*(*J*, *X*) →

{*CE*(*J*), *IM*(*J*)} →

Variant Measures VM

*PM*(*J*, *X*) → *SM* →

*ME*(*J*, *X*) → *IP*(*J*, *X*) → ∀*X* →

{*CE*(*J*), *IM*(*J*)} →

and Interactive Map; (d) GEM Global Ensemble Matrix

*CEM* one CE matrix under FC condition

**2.2. CME creating micro ensemble**

**Output group:**

∀*J* →

*IMM* a set of eight IM matrices under FC condition

The various parameter can be described as follows:

∀*X* →

(a) Architecture

→

(b) CME Creating Micro Ensemble Component

(c) CEIM Canonical Ensemble and Interactive Map Component

(d) GEM Global Ensemble Matrix Component

**Figure 1.** (a-d) Variant Phase Space System; (a) Architecture; (b) CME Creating Micro Ensemble; (c) CEIM Canonical Ensemble

The CME component as shown in Figure 1(b) is composed of four modules: *VM Variant Measures, PM Probability Measurements, ME Micro Ensemble and IP Interactive Projection*. Five distinct parameters are shown as input signals {*N*, *n*, *X*, *J*, *SM*} and two groups of vector measurements are performed as a group of output signals {*ME*(*J*, *X*), *IP*(*J*, *X*)} respectively.

→

Sets of {*CE*(*J*)}, {*IM*(*J*)} SCEIM

→ *VM*(*J*, *X*) →

Micro Ensemble ME

Canonical Ensemble CE

∀*J* → *FC* →

Canonical Ensemble & Interactive Maps CEIM

> → *CEM* → *IMM*

Probability Measurements PM

Interactive Projection IP

Interactive Maps IM

> → *SCE* → → *SIM* → *FC* →

Global Ensemble Matrices GEM

→ {*CE*(*J*), *IM*(*J*)}

→ *PM*(*J*, *X*)

→ *CEM* → *IMM*

→ *ME*(*J*, *X*) → *IP*(*J*, *X*)

→ {*CE*(*J*), *IM*(*J*)}

CE & IM Matrices CIM

*J* ∈ *B*2*<sup>n</sup>* <sup>2</sup> <sup>→</sup> *SM* → *N* an integer indicating a 0-1 vector with *N* elements

*n* an integer indicating *n* variables for a function

*X* a 0-1 vector with *N* elements, *X* ∈ *B<sup>N</sup>* 2

*J* a function with *n* variables, *J* ∈ *B*2*<sup>n</sup>* 2

*SM* a selection on a pair of measurements

#### **Output:**

*ME*(*J*, *X*) a micro ensemble under either multiple or conditional probability measurements

*IP*(*J*, *X*) a set of eight interactive projections under the SM condition

A point in variant phase space can be determined under a set of conditions. A set of relevant projections can be associated with an interactive environment. The operation of this module transfers each set of input parameters to one micro ensemble signal and its distinct interactive projections subject to certain restrictions.

#### **2.3. CEIM canonical ensemble and interactive map**

The CEIM component as shown in Figure 1(c) is composed of two modules: *CE Canonical Ensemble and IP Interactive Projection*. This component inputs three groups of parameters {*ME*(*J*, *X*), *IP*(*J*, *X*), ∀*X*} from the CME component and outputs two sets of canonical ensembles together with its interactive maps {*CE*(*J*), *IM*(*J*)} as distinct distributions under certain environments. One additional input and two output parameters are described as follows:

#### **Adding Input:**

∀*X* exhaustive set of all states of *N* bit vectors with 2*<sup>N</sup>* elements

#### **Output:**

*CE*(*J*) a canonical ensemble for an *N* bit vector under an *n* variable function *J*

*IM*(*J*) a set of eight interactive maps associated with *CE*(*J*)

The CEIM component collects all possible micro ensembles for a given function to form a canonical ensemble on variant phase space. Meanwhile, different interactive maps associated with this CE can be calculated to output as a set of *IM*(*J*) as distinct distributions under certain environment.

#### **2.4. GIM global ensemble and interactive map matrix**

The GIM component as shown in Figure 1(d) is composed of two modules: one for the *SCEIM Set of Cannonical Ensembles together with Interactive Maps, and the other for the CIM CE & IM Matrices*.

Two outputs {*CE*(*J*), *IM*(*J*)} from CEIM are taken as inputs, while another two parameters {∀*J*, *FC*} and two outputs can be described as follows:

#### **Adding Input:**

∀*J* exhaustive set of all functions of *n* variables with 22*<sup>n</sup>* elements

*FC* a given configuration for variant logic functions: a 22*n*−<sup>1</sup> × 22*<sup>n</sup>*−<sup>1</sup> matrix

#### **Output:**

*CEM* a CE matrix under FC condition

*IMM* a set of IM matrices under FC condition

The SCEIM module processes an exhaustive operation on all possible values of function *J* to generate sets of {{*CE*(*J*)}, {*IM*(*J*)}}∀*<sup>J</sup>* as the output. The CIM module further organizes the data to arrange each set as a 22*<sup>n</sup>*−<sup>1</sup> × <sup>2</sup>2*n*−<sup>1</sup> matrix with 22*<sup>n</sup>* elements and with the specific arrangement determined by *FC* condition.

*3.1.2. Four variant functions*

functions { *f*⊥, *f*+, *f*−, *f*⊤}.

*3.1.3. Two polarized functions*

*3.1.4. n* = 2 *representation*

  ⊥, *x* = 0, *y* = 0; +, *x* = 0, *y* = 1; −, *x* = 1, *y* = 0; ⊤, *x* = 1, *y* = 1.

*3.1.5. Variant measure functions*

Let *x<sup>v</sup>* =

0.

 

For a given logic function *f* , input and output pair relationships define four variant logic

*f*⊥(*x*) = { *f*(*x*)|*x* ∈ *S*0(*n*), *y* = 0} *f*+(*x*) = { *f*(*x*)|*x* ∈ *S*0(*n*), *y* = 1} *f*−(*x*) = { *f*(*x*)|*x* ∈ *S*1(*n*), *y* = 0} *f*⊤(*x*) = { *f*(*x*)|*x* ∈ *S*1(*n*), *y* = 1}

Considering two standard logic canonical expressions: AND-OR form is selected from { *f*+(*x*), *f*⊤(*x*)} as *y* = 1 items, and OR-AND form is selected from { *f*−(*x*), *f*⊥(*x*)} as *y* = 0

To select { *f*+(*x*), *f*−(*x*)}, *xj* �= *y* in forming a variant logic expression. Let *f*(*x*) = �*f*+|*x*| *f*−� be the variant logic expression. Any logic function can be expressed as a variant logic form. In �*f*+|*x*| *f*−� structure, *f*<sup>+</sup> selected 1 items in *S*0(*n*) as same as the AND-OR standard

For a convenient understanding of the variant representation, 2-variable logic structures are

For a pair of { *f*+, *f*−} functions selected from the structure, relevant representations are illustrated in Table 3 to show the variant capacity on the full expression of all logic functions.

> ∆*J*(*x*) = �∆<sup>⊥</sup> *J*(*x*), ∆<sup>+</sup> *J*(*x*), ∆<sup>−</sup> *J*(*x*), ∆<sup>⊤</sup> *J*(*x*)� <sup>∆</sup>*<sup>α</sup> <sup>J</sup>*(*x*) = � 1, *<sup>J</sup>*(*x*) <sup>∈</sup> *<sup>J</sup>α*(*x*), *<sup>α</sup>* ∈ {⊥, <sup>+</sup>, <sup>−</sup>, ⊤}

For any given *n*-variable state there is one position in ∆*J*(*x*) to be 1 and other 3 positions are

� *x*, *δ* = 1; *x*¯, *δ* = 0.

items. Considering { *f*⊤(*x*), *f*⊥(*x*)}, *xj* = *y* items, they are invariant themselves.

expression, and *f*<sup>−</sup> selected 0 items in *S*1(*n*) as same as OR-AND expression.

illustrated in Table 2 for its 16 functions in four variant functions as follows.

∆ = �∆⊥, ∆+, ∆−, ∆⊤�

0, others

and *x<sup>δ</sup>* =

Checking two functions *f* = 3 and *f* = 6 respectively. { *f* = 3 := {1, 0}, *f*<sup>+</sup> = 11 := �0|∅�, *f*<sup>−</sup> = 2 := �∅|3�}; { *f* = 6 := {2, 1}, *f*<sup>+</sup> = 14 := �2|∅�, *f*<sup>−</sup> = 2 := �∅|3�}.

Let ∆ be the variant measure function [1, 35]-[42]

(4)

125

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635

(7)

After two exhaustive processes through CEIM and GIM activities, a CE matrix and the relevant IM Matrices are generated. Each matrix contains 22*<sup>n</sup>* elements as distributions. Further symmetry properties can be identified from each specific configuration.

Since specific components and modules are relevant to the detail of the complex mechanisms, further explanations on each component are presented in Sections 3 through 5.

#### **3. Creating micro ensemble**

The first part of the system is the CME component composed of four modules: VM Variant Measures, PM Probability Measurements, ME Micro Ensemble and IP Interactive Projection respectively.

It is necessary to clearly describe these four modules in order to understand the measurement properties of variant construction [35]-[44]. Relevant information and supporting materials on fundamental levels of variant construction are briefly descried in Section 3.1 and the four modules are investigated in Sections 3.2 through 3.5.

#### **3.1. Initial preparation on variant measurements**

The variant measurement construction is based on *n*-variable logic functions and *N* bit vectors taken as input and output results [40, 43, 44].

#### *3.1.1. Two sets of states*

For n-variables where *x* = *xn*−1...*xi*...*x*0, 0 ≤ *i* < *n*, *xi* ∈ {0, 1} = *B*2, let a position *j* be the selected variable 0 ≤ *j* < *n*, *xj* be the selected variable. Let output variable *y* and *n*-variable function *f* , *y* = *f*(*x*), *y* ∈ *B*2, *x* ∈ *B<sup>n</sup>* <sup>2</sup> . For all states of *<sup>x</sup>*, a set *<sup>S</sup>*(*n*) composed of the 2*<sup>n</sup>* states can be divided into two sets: *S*0(*n*) and *S*1(*n*).

$$\begin{cases} S\_0(n) = \{ \mathbf{x} | \mathbf{x}\_j = \mathbf{0}, \forall \mathbf{x} \in \mathcal{B}\_2^n \} \\ S\_1(n) = \{ \mathbf{x} | \mathbf{x}\_j = \mathbf{1}, \forall \mathbf{x} \in \mathcal{B}\_2^n \} \\ S(n) = \{ S\_0(n), S\_1(n) \} \end{cases} \tag{3}$$

#### *3.1.2. Four variant functions*

12 Cellular Automata

**Adding Input:**

**Output:**

∀*J* exhaustive set of all functions of *n* variables with 22*<sup>n</sup>*

*CEM* a CE matrix under FC condition

the data to arrange each set as a 22*<sup>n</sup>*−<sup>1</sup>

**3. Creating micro ensemble**

modules are investigated in Sections 3.2 through 3.5.

**3.1. Initial preparation on variant measurements**

vectors taken as input and output results [40, 43, 44].

respectively.

*3.1.1. Two sets of states*

function *f* , *y* = *f*(*x*), *y* ∈ *B*2, *x* ∈ *B<sup>n</sup>*

can be divided into two sets: *S*0(*n*) and *S*1(*n*).

 

arrangement determined by *FC* condition.

*IMM* a set of IM matrices under FC condition

*FC* a given configuration for variant logic functions: a 22*n*−<sup>1</sup>

relevant IM Matrices are generated. Each matrix contains 22*<sup>n</sup>*

elements

The SCEIM module processes an exhaustive operation on all possible values of function *J* to generate sets of {{*CE*(*J*)}, {*IM*(*J*)}}∀*<sup>J</sup>* as the output. The CIM module further organizes

After two exhaustive processes through CEIM and GIM activities, a CE matrix and the

Since specific components and modules are relevant to the detail of the complex mechanisms,

The first part of the system is the CME component composed of four modules: VM Variant Measures, PM Probability Measurements, ME Micro Ensemble and IP Interactive Projection

It is necessary to clearly describe these four modules in order to understand the measurement properties of variant construction [35]-[44]. Relevant information and supporting materials on fundamental levels of variant construction are briefly descried in Section 3.1 and the four

The variant measurement construction is based on *n*-variable logic functions and *N* bit

For n-variables where *x* = *xn*−1...*xi*...*x*0, 0 ≤ *i* < *n*, *xi* ∈ {0, 1} = *B*2, let a position *j* be the selected variable 0 ≤ *j* < *n*, *xj* be the selected variable. Let output variable *y* and *n*-variable

*S*0(*n*) = {*x*|*xj* = 0, ∀*x* ∈ *B<sup>n</sup>*

*S*1(*n*) = {*x*|*xj* = 1, ∀*x* ∈ *B<sup>n</sup>*

*S*(*n*) = {*S*0(*n*), *S*1(*n*)}

<sup>2</sup> . For all states of *<sup>x</sup>*, a set *<sup>S</sup>*(*n*) composed of the 2*<sup>n</sup>* states

2 }

2 }

matrix with 22*<sup>n</sup>*

× 22*<sup>n</sup>*−<sup>1</sup>

Further symmetry properties can be identified from each specific configuration.

further explanations on each component are presented in Sections 3 through 5.

× <sup>2</sup>2*n*−<sup>1</sup>

matrix

elements and with the specific

elements as distributions.

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For a given logic function *f* , input and output pair relationships define four variant logic functions { *f*⊥, *f*+, *f*−, *f*⊤}.

$$\begin{cases} f\_{\perp}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathbb{S}\_{0}(n), y = 0 \} \\ f\_{+}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathbb{S}\_{0}(n), y = 1 \} \\ f\_{-}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathbb{S}\_{1}(n), y = 0 \} \\ f\_{\top}(\mathbf{x}) = \{ f(\mathbf{x}) | \mathbf{x} \in \mathbb{S}\_{1}(n), y = 1 \} \end{cases} \tag{4}$$

#### *3.1.3. Two polarized functions*

Considering two standard logic canonical expressions: AND-OR form is selected from { *f*+(*x*), *f*⊤(*x*)} as *y* = 1 items, and OR-AND form is selected from { *f*−(*x*), *f*⊥(*x*)} as *y* = 0 items. Considering { *f*⊤(*x*), *f*⊥(*x*)}, *xj* = *y* items, they are invariant themselves.

To select { *f*+(*x*), *f*−(*x*)}, *xj* �= *y* in forming a variant logic expression. Let *f*(*x*) = �*f*+|*x*| *f*−� be the variant logic expression. Any logic function can be expressed as a variant logic form. In �*f*+|*x*| *f*−� structure, *f*<sup>+</sup> selected 1 items in *S*0(*n*) as same as the AND-OR standard expression, and *f*<sup>−</sup> selected 0 items in *S*1(*n*) as same as OR-AND expression.

#### *3.1.4. n* = 2 *representation*

For a convenient understanding of the variant representation, 2-variable logic structures are illustrated in Table 2 for its 16 functions in four variant functions as follows.

$$\text{Let } \mathbf{x}^{\upsilon} = \begin{cases} \bot, \; \mathbf{x} = \mathbf{0}, y = \mathbf{0}; \\ +, \; \mathbf{x} = \mathbf{0}, y = \mathbf{1}; \\ -, \; \mathbf{x} = \mathbf{1}, y = \mathbf{0}; \\ \top, \; \mathbf{x} = \mathbf{1}, y = \mathbf{1}. \end{cases} \text{ and } \mathbf{x}^{\delta} = \begin{cases} \mathbf{x}, \; \delta = \mathbf{1}; \\ \bar{\mathbf{x}}, \; \delta = \mathbf{0}. \end{cases}$$

For a pair of { *f*+, *f*−} functions selected from the structure, relevant representations are illustrated in Table 3 to show the variant capacity on the full expression of all logic functions.

Checking two functions *f* = 3 and *f* = 6 respectively. { *f* = 3 := {1, 0}, *f*<sup>+</sup> = 11 := �0|∅�, *f*<sup>−</sup> = 2 := �∅|3�}; { *f* = 6 := {2, 1}, *f*<sup>+</sup> = 14 := �2|∅�, *f*<sup>−</sup> = 2 := �∅|3�}.

#### *3.1.5. Variant measure functions*

Let ∆ be the variant measure function [1, 35]-[42]

$$\begin{array}{l} \Delta = \langle \Delta\_{\perp}, \Delta\_{+}, \Delta\_{-}, \Delta\_{\top} \rangle \\ \Delta f(\mathbf{x}) = \langle \Delta\_{\perp} f(\mathbf{x}), \Delta\_{+} f(\mathbf{x}), \Delta\_{-} f(\mathbf{x}), \Delta \top f(\mathbf{x}) \rangle \\ \Delta\_{\mathbf{d}} f(\mathbf{x}) = \begin{cases} 1, f(\mathbf{x}) \in f\_{\mathbf{d}}(\mathbf{x}), \mathbf{a} \in \{\perp, +, -, \top\} \\ 0, \text{others} \end{cases} \end{array} \tag{7}$$

For any given *n*-variable state there is one position in ∆*J*(*x*) to be 1 and other 3 positions are 0.


**Table 2.** Four Variant Functions in 2-variable logic


**Table 3.** A pair of selected functions and their full expression

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*3.1.6. Variant measures on vector*

E.g. *N* = 12, given *J*,*Y* = *J*(*X*).

*N*,*Yj* ∈ *B*2,*Y* ∈ *B<sup>N</sup>*

quaternion

*3.1.7. Example*

For any *N* bit 0-1 vector *X*, *X* = *XN*−1...*Xj*...*X*0, 0 ≤ *j* < *N*, *Xj* ∈ *B*2, *X* ∈ *B<sup>N</sup>*

<sup>2</sup> . For the *<sup>j</sup>*-th position *<sup>x</sup><sup>j</sup>* = [...*Xj*...] <sup>∈</sup> *<sup>B</sup><sup>n</sup>*

∆(*X* : *Y*) = ∆*J*(*X*) =

�*N*⊥, *N*+, *N*−, *N*⊤�, *N* = *N*<sup>⊥</sup> + *N*<sup>+</sup> + *N*<sup>−</sup> + *N*⊤.

∆*J*(*X*) = �*N*⊥, *N*+, *N*−, *N*⊤� = �3, 1, 4, 4�, *N* = 12.

sequences. Four meta measures are determined.

*3.1.8. Basic Properties of Variant Logic*

can be described [35]-[44].

(*f*⊥, *f*+, *f*−, *f*⊤) respectively.

**3.2. VM variant measures**

�*N*⊥, *N*+, *N*−, *N*⊤�.

follows.

function *J*, *n* bit 0-1 output vector *Y*, *Y* = *J*(*X*) = �*J*+|*X*|*J*−�, *Y* = *YN*−1...*Yj*...*Y*0, 0 ≤ *j* <

Let *N* bit positions be cyclic linked. Variant measures of *J*(*X*) can be decomposed as a

∆*J*(*x<sup>j</sup>*

*X* = 101110111001 *Y* = 001010101100 ∆(*X* : *Y*) = −⊥⊤−⊤⊥⊤−⊤ + ⊥ −

Input and output pairs are 0-1 variables for only four combinations. For any given function *J*, the quantitative relationship of {⊥, +, −, ⊤} is directly derived from the input/output

For given *n* 0-1 variables, a given function *J* and an *N* bit vector *X*, the following corollaries

**Corollary 3.3:** For an *N* bit vector *X*, the phase space is composed of a total of 2*<sup>N</sup>* vectors. **Corollary 3.4:** A logic function *f* can be partitioned as four variant functions as *f* =

**Corollary 3.5:** For a given vector *X* and a given function *J*, a measure vector of four meta measures for variant measures can be determined as a quaternion ∆*J*(*X*) =

Using defined variant functions, it is possible to describe the VM module in Fig. 1(b) as

**Corollary 3.1:** For *n* 0-1 variables, the state set contains a total of 2*<sup>n</sup>* states. **Corollary 3.2:** For *n* 0-1 variables, the function set contains a total of 22*<sup>n</sup>*

*N*−1 ∑ *j*=0

<sup>2</sup> under *n*-variable

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635

) = �*J*+|*x<sup>j</sup>*

functions.


127

<sup>2</sup> to form *Yj* <sup>=</sup> *<sup>J</sup>*(*x<sup>j</sup>*

) = �*N*⊥, *N*+, *N*−, *N*⊤� (8)

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#### *3.1.6. Variant measures on vector*

For any *N* bit 0-1 vector *X*, *X* = *XN*−1...*Xj*...*X*0, 0 ≤ *j* < *N*, *Xj* ∈ *B*2, *X* ∈ *B<sup>N</sup>* <sup>2</sup> under *n*-variable function *J*, *n* bit 0-1 output vector *Y*, *Y* = *J*(*X*) = �*J*+|*X*|*J*−�, *Y* = *YN*−1...*Yj*...*Y*0, 0 ≤ *j* < *N*,*Yj* ∈ *B*2,*Y* ∈ *B<sup>N</sup>* <sup>2</sup> . For the *<sup>j</sup>*-th position *<sup>x</sup><sup>j</sup>* = [...*Xj*...] <sup>∈</sup> *<sup>B</sup><sup>n</sup>* <sup>2</sup> to form *Yj* <sup>=</sup> *<sup>J</sup>*(*x<sup>j</sup>* ) = �*J*+|*x<sup>j</sup>* |*J*−�. Let *N* bit positions be cyclic linked. Variant measures of *J*(*X*) can be decomposed as a quaternion

$$\Delta(X:Y) = \Delta f(X) = \sum\_{j=0}^{N-1} \Delta f(\mathbf{x}^j) = \langle N\_{\perp}, N\_{+\prime}, N\_{-\prime}, N\_{\top} \rangle \tag{8}$$

�*N*⊥, *N*+, *N*−, *N*⊤�, *N* = *N*<sup>⊥</sup> + *N*<sup>+</sup> + *N*<sup>−</sup> + *N*⊤.

*3.1.7. Example*

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14 Cellular Automata

**Table 2.** Four Variant Functions in 2-variable logic

**Table 3.** A pair of selected functions and their full expression

*f f* ∈ 3210 3*<sup>v</sup>* 2*<sup>v</sup>* 1*<sup>v</sup>* 0*<sup>v</sup> f*<sup>⊥</sup> ∈ *f*<sup>+</sup> ∈ *f*<sup>−</sup> ∈ *f*<sup>⊤</sup> ∈ *No*. *S*(*n*) 11 10 01 00 11*<sup>v</sup>* 10*<sup>v</sup>* 01*<sup>v</sup>* 00*<sup>v</sup> S*0(*n*) *S*0(*n*) *S*1(*n*) *S*1(*n*) {∅} 0000 −⊥−⊥ {2, 0} {∅} {3, 1} {∅} {0} 0001 −⊥− + {2} {0} {3, 1} {∅} {1} 0010 −⊥⊤⊥ {2.0} {∅} {3} {1} {1, 0} 0011 −⊥⊤ + {2} {0} {3} {1} {2} 0100 − + − ⊥ {0} {2} {3, 1} {∅} {2, 0} 0101 − + − + {∅} {2, 0} {3, 1} {∅} {2, 1} 0110 − + ⊤ ⊥ {0} {2} {3} {1} {2, 1, 0} 0111 − + ⊤ + {∅} {2, 0} {3} {1} {3} 1000 ⊤⊥−⊥ {2, 0} {∅} {1} {3} {3, 0} 1001 ⊤⊥− + {2} {0} {1} {3} {3, 1} 1010 ⊤⊥⊤⊥ {2, 0} {∅} {∅} {3, 1} {3, 1, 0} 1011 ⊤⊥⊤ + {2} {0} {∅} {3, 1} {3, 2} 1100 ⊤ + − ⊥ {0} {2} {1} {3} {3, 2, 0} 1101 ⊤ + − + {∅} {2, 0} {1} {3} {3, 2, 1} 1110 ⊤ + ⊤ ⊥ {0} {2} {∅} {3, 1} {3, 2, 1, 0} 1111 ⊤ + ⊤ + {∅} {2, 0} {∅} {3, 1}

*f f* ∈ 3210 *f*<sup>+</sup> ∈ 3<sup>0</sup> 2<sup>1</sup> 10 01 *f*<sup>−</sup> ∈ *No*. *S*(*n*) 11 10 01 00 *S*0(*n*) 110 101 01<sup>0</sup> 00<sup>1</sup> *S*1(*n*) {∅} 0000 �∅| 1010 |3, 1� {0} 0001 �0| 1011 |3, 1� {1} 0010 �∅| 1000 |3� {1, 0} 0011 �0| 1001 |3� {2} 0100 �2| 1110 |3, 1� {2, 0} 0101 �2, 0| 1111 |3, 1� {2, 1} 0110 �2| 1100 |3� {2, 1, 0} 0111 �2, 0| 1101 |3� {3} 1000 �∅| 0010 |1� {3, 0} 1001 �0| 0011 |1� {3, 1} 1010 �∅| 0000 |∅� {3, 1, 0} 1011 �0| 0001 |∅� {3, 2} 1100 �2| 0110 |1� {3, 2, 0} 1101 �2, 0| 0111 |1� {3, 2, 1} 1110 �2| 0100 |∅� {3, 2, 1, 0} 1111 �2, 0| 0101 |∅�

E.g. *N* = 12, given *J*,*Y* = *J*(*X*).

*X* = 101110111001 *Y* = 001010101100 ∆(*X* : *Y*) = −⊥⊤−⊤⊥⊤−⊤ + ⊥ −

$$
\Delta J(X) = \langle N\_{\perp \prime} N\_{+\prime} N\_{-\prime} N\_{\top} \rangle = \langle 3, 1, 4, 4 \rangle, N = 12.1
$$

Input and output pairs are 0-1 variables for only four combinations. For any given function *J*, the quantitative relationship of {⊥, +, −, ⊤} is directly derived from the input/output sequences. Four meta measures are determined.

#### *3.1.8. Basic Properties of Variant Logic*

For given *n* 0-1 variables, a given function *J* and an *N* bit vector *X*, the following corollaries can be described [35]-[44].

**Corollary 3.1:** For *n* 0-1 variables, the state set contains a total of 2*<sup>n</sup>* states.

**Corollary 3.2:** For *n* 0-1 variables, the function set contains a total of 22*<sup>n</sup>* functions.

**Corollary 3.3:** For an *N* bit vector *X*, the phase space is composed of a total of 2*<sup>N</sup>* vectors.

**Corollary 3.4:** A logic function *f* can be partitioned as four variant functions as *f* = (*f*⊥, *f*+, *f*−, *f*⊤) respectively.

**Corollary 3.5:** For a given vector *X* and a given function *J*, a measure vector of four meta measures for variant measures can be determined as a quaternion ∆*J*(*X*) = �*N*⊥, *N*+, *N*−, *N*⊤�.

#### **3.2. VM variant measures**

Using defined variant functions, it is possible to describe the VM module in Fig. 1(b) as follows.

Under variant construction, *N* bits of 0-1 vector *X* under a function *J* produce seven Meta measures composed of a measure vector *VM*(*J*, *X*).

$$\begin{array}{c} (X:J(X)) \to (N\_\perp, N\_+, N\_-, N\_\top)\_\prime\\ \text{N}\_0 = \text{N}\_\perp + \text{N}\_+,\\ \text{N}\_1 = \text{N}\_- + \text{N}\top,\\ \text{N} = \text{N}\_0 + \text{N}\_1. \end{array} \tag{9}$$

corresponds to conditional probability measurements. In this Chapter, only two quaternion measurements are used in order to focus attention on the simplest interactive combinations

Under such condition, the output signals of the PM module can be expressed as a pair of

The ME module has two inputs. *PM*(*J*, *X*) provides probability measurement vectors to provide the basis of the measurement. The input parameter *SM* indicates Selected

In this paper, two cases for a pair of measurement selections are restricted to permit an investigation of possible configurations of interactive distributions in their variant phase

Under these conditions, each (*pi*(*J*, *X*), *pj*(*J*, *X*)) or (*p*˜*i*(*J*, *X*), *p*˜*j*(*J*, *X*)) determines a fixed position on variant phase space as a Micro Ensemble. The output of the ME module can be expressed as *ME*(*J*, *X*)=(*pi*(*J*, *X*), *pj*(*J*, *X*))|(*p*˜*i*(*J*, *X*), *p*˜*j*(*J*, *X*)) under a given function *J*,

Since each ME must be located on a certain position in a square area on variant phase space, it is convenient to show the restrictions and specific properties according to the following

**Proposition 3.1:** In the Case A condition, a total of six configurations can be identified in different *P* selections. For each configuration, its pair of probability measurements can be

**Proof:** Any selection of two elements (*pi*(*J*, *X*), *pj*(*J*, *X*)) from *P*, it satisfies 0 ≤ *pi*(*J*, *X*) +

**Proposition 3.2:** In the Case B condition, a total of six configurations can be identified into two groups in different *P*˜ selections, four configurations in the first group are restricted in a square area and two configurations of the second group are restricted on a diagonal line.

For the first group, two selected components can satisfy 0 ≤ *p*˜*i*, *p*˜*<sup>j</sup>* ≤ 1, four distinct selections

can satisfy 0 ≤ *p*˜*i*, *p*˜*<sup>j</sup>* ≤ 1 and *p*˜*<sup>i</sup>* + *p*˜*<sup>j</sup>* = 1, only two distinct selections are identified on a

Under this arrangement, all measurements are relevant to variant construction. Now, let this type of phase spaces be *Variant Phase Space VPS*. For an *N* bit vector *X*, a pair of probability

<sup>2</sup> region.

measurements determines a micro ensemble to be a specific position in VPS.

<sup>2</sup> restricted region. For the second group, two selected components

<sup>⊥</sup> + *ρ*˜

<sup>+</sup> = *ρ*˜

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 129

<sup>−</sup> + *ρ*˜

⊤ = 1.

<sup>2</sup> region.

**Proof:** Since two equations in *ρ*˜ quaternion are in the conditions of *ρ*˜

without further measurements of {*ρ*0, *ρ*1} and {*ρ*˜0, *ρ*˜1} involved.

**Case A:** (*pi*, *pj*) or (*p*+, *p*−) ∈ *P* ⊂ *ρ* with two measurements from *ρ*; **Case B:** (*p*˜*i*, *p*˜*j*) or (*p*˜+, *p*˜−) ∈ *P*˜ ⊂ *ρ*˜ with two measurements from *ρ*˜.

probability vectors in quaternion forms *PM*(*J*, *X*) = {*ρ*, *ρ*˜}.

**3.4. ME micro ensemble**

Measurements from *PM*(*J*, *X*).

spaces under simple conditions.

an *N* bit vector *X* and *SM* conditions.

restricted in a triangle area of a [0, 1]

*pj*(*J*, *<sup>X</sup>*) ≤ 1, there are six distinct selections.

*3.4.1. Variant Phase Space*

are identified in a [0, 1]

diagonal line distributed in a [0, 1]

propositions.

From a measuring viewpoint, there are seven measures identified in this set of parameters. They can be expressed in three levels.

$$\begin{array}{ccccc} N & & & N \\ N\_0 & & N\_1 & & \\ N\_\perp & N\_+ & & N\_\top \end{array} \tag{10}$$

In the current system, the output of the VM module is expressed as *VM*(*J*, *X*) = {*N*⊥, *N*+, *N*−, *N*⊤, *N*0, *N*1, *N*}.

#### **3.3. PM probability measurements**

Measures of *VM*(*J*, *X*) are input as numeric vectors into the PM module. Using variant quaternion and other three core measures, local measurements of probability signals are calculated as eight meta measurements in two groups by following the given equations. For any *N* bit 0-1 vector *X*, function *J*, under ∆ measurement: ∆*J*(*X*) = �*N*⊥, *N*+, *N*−, *N*⊤�, *N*<sup>0</sup> = *N*<sup>⊥</sup> + *N*+, *N*<sup>1</sup> = *N*<sup>−</sup> + *N*⊤, *N* = *N*<sup>0</sup> + *N*<sup>1</sup>

The first group of probability signal vectors *ρ* and {*ρ*0, *ρ*1} are defined by

$$\begin{cases} \rho = \frac{\Delta f(X)}{N} = (\rho\_{\perp'}\rho\_{+'}\rho\_{-'}\rho\_{\top'})\\ \rho\_{\mathfrak{a}} = \frac{N\_{\mathfrak{a}}}{N}, \mathfrak{a} \in \{\perp, +, -, \top\};\\ \rho\_{\mathfrak{0}} = N\_{\mathfrak{0}}/N, \\ \rho\_{\mathfrak{1}} = N\_{\mathfrak{1}}/N. \end{cases} \tag{11}$$

The second group of probability signal vectors *ρ*˜ and {*ρ*˜0, *ρ*˜1} is defined by

$$\begin{cases} \begin{array}{l} \tilde{\rho} = \frac{\Lambda f(X)}{N\_0|N\_1} = (\tilde{\rho}\_{\perp}, \tilde{\rho}\_{+}, \tilde{\rho}\_{-}, \tilde{\rho}\_{\top}), \\ \tilde{\rho}\_{\perp} = \frac{N\_{\perp}}{N\_0}, \\ \tilde{\rho}\_{+} = \frac{N\_{\perp}}{N\_0}, \\ \tilde{\rho}\_{-} = \frac{N\_{\perp}}{N\_1}, \\ \tilde{\rho}\_{\top} = \frac{N\_{\perp}}{N\_1}; \\ \tilde{\rho}\_0 = N\_0/N\_{\perp} \\ \tilde{\rho}\_1 = N\_1/N. \end{array} \tag{12}$$

The two groups of probability measurements are key components in variant measurement. The first group corresponds to multiple probability measurements and the second group corresponds to conditional probability measurements. In this Chapter, only two quaternion measurements are used in order to focus attention on the simplest interactive combinations without further measurements of {*ρ*0, *ρ*1} and {*ρ*˜0, *ρ*˜1} involved.

Under such condition, the output signals of the PM module can be expressed as a pair of probability vectors in quaternion forms *PM*(*J*, *X*) = {*ρ*, *ρ*˜}.

#### **3.4. ME micro ensemble**

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(10)

(11)

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16 Cellular Automata

Under variant construction, *N* bits of 0-1 vector *X* under a function *J* produce seven Meta

(*X* : *J*(*X*)) → (*N*⊥, *N*+, *N*−, *N*⊤), *N*<sup>0</sup> = *N*<sup>⊥</sup> + *N*+, *N*<sup>1</sup> = *N*<sup>−</sup> + *N*⊤, *N* = *N*<sup>0</sup> + *N*1.

From a measuring viewpoint, there are seven measures identified in this set of parameters.

*N N*<sup>0</sup> *N*<sup>1</sup> *N*⊥ *N*+ *N*− *N*⊤

In the current system, the output of the VM module is expressed as *VM*(*J*, *X*) =

Measures of *VM*(*J*, *X*) are input as numeric vectors into the PM module. Using variant quaternion and other three core measures, local measurements of probability signals are calculated as eight meta measurements in two groups by following the given equations. For any *N* bit 0-1 vector *X*, function *J*, under ∆ measurement: ∆*J*(*X*) = �*N*⊥, *N*+, *N*−, *N*⊤�,

*<sup>N</sup>* <sup>=</sup> (*ρ*⊥, *<sup>ρ</sup>*+, *<sup>ρ</sup>*−, *<sup>ρ</sup>*⊤,)

*<sup>N</sup>*0|*N*<sup>1</sup> <sup>=</sup> (*ρ*˜⊥, *<sup>ρ</sup>*˜+, *<sup>ρ</sup>*˜−, *<sup>ρ</sup>*˜⊤),

The two groups of probability measurements are key components in variant measurement. The first group corresponds to multiple probability measurements and the second group

*<sup>N</sup>* , *<sup>α</sup>* ∈ {⊥, <sup>+</sup>, <sup>−</sup>, ⊤};

The first group of probability signal vectors *ρ* and {*ρ*0, *ρ*1} are defined by

*<sup>ρ</sup>* <sup>=</sup> <sup>∆</sup>*J*(*X*)

*ρ*<sup>0</sup> = *N*0/*N*, *ρ*<sup>1</sup> = *N*1/*N*.

The second group of probability signal vectors *ρ*˜ and {*ρ*˜0, *ρ*˜1} is defined by

*<sup>ρ</sup>*˜ <sup>=</sup> <sup>∆</sup>*J*(*X*)

*ρ*˜<sup>⊥</sup> = *<sup>N</sup>*<sup>⊥</sup> *<sup>N</sup>*<sup>0</sup> , *<sup>ρ</sup>*˜<sup>+</sup> = *<sup>N</sup>*<sup>+</sup> *<sup>N</sup>*<sup>0</sup> , *ρ*˜<sup>−</sup> = *<sup>N</sup>*<sup>−</sup> *<sup>N</sup>*<sup>1</sup> , *ρ*˜<sup>⊤</sup> = *<sup>N</sup>*<sup>⊤</sup> *<sup>N</sup>*<sup>1</sup> ; *ρ*˜0 = *N*0/*N*, *ρ*˜1 = *N*1/*N*.

*ρα* = *<sup>N</sup><sup>α</sup>*

 

measures composed of a measure vector *VM*(*J*, *X*).

They can be expressed in three levels.

**3.3. PM probability measurements**

*N*<sup>0</sup> = *N*<sup>⊥</sup> + *N*+, *N*<sup>1</sup> = *N*<sup>−</sup> + *N*⊤, *N* = *N*<sup>0</sup> + *N*<sup>1</sup>

{*N*⊥, *N*+, *N*−, *N*⊤, *N*0, *N*1, *N*}.

The ME module has two inputs. *PM*(*J*, *X*) provides probability measurement vectors to provide the basis of the measurement. The input parameter *SM* indicates Selected Measurements from *PM*(*J*, *X*).

In this paper, two cases for a pair of measurement selections are restricted to permit an investigation of possible configurations of interactive distributions in their variant phase spaces under simple conditions.

**Case A:** (*pi*, *pj*) or (*p*+, *p*−) ∈ *P* ⊂ *ρ* with two measurements from *ρ*; **Case B:** (*p*˜*i*, *p*˜*j*) or (*p*˜+, *p*˜−) ∈ *P*˜ ⊂ *ρ*˜ with two measurements from *ρ*˜.

Under these conditions, each (*pi*(*J*, *X*), *pj*(*J*, *X*)) or (*p*˜*i*(*J*, *X*), *p*˜*j*(*J*, *X*)) determines a fixed position on variant phase space as a Micro Ensemble. The output of the ME module can be expressed as *ME*(*J*, *X*)=(*pi*(*J*, *X*), *pj*(*J*, *X*))|(*p*˜*i*(*J*, *X*), *p*˜*j*(*J*, *X*)) under a given function *J*, an *N* bit vector *X* and *SM* conditions.

#### *3.4.1. Variant Phase Space*

Since each ME must be located on a certain position in a square area on variant phase space, it is convenient to show the restrictions and specific properties according to the following propositions.

**Proposition 3.1:** In the Case A condition, a total of six configurations can be identified in different *P* selections. For each configuration, its pair of probability measurements can be restricted in a triangle area of a [0, 1] <sup>2</sup> region.

**Proof:** Any selection of two elements (*pi*(*J*, *X*), *pj*(*J*, *X*)) from *P*, it satisfies 0 ≤ *pi*(*J*, *X*) + *pj*(*J*, *<sup>X</sup>*) ≤ 1, there are six distinct selections.

**Proposition 3.2:** In the Case B condition, a total of six configurations can be identified into two groups in different *P*˜ selections, four configurations in the first group are restricted in a square area and two configurations of the second group are restricted on a diagonal line.

**Proof:** Since two equations in *ρ*˜ quaternion are in the conditions of *ρ*˜ <sup>⊥</sup> + *ρ*˜ <sup>+</sup> = *ρ*˜ <sup>−</sup> + *ρ*˜ ⊤ = 1. For the first group, two selected components can satisfy 0 ≤ *p*˜*i*, *p*˜*<sup>j</sup>* ≤ 1, four distinct selections are identified in a [0, 1] <sup>2</sup> restricted region. For the second group, two selected components can satisfy 0 ≤ *p*˜*i*, *p*˜*<sup>j</sup>* ≤ 1 and *p*˜*<sup>i</sup>* + *p*˜*<sup>j</sup>* = 1, only two distinct selections are identified on a diagonal line distributed in a [0, 1] <sup>2</sup> region.

Under this arrangement, all measurements are relevant to variant construction. Now, let this type of phase spaces be *Variant Phase Space VPS*. For an *N* bit vector *X*, a pair of probability measurements determines a micro ensemble to be a specific position in VPS.

In order to distinguish between the two types of VPS, let us name a subset of VPS under Case A conditions as *a Multiple Phase Space MPS* while we name a subset of VPS under Case B conditions as *a Conditional Phase Space CPS*. Samples of canonical ensembles of the six combinations under a given SM condition under a function in both MPS & CPS are shown in Figure 2(I,II) respectively.

#### **3.5. IP interactive projections**

Using a micro ensemble *ME*(*J*, *X*), different projections can be identified in an IP module under various interactive conditions. Based on the input micro ensemble for each Case, two groups of eight interactive projections can be distinguished by symmetry/anti-symmetry and synchronous/asynchronous conditions.

#### *3.5.1. Synchronous and Asynchronous Operations*

Each *ME*(*J*, *X*) is a pair of probability measurements and it is essential to establish corresponding rules to place their interactive projection in the same probability region i.e. [0, 1] segment.

We can distinguish between Synchronous and Asynchronous time-related operations.

Under a synchronous operation {+, −, ×, /, }, only one merged measurement is located in [0, 1] region to express one activity from a ME.

(I) Type A

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 131

(II) Type B

**Figure 2.** (I-II) Six combinations of two selected measurements for a function on VPS of two probability models (I) Type A:(a-f)

Six combinations in MPS; (II) Type B:(a-f) Six combinations in CPS

However, under an asynchronous operation ⊕, two input measurements *p*<sup>+</sup> �= *p*−, generate an output result as a vector that has two positions of *p*+ and *p*− with a weighted value 1 on each position; when *p* = *p*+ = *p*− there is a weighted value of 2 on the position *p*.

Under asynchronous operations, merged results may be distinguished by their position or overlap each other with a cumulative weight value of 2. However, under synchronous operations, two measurements are merged as a unit weight to shift interactive measurements to one position in the [0, 1] region.

From an integrative viewpoint, the two types of operations may be considered capable of either merging two particles (asynchronous) on two positions or integrating two waves (synchronous) on a position.

#### *3.5.2. Case A: Multiple Probability Interactive Projections*

For each *ME*(*J*, *P*)=(*pi*(*J*, *X*), *pj*(*J*, *X*)) has a position on a unit square [0, 1] 2.

Let *P* = {*p*+, *p*−} (or {*px*, *py*}) locate a pair of measurements, the IP module projects two measurements and its weight into four conditions in different symmetric properties to form two groups of eight weight vectors as interactive projections.

Using *P* = {*p*+, *p*−}, a pair of measurement vectors {*u*, *v*} are formulated:

18 Cellular Automata

[0, 1] segment.

in Figure 2(I,II) respectively.

**3.5. IP interactive projections**

synchronous/asynchronous conditions.

*3.5.1. Synchronous and Asynchronous Operations*

[0, 1] region to express one activity from a ME.

*3.5.2. Case A: Multiple Probability Interactive Projections*

two groups of eight weight vectors as interactive projections.

to one position in the [0, 1] region.

(synchronous) on a position.

In order to distinguish between the two types of VPS, let us name a subset of VPS under Case A conditions as *a Multiple Phase Space MPS* while we name a subset of VPS under Case B conditions as *a Conditional Phase Space CPS*. Samples of canonical ensembles of the six combinations under a given SM condition under a function in both MPS & CPS are shown

Using a micro ensemble *ME*(*J*, *X*), different projections can be identified in an IP module under various interactive conditions. Based on the input micro ensemble for each Case, two groups of eight interactive projections can be distinguished by symmetry/anti-symmetry and

Each *ME*(*J*, *X*) is a pair of probability measurements and it is essential to establish corresponding rules to place their interactive projection in the same probability region i.e.

Under a synchronous operation {+, −, ×, /, }, only one merged measurement is located in

However, under an asynchronous operation ⊕, two input measurements *p*<sup>+</sup> �= *p*−, generate an output result as a vector that has two positions of *p*+ and *p*− with a weighted value 1 on

Under asynchronous operations, merged results may be distinguished by their position or overlap each other with a cumulative weight value of 2. However, under synchronous operations, two measurements are merged as a unit weight to shift interactive measurements

From an integrative viewpoint, the two types of operations may be considered capable of either merging two particles (asynchronous) on two positions or integrating two waves

Let *P* = {*p*+, *p*−} (or {*px*, *py*}) locate a pair of measurements, the IP module projects two measurements and its weight into four conditions in different symmetric properties to form

2.

We can distinguish between Synchronous and Asynchronous time-related operations.

each position; when *p* = *p*+ = *p*− there is a weighted value of 2 on the position *p*.

For each *ME*(*J*, *P*)=(*pi*(*J*, *X*), *pj*(*J*, *X*)) has a position on a unit square [0, 1]

Using *P* = {*p*+, *p*−}, a pair of measurement vectors {*u*, *v*} are formulated:

**Figure 2.** (I-II) Six combinations of two selected measurements for a function on VPS of two probability models (I) Type A:(a-f) Six combinations in MPS; (II) Type B:(a-f) Six combinations in CPS

$$\begin{cases} u = (u\_{+}, u\_{-}, u\_{0}, u\_{1}) = \{u\_{\beta}\} \\ v = (v\_{+}, v\_{-}, v\_{0}, v\_{1}) = \{v\_{\beta}\} \\ \quad \beta \in \{+, -, 0, 1\} \end{cases} \tag{13}$$

For the two projections in a diagonal line, the following equations are satisfied.

*u*˜<sup>+</sup> = *p*˜*<sup>i</sup> u*˜<sup>−</sup> = *p*˜*<sup>j</sup> u*˜0 = *u*˜<sup>+</sup> ⊕ *u*˜<sup>−</sup> *u*˜1 = *u*˜<sup>+</sup> + *u*˜<sup>−</sup> *<sup>v</sup>*˜<sup>+</sup> <sup>=</sup> <sup>1</sup>+*p*˜*<sup>i</sup>* 2 *<sup>v</sup>*˜<sup>−</sup> <sup>=</sup> <sup>1</sup>−*p*˜*<sup>j</sup>* 2 *v*˜0 = *v*˜<sup>+</sup> ⊕ *v*˜<sup>−</sup> *v*˜1 = *v*˜<sup>+</sup> + *v*˜<sup>−</sup> − 0.5

where 0 ≤ *u*˜*β*, *v*˜*<sup>β</sup>* ≤ 1, *β* ∈ {+, −, 0, 1}, ⊕ : Asynchronous addition, + : Synchronous

Under Symmetry/Anti-symmetry and Synchronous/Asynchronous conditions, one ME

**Proposition 3.3:** Two types of distinguished projections can be identified under

**Proof:** For two projection vectors {*u*, *v*}, we can note that *u* represents a symmetry condition

**Proposition 3.4:** Synchronous and Asynchronous conditions lead to significant different

**Proof:** In a synchronous operation, only one unit weight is output as {*u*1, *v*1}. However, in an asynchronous operation, two positions may be seen to have a combined weight {*u*0, *v*0}.

**Proposition 3.5:** Other projections are simple ones corresponding to relevant measurement

**Proof:** Other output results are {*u*+, *u*−, *v*+, *v*−}, each parameter is only dependent on one measurement to get a similar distribution from a projection viewpoint. There is no real

**Proposition 3.6:** If two probability measurements are required to satisfy *pi* + *pj* ≤ 1, then

**Proposition 3.7:** If two probability measurements independently have 0 ≤ *pi*, *pj* ≤ 1, then

**Proposition 3.8** From a selected ME, eight interactive projections can be formulated.

corresponds to eight interactive projections to express their selected characteristics.

(17)

133

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635

addition.

output results.

projections.

*3.5.4. Key Properties in IP Module*

interactive activity involved.

**Proof:** By Propositions 3.3-3.7.

symmetry/anti-symmetry conditions for each ME.

and *v* represents an anti-symmetry condition.

their symmetry interactive projection result is equal to *pi* + *pj*.

**Proof:** Under this condition, merged results are in the [0, 1] region.

**Proof:** Merged result is in the [0, 1] region.

their symmetry interactive projection is (*pi* + *pj*)/2.

$$\begin{cases} u\_+ = p\_+ \\ u\_- = p\_- \\ u\_0 = u\_+ \oplus u\_- \\ u\_1 = u\_+ + u\_- \\ v\_+ = \frac{1+p\_+}{2} \\ v\_- = \frac{1-p\_-}{2} \\ v\_0 = v\_+ \oplus v\_- \\ v\_1 = v\_+ + v\_- - 0.5 \end{cases} \tag{14}$$

where 0 ≤ *uβ*, *v<sup>β</sup>* ≤ 1, *β* ∈ {+, −, 0, 1}, ⊕ : Asynchronous addition, + : Synchronous addition.

#### *3.5.3. Case B: Conditional Probability Interactive Projections*

For each *ME*(*J*, *P*˜)=(*p*˜*i*(*J*, *X*), *p*˜*j*(*J*, *X*)) we can note that it has a position on a unit square [0, 1] 2.

Let *P*˜ = {*p*˜+, *p*˜−} locate a pair of measurements, the IP module projects two measurements and its weight into four conditions in different symmetric properties to form two groups of eight weights as interactive projections.

Using *P*˜ = {*p*˜+, *p*˜−}, a pair of vectors {*u*˜, *v*˜} are formulated:

$$\begin{cases} \vec{u} = (\vec{u}\_{+}, \vec{u}\_{-}, \vec{u}\_{0}, \vec{u}\_{1}) = \{\vec{u}\_{\beta}\} \\\\ \vec{v} = (\vec{v}\_{+}, \vec{v}\_{-}, \vec{v}\_{0}, \vec{v}\_{1}) = \{\vec{v}\_{\beta}\} \\\\ \beta \in \{+, -, 0, 1\} \end{cases} \tag{15}$$

For the four projections in a square area, the following equations are required.

$$\begin{cases} \begin{aligned} \vec{u}\_{+} &= \vec{p}\_{+} \\ \vec{u}\_{-} &= \vec{p}\_{-} \\ \vec{u}\_{0} &= \vec{u}\_{+} \oplus \vec{u}\_{-} \\ \vec{u}\_{1} &= \frac{\vec{u}\_{+} + \vec{u}\_{-}}{2} \\ \vec{v}\_{+} &= \frac{1 + \vec{p}\_{+}}{2} \\ \vec{v}\_{-} &= \frac{1 - \vec{p}\_{-}}{2} \\ \vec{v}\_{0} &= \vec{v}\_{+} \oplus \vec{v}\_{-} \\ \vec{v}\_{1} &= \vec{v}\_{+} + \vec{v}\_{-} - 0.5 \end{aligned} \tag{16}$$

For the two projections in a diagonal line, the following equations are satisfied.

$$\begin{cases} \begin{aligned} \vec{u}\_{+} &= \vec{p}\_{i} \\ \vec{u}\_{-} &= \vec{p}\_{j} \\ \vec{u}\_{0} &= \vec{u}\_{+} \oplus \vec{u}\_{-} \\ \vec{u}\_{1} &= \vec{u}\_{+} + \vec{u}\_{-} \\ \vec{v}\_{+} &= \frac{1+\rho\_{i}}{2} \\ \vec{v}\_{-} &= \frac{1-\rho\_{j}}{2} \\ \vec{v}\_{0} &= \vec{v}\_{+} \oplus \vec{v}\_{-} \\ \vec{v}\_{1} &= \vec{v}\_{+} + \vec{v}\_{-} - 0.5 \end{aligned} \tag{17}$$

where 0 ≤ *u*˜*β*, *v*˜*<sup>β</sup>* ≤ 1, *β* ∈ {+, −, 0, 1}, ⊕ : Asynchronous addition, + : Synchronous addition.

#### *3.5.4. Key Properties in IP Module*

20 Cellular Automata

addition.

[0, 1] 2.   *u* = (*u*+, *u*−, *u*0, *u*1) = {*uβ*} *v* = (*v*+, *v*−, *v*0, *v*1) = {*vβ*} *β* ∈ {+, −, 0, 1}

(13)

(14)

(15)

(16)

*u*+ = *p*+ *u*− = *p*− *u*<sup>0</sup> = *u*<sup>+</sup> ⊕ *u*<sup>−</sup> *u*<sup>1</sup> = *u*<sup>+</sup> + *u*<sup>−</sup> *<sup>v</sup>*<sup>+</sup> <sup>=</sup> <sup>1</sup>+*p*<sup>+</sup> 2 *<sup>v</sup>*<sup>−</sup> <sup>=</sup> <sup>1</sup>−*p*<sup>−</sup> 2 *v*<sup>0</sup> = *v*<sup>+</sup> ⊕ *v*<sup>−</sup> *v*<sup>1</sup> = *v*<sup>+</sup> + *v*<sup>−</sup> − 0.5

where 0 ≤ *uβ*, *v<sup>β</sup>* ≤ 1, *β* ∈ {+, −, 0, 1}, ⊕ : Asynchronous addition, + : Synchronous

For each *ME*(*J*, *P*˜)=(*p*˜*i*(*J*, *X*), *p*˜*j*(*J*, *X*)) we can note that it has a position on a unit square

Let *P*˜ = {*p*˜+, *p*˜−} locate a pair of measurements, the IP module projects two measurements and its weight into four conditions in different symmetric properties to form two groups of

> *u*˜ = (*u*˜+, *u*˜−, *u*˜0, *u*˜1) = {*u*˜*β*} *v*˜ = (*v*˜+, *v*˜−, *v*˜0, *v*˜1) = {*v*˜*β*} *β* ∈ {+, −, 0, 1}

*3.5.3. Case B: Conditional Probability Interactive Projections*

Using *P*˜ = {*p*˜+, *p*˜−}, a pair of vectors {*u*˜, *v*˜} are formulated:

 

For the four projections in a square area, the following equations are required.

*u*˜+ = *p*˜+ *u*˜− = *p*˜− *u*˜0 = *u*˜<sup>+</sup> ⊕ *u*˜<sup>−</sup> *<sup>u</sup>*˜1 <sup>=</sup> *<sup>u</sup>*˜++*u*˜<sup>−</sup> 2 *<sup>v</sup>*˜<sup>+</sup> <sup>=</sup> <sup>1</sup>+*p*˜<sup>+</sup> 2 *<sup>v</sup>*˜<sup>−</sup> <sup>=</sup> <sup>1</sup>−*p*˜<sup>−</sup> 2 *v*˜0 = *v*˜<sup>+</sup> ⊕ *v*˜<sup>−</sup> *v*˜1 = *v*˜<sup>+</sup> + *v*˜<sup>−</sup> − 0.5

 

eight weights as interactive projections.

Under Symmetry/Anti-symmetry and Synchronous/Asynchronous conditions, one ME corresponds to eight interactive projections to express their selected characteristics.

**Proposition 3.3:** Two types of distinguished projections can be identified under symmetry/anti-symmetry conditions for each ME.

**Proof:** For two projection vectors {*u*, *v*}, we can note that *u* represents a symmetry condition and *v* represents an anti-symmetry condition.

**Proposition 3.4:** Synchronous and Asynchronous conditions lead to significant different output results.

**Proof:** In a synchronous operation, only one unit weight is output as {*u*1, *v*1}. However, in an asynchronous operation, two positions may be seen to have a combined weight {*u*0, *v*0}. 

**Proposition 3.5:** Other projections are simple ones corresponding to relevant measurement projections.

**Proof:** Other output results are {*u*+, *u*−, *v*+, *v*−}, each parameter is only dependent on one measurement to get a similar distribution from a projection viewpoint. There is no real interactive activity involved.

**Proposition 3.6:** If two probability measurements are required to satisfy *pi* + *pj* ≤ 1, then their symmetry interactive projection result is equal to *pi* + *pj*.

**Proof:** Merged result is in the [0, 1] region.

**Proposition 3.7:** If two probability measurements independently have 0 ≤ *pi*, *pj* ≤ 1, then their symmetry interactive projection is (*pi* + *pj*)/2.

**Proof:** Under this condition, merged results are in the [0, 1] region.

**Proposition 3.8** From a selected ME, eight interactive projections can be formulated.

**Proof:** By Propositions 3.3-3.7.

Under different conditions, one pair of probability measurements can be interactively projected into eight distinct results. However, from a variant viewpoint, it is not sufficient for a serious analysis to use only a single set of measurements from a ME, further extensions are required.

Using equation *CE*(*J*), a canonical ensemble of variant phase space is produced. Each non-zero position has a numeric weight as a value to indicate numbers of MEs collected

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 135

**Proposition 4.1:** Under Case A conditions, *O*(*N*2/2) points may be identified on a *CEL*(*J*)

**Proof:** For each probability measurement, *N* + 1 values may be distinguished; points are located in a triangular area and a total of (*N* + 1)*N*/2 points may be distinguished.

**Proposition 4.2:** Under Case B conditions, *O*(*N*) − *O*(*N*2) points may be distinguished on a

**Proof:** For each probability measurement, *N* + 1 values may be distinguished; points are located in a square area and (*N* + 1)<sup>2</sup> points may be distinguished for four square distributions and *N* + 1 points may be distinguished for two diagonal line distributions.

**Proposition 4.3:** In Case A or Case B, values of all possible points of *CE*(*J*) collected are

**Proposition 4.4:** For a given *SM* condition, *CE*(*J*) is a statistical canonical ensemble on

**Proof:** For a given SM condition, a *CE*(*J*) distribution is independent of special sequences of collection. Its detailed configuration is relevant to {*n*, *N*} and *SM* respectively. All valid

Under this organization, each *CE*(*J*) has a fixed plane lattice with a distinct distribution. This

In an IM module, all possible IP projections of either {*u*, *v*} or {*u*˜, *v*˜} are collected. Each

The IM module provides a statistical means to accumulate all possible vectors of *N* bits for a selected signal and generate a histogram. Eight signals correspond to eight histograms respectively. Among these, four histograms exhibit properties of symmetry and another four

For a function *J*, all measurement signals are collected from the *IP* and the relevant histogram

Using *u* and *v* signals, each *uβ* or *vβ* determines a fixed position in the relevant histogram to make vector *X* on a position. After completing 2*<sup>N</sup>* data sequences, eight

**Proof:** This is the total number of vectors that may be distinguished for ∀*X*.

in a position.

*CEL*(*J*) lattice.

equal to 2*N*.

variant phase space.

**4.2. IM interactive map**

*4.2.1. Statistical distributions*

positions can be statistically generated.

invariant property is useful for our further explorations.

projection corresponds to a specific IM distribution.

histograms exhibit properties of anti-symmetry.

represents a complete statistical distribution as an IP map.

lattice.

*4.1.1. Key Properties in CE*

To distinguish among different measurements in interactive projections, four types of measurements are defined as *Left, Right, D-P and D-W* , where {*u*+, *v*+, *u*˜+, *v*˜+} are *Left* measurements, {*u*−, *v*−, *u*˜−, *v*˜−} are *Right* measurements, {*u*0, *v*0, *u*˜0, *v*˜0} are *D-P* measurements and {*u*1, *v*1, *u*˜1, *v*˜1} are *D-W* measurements respectively.

#### **4. CEIM canonical ensemble and interactive maps**

It is a basic step to generate a micro ensemble and eight interactive projections on variant phase space. For a given function *J*, it is necessary to determine the specific positions of all possible vectors of ∀*X* to form a canonical ensemble on variant phase space.

The CEIM component is composed of two modules: the CE Canonical Ensemble and the IM Interactive Map.

The CE module collects all possible MEs into a canonical ensemble. In addition, the IM module makes relevant interactive projections via IP's outputs to generate a list of interactive distributions in the relevant maps as output results.

#### **4.1. CE canonical ensemble**

In the CE module, all the MEs are collected to form a CE in variant phase space according to the following equations.

For a function *J* and all 2*<sup>N</sup>* vectors of ∀*X*, let *CEL*(*J*, *X*) be a point of *CEL*(*J*) on a plane lattice

$$\text{CEL}(f, X) = \begin{cases} \, ^\prime \boldsymbol{T} \, ^\prime \boldsymbol{M} \text{E}(f, X) = \mathcal{P} | \vec{\mathcal{P}} \\ \, ^\prime \boldsymbol{F} \text{, otherwise} \end{cases} \tag{18}$$

$$\text{CEL}(f) = \bigcup\_{\forall X} \text{CEL}(f, X) \tag{19}$$

Applying the equation for *CEL*(*J*), a canonical lattice *CEL* can be established to indicate a specific distribution from a logic viewpoint.

Since different *CEL*(*J*, *X*) may have the same position, let *CE*(*J*, *X*) be a point of *CE*(*J*) in a canonical ensemble

$$CE(f, X) = \begin{cases} 1, \text{CEL}(f, X) = T \\ 0, \text{Otherwise} \end{cases} \tag{20}$$

$$CE(f) = \sum\_{\forall X} CE(f, X) \tag{21}$$

Using equation *CE*(*J*), a canonical ensemble of variant phase space is produced. Each non-zero position has a numeric weight as a value to indicate numbers of MEs collected in a position.

#### *4.1.1. Key Properties in CE*

22 Cellular Automata

are required.

Interactive Map.

**4.1. CE canonical ensemble**

the following equations.

canonical ensemble

lattice

Under different conditions, one pair of probability measurements can be interactively projected into eight distinct results. However, from a variant viewpoint, it is not sufficient for a serious analysis to use only a single set of measurements from a ME, further extensions

To distinguish among different measurements in interactive projections, four types of measurements are defined as *Left, Right, D-P and D-W* , where {*u*+, *v*+, *u*˜+, *v*˜+} are *Left* measurements, {*u*−, *v*−, *u*˜−, *v*˜−} are *Right* measurements, {*u*0, *v*0, *u*˜0, *v*˜0} are *D-P*

It is a basic step to generate a micro ensemble and eight interactive projections on variant phase space. For a given function *J*, it is necessary to determine the specific positions of all

The CEIM component is composed of two modules: the CE Canonical Ensemble and the IM

The CE module collects all possible MEs into a canonical ensemble. In addition, the IM module makes relevant interactive projections via IP's outputs to generate a list of interactive

In the CE module, all the MEs are collected to form a CE in variant phase space according to

For a function *J* and all 2*<sup>N</sup>* vectors of ∀*X*, let *CEL*(*J*, *X*) be a point of *CEL*(*J*) on a plane

∀*X*

Applying the equation for *CEL*(*J*), a canonical lattice *CEL* can be established to indicate a

Since different *CEL*(*J*, *X*) may have the same position, let *CE*(*J*, *X*) be a point of *CE*(*J*) in a

∀*X*

1, *CEL*(*J*, *X*) = *T*

*T*, *ME*(*J*, *X*) = *P*|*P*˜

*<sup>F</sup>*, Otherwise (18)

*CEL*(*J*, *X*) (19)

0, Otherwise (20)

*CE*(*J*, *X*) (21)

measurements and {*u*1, *v*1, *u*˜1, *v*˜1} are *D-W* measurements respectively.

possible vectors of ∀*X* to form a canonical ensemble on variant phase space.

*CEL*(*J*, *X*) =

*CE*(*J*, *X*) =

*CEL*(*J*) =

*CE*(*J*) = ∑

**4. CEIM canonical ensemble and interactive maps**

distributions in the relevant maps as output results.

specific distribution from a logic viewpoint.

**Proposition 4.1:** Under Case A conditions, *O*(*N*2/2) points may be identified on a *CEL*(*J*) lattice.

**Proof:** For each probability measurement, *N* + 1 values may be distinguished; points are located in a triangular area and a total of (*N* + 1)*N*/2 points may be distinguished.

**Proposition 4.2:** Under Case B conditions, *O*(*N*) − *O*(*N*2) points may be distinguished on a *CEL*(*J*) lattice.

**Proof:** For each probability measurement, *N* + 1 values may be distinguished; points are located in a square area and (*N* + 1)<sup>2</sup> points may be distinguished for four square distributions and *N* + 1 points may be distinguished for two diagonal line distributions. 

**Proposition 4.3:** In Case A or Case B, values of all possible points of *CE*(*J*) collected are equal to 2*N*.

**Proof:** This is the total number of vectors that may be distinguished for ∀*X*.

**Proposition 4.4:** For a given *SM* condition, *CE*(*J*) is a statistical canonical ensemble on variant phase space.

**Proof:** For a given SM condition, a *CE*(*J*) distribution is independent of special sequences of collection. Its detailed configuration is relevant to {*n*, *N*} and *SM* respectively. All valid positions can be statistically generated.

Under this organization, each *CE*(*J*) has a fixed plane lattice with a distinct distribution. This invariant property is useful for our further explorations.

#### **4.2. IM interactive map**

In an IM module, all possible IP projections of either {*u*, *v*} or {*u*˜, *v*˜} are collected. Each projection corresponds to a specific IM distribution.

The IM module provides a statistical means to accumulate all possible vectors of *N* bits for a selected signal and generate a histogram. Eight signals correspond to eight histograms respectively. Among these, four histograms exhibit properties of symmetry and another four histograms exhibit properties of anti-symmetry.

#### *4.2.1. Statistical distributions*

For a function *J*, all measurement signals are collected from the *IP* and the relevant histogram represents a complete statistical distribution as an IP map.

Using *u* and *v* signals, each *uβ* or *vβ* determines a fixed position in the relevant histogram to make vector *X* on a position. After completing 2*<sup>N</sup>* data sequences, eight symmetry/anti-symmetry histograms of {*H*(*uβ*|*J*)}, {*H*(*vβ*|*J*)}|{*H*(*u*˜*β*|*J*)}, {*H*(*v*˜*β*|*J*)} are generated.

Under the multiple probability condition, *β* ∈ {+, −, 0, 1}

$$\begin{cases} H(u\_{\beta}|I) = \sum\_{\forall X \in B\_2^N} H(u\_{\beta}|f(X)) \\ H(v\_{\beta}|f) = \sum\_{\forall X \in B\_2^N} H(v\_{\beta}|f(X))\_{\prime} \end{cases} \tag{22}$$

⇒

CEL *J* = 3 in MPS Eight Interactive Maps (I) Representative patterns of Histograms for function *J* (a-d) symmetric cases; (e-h) antisymmetric cases

(a) Left (b) Right

(c) D-P (d) D-W

(e) Left (f) Right

(g) D-P (h) D-W (II) *N* = {12}, *n* = 2, *J* = 3 Two groups of results in eight histograms **Figure 3.** (I-II) *N* = {12}, *n* = 2, *J* = 3 Simulation results ; (I) Representative Patterns for *PH* (*u*+|*J*) = *PH* (*u*−|*J*) and *PH* (*v*+|*J*) = *PH* (1 − *v*−|*J*) conditions; (II) *N* = {12}, *n* = 2, *J* = 3 Two groups of eight interactive histograms on MPS

*PH*(*u*+|*J*) *PH*(*u*−|*J*) (a) Left (b) Right *PH*(*u*0|*J*) *PH*(*u*1|*J*) (c) D-P (d) D-W *PH*(*v*+|*J*) *PH*(*v*−|*J*) (e) Left (f) Right *PH*(*v*0|*J*) *PH*(*v*1|*J*) (g) D-P (h) D-W

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 137

Under the conditional probability condition, *β* ∈ {+, −, 0, 1}

$$\begin{cases} H(\mathfrak{u}\_{\mathfrak{\beta}}|I) = \sum\_{\forall X \in B\_2^N} H(\mathfrak{u}\_{\mathfrak{\beta}}|I(X)) \\ H(\mathfrak{v}\_{\mathfrak{\beta}}|I) = \sum\_{\forall X \in B\_2^N} H(\mathfrak{v}\_{\mathfrak{\beta}}|I(X)), I \in B\_2^{2^u} \end{cases} \tag{23}$$

#### *4.2.2. Normalized probability histograms*

Let |*H*(..)| denote the total number in the histogram *H*(..), a normalized probability histogram (*PH*(..)) can be expressed as

$$\begin{cases} P\_H(u\_\beta|J) = \frac{H(u\_\beta|I)}{|H(u\_\beta|I)|}\\ P\_H(v\_\beta|J) = \frac{H(v\_\beta|I)}{|H(v\_\beta|I)|}, J \in \mathbb{B}\_2^{2^u} \\ P\_H(\vec{u}\_\beta|J) = \frac{H(\vec{u}\_\beta|I)}{|H(\vec{u}\_\beta|I)|}\\ P\_H(\vec{v}\_\beta|J) = \frac{H(\vec{v}\_\beta|I)}{|H(\vec{v}\_\beta|I)|}, J \in \mathbb{B}\_2^{2^u} \end{cases} \tag{24}$$

Here, all interactive maps are also restricted in [0, 1] <sup>2</sup> areas respectively.

Distributions are dependant on the data set as a whole and are not sensitive to varying under special sequences. Under this condition, when the data set has been exhaustively listed, then the same distributions are always linked to the given signal set.

Let *IM*(*J*) = {*PH*(*u*|*J*), *PH*(*v*|*J*)} or {*PH*(*u*˜|*J*), *PH*(*v*˜|*J*)} be the output results of an IM module. Then the eight histogram distributions provide invariant spectrums to represent properties among different interactive conditions.

Using such descriptions, the output results of the CEIM component are {*CE*(*J*), *IM*(*J*)}.

From a given function, a set of histograms can be generated as a group of eight probability histograms in variant phase space. Two groups of sixteen histograms are required. Sample cases are shown in Figures 3-4(I-II).

24 Cellular Automata

generated.

symmetry/anti-symmetry histograms of {*H*(*uβ*|*J*)}, {*H*(*vβ*|*J*)}|{*H*(*u*˜*β*|*J*)}, {*H*(*v*˜*β*|*J*)} are

<sup>2</sup> *<sup>H</sup>*(*uβ*|*J*(*X*))

<sup>2</sup> *<sup>H</sup>*(*u*˜*β*|*J*(*X*))

Let |*H*(..)| denote the total number in the histogram *H*(..), a normalized probability





Distributions are dependant on the data set as a whole and are not sensitive to varying under special sequences. Under this condition, when the data set has been exhaustively listed, then

Let *IM*(*J*) = {*PH*(*u*|*J*), *PH*(*v*|*J*)} or {*PH*(*u*˜|*J*), *PH*(*v*˜|*J*)} be the output results of an IM module. Then the eight histogram distributions provide invariant spectrums to represent

From a given function, a set of histograms can be generated as a group of eight probability histograms in variant phase space. Two groups of sixteen histograms are required. Sample

Using such descriptions, the output results of the CEIM component are {*CE*(*J*), *IM*(*J*)}.

*PH*(*uβ*|*J*) = *<sup>H</sup>*(*uβ*|*J*)

*PH*(*vβ*|*J*) = *<sup>H</sup>*(*vβ*|*J*)

*PH*(*u*˜*β*|*J*) = *<sup>H</sup>*(*u*˜*β*|*J*)

*PH*(*v*˜*β*|*J*) = *<sup>H</sup>*(*v*˜*β*|*J*)

<sup>2</sup> *<sup>H</sup>*(*vβ*|*J*(*X*)), *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

<sup>2</sup> *<sup>H</sup>*(*v*˜*β*|*J*(*X*)), *<sup>J</sup>* <sup>∈</sup> *<sup>B</sup>*2*<sup>n</sup>*

, *J* ∈ *B*2*<sup>n</sup>* 2

, *J* ∈ *B*2*<sup>n</sup>* 2

<sup>2</sup> areas respectively.

2

2

(22)

(23)

(24)

Under the multiple probability condition, *β* ∈ {+, −, 0, 1}

Under the conditional probability condition, *β* ∈ {+, −, 0, 1}

 

the same distributions are always linked to the given signal set.

Here, all interactive maps are also restricted in [0, 1]

properties among different interactive conditions.

cases are shown in Figures 3-4(I-II).

*4.2.2. Normalized probability histograms*

histogram (*PH*(..)) can be expressed as

� *<sup>H</sup>*(*uβ*|*J*) = <sup>∑</sup>∀*X*∈*B<sup>N</sup>*

� *<sup>H</sup>*(*u*˜*β*|*J*) = <sup>∑</sup>∀*X*∈*B<sup>N</sup>*

*<sup>H</sup>*(*v*˜*β*|*J*) = <sup>∑</sup>∀*X*∈*B<sup>N</sup>*

*<sup>H</sup>*(*vβ*|*J*) = <sup>∑</sup>∀*X*∈*B<sup>N</sup>*

**Figure 3.** (I-II) *N* = {12}, *n* = 2, *J* = 3 Simulation results ; (I) Representative Patterns for *PH* (*u*+|*J*) = *PH* (*u*−|*J*) and *PH* (*v*+|*J*) = *PH* (1 − *v*−|*J*) conditions; (II) *N* = {12}, *n* = 2, *J* = 3 Two groups of eight interactive histograms on MPS

**5. GEIM global ensemble and interactive map matrices**

**5.1. SCEIM sets of canonical ensembles and interactive maps**

*SCE* = {∀*J*, *CE*(*J*)|*J* ∈ *B*2*<sup>n</sup>*

Matrices.

configuration of output matrices.

under exhaustive conditions.

detailed configuration for each matrix.

*5.2.1. Global Matrix Representations*

*5.2.2. The Matrix and Its elements*

*5.2.3. Representative patterns of matrices*

of 22*<sup>n</sup>*

all 22*<sup>n</sup>*

The GEIM component is composed of two modules: SCEIM Sets of CE&IM, and CIM CE&IM

{*CE*(*J*), *IM*(*J*)} and ∀*J* are put in the SCEIM module to generate sets of CEs and IMs on each given function exhaustively. All generated CEs and IMs are organized by the CIM module under the FC condition in which a special variant coding scheme is applied for a global

The SCEIM module produces {*SCE*, *SIM*} composed of all possible CE and IM sets of ∀*J*

*SIM* = {∀*J*, {*PH*(*u*|*J*), *PH*(*v*|*J*)} or {*PH*(*u*˜|*J*), *PH*(*v*˜|*J*)}|*<sup>J</sup>* ∈ *<sup>B</sup>*2*<sup>n</sup>*

In addition to using {*SCE*, *SIM*} as inputs, the *FC* also inputs a code scheme to determine a

In the CIM module, {*SCE*, *SIM*} inputs have nine sets of CEs and IMs. Each set is composed

*<sup>M</sup>*�*J*<sup>1</sup>|*J*<sup>0</sup>�(*uβ*|*J*) = *PH*(*uβ*|*J*) *<sup>M</sup>*�*J*<sup>1</sup>|*J*<sup>0</sup>�(*vβ*|*J*) = *PH*(*vβ*|*J*) *J* ∈ *B*2*<sup>n</sup>*

Four cases of FC codes are selected for illustrations in this Chapter. Further discussion on

the details of variant coding scheme has been previously published in [40, 44]. For example, four sample cases are shown in Figure 5 under relevant conditions.

<sup>2</sup> ; *<sup>J</sup>*1, *<sup>J</sup>*<sup>0</sup> <sup>∈</sup> *<sup>B</sup>*2*n*−<sup>1</sup>

2

elements and each element is a histogram or a plane lattice. The CIM module arranges

<sup>2</sup> } (25)

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 139

(26)

2 }

Meanwhile, the SCE and the SIM provide output results to the CIM module.

**5.2. CIM canonical ensemble and interactive map matrices**

elements generated as a matrix by a given FC code scheme.

For a given FC scheme, let *FC*(*J*) = �*J*1|*J*0�, each element

 

**Figure 4.** (I-II) *N* = {12, }, *n* = 2, *J* = 3 Simulation results; (I) Representative Patterns for *PH* (*u*˜+|*J*) = *PH* (*u*˜−|*J*) and *PH* (*v*˜+|*J*) = *PH* (1 − *v*˜−|*J*) conditions; (II) *N* = {12}, *n* = 2, *J* = 3 Two groups of eight interactive histograms on CPS

#### **5. GEIM global ensemble and interactive map matrices**

The GEIM component is composed of two modules: SCEIM Sets of CE&IM, and CIM CE&IM Matrices.

{*CE*(*J*), *IM*(*J*)} and ∀*J* are put in the SCEIM module to generate sets of CEs and IMs on each given function exhaustively. All generated CEs and IMs are organized by the CIM module under the FC condition in which a special variant coding scheme is applied for a global configuration of output matrices.

#### **5.1. SCEIM sets of canonical ensembles and interactive maps**

The SCEIM module produces {*SCE*, *SIM*} composed of all possible CE and IM sets of ∀*J* under exhaustive conditions.

$$\begin{array}{l} \text{SCE} = \{ \forall f, \text{CE}(f) | f \in B\_2^{2^\*} \} \\ \text{SIM} = \{ \forall | f, \{ P\_H(u | f), P\_H(v | f) \} \text{ or } \{ P\_H(\tilde{u} | f), P\_H(\tilde{v} | f) \} | f \in B\_2^{2^\*} \} \end{array} \tag{25}$$

Meanwhile, the SCE and the SIM provide output results to the CIM module.

#### **5.2. CIM canonical ensemble and interactive map matrices**

In addition to using {*SCE*, *SIM*} as inputs, the *FC* also inputs a code scheme to determine a detailed configuration for each matrix.

#### *5.2.1. Global Matrix Representations*

26 Cellular Automata

⇒

CEL *J* = 3 in CPS Eight Interactive Maps (I) Representative patterns of Histograms for function *J* (a-d) symmetric cases; (e-h) antisymmetric cases

(a) Left (b) Right

(c) D-P (d) D-W

(e) Left (f) Right

(g) D-P (h) D-W (II) *N* = {12}, *n* = 2, *J* = 3 Two groups of results in eight histograms **Figure 4.** (I-II) *N* = {12, }, *n* = 2, *J* = 3 Simulation results; (I) Representative Patterns for *PH* (*u*˜+|*J*) = *PH* (*u*˜−|*J*) and *PH* (*v*˜+|*J*) = *PH* (1 − *v*˜−|*J*) conditions; (II) *N* = {12}, *n* = 2, *J* = 3 Two groups of eight interactive histograms on CPS

*PH*(*u*˜+|*J*) *PH*(*u*˜−|*J*) (a) Left (b) Right *PH*(*u*˜0|*J*) *PH*(*u*˜1|*J*) (c) D-P (d) D-W *PH*(*v*˜+|*J*) *PH*(*v*˜−|*J*) (e) Left (f) Right *PH*(*v*˜0|*J*) *PH*(*v*˜1|*J*) (g) D-P (h) D-W

> In the CIM module, {*SCE*, *SIM*} inputs have nine sets of CEs and IMs. Each set is composed of 22*<sup>n</sup>* elements and each element is a histogram or a plane lattice. The CIM module arranges all 22*<sup>n</sup>* elements generated as a matrix by a given FC code scheme.

#### *5.2.2. The Matrix and Its elements*

For a given FC scheme, let *FC*(*J*) = �*J*1|*J*0�, each element

$$\begin{cases} M\_{\langle f^1 | f^0 \rangle} (u\_\beta | f) = P\_H (u\_\beta | f) \\ M\_{\langle f^1 | f^0 \rangle} (v\_\beta | f) = P\_H (v\_\beta | f) \\ \qquad J \in B\_2^{2^n}; J^1, J^0 \in B\_2^{2^{n-1}} \end{cases} \tag{26}$$

#### *5.2.3. Representative patterns of matrices*

Four cases of FC codes are selected for illustrations in this Chapter. Further discussion on the details of variant coding scheme has been previously published in [40, 44].

For example, four sample cases are shown in Figure 5 under relevant conditions.


2*<sup>N</sup>* bits

 

 2*<sup>n</sup>* bits *GCEIM* ⇒

**Figure 7.** Illustrations of Representative Model of the VPS for GCEIM and GEM results

CE(J) in MPS CE(J) in CPS

... ... Eight IM(J) Eight IM(J) ... ...

⇒ *GEM*

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 141

IM(J) in MPS IM(J) in CPS

CEM in MPS CEM in CPS

IMM in MPS IMM in CPS

... ... Eight IM Matrices in MPS Eight IM Matrices in CPS ... ...

IMM in MPS IMM in CPS

0 ... 0 ... 1 ... 1

0 ... 0 ... 1 ... 1

∀*X* ∈

*J* ∈ 

*GEM* ⇒

∀*J* →

 

**Figure 5.** (a-c) Four Cases of Matrix configurations for *n* = 2 on FC (a) Case 1. SL code (b) Case 2. W code (c) Case 3. F code (d) Case 4. C code

**Case 1:** *FC* = {*n* = 2, *P* = (3210)} a SL code; **Case 2:** *FC* = {*n* = 2, *P* = (2103)} a W code; **Case 3:** *FC* = {*n* = 2, *P* = (3201)} a F code; **Case 4:** *FC* = {*n* = 2, *P* = (3102)} a C code.

Under each condition, each FC code is a special configuration to make sixteen elements arranged as a 4 × 4 matrix.

For the matrices in this chapter, four configurations are applied to construct sample matrices with elements arranged for illustration purposes.

#### **6. Representation model**

Figure 6 presents a graphical summary of the above. Further representations are offered in Figure 7 to show the main steps in creating a CE in the MPS or CPS and IMs relevant to global CEM and IMM procedures.

$$\begin{aligned} \forall X \in \mathcal{B}\_2^N &\to \begin{bmatrix} \text{Generating Canada} \\ \text{Ensemble } \& \\ \text{Ensemble } \& \\ \text{Interactive Maps} \end{bmatrix} &\to \begin{Bmatrix} \text{CE}(\!\!/ ), IM(\!\!/ ) \end{Bmatrix} &\to \begin{Bmatrix} \text{Global} \\ \text{Ensemble} \\ \text{Ensemble} \\ \text{Matrices} \\ \text{GEM} \end{Bmatrix} &\to \begin{Bmatrix} \text{Mobile} \\ \text{Matrices} \\ \text{GEM} \end{Bmatrix} &\to \begin{Bmatrix} \text{MARM} \end{Bmatrix} \end{aligned}$$

**Figure 6.** Diagrammatical Representation of VPS Model

#### **7. Symbolic representations on selected cases**

Using a representational model, for a given condition, there are two sets of CEM in both MPS and CPS. Each set contains a CEM and eight IMMs. Since each matrix contains 22*<sup>n</sup>* elements, the existence of so many possible configurations adds to the difficulties in reaching a satisfactory understanding of the data sets. In this section, symbolic representations are applied to show more clearly the essential symmetric properties of various matrices. Using variant logic, the following equations can be established for an *n* = 2 condition to apply (*a*, *b*, *c*, *d*)=(10, 8, 2, 0) and (*a*˜, ˜ *b*, *c*˜, ˜*d*)=(10, 14, 11, 15) for each meta function.

**Figure 7.** Illustrations of Representative Model of the VPS for GCEIM and GEM results

28 Cellular Automata

(d) Case 4. C code

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

arranged as a 4 × 4 matrix.

**6. Representation model**

global CEM and IMM procedures.

**Figure 6.** Diagrammatical Representation of VPS Model

<sup>2</sup> <sup>→</sup>

(*a*, *b*, *c*, *d*)=(10, 8, 2, 0) and (*a*˜, ˜

∀*X* ∈ *B<sup>N</sup>*

*J* ∈ *B*2*<sup>n</sup>* <sup>2</sup> <sup>→</sup>

**Case 1:** *FC* = {*n* = 2, *P* = (3210)} a SL code; **Case 2:** *FC* = {*n* = 2, *P* = (2103)} a W code; **Case 3:** *FC* = {*n* = 2, *P* = (3201)} a F code; **Case 4:** *FC* = {*n* = 2, *P* = (3102)} a C code.

with elements arranged for illustration purposes.

Generating Canonical Ensemble & Interactive Maps GCEIM

**7. Symbolic representations on selected cases**

0 8 1 9 2 10 3 11 4 12 5 13 6 14 7 15

(a) SL code (b) W code (c) F code (d) C code

**Figure 5.** (a-c) Four Cases of Matrix configurations for *n* = 2 on FC (a) Case 1. SL code (b) Case 2. W code (c) Case 3. F code

Under each condition, each FC code is a special configuration to make sixteen elements

For the matrices in this chapter, four configurations are applied to construct sample matrices

Figure 6 presents a graphical summary of the above. Further representations are offered in Figure 7 to show the main steps in creating a CE in the MPS or CPS and IMs relevant to

Using a representational model, for a given condition, there are two sets of CEM in both MPS and CPS. Each set contains a CEM and eight IMMs. Since each matrix contains 22*<sup>n</sup>* elements, the existence of so many possible configurations adds to the difficulties in reaching a satisfactory understanding of the data sets. In this section, symbolic representations are applied to show more clearly the essential symmetric properties of various matrices. Using variant logic, the following equations can be established for an *n* = 2 condition to apply

→ {*CE*(*J*), *IM*(*J*)} →

*b*, *c*˜, ˜*d*)=(10, 14, 11, 15) for each meta function.

∀*J* →

Global Ensemble Matrices GEM

→ *CEM*

→ *IMM*

0 2 1 3 4 6 5 7 8 10 9 11 12 14 13 15

0 4 1 5 2 6 3 7 8 12 9 13 10 14 11 15

$$\begin{array}{llll} 0 = \langle 0|10 \rangle = \langle d|\overline{a} \rangle = d; & 1 = \langle 0|11 \rangle = \langle d|\overline{c} \rangle;\\ 2 = \langle 2|10 \rangle = \langle c|\overline{a} \rangle = c; & 3 = \langle 2|11 \rangle = \langle c|\overline{c} \rangle;\\ 4 = \langle 0|14 \rangle = \langle d|\overline{b} \rangle; & 5 = \langle 0|15 \rangle = \langle d|\overline{d} \rangle;\\ 6 = \langle 2|14 \rangle = \langle c|\overline{b} \rangle; & 7 = \langle 2|15 \rangle = \langle c|\overline{d} \rangle;\\ 8 = \langle 8|10 \rangle = \langle b|\overline{a} \rangle = b; & 9 = \langle 8|11 \rangle = \langle b|\overline{c} \rangle;\\ 10 = \langle 10|10 \rangle = \langle a|\overline{a} \rangle = a = \overline{a}; & 11 = \langle 10|11 \rangle = \langle a|\overline{c} \rangle = \overline{c};\\ 12 = \langle 8|14 \rangle = \langle b|\overline{b} \rangle; & 13 = \langle 8|15 \rangle = \langle b|\overline{d} \rangle;\\ 14 = \langle 10|14 \rangle = \langle a|\overline{b} \rangle = \overline{b}; & 15 = \langle 10|15 \rangle = \langle a|\overline{d} \rangle = \overline{d}. \end{array} \tag{27}$$

$$
\begin{aligned}
\begin{pmatrix}
0 & 2 & 1 & 3 \\
4 & 6 & 5 & 7 \\
8 & 10 & 9 & 11 \\
12 & 14 & 13 & 15
\end{pmatrix} &= \left\langle \begin{pmatrix} 0 & 2 & 0 & 2 \\ 0 & 2 & 0 & 2 \\ 8 & 10 & 8 & 10 \\ 8 & 10 & 10 & 11 \\ 14 & 14 & 15 & 15
\end{pmatrix} \right\rangle \\ &= \left\langle \begin{pmatrix} d & c & d & c \\ d & c & d & c \\ b & a & b & a \\ b & a & b & a \end{pmatrix} \mid \begin{pmatrix} \bar{a} \ \bar{a} \ \bar{c} \ \bar{c} \\ \bar{b} \ \bar{b} \ \bar{d} \ \bar{d} \\ \bar{b} \ \bar{b} \ \bar{d} \ \bar{d}
\end{pmatrix} \right\rangle \\ &= \begin{pmatrix} \langle d | \bar{a} \rangle \langle c | \bar{a} \rangle \langle d | \bar{c} \rangle \langle c | \bar{c} \rangle \\ \langle d | \bar{b} \rangle \langle c | \bar{b} \rangle \langle d | \bar{d} \rangle \langle c | \bar{d} \rangle \\ \langle d | \bar{b} \rangle \langle a | \bar{b} \rangle \langle b | \bar{c} \rangle \langle a | \bar{c} \rangle \\ \langle b | \bar{b} \rangle \langle a | \bar{b} \rangle \langle b | \bar{d} \rangle \langle a | \bar{d} \rangle \\ \langle b | \bar{b} \rangle \langle a | \bar{b} \rangle \langle b | \bar{d} \rangle \langle a | \bar{d} \rangle \\ \langle \text{V-Run} \text{H-Run} \rangle \end{pmatrix},
\end{aligned} \tag{30}$$

$$
\begin{aligned}
\begin{pmatrix} 0 & 4 & 1 & 5 \\ 2 & 6 & 3 & 7 \\ 8 & 12 & 9 & 13 \\ 10 & 14 & 11 & 15 \end{pmatrix} &= \left\langle \begin{pmatrix} 0 & 0 & 0 & 0 \\ 2 & 2 & 2 & 2 \\ 8 & 8 & 8 & 8 \\ 10 & 10 & 10 & 10 \end{pmatrix} \mid \begin{pmatrix} 10 \ 14 \ 11 \ 15 \\ 10 \ 14 \ 11 \ 15 \\ 10 \ 14 \ 11 \ 15 \end{pmatrix} \right\rangle \\ &= \left\langle \begin{pmatrix} d \ d \ d \ d \ d \end{pmatrix} \mid \begin{pmatrix} \bar{a} \ \ \bar{b} \ \ \bar{c} \ \ \bar{d} \\ \bar{a} \ \ \bar{b} \ \ \bar{c} \ \bar{d} \\ \bar{a} \ \bar{b} \ \bar{c} \ \bar{d} \end{pmatrix} \right\rangle \\ &= \begin{pmatrix} \langle d | \bar{a} \rangle \ \langle d | \bar{b} \rangle \ \langle d | \bar{c} \rangle \ \langle d | \bar{c} \rangle \\ \langle c | \bar{a} \rangle \ \langle c | \bar{b} \rangle \ \langle c | \bar{d} \rangle \\ \langle b | \bar{a} \rangle \ \langle b | \bar{c} \rangle \ \langle b | \bar{c} \rangle \ \langle a | \bar{d} \rangle \\ \langle b | \bar{a} \rangle \ \langle b | \bar{b} \rangle \ \langle a | \bar{c} \rangle \ \langle a | \bar{d} \rangle \\ \langle a | \bar{a} \rangle \ \langle a | \bar{b} \rangle \ \langle a | \bar{c} \rangle \ \langle a | \bar{d} \rangle \\ \end{pmatrix} \end{aligned} \tag{31}$$

$$\begin{array}{l} 0 : 15 = \langle d|\vec{a}\rangle : \langle a|\vec{d}\rangle; \\ 1 : 7 = \langle d|\vec{c}\rangle : \langle c|\vec{d}\rangle; \\ 2 : 11 = \langle c|\vec{a}\rangle : \langle a|\vec{c}\rangle; \\ 4 : 13 = \langle d|\vec{b}\rangle : \langle b|\vec{d}\rangle; \\ 6 : 9 = \langle c|\vec{b}\rangle : \langle b|\vec{c}\rangle; \\ 8 : 14 = \langle b|\vec{a}\rangle : \langle a|\vec{b}\rangle. \end{array} \tag{32}$$

Six pairs {1 : 8, 2 : 4, 3 : 12, 5 : 10, 7 : 14, 11 : 13} of distributions may have anti-symmetry properties

$$\begin{array}{l} 1 : 8 &= \langle d|\vec{c}\rangle : \langle b|\vec{a}\rangle; \\ 2 : 4 &= \langle c|\vec{a}\rangle : \langle d|\vec{b}\rangle; \\ 3 : 12 &= \langle c|\vec{c}\rangle : \langle b|\vec{b}\rangle; \\ 5 : 10 &= \langle d|\vec{d}\rangle : \langle a|\vec{a}\rangle; \\ 7 : 14 &= \langle c|\vec{d}\rangle : \langle a|\vec{b}\rangle; \\ 11 : 13 &= \langle a|\vec{c}\rangle : \langle b|\vec{d}\rangle. \end{array} \tag{33}$$

Two pairs {3 : 12, 5 : 10} of distributions may have self-conjugate properties with both symmetry and anti-symmetry properties.

$$\begin{array}{l} \mathbf{3}: 12 \ = \langle \mathbf{c} | \vec{\varepsilon} \rangle : \langle \mathbf{b} | \vec{b} \rangle; \\ \mathbf{5}: 10 \ = \langle d | \vec{d} \rangle : \langle a | \vec{a} \rangle. \end{array} \tag{34}$$

Four pairs {0 : 15, 3 : 12, 5 : 10, 6 : 9} of distributions may have special properties.

$$\begin{array}{l} 0 : 15 = \langle d|\vec{a}\rangle : \langle a|\vec{d}\rangle; \\ 3 : 12 = \langle c|\vec{c}\rangle : \langle b|\vec{b}\rangle; \\ 5 : 10 = \langle d|\vec{d}\rangle : \langle a|\vec{a}\rangle; \\ 6 : 9 : = \langle c|\vec{b}\rangle : \langle b|\vec{c}\rangle. \end{array} \tag{35}$$

(a) SL in MPS

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 145

(b) W in MPS

Regions of Measurements in MPS can be illustrated as

$$\begin{array}{ccccc}\text{MPS}: \begin{pmatrix} 0 & 5\\ & \cdots\\ & \cdots\\ 10 & 15 \end{pmatrix} \Rightarrow \begin{pmatrix} \begin{bmatrix} 1,0\end{bmatrix} & \begin{pmatrix} -,-\end{pmatrix} & \cdots & \begin{bmatrix} 1/2,1/2 \end{bmatrix} \\\end{pmatrix} \\\ \text{10} \end{array} \begin{array}{ccccc}\text{-(} & \text{0}\text{)} & \begin{array}{ccccc} \text{-(} & \text{1}\text{/} \text{2},1/\text{2} \text{)} \\\hline \text{(} & \text{0}\text{)} & \cdots & \text{(} & \text{-}/\text{-} \text{)} \\\text{(} & \text{0}\text{)} & \cdots & \text{(} & \text{0}\text{-} \text{)} \end{array} \end{array} \tag{36}$$

Regions of Measurements in CPS can be illustrated as

$$\text{CPS}: \begin{pmatrix} 0 & 5 \\ & \cdots \\ & & \cdots \\ 10 & & 15 \end{pmatrix} \Rightarrow \begin{pmatrix} [1,0] & (1,-) & \cdots & [1,1] \\ (-,0) & \cdots & & \cdots \\ \cdots & & \cdots & (-,1) \\ (0,0) & \cdots & (0,-) & [0,1] \end{pmatrix} \tag{37}$$

#### **8. Sample results**

#### **8.1. CEM groups**

Using *n* = 2 configurations, relevant CEMs on either MPS or CPS are shown in Figure 8(a-h).

32 Cellular Automata

properties

symmetry and anti-symmetry properties.

Regions of Measurements in MPS can be illustrated as

Regions of Measurements in CPS can be illustrated as

 ⇒

 ⇒

*MPS* :

*CPS* :

**8. Sample results**

**8.1. CEM groups**

Six pairs {1 : 8, 2 : 4, 3 : 12, 5 : 10, 7 : 14, 11 : 13} of distributions may have anti-symmetry

1:8 = �*d*|*c*˜� : �*b*|*a*˜�; 2:4 = �*c*|*a*˜� : �*d*|˜

3 : 12 = �*c*|*c*˜� : �*b*|˜

5 : 10 = �*d*| ˜*d*� : �*a*|*a*˜�; 7 : 14 = �*c*| ˜*d*� : �*a*|˜

11 : 13 = �*a*|*c*˜� : �*b*| ˜*d*�.

Two pairs {3 : 12, 5 : 10} of distributions may have self-conjugate properties with both

3 : 12 = �*c*|*c*˜� : �*b*|˜

0 : 15 = �*d*|*a*˜� : �*a*| ˜*d*�; 3 : 12 = �*c*|*c*˜� : �*b*|˜

5 : 10 = �*d*| ˜*d*� : �*a*|*a*˜�;

Four pairs {0 : 15, 3 : 12, 5 : 10, 6 : 9} of distributions may have special properties.

6:9 = �*c*|˜

Using *n* = 2 configurations, relevant CEMs on either MPS or CPS are shown in Figure 8(a-h).

*b*�;

*b*�;

(33)

(35)

(36)

(37)

*b*�;

*b*�;

*b*�;

[1, 0] (−, −) ... [1/2, 1/2] (−, 0) ... ... ... ... (−, −) (0, 0) ... (0, −) [0, 1]

[1, 0] (1, −) ... [1, 1] (−, 0) ... ... ... ... (−, 1) (0, 0) ... (0, −) [0, 1]

*b*� : �*b*|*c*˜�.

5 : 10 <sup>=</sup> �*d*<sup>|</sup> ˜*d*� : �*a*|*a*˜�. (34)

(b) W in MPS

(e) SL in CPS

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 147

(f) W in CPS

34 Cellular Automata

(c) F in MPS

(d) C in MPS

(f) W in CPS

**8.2. IMM groups**

shown in Figures 10(C1-C32).

**9.1. VPS organization**

*9.1.1. MPS Structures*

**SL in MPS**

**W in MPS**

properties. **F in MPS**

**C in MPS**

symmetry properties.

*9.1.2. CPS Structures*

**SL in CPS**

evident top-right in a 3 × 3 matrix.

**9. Analysis of visual distributions**

CE through either rotation or reflection.

IM Matrices under MPS are shown in Figures 9(M1-M32) and IM Matrices under CPS are

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 149

Two groups of matrices are shown in Figure 8. The four matrices shown as 8 (a-d) illustrate MPS conditions and the four matrices shown as 8 (e-h) illustrate CPS conditions. Considering various CEs exhibiting conjugate symmetry properties, such arrangements may be noted to have similar distributions along the diagonal and anti-diagonal directions so that it is possible to find a pair of CEs with each CE pair-matched by a geometric transformation to another

The four CE matrices in the MPS group as shown in Figure 8(a-d) can be analyzed as follows.

For the matrix in Figure 8(a) showing SL in MPS, 16 CEs are arranged in linear order from 0-15 in the 4×4 matrix. Only two pairs of {0:15, 6:9} CEs have conjugate symmetry properties.

For the matrix in Figure 8(b) showing W in MPS, 16 CEs are not arranged in linear order in {0,...,15} in the 4 × 4 matrix. Only two pairs of {0:15, 6:9} CEs have conjugate symmetry

However, for the matrix in Figure 8(c) showing F in MPS, 16 CEs exhibit more conjugate pairs in the 4 × 4 matrix. Here, six pairs of CEs {0:15, 1:7, 2:11, 4:13, 6:9, 8:14} show conjugate

Also, for the matrix in Figure 8(d) showing C in MPS, we find the same number of conjugate pairs as with the F condition. Moreover, not only do six pairs of CEs {0:15, 1:7, 2:11, 4:13, 6:9, 8:14} exhibit conjugate symmetry properties, but also four CEs {10,8,2,0} are polarized on the vertical as per the left column and four CEs {10,14,11,15} are polarized on the horizontal direction as per the bottom row. In addition, nine CEs showing interactive properties are

The four CE matrices shown in Figure 8(e-h) in the CPS group can be analyzed as follows.

**Figure 8.** (a-f) Matrices of Plane Lattices of VPS for MPS and CPS in {*SL*, *W*, *F*, *C*} codes, (a-d) MPS, (e-h) CPS; (a,e) SL code, (b,f) W code,(c,g) F code, (d,h) C code.

#### **8.2. IMM groups**

36 Cellular Automata

(g) F in CPS

(h) C in CPS **Figure 8.** (a-f) Matrices of Plane Lattices of VPS for MPS and CPS in {*SL*, *W*, *F*, *C*} codes, (a-d) MPS, (e-h) CPS; (a,e) SL code,

(b,f) W code,(c,g) F code, (d,h) C code.

IM Matrices under MPS are shown in Figures 9(M1-M32) and IM Matrices under CPS are shown in Figures 10(C1-C32).

#### **9. Analysis of visual distributions**

#### **9.1. VPS organization**

Two groups of matrices are shown in Figure 8. The four matrices shown as 8 (a-d) illustrate MPS conditions and the four matrices shown as 8 (e-h) illustrate CPS conditions. Considering various CEs exhibiting conjugate symmetry properties, such arrangements may be noted to have similar distributions along the diagonal and anti-diagonal directions so that it is possible to find a pair of CEs with each CE pair-matched by a geometric transformation to another CE through either rotation or reflection.

#### *9.1.1. MPS Structures*

The four CE matrices in the MPS group as shown in Figure 8(a-d) can be analyzed as follows.

#### **SL in MPS**

For the matrix in Figure 8(a) showing SL in MPS, 16 CEs are arranged in linear order from 0-15 in the 4×4 matrix. Only two pairs of {0:15, 6:9} CEs have conjugate symmetry properties.

#### **W in MPS**

For the matrix in Figure 8(b) showing W in MPS, 16 CEs are not arranged in linear order in {0,...,15} in the 4 × 4 matrix. Only two pairs of {0:15, 6:9} CEs have conjugate symmetry properties.

#### **F in MPS**

However, for the matrix in Figure 8(c) showing F in MPS, 16 CEs exhibit more conjugate pairs in the 4 × 4 matrix. Here, six pairs of CEs {0:15, 1:7, 2:11, 4:13, 6:9, 8:14} show conjugate symmetry properties.

#### **C in MPS**

Also, for the matrix in Figure 8(d) showing C in MPS, we find the same number of conjugate pairs as with the F condition. Moreover, not only do six pairs of CEs {0:15, 1:7, 2:11, 4:13, 6:9, 8:14} exhibit conjugate symmetry properties, but also four CEs {10,8,2,0} are polarized on the vertical as per the left column and four CEs {10,14,11,15} are polarized on the horizontal direction as per the bottom row. In addition, nine CEs showing interactive properties are evident top-right in a 3 × 3 matrix.

#### *9.1.2. CPS Structures*

The four CE matrices shown in Figure 8(e-h) in the CPS group can be analyzed as follows.

#### **SL in CPS**

For the matrix in Figure 8(e) showing SL in CPS, 16 CEs are arranged in linear order from 0-15 as a 4 × 4 matrix. Four pairs of CEs {0:15, 3:12, 5:10, 6:9} have conjugate symmetry properties.

In *u*<sup>0</sup> matrix M3, each element shows simple additions from elements in *u*+ and *u*− respectively. It it interesting to note that only two pairs of positions {0:15, 6:9} are similarly

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 151

However, in *u*<sup>1</sup> matrix M4, significant symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are different from the

**M4-M8:** Let us now consider elements M5-M8 where the four matrices of (*v*+, *v*−, *v*0, *v*1) are

In *v*+ matrix M5, elements in the columns and rows are arranged as periodic crossing

In *v*<sup>−</sup> matrix M6, four elements with the same IMs are arranged in a 2 × 2 block with four

In *v*<sup>0</sup> matrix M7, each element shows simple additions from elements in *v*+ and *v*− respectively. Only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant

In *v*<sup>1</sup> matrix M8, significant symmetry properties can be observed. Two pairs {0:15, 6:9} have

For the W group in Figure 9 (M9-M16), the two matrix vectors {*u*, *v*} =

**M9-M12:** Let us now consider elements M9-M12 where the four matrices (*u*+, *u*−, *u*0, *u*1) are

In *u*<sup>+</sup> matrix M9, four elements with the same IMs are arranged in a 2 × 2 block with four

In *u*− matrix M10, elements in the columns and rows are arranged as a periodic crossing

In *u*<sup>0</sup> matrix M11, each element shows simple additions with elements in *u*+ and *u*− respectively. It it interesting to note that only two pairs of positions {0:15, 6:9} are similarly

However, in *u*<sup>1</sup> matrix M12, significant symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are different from the

**M13-M16:** Let us now consider elements M13-M16 where the four matrices (*v*+, *v*,*v*0, *v*1) are

In *v*<sup>+</sup> matrix M13, four elements with the same IMs are arranged in a 2 × 2 block with four

In *v*− matrix M14, elements in the columns and rows are arranged as a periodic crossing

anti-symmetry properties that are the same as the *v*<sup>0</sup> condition.

{(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

distributed in the relevant MPS matrix.

*u*<sup>0</sup> condition.

structures.

MPS matrix.

**W group in MPS**

in a symmetry group,

structure.

*u*<sup>0</sup> condition.

structure.

distinct distributions observed.

in an anti-symmetry group,

distinct distributions observed.

distributed in the relevant MPS matrix.

in an anti-symmetry group,

distinct distributions observed.

#### **W in CPS**

For the matrix in Figure 8(f) showing W in CPS, 16 CEs are not arranged in linear order. However, four pairs of CEs {0:15, 3:12, 5:10, 6:9} have conjugate symmetry properties.

#### **F in CPS**

For the matrix in Figure 8(g) showing F in CPS, we can recognize two CE groups where each group has six pairs of CEs with conjugate symmetry properties {0:15, 1:7, 2:11, 4:13, 6:9, 8:14} and {2:4, 1:8, 3:12, 5:10, 7:14, 11:13}.

#### **C in CPS**

For the matrix in Figure 8(h) showing C in CPS, 12 CEs (out of 16) show conjugate pairing. This is the same number of conjugate pairs as is evident with the F condition in CPS. Also, if we look at polarization, the matrix for C in CPS is very different from the other coding matrices. It has significant polarized properties connecting the outer elements of the matrix. Here, four CEs {10,8,2,0} are polarized on the vertical as per the left column, and another four CEs {5,7,13,15} as per the right column. Also, four CEs {0,4,1,8} are polarized on the horizontal as per the top-row and another four CEs {10,14,11,15} as per the bottom-row. Four more CEs {3,6,9,12} exhibit interactive properties in a 2 x 2 central grid. In all, five distinct regions can be identified as significant. It is interesting to note such remarkable symmetry illustrating interactions between and among these meta functions.

#### **9.2. IMM organization**

From one matrix in VPS, eight matrices corresponding to the two vectors {*u*, *v*} = {(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} can be generated showing interactive properties under symmetry/anti-symmetry, and synchronous/asynchronous conditions respectively. A total of 64 matrices are shown in two groups in Figures 9 (M1-M32) for MPS and in Figures 10 (C1-C32) for CPS, respectively.

#### *9.2.1. MPS Structures*

#### **SL group in MPS**

For the SL group in Figure 9 (M1-M8), the two matrix vectors {*u*, *v*} = {(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

**M1-M4:** Let us first consider elements M1-M4 where the four matrices of (*u*+, *u*−, *u*0, *u*1) are in a symmetry group,

In *u*+ matrix M1, elements in the columns and rows are arranged in what may be described as a periodic crossing structure.

In *u*<sup>−</sup> matrix M2, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions being observed.

In *u*<sup>0</sup> matrix M3, each element shows simple additions from elements in *u*+ and *u*− respectively. It it interesting to note that only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.

However, in *u*<sup>1</sup> matrix M4, significant symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are different from the *u*<sup>0</sup> condition.

**M4-M8:** Let us now consider elements M5-M8 where the four matrices of (*v*+, *v*−, *v*0, *v*1) are in an anti-symmetry group,

In *v*+ matrix M5, elements in the columns and rows are arranged as periodic crossing structures.

In *v*<sup>−</sup> matrix M6, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.

In *v*<sup>0</sup> matrix M7, each element shows simple additions from elements in *v*+ and *v*− respectively. Only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.

In *v*<sup>1</sup> matrix M8, significant symmetry properties can be observed. Two pairs {0:15, 6:9} have anti-symmetry properties that are the same as the *v*<sup>0</sup> condition.

#### **W group in MPS**

38 Cellular Automata

properties. **W in CPS**

**F in CPS**

**C in CPS**

and {2:4, 1:8, 3:12, 5:10, 7:14, 11:13}.

**9.2. IMM organization**

(C1-C32) for CPS, respectively.

*9.2.1. MPS Structures* **SL group in MPS**

in a symmetry group,

as a periodic crossing structure.

distinct distributions being observed.

For the matrix in Figure 8(e) showing SL in CPS, 16 CEs are arranged in linear order from 0-15 as a 4 × 4 matrix. Four pairs of CEs {0:15, 3:12, 5:10, 6:9} have conjugate symmetry

For the matrix in Figure 8(f) showing W in CPS, 16 CEs are not arranged in linear order. However, four pairs of CEs {0:15, 3:12, 5:10, 6:9} have conjugate symmetry properties.

For the matrix in Figure 8(g) showing F in CPS, we can recognize two CE groups where each group has six pairs of CEs with conjugate symmetry properties {0:15, 1:7, 2:11, 4:13, 6:9, 8:14}

For the matrix in Figure 8(h) showing C in CPS, 12 CEs (out of 16) show conjugate pairing. This is the same number of conjugate pairs as is evident with the F condition in CPS. Also, if we look at polarization, the matrix for C in CPS is very different from the other coding matrices. It has significant polarized properties connecting the outer elements of the matrix. Here, four CEs {10,8,2,0} are polarized on the vertical as per the left column, and another four CEs {5,7,13,15} as per the right column. Also, four CEs {0,4,1,8} are polarized on the horizontal as per the top-row and another four CEs {10,14,11,15} as per the bottom-row. Four more CEs {3,6,9,12} exhibit interactive properties in a 2 x 2 central grid. In all, five distinct regions can be identified as significant. It is interesting to note such remarkable symmetry

From one matrix in VPS, eight matrices corresponding to the two vectors {*u*, *v*} = {(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} can be generated showing interactive properties under symmetry/anti-symmetry, and synchronous/asynchronous conditions respectively. A total of 64 matrices are shown in two groups in Figures 9 (M1-M32) for MPS and in Figures 10

For the SL group in Figure 9 (M1-M8), the two matrix vectors {*u*, *v*} =

**M1-M4:** Let us first consider elements M1-M4 where the four matrices of (*u*+, *u*−, *u*0, *u*1) are

In *u*+ matrix M1, elements in the columns and rows are arranged in what may be described

In *u*<sup>−</sup> matrix M2, four elements with the same IMs are arranged in a 2 × 2 block with four

illustrating interactions between and among these meta functions.

{(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

For the W group in Figure 9 (M9-M16), the two matrix vectors {*u*, *v*} = {(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

**M9-M12:** Let us now consider elements M9-M12 where the four matrices (*u*+, *u*−, *u*0, *u*1) are in a symmetry group,

In *u*<sup>+</sup> matrix M9, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.

In *u*− matrix M10, elements in the columns and rows are arranged as a periodic crossing structure.

In *u*<sup>0</sup> matrix M11, each element shows simple additions with elements in *u*+ and *u*− respectively. It it interesting to note that only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.

However, in *u*<sup>1</sup> matrix M12, significant symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are different from the *u*<sup>0</sup> condition.

**M13-M16:** Let us now consider elements M13-M16 where the four matrices (*v*+, *v*,*v*0, *v*1) are in an anti-symmetry group,

In *v*<sup>+</sup> matrix M13, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.

In *v*− matrix M14, elements in the columns and rows are arranged as a periodic crossing structure.

In *v*<sup>0</sup> matrix M15, each element shows simple additions with elements in *v*+ and *v*− respectively. Only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant MPS matrix.

In *u*+ matrix M25, the horizontal elements are in a periodic crossing structure and the vertical

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 153

In *u*− matrix M26, the horizontal elements are arranged in H-4R patterns and the vertical

In *u*<sup>0</sup> matrix M27, each element shows simple additions with elements in *u*+ and *u*− respectively. It it interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9,

However, in *u*<sup>1</sup> matrix M28, significant symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have symmetry but another six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } have anti-symmetry properties, all significantly different from the *u*<sup>0</sup> condition. **M29-M32:** Let us now consider elements M29-M32 where the four matrices of (*v*+, *v*,*v*0, *v*1)

In *v*+ matrix M29, the horizontal elements are arranged in H-4R patterns and the vertical

In *v*− matrix M30, the horizontal elements are arranged in H-4R patterns and the vertical

In *v*<sup>0</sup> matrix M31, each element shows simple additions with elements in *v*+ and *v*− respectively. The distribution of six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} exhibit

In *v*<sup>1</sup> matrix M32, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and two pairs {2:4, 11:13} have symmetry

Four groups of different configurations shown in Figure 10 (C1-C32) are discussed separately

For the SL group in Figure 10 (C1-C8), he two matrix vectors {*u*˜, *v*˜} =

**C1-C4:** Let us now consider elements C1-C4 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1) are

In *u*˜0 matrix C3, each element shows simple additions from elements in *u*˜+ and *u*˜− respectively. It it interesting to note that four pairs of positions {0:15, 3:12, 5:10, 6:9} are

In *u*˜1 matrix C4, similar symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are the same as for the *u*˜0 condition.

In *u*˜+ matrix C1, elements in the columns and rows are in a periodic crossing structure. In *u*˜<sup>−</sup> matrix C2, four elements with the same IMs are arranged in a 2 × 2 block with four

{(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

elements are arranged in V-4R patterns

elements as a periodic crossing structure.

8:14} are similarly distributed.

are in an anti-symmetry group,

anti-symmetry.

*9.2.2. CPS Structures*

**SL group in CPS**

in a symmetry group,

distinct distributions observed.

similarly distributed in the relevant CPS matrix.

properties.

as follows.

elements as a periodic crossing structure.

elements as a periodic crossing structure.

In *v*<sup>1</sup> matrix M16, significant symmetry properties can be observed. Two pairs {0:15, 6:9} have anti-symmetry properties the same as under the *v*<sup>0</sup> condition.

#### **F group in MPS**

For the F group in Figure 9 (M17-M24), the two matrix vectors {*u*, *v*} = {(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

**M17-M20:** Let us now consider elements M17-M20 where the four matrices (*u*+, *u*−, *u*0, *u*1) are in a symmetry group,

In *u*+ matrix M17, the horizontal elements are arranged in H-2R patterns and vertical elements are in a periodic crossing structure.

In *u*− matrix M18, vertical elements are arranged in V-2R patterns and the horizontal elements as a periodic crossing structure.

In *u*<sup>0</sup> matrix M19, each element shows simple additions with elements in *u*+ and *u*− respectively. It it interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are similarly distributed.

However, in *u*<sup>1</sup> matrix M20, significant symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have symmetry properties and another six pairs of {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } with anti-symmetry properties but there are also significantly differences compared with the *u*<sup>0</sup> condition.

**M21-M24:** Let us now consider elements M17-M20 where the four matrices of (*v*+, *v*,*v*0, *v*1) are in an anti-symmetry group,

In *v*+ matrix M21, the horizontal elements are arranged in H-2R patterns and the vertical elements as a periodic crossing structure.

In *v*− matrix M22, the vertical elements are arranged in V-2R patterns and the horizontal elements as a periodic crossing structure.

In *v*<sup>0</sup> matrix M23, each element shows simple additions with elements in *v*+ and *v*− respectively. It it interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are in the anti-symmetry distribution.

In *v*<sup>1</sup> matrix M24, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and two pairs {2:4, 11:13} have symmetry properties.

#### **C group in MPS**

For the C group in Figure 9 (M25-M32), two matrix vectors {*u*, *v*} = {(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

**M25-M28:** Let us now consider elements M25-M28 where the four matrices of (*u*+, *u*−, *u*0, *u*1) are in a symmetry group,

In *u*+ matrix M25, the horizontal elements are in a periodic crossing structure and the vertical elements are arranged in V-4R patterns

In *u*− matrix M26, the horizontal elements are arranged in H-4R patterns and the vertical elements as a periodic crossing structure.

In *u*<sup>0</sup> matrix M27, each element shows simple additions with elements in *u*+ and *u*− respectively. It it interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are similarly distributed.

However, in *u*<sup>1</sup> matrix M28, significant symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have symmetry but another six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } have anti-symmetry properties, all significantly different from the *u*<sup>0</sup> condition.

**M29-M32:** Let us now consider elements M29-M32 where the four matrices of (*v*+, *v*,*v*0, *v*1) are in an anti-symmetry group,

In *v*+ matrix M29, the horizontal elements are arranged in H-4R patterns and the vertical elements as a periodic crossing structure.

In *v*− matrix M30, the horizontal elements are arranged in H-4R patterns and the vertical elements as a periodic crossing structure.

In *v*<sup>0</sup> matrix M31, each element shows simple additions with elements in *v*+ and *v*− respectively. The distribution of six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} exhibit anti-symmetry.

In *v*<sup>1</sup> matrix M32, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and two pairs {2:4, 11:13} have symmetry properties.

#### *9.2.2. CPS Structures*

40 Cellular Automata

MPS matrix.

**F group in MPS**

are in a symmetry group,

8:14} are similarly distributed.

are in an anti-symmetry group,

properties.

**C group in MPS**

are in a symmetry group,

elements are in a periodic crossing structure.

elements as a periodic crossing structure.

differences compared with the *u*<sup>0</sup> condition.

elements as a periodic crossing structure.

elements as a periodic crossing structure.

8:14} are in the anti-symmetry distribution.

In *v*<sup>0</sup> matrix M15, each element shows simple additions with elements in *v*+ and *v*− respectively. Only two pairs of positions {0:15, 6:9} are similarly distributed in the relevant

In *v*<sup>1</sup> matrix M16, significant symmetry properties can be observed. Two pairs {0:15, 6:9} have

For the F group in Figure 9 (M17-M24), the two matrix vectors {*u*, *v*} =

**M17-M20:** Let us now consider elements M17-M20 where the four matrices (*u*+, *u*−, *u*0, *u*1)

In *u*+ matrix M17, the horizontal elements are arranged in H-2R patterns and vertical

In *u*− matrix M18, vertical elements are arranged in V-2R patterns and the horizontal

In *u*<sup>0</sup> matrix M19, each element shows simple additions with elements in *u*+ and *u*− respectively. It it interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9,

However, in *u*<sup>1</sup> matrix M20, significant symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have symmetry properties and another six pairs of {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } with anti-symmetry properties but there are also significantly

**M21-M24:** Let us now consider elements M17-M20 where the four matrices of (*v*+, *v*,*v*0, *v*1)

In *v*+ matrix M21, the horizontal elements are arranged in H-2R patterns and the vertical

In *v*− matrix M22, the vertical elements are arranged in V-2R patterns and the horizontal

In *v*<sup>0</sup> matrix M23, each element shows simple additions with elements in *v*+ and *v*− respectively. It it interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9,

In *v*<sup>1</sup> matrix M24, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and two pairs {2:4, 11:13} have symmetry

For the C group in Figure 9 (M25-M32), two matrix vectors {*u*, *v*} =

**M25-M28:** Let us now consider elements M25-M28 where the four matrices of (*u*+, *u*−, *u*0, *u*1)

{(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

anti-symmetry properties the same as under the *v*<sup>0</sup> condition.

{(*u*+, *u*−, *u*0, *u*1),(*v*+, *v*−, *v*0, *v*1)} are best considered separately.

Four groups of different configurations shown in Figure 10 (C1-C32) are discussed separately as follows.

#### **SL group in CPS**

For the SL group in Figure 10 (C1-C8), he two matrix vectors {*u*˜, *v*˜} = {(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

**C1-C4:** Let us now consider elements C1-C4 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1) are in a symmetry group,

In *u*˜+ matrix C1, elements in the columns and rows are in a periodic crossing structure.

In *u*˜<sup>−</sup> matrix C2, four elements with the same IMs are arranged in a 2 × 2 block with four distinct distributions observed.

In *u*˜0 matrix C3, each element shows simple additions from elements in *u*˜+ and *u*˜− respectively. It it interesting to note that four pairs of positions {0:15, 3:12, 5:10, 6:9} are similarly distributed in the relevant CPS matrix.

In *u*˜1 matrix C4, similar symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are the same as for the *u*˜0 condition.

**C5-C8:** Let us now consider elements C5-C8 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are in an anti-symmetry group,

For the F group in Figure 10 (C17-C24), the two matrix vectors {*u*˜, *v*˜} =

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 155

**C17-C20:** Let us now consider elements C17-C20 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1)

In *u*˜+ matrix C17, horizontal elements are arranged in H-2R patterns and vertical elements

In *u*˜− matrix C18, vertical elements are arranged in V-2R patterns and horizontal elements

In *u*˜0 matrix C19, each element shows simple additions from elements in *u*˜+ and *u*˜− respectively. It interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14}

In *u*˜1 matrix C20, similar symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have symmetry but also another six pairs {1:8, 2:4, 3:12, 5:10, 7:14,

**C21-C24:** Let us now consider elements C21-C24 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are

In *v*˜+ matrix C21, horizontal elements are arranged in H-2R patterns and vertical elements

In *v*˜− matrix C22, vertical elements are arranged in V-2R patterns and horizontal elements as

In *v*˜0 matrix C23, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. It is interesting to note that only six pairs of positions {0:15, 1:7. 2:11, 4:13,

In *v*˜1 matrix C24, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 }

For the C group in Figure 10 (C25-C32), the two matrix vectors {*u*˜, *v*˜} =

**C25-C28:** Let us now consider elements C25-C28 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1)

In *u*˜+ matrix C25, horizontal elements are arranged as a periodic crossing structure. and

In *u*˜− matrix C26, horizontal elements are arranged in H-4R patterns and vertical elements

In *u*˜0 matrix C27, each element shows simple additions from elements in *u*+ and *u*− respectively. It is interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are similarly distributed and six pairs of positions {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } show

11:13 } exhibit anti-symmetry properties that are the same as under *u*˜0 conditions.

{(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

and {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are in the similar distributions.

{(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

are in a symmetry group,

are in a periodic crossing structure.

as a periodic crossing structure.

in an anti-symmetry group,

as periodic crossing structures.

a periodic crossing structure.

have symmetry properties.

are in a symmetry group,

as a periodic crossing structure.

anti-symmetry properties.

**C group in CPS**

6:9, 8:14} show anti-symmetry distributions.

vertical elements are arranged in V-4R patterns

In *v*˜+ matrix C5, elements in the columns and rows are arranged in a periodic crossing structure.

In *v*˜<sup>−</sup> matrix C6, four elements with same IMs are arranged in a 2 × 2 block and four distinct distributions are observed.

In *v*˜0 matrix C7, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. Only two pairs of positions {0:15, 6:9} are in the same distribution in similar arrangements.

In *v*˜1 matrix C8, similar symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have anti-symmetry properties significantly different from those in the *v*˜0 condition.

#### **W group in CPS**

For the W group in Figure 10 (C9-C16), the two matrix vectors {*u*˜, *v*˜} = {(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

**C9-C12:** Let us now consider elements C9-C12 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1) are in a symmetry group,

In *u*˜<sup>+</sup> matrix C9, four elements with the same IMs are arranged in a 2 × 2 block and four distinct distributions are observed.

In *u*˜− matrix C10, elements in the columns and rows are arranged as a periodic crossing structure.

In *u*˜0 matrix C11, each element shows simple additions from elements in *u*˜+ and *u*˜− respectively. It it interesting to note that four pairs of positions {0:15, 3:12, 5:10, 6:9} are distributed in similar arrangements in the relevant CPS matrix.

In *u*˜1 matrix C12, similar symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are the same as under the *u*˜0 condition.

**C13-C16:** Let us now consider elements C13-C16 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are in an anti-symmetry group,

In *v*˜<sup>+</sup> matrix C13, four elements with the same IMs are arranged in a 2 × 2 block and four distinct distributions are observed.

In *v*˜− matrix C14, elements in the columns and rows are arranged as a periodic crossing structure.

In *v*˜0 matrix C15, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. Only two pairs of positions {0:15, 6:9} show the same distribution in similar arrangements.

In *v*˜1 matrix C16, significant symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have anti-symmetry properties that are different from those under *v*˜0 conditions.

**F group in CPS**

For the F group in Figure 10 (C17-C24), the two matrix vectors {*u*˜, *v*˜} = {(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

**C17-C20:** Let us now consider elements C17-C20 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1) are in a symmetry group,

In *u*˜+ matrix C17, horizontal elements are arranged in H-2R patterns and vertical elements are in a periodic crossing structure.

In *u*˜− matrix C18, vertical elements are arranged in V-2R patterns and horizontal elements as a periodic crossing structure.

In *u*˜0 matrix C19, each element shows simple additions from elements in *u*˜+ and *u*˜− respectively. It interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} and {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are in the similar distributions.

In *u*˜1 matrix C20, similar symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have symmetry but also another six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } exhibit anti-symmetry properties that are the same as under *u*˜0 conditions.

**C21-C24:** Let us now consider elements C21-C24 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are in an anti-symmetry group,

In *v*˜+ matrix C21, horizontal elements are arranged in H-2R patterns and vertical elements as periodic crossing structures.

In *v*˜− matrix C22, vertical elements are arranged in V-2R patterns and horizontal elements as a periodic crossing structure.

In *v*˜0 matrix C23, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. It is interesting to note that only six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} show anti-symmetry distributions.

In *v*˜1 matrix C24, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } have symmetry properties.

#### **C group in CPS**

42 Cellular Automata

structure.

arrangements.

**W group in CPS**

in a symmetry group,

structure.

structure.

arrangements.

**F group in CPS**

distinct distributions are observed.

in an anti-symmetry group,

distinct distributions are observed.

an anti-symmetry group,

distributions are observed.

**C5-C8:** Let us now consider elements C5-C8 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are in

In *v*˜+ matrix C5, elements in the columns and rows are arranged in a periodic crossing

In *v*˜<sup>−</sup> matrix C6, four elements with same IMs are arranged in a 2 × 2 block and four distinct

In *v*˜0 matrix C7, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. Only two pairs of positions {0:15, 6:9} are in the same distribution in similar

In *v*˜1 matrix C8, similar symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have anti-symmetry properties significantly different from those in the *v*˜0 condition.

For the W group in Figure 10 (C9-C16), the two matrix vectors {*u*˜, *v*˜} =

**C9-C12:** Let us now consider elements C9-C12 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1) are

In *u*˜<sup>+</sup> matrix C9, four elements with the same IMs are arranged in a 2 × 2 block and four

In *u*˜− matrix C10, elements in the columns and rows are arranged as a periodic crossing

In *u*˜0 matrix C11, each element shows simple additions from elements in *u*˜+ and *u*˜− respectively. It it interesting to note that four pairs of positions {0:15, 3:12, 5:10, 6:9} are

In *u*˜1 matrix C12, similar symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have symmetry or anti-symmetry properties that are the same as under the *u*˜0 condition. **C13-C16:** Let us now consider elements C13-C16 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are

In *v*˜<sup>+</sup> matrix C13, four elements with the same IMs are arranged in a 2 × 2 block and four

In *v*˜− matrix C14, elements in the columns and rows are arranged as a periodic crossing

In *v*˜0 matrix C15, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. Only two pairs of positions {0:15, 6:9} show the same distribution in similar

In *v*˜1 matrix C16, significant symmetry properties can be observed. Four pairs {0:15, 3:12, 5:10, 6:9} have anti-symmetry properties that are different from those under *v*˜0 conditions.

{(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

distributed in similar arrangements in the relevant CPS matrix.

For the C group in Figure 10 (C25-C32), the two matrix vectors {*u*˜, *v*˜} = {(*u*˜+, *u*˜−, *u*˜0, *u*˜1),(*v*˜+, *v*˜−, *v*˜0, *v*˜1)} are best considered separately.

**C25-C28:** Let us now consider elements C25-C28 where the four matrices of (*u*˜+, *u*˜−, *u*˜0, *u*˜1) are in a symmetry group,

In *u*˜+ matrix C25, horizontal elements are arranged as a periodic crossing structure. and vertical elements are arranged in V-4R patterns

In *u*˜− matrix C26, horizontal elements are arranged in H-4R patterns and vertical elements as a periodic crossing structure.

In *u*˜0 matrix C27, each element shows simple additions from elements in *u*+ and *u*− respectively. It is interesting to note that six pairs of positions {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} are similarly distributed and six pairs of positions {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } show anti-symmetry properties.


**Type Case Left Right SP(D-P) ASP(D-P) SP(D-W) ASP(D-W) GS** SL P=(3210) Cross 2x2Block Weak *u* 2 (a) 0 3 (b) 1 (c) *v* 0 2 (a) 0 2 (a) *u*˜ 2 (a) 2 (d) 2 (a) 2 (d) *v*˜ 0 2 (a) 2 (d) 2 (a) W P=(2103) 2x2Block Cross Weak *u* 2 (a) 0 3 (b) 1 (c) *v* 0 2 (a) 0 2 (a) *u*˜ 2 (a) 2 (d) 2 (a) 2 (d) *v*˜ 0 2 (a) 2 (d) 2 (a) F P=(3201) V-2R H-2R Stronger *u* 6 (e) 0 6 (e) 6 (f) *v* 0 6 (e) 2 (g) 6 (e) *u*˜ 6 (e) 6 (f) 6 (e) 6 (f) *v*˜ 0 6 (e) 6 (e) 6 (f) C P=(3102) V-4R H-4R Strongest *u* 6 (e) 0 6 (e) 6 (f) *v* 0 6 (e) 2 (g) 6 (e) *u*˜ 6 (e) 6 (f) 6 (e) 6 (f) *v*˜ 0 6 (e) 6 (e) 6 (f)

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 157

**Table 5.** Global Symmetry Properties on IM Matrices

**10.1. Comparison of variant phase space and statistical mechanics**

Mechanics (CSM), two types of systems are compared in Table 6.

mechanics has no computational mechanism for GEM capacities.

further distinguished on CE(VPS) and IM(VPS) levels.

Both Maxwell-Boltzmann and Darwin-Fowler schemes are considered suitable for processing isolated systems. Meanwhile, a Gibbs scheme can be applied to several different systems namely, an isolated system on a micro canonical ensemble, a closed system on a canonical ensemble, and an open system on a grand canonical ensemble [20, 23, 24, 31, 33]. Such significant differences can offer useful comparisons when considering *Variant Phase Space*. Using Variant Phase Space (VPS) components and key properties of Classical Statistical

Table 6 shows some key differences that may be distinguished between VPS and CSM. Both approaches use parameters {*n*, *N*, *X*} on a selected function. However, there is a distinct difference for ME with a split into non-interactive and interactive activities between Maxwell-Boltzmann on ME(VPS) and Gibbs on IP(VPS), respectively. This difference is

Normally statistical mechanics is not based on all possible functions Instead, one function with the most probable properties is selected. Only the Maxwell demon mechanism provides any possible function for potential applications, under such restriction, modern statistical

GEM capacities do not cover a Gibbs grand canonical ensemble. However, using a given configuration of variant logic function to arrange full sets of distributions similar to variation, functional capacities can be associated with a truly large number of configurations: 2*n*! × 22*<sup>n</sup>*

This provides an opportunity to exhaust distributions for possible functions on a scale that

goes way beyond the conventional framework of modern statistical mechanics.

.

**Table 4.** Global Symmetry Properties on CE Matrices

In *u*˜1 matrix C28, significant symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} show symmetry but also another six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } exhibit anti-symmetry properties similar to those under *u*˜0 conditions.

**C29-C32:** Let us now consider elements C29-C32 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are in an anti-symmetry group,

In *v*˜+ matrix C29, horizontal elements are arranged in H-4R patterns and vertical elements are arranged as a periodic crossing structure.

In *v*˜− matrix C30, horizontal elements are arranged in H-4R patterns and vertical elements are arranged as a periodic crossing structure.

In *v*˜0 matrix C31, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. Two pairs of positions {0:15, 4:13, 6:9, 8:14} exhibit anti-symmetry distributions.

In *v*˜1 matrix C32, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and another six pairs {1:8, 2:4, 3:12, 5:10, 7:14, 11:13 } have symmetry properties that are different from those under *v*˜0 condition.

#### **10. Global symmetric properties**

Working from four sets of CEM and IMM results, key global symmetry properties are presented and summarized in Table 4 for CEMs and in Table 5 for IMMs as follows.

Where CP is a conjugate pair, GP is global polarization and a:(0:15,6:9), d:(3:12,5:10), e:(0:15,1:7,2:11,4:13,6:9,8:14), f:(1:8,2:4,3:12,5:10,7:14,11:13), are pair functions.

It is interesting to note that significant differences in symmetry properties between MPS and CPS can be observed for CEM conjugate pairs.

In general, we find double the number of incidences of symmetry properties with CPS compared with MPS shown in Table 4.

Where SP is a Symmetric Pair, ASP is an Anti-symmetric Pair, GS is Global Symmetry and a:(0:15,6:9), b:(0:15,6:9,3:12), c:(5:10), d:(3:12,5:10), e:(0:15,1:7,2:11,4:13,6:9,8:14), f:(1:8,2:4,3:12,5:10,7:14,11:13), g:(2:4,11:13) are pair functions.

It is interesting to note that symmetry properties evident in IMM groups in Table 5 are more refined than the original configurations under MPS and CPS conditions.

The classification of different projections and polarized properties can be further refined to show their various interactive activities in relevant sub-categories. Further details for conjugate pairs can be distinguished under symmetry/anti-symmetry and synchronous/asynchronous configurations. Conjugate pairs can be further differentiated as being either symmetric or anti-symmetric pairs.


**Table 5.** Global Symmetry Properties on IM Matrices

44 Cellular Automata

**Table 4.** Global Symmetry Properties on CE Matrices

are arranged as a periodic crossing structure.

are arranged as a periodic crossing structure.

**10. Global symmetric properties**

CPS can be observed for CEM conjugate pairs.

f:(1:8,2:4,3:12,5:10,7:14,11:13), g:(2:4,11:13) are pair functions.

as being either symmetric or anti-symmetric pairs.

refined than the original configurations under MPS and CPS conditions.

compared with MPS shown in Table 4.

in an anti-symmetry group,

**Type Case CP** ∈ **MPS CP** ∈ **CPS GP Notes**

SL P=(3210) 2(a) 4(a,d) N Limited conjugate symmetry W P=(2103) 2(a) 4(a,d) N Limited conjugate symmetry F P=(3201) 6(e) 12(e,f) N Pairs conjugate symmetry

In *u*˜1 matrix C28, significant symmetry properties can be observed. Not only do six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} show symmetry but also another six pairs {1:8, 2:4, 3:12, 5:10,

**C29-C32:** Let us now consider elements C29-C32 where the four matrices of (*v*˜+, *v*˜,*v*˜0, *v*˜1) are

In *v*˜+ matrix C29, horizontal elements are arranged in H-4R patterns and vertical elements

In *v*˜− matrix C30, horizontal elements are arranged in H-4R patterns and vertical elements

In *v*˜0 matrix C31, each element shows simple additions from elements in *v*˜+ and *v*˜− respectively. Two pairs of positions {0:15, 4:13, 6:9, 8:14} exhibit anti-symmetry distributions. In *v*˜1 matrix C32, significant symmetry properties can be observed. Six pairs {0:15, 1:7. 2:11, 4:13, 6:9, 8:14} have anti-symmetry properties and another six pairs {1:8, 2:4, 3:12, 5:10, 7:14,

Working from four sets of CEM and IMM results, key global symmetry properties are

Where CP is a conjugate pair, GP is global polarization and a:(0:15,6:9), d:(3:12,5:10),

It is interesting to note that significant differences in symmetry properties between MPS and

In general, we find double the number of incidences of symmetry properties with CPS

Where SP is a Symmetric Pair, ASP is an Anti-symmetric Pair, GS is Global Symmetry and a:(0:15,6:9), b:(0:15,6:9,3:12), c:(5:10), d:(3:12,5:10), e:(0:15,1:7,2:11,4:13,6:9,8:14),

It is interesting to note that symmetry properties evident in IMM groups in Table 5 are more

The classification of different projections and polarized properties can be further refined to show their various interactive activities in relevant sub-categories. Further details for conjugate pairs can be distinguished under symmetry/anti-symmetry and synchronous/asynchronous configurations. Conjugate pairs can be further differentiated

11:13 } have symmetry properties that are different from those under *v*˜0 condition.

presented and summarized in Table 4 for CEMs and in Table 5 for IMMs as follows.

e:(0:15,1:7,2:11,4:13,6:9,8:14), f:(1:8,2:4,3:12,5:10,7:14,11:13), are pair functions.

C P=(3102) 6(e) 12(e,f) Y Global symmetry

7:14, 11:13 } exhibit anti-symmetry properties similar to those under *u*˜0 conditions.

#### **10.1. Comparison of variant phase space and statistical mechanics**

Both Maxwell-Boltzmann and Darwin-Fowler schemes are considered suitable for processing isolated systems. Meanwhile, a Gibbs scheme can be applied to several different systems namely, an isolated system on a micro canonical ensemble, a closed system on a canonical ensemble, and an open system on a grand canonical ensemble [20, 23, 24, 31, 33]. Such significant differences can offer useful comparisons when considering *Variant Phase Space*.

Using Variant Phase Space (VPS) components and key properties of Classical Statistical Mechanics (CSM), two types of systems are compared in Table 6.

Table 6 shows some key differences that may be distinguished between VPS and CSM. Both approaches use parameters {*n*, *N*, *X*} on a selected function. However, there is a distinct difference for ME with a split into non-interactive and interactive activities between Maxwell-Boltzmann on ME(VPS) and Gibbs on IP(VPS), respectively. This difference is further distinguished on CE(VPS) and IM(VPS) levels.

Normally statistical mechanics is not based on all possible functions Instead, one function with the most probable properties is selected. Only the Maxwell demon mechanism provides any possible function for potential applications, under such restriction, modern statistical mechanics has no computational mechanism for GEM capacities.

GEM capacities do not cover a Gibbs grand canonical ensemble. However, using a given configuration of variant logic function to arrange full sets of distributions similar to variation, functional capacities can be associated with a truly large number of configurations: 2*n*! × 22*<sup>n</sup>* . This provides an opportunity to exhaust distributions for possible functions on a scale that goes way beyond the conventional framework of modern statistical mechanics.


For an *n* variable function *J* ∈ *B*2*<sup>n</sup>*

pair has eight interactive projections.

supported by Propositions 11.1 to 11.2.

corresponds to one IM distribution.

reference to thermodynamic issues. **Proposition 11.6:** Exhausting ∀*J* ∈ *B*2*<sup>n</sup>*

structures on its columns and/or rows.

operations on the vector with 22*<sup>n</sup>*

be established.

create CPS.

eight IP modes.

Demon mechanism.

each set contains 22*<sup>n</sup>*

matrix.

**11.1. Propositions**

<sup>2</sup> and an *<sup>N</sup>* bit vector *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>*

**Proposition 11.1:** Two types of probability measurements, Multiple and Conditional

**Proof:** In a PM module, multiple probabilities generates MPS and conditional probabilities

**Proposition 11.2:** Two types of operations: symmetry/anti-symmetry and

**Proof:** Two pairs of measurement vectors {*u*, *v*} or {*u*˜, *v*˜} are involved in projections, where *u* = (*u*+, *u*−, *u*0, *u*1), *v* = (*v*+, *v*−, *v*0, *v*1) and *u*˜ = (*u*˜+, *u*˜−, *u*˜0, *u*˜1), *v*˜ = (*v*˜+, *v*˜−, *v*˜0, *v*˜1), each

**Proposition 11.3:** Following a bottom-up approach, two CE and 16 IMs can be generated to

**Proof:** Results may be generated using a CEIM module and Proposition 11.3 is further

**Proposition 11.4:** Each CE is a statistical distribution and each IM corresponds to one of

**Proof:** A pair of probability measurements has one fixed CE combination and each IP mode

**Proposition 11.5:** Both Proposition 11.3 and Proposition 11.4 provide a general Maxwell

**Proof:** For any function, CE and IMs can be fully and exhaustively generated without

**Proof:** Since each IMM has the same organization as the CEM, a total of 2*n*! × 22*<sup>n</sup>* configurations can be distinguished and each configuration corresponds to a variant logic

**Proposition 11.8:** With a top-down approach, either a CEM or an IMM on a proper configuration can be composed of two polarized matrices. Each polarized matrix has periodic

**Proof:** Since a proper configuration is based on *n* periodic meta vectors and their combinations, its relative arrangements are invariant under permutation and complementary

bits that determine each polarized structure.

elements and each element is a distribution.

**Proof:** Using the SCEIM module, they are natural outputs.

**Proposition 11.7:** In a variant logic framework, there are 2*n*! × 22*<sup>n</sup>*

arranging a set of {CE} and eight sets of {IM} into a CEM and eight IMMs.

<sup>2</sup> , two sets of {CE} and 16 sets of {IM} can be generated,

configurations for

probabilities determine two distinct phase spaces, MPS and CPS.

synchronous/asynchronous generate eight interactive projections.

exhaust all 2*<sup>N</sup>* input vectors for the relevant ME and IP measurements.

<sup>2</sup> , following propositions can

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 159

**Table 6.** Comparison between VPS and CSM


**Table 7.** Operation, strategy and expression of VPS

#### **10.2. Corresponding structures on variant phase space**

Top-down and bottom-up strategies can both be applied to Variant Phase Space. See Table 7.

Top-down and bottom-up strategies can each open a window through which to glimpse the mysteries of Variant Phase Space. Such glimpses do not yet provide a complete picture and further investigation is clearly required.

#### **11. Main results**

It is appropriate to present the results as a series of detailed propositions and predictions as follows.

For an *n* variable function *J* ∈ *B*2*<sup>n</sup>* <sup>2</sup> and an *<sup>N</sup>* bit vector *<sup>X</sup>* <sup>∈</sup> *<sup>B</sup><sup>N</sup>* <sup>2</sup> , following propositions can be established.

#### **11.1. Propositions**

46 Cellular Automata

**Component VPS Meaning CSM Notes Parameter** *n* Variables Local unit Cell unit on rule space

**CME** VM Variant Measures Classes of events Types of vector elements

**CEIM** CE Canonical Canonical Ensemble Non-interactive

**GEM** SCEIM Sets of CE&IM Full set of Full set of

**Key Output Operation Strategy Expression CIM** Matrices for Global organization of Top Hilbert space, Dynamic

**SCEIM** Sets of Global Integration on Meta distribution, distribution

**CEIM** CE&IMs Integration of distributions UP Maxwell-Boltzmann, Gibbs,

**CME** ME&IPs From local measures to Hamilton,Lagrange, Uncertainty,

Top-down and bottom-up strategies can both be applied to Variant Phase Space. See Table 7. Top-down and bottom-up strategies can each open a window through which to glimpse the mysteries of Variant Phase Space. Such glimpses do not yet provide a complete picture and

It is appropriate to present the results as a series of detailed propositions and predictions as

CE&IMs distributions for a configuration systems, Variation functional

CE&IMs distributions for all functions Down function, periodic distribution

for a function Euler, Canonical ensemble

micro ensemble and projections Bottom Fourier pairs, Phase point

*X* ∈ *B<sup>N</sup>* 2

*J* ∈ *B*2*<sup>n</sup>* 2

**Table 6.** Comparison between VPS and CSM

**Table 7.** Operation, strategy and expression of VPS

further investigation is clearly required.

**11. Main results**

follows.

**10.2. Corresponding structures on variant phase space**

*N* Dimension Dimension Vector Dimension on value space *X N*bit vector Random events I/O vectors

*J n*-function Probable function Selected function

of measurements of measurements

Ensemble Maxwell-Boltzmann distribution IM Interactive Canonical Ensemble Interactive Maps Gibbs distributions

Maxwell demons possible distributions

Matrices and interactive distributions

⇓ Top-down

⇑ Bottom-up

{*p*+, *p*−} Probability pairs {*q*, *p*} Conjugate pairs

PM Probability Density Probability Measurements Probability on each class ME Micro Phase Point Unit in Phase Space for Ensemble Maxwell-Boltzmann non interaction IP Interactive Micro Canonical Unit in Ensemble Projections Ensemble Gibbs with interaction

CIM CE&IM Matrices Global distribution Matrices for non-interactive

**Proposition 11.1:** Two types of probability measurements, Multiple and Conditional probabilities determine two distinct phase spaces, MPS and CPS.

**Proof:** In a PM module, multiple probabilities generates MPS and conditional probabilities create CPS.

**Proposition 11.2:** Two types of operations: symmetry/anti-symmetry and synchronous/asynchronous generate eight interactive projections.

**Proof:** Two pairs of measurement vectors {*u*, *v*} or {*u*˜, *v*˜} are involved in projections, where *u* = (*u*+, *u*−, *u*0, *u*1), *v* = (*v*+, *v*−, *v*0, *v*1) and *u*˜ = (*u*˜+, *u*˜−, *u*˜0, *u*˜1), *v*˜ = (*v*˜+, *v*˜−, *v*˜0, *v*˜1), each pair has eight interactive projections.

**Proposition 11.3:** Following a bottom-up approach, two CE and 16 IMs can be generated to exhaust all 2*<sup>N</sup>* input vectors for the relevant ME and IP measurements.

**Proof:** Results may be generated using a CEIM module and Proposition 11.3 is further supported by Propositions 11.1 to 11.2.

**Proposition 11.4:** Each CE is a statistical distribution and each IM corresponds to one of eight IP modes.

**Proof:** A pair of probability measurements has one fixed CE combination and each IP mode corresponds to one IM distribution.

**Proposition 11.5:** Both Proposition 11.3 and Proposition 11.4 provide a general Maxwell Demon mechanism.

**Proof:** For any function, CE and IMs can be fully and exhaustively generated without reference to thermodynamic issues.

**Proposition 11.6:** Exhausting ∀*J* ∈ *B*2*<sup>n</sup>* <sup>2</sup> , two sets of {CE} and 16 sets of {IM} can be generated, each set contains 22*<sup>n</sup>* elements and each element is a distribution.

**Proof:** Using the SCEIM module, they are natural outputs.

**Proposition 11.7:** In a variant logic framework, there are 2*n*! × 22*<sup>n</sup>* configurations for arranging a set of {CE} and eight sets of {IM} into a CEM and eight IMMs.

**Proof:** Since each IMM has the same organization as the CEM, a total of 2*n*! × 22*<sup>n</sup>* configurations can be distinguished and each configuration corresponds to a variant logic matrix.

**Proposition 11.8:** With a top-down approach, either a CEM or an IMM on a proper configuration can be composed of two polarized matrices. Each polarized matrix has periodic structures on its columns and/or rows.

**Proof:** Since a proper configuration is based on *n* periodic meta vectors and their combinations, its relative arrangements are invariant under permutation and complementary operations on the vector with 22*<sup>n</sup>* bits that determine each polarized structure.

**Proposition 11.9:** For MPS on C code conditions, a pair of measurements in a CEM can be arranged in a square with corners having values {[0, 0], [1, 0], [1/2, 1/2], [0, 1]}.

**Proof:** Under a C code configuration, the possible regions of measurements for a CEM in MPS can be shown in

$$\begin{array}{ccccc}\hline\\\text{MPS}: \text{CEM} = \begin{pmatrix} [1,0] & \dots \ (- ,-) & \dots \ [1/2, 1/2] \\ \dots & & \dots \\ (- ,0) & \dots & (- ,-) \\ \dots & & \dots \\ (0,0) & \dots & (0,-) & \dots \end{pmatrix} \end{array}$$

**Proposition 11.10:** For CPS on C code conditions, a pair of measurements in a CEM can be arranged in a square with corners having values {[0, 0], [1, 0], [1, 1], [0, 1]}.

**Proof:** Under a C code configuration, the possible regions of measurements for a CEM in CPS can be shown in

$$\text{CPS}: \text{CEM} = \begin{pmatrix} [1,0] & \dots (1,-) & \dots & [1,1] \\ \dots & & \dots \\ (- ,0) & \dots & (- ,1) \\ \dots & & \dots \\ (0,0) & \dots (0,-) & \dots & [0,1] \end{pmatrix}$$

(M1) *u*+ SL group

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(M2) *u*− SL group

#### **11.2. Predictions**

**Prediction 11.1:** Following a bottom-up strategy, it is not possible to determine CE properties using limited numbers of ME.

This prediction points towards a more general intrinsic restriction on uncertainty effects for incomplete procedures applied to random events.

**Prediction 11.2:** For a configuration that is not in a variant logic framework, there may be a square integral configuration capable of providing an approximate solution.

Periodic matrices could play a key role as core components of approximation procedures.

**Prediction 11.3:** A sound statistical interpretation of quantum mechanics can be established using VPS construction.

Since both top-down and bottom-up strategies are included, further exploration is feasible.

**Prediction 11.4:** VPS construction can provide a foundation based on logic and hierarchies of measurement levels for complex dynamic systems, statistical mechanics, and cellular automata.

Through VPS construction clearly offers significant potential, this prediction needs to be tested by solid experimental and theoretical results backed by evidence.

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48 Cellular Automata

MPS can be shown in

CPS can be shown in

**11.2. Predictions**

using limited numbers of ME.

using VPS construction.

automata.

incomplete procedures applied to random events.

**Proposition 11.9:** For MPS on C code conditions, a pair of measurements in a CEM can be

**Proof:** Under a C code configuration, the possible regions of measurements for a CEM in

**Proposition 11.10:** For CPS on C code conditions, a pair of measurements in a CEM can be

**Proof:** Under a C code configuration, the possible regions of measurements for a CEM in

**Prediction 11.1:** Following a bottom-up strategy, it is not possible to determine CE properties

This prediction points towards a more general intrinsic restriction on uncertainty effects for

**Prediction 11.2:** For a configuration that is not in a variant logic framework, there may be a

Periodic matrices could play a key role as core components of approximation procedures. **Prediction 11.3:** A sound statistical interpretation of quantum mechanics can be established

Since both top-down and bottom-up strategies are included, further exploration is feasible. **Prediction 11.4:** VPS construction can provide a foundation based on logic and hierarchies of measurement levels for complex dynamic systems, statistical mechanics, and cellular

Through VPS construction clearly offers significant potential, this prediction needs to be

square integral configuration capable of providing an approximate solution.

tested by solid experimental and theoretical results backed by evidence.

[1, 0] ... (−, −) ... [1/2, 1/2] ... ... (−, 0) ... (−, −) ... ... (0, 0) ... (0, −) ... [0, 1]

[1, 0] ... (1, −) ... [1, 1] ... ... (−, 0) ... (−, 1) ... ... (0, 0) ... (0, −) ... [0, 1]

arranged in a square with corners having values {[0, 0], [1, 0], [1/2, 1/2], [0, 1]}.

arranged in a square with corners having values {[0, 0], [1, 0], [1, 1], [0, 1]}.

*CPS* : *CEM* =

*MPS* : *CEM* =

(M5) *v*+ SL group

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(M6) *v*− SL group

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50 Cellular Automata

(M3) *u*<sup>0</sup> SL group

(M4) *u*<sup>1</sup> SL group

163

(M9) *u*+ W group

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(M10) *u*− W group

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52 Cellular Automata

(M7) *v*<sup>0</sup> SL group

(M8) *v*<sup>1</sup> SL group

165

(M13) *v*+ W group

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(M14) *v*− W group

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54 Cellular Automata

(M11) *u*<sup>0</sup> W group

(M12) *u*<sup>1</sup> W group

(M17) *u*+ F group

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(M18) *u*− F group

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56 Cellular Automata

(M15) *v*<sup>0</sup> W group

(M16) *v*<sup>1</sup> W group

169

(M21) *v*+ F group

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(M22) *v*− F group

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58 Cellular Automata

(M19) *u*<sup>0</sup> F group

(M20) *u*<sup>1</sup> F group

(M25) *u*+ C group

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(M26) *u*− C group

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60 Cellular Automata

(M23) *v*<sup>0</sup> F group

(M24) *v*<sup>1</sup> F group

(M29) *v*+ C group

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(M30) *v*− C group

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62 Cellular Automata

(M27) *u*<sup>0</sup> C group

(M28) *u*<sup>1</sup> C group

(C1) *u*˜+ SL group

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(C2) *u*˜− SL group

**Figure 9.** (M1-M32) IMM for MPS; (M1-M8) SL group; (M9-M16) W group; (M17-M24) F group; (M25-M-32) C group.

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64 Cellular Automata

(M31) *v*<sup>0</sup> C group

(M32) *v*<sup>1</sup> C group **Figure 9.** (M1-M32) IMM for MPS; (M1-M8) SL group; (M9-M16) W group; (M17-M24) F group; (M25-M-32) C group.

(C5) *v*˜+ SL group

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(C6) *v*˜− SL group

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66 Cellular Automata

(C3) *u*˜0 SL group

(C4) *u*˜1 SL group

(C9) *u*˜+ W group

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(C10) *u*˜− W group

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68 Cellular Automata

(C7) *v*˜0 SL group

(C8) *v*˜1 SL group

181

(C13) *v*˜+ W group

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(C14) *v*˜− W group

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70 Cellular Automata

(C11) *u*˜0 W group

(C12) *u*˜1 W group

183

(C17) *u*˜+ F group

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(C18) *u*˜− F group

(C16) *v*˜1 W group

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72 Cellular Automata

(C15) *v*˜0 W group

(C16) *v*˜1 W group

185

(C21) *v*˜+ F group

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(C22) *v*˜− F group

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74 Cellular Automata

(C19) *u*˜0 F group

(C20) *u*˜1 F group

187

(C25) *u*˜+ C group

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(C26) *u*˜− C group

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76 Cellular Automata

(C23) *v*˜0 F group

(C24) *v*˜1 F group

189

(C29) *v*˜+ C group

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(C30) *v*˜− C group

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78 Cellular Automata

(C27) *u*˜0 C group

(C28) *u*˜1 C group

191

**12. Conclusion**

approach.

a 22*n*−<sup>1</sup>

future.

2010KS06).

**Author details**

× 22*<sup>n</sup>*−<sup>1</sup>

transformation.

This chapter provides a brief investigation into Variant Phase Space (VPS) construction. Using an *n* variable 0-1 function and an *N* bit vector, a VPS hierarchy can be progressively established via variant measures, multiple or conditional probability measurements, and selected pair of measurements to determine a Micro Ensemble (ME) and its eight interactive projections. Collecting all possible 2*<sup>N</sup>* pairs of probability measurements, a Canonical Ensemble (CE) and its eight Interactive Maps (IMs) are generated following a bottom-up

a result comprising a {CE} and eight sets of {IM}. Using either a CE or an IM as an element, it is possible to use a variant logic configuration to organize each set of distributions to be

a top-down approach, a CEM or IMM can be decomposed into two polarized matrices with each matrix having periodic properties that meet the requirements of a Fourier-like

The main results are presented as ten propositions and four predictions to provide a foundation for further exploration of quantum interpretations, statistical mechanics, complex

The chapter does not explore global properties in detail, and further detailed investigations

Anticipating that the principles put forward in this chapter will prove to be well founded, we look forward to exploring advanced scientific and technological applications in the near

Thanks to Professor Hui C. Shen of USTC for the selected works of de Broglie, and a historical review of statistical interpretation and modern development of statistical mechanics, to Colin W. Campbell for help with the English edition, to Jie Wan for MPS and CPS figures, to The School of Software Engineering, Yunnan University, The Key Laboratory of Software Engineering of Yunnan Province, and The Yunnan Advanced Overseas Scholar Project (W8110305) for financial support to the Information Security research projects (2010EI02,

matrix as a CE Matrix (CEM) or IM Matrix (IMM), respectively. Following

functions can be calculated to create

Interactive Maps on Variant Phase Spaces http://dx.doi.org/10.5772/51635 193

Applying a Maxwell demon mechanism, all possible 22*<sup>n</sup>*

dynamic systems, and cellular automata.

Jeffrey Zheng1, Christian Zheng2 and Tosiyasu Kunii<sup>3</sup>

2 University of Melbourne, Australia

3 University of Tokyo, Japan

1 Yunnan University, Key Lab of Yunnan Software Engineering, P.R. China

and expansions are necessary.

**Acknowledgements**

**Figure 10.** (C1-C32) IMM for CPS; (C1-C8) SL group; (C9-C16) W group; (C17-C24) F group; (C25-C32) C group.

### **12. Conclusion**

80 Cellular Automata

(C31) *v*˜0 C group

(C32) *v*˜1 C group

**Figure 10.** (C1-C32) IMM for CPS; (C1-C8) SL group; (C9-C16) W group; (C17-C24) F group; (C25-C32) C group.

This chapter provides a brief investigation into Variant Phase Space (VPS) construction. Using an *n* variable 0-1 function and an *N* bit vector, a VPS hierarchy can be progressively established via variant measures, multiple or conditional probability measurements, and selected pair of measurements to determine a Micro Ensemble (ME) and its eight interactive projections. Collecting all possible 2*<sup>N</sup>* pairs of probability measurements, a Canonical Ensemble (CE) and its eight Interactive Maps (IMs) are generated following a bottom-up approach.

Applying a Maxwell demon mechanism, all possible 22*<sup>n</sup>* functions can be calculated to create a result comprising a {CE} and eight sets of {IM}. Using either a CE or an IM as an element, it is possible to use a variant logic configuration to organize each set of distributions to be a 22*n*−<sup>1</sup> × 22*<sup>n</sup>*−<sup>1</sup> matrix as a CE Matrix (CEM) or IM Matrix (IMM), respectively. Following a top-down approach, a CEM or IMM can be decomposed into two polarized matrices with each matrix having periodic properties that meet the requirements of a Fourier-like transformation.

The main results are presented as ten propositions and four predictions to provide a foundation for further exploration of quantum interpretations, statistical mechanics, complex dynamic systems, and cellular automata.

The chapter does not explore global properties in detail, and further detailed investigations and expansions are necessary.

Anticipating that the principles put forward in this chapter will prove to be well founded, we look forward to exploring advanced scientific and technological applications in the near future.

#### **Acknowledgements**

Thanks to Professor Hui C. Shen of USTC for the selected works of de Broglie, and a historical review of statistical interpretation and modern development of statistical mechanics, to Colin W. Campbell for help with the English edition, to Jie Wan for MPS and CPS figures, to The School of Software Engineering, Yunnan University, The Key Laboratory of Software Engineering of Yunnan Province, and The Yunnan Advanced Overseas Scholar Project (W8110305) for financial support to the Information Security research projects (2010EI02, 2010KS06).

#### **Author details**

Jeffrey Zheng1, Christian Zheng2 and Tosiyasu Kunii<sup>3</sup>

1 Yunnan University, Key Lab of Yunnan Software Engineering, P.R. China


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*Education Press and Springer* 5(2): 163–172.

## *Edited by Alejandro Salcido*

Cellular automata have become a core subject in the sciences of complexity due to their conceptual simplicity, easiness of implementation for computer simulation, and ability to exhibit a wide variety of amazingly complex behavior. These features of cellular automata have attracted the researchers' attention from a wide range of divergent fields of science.

In this book, six outstanding emerging cellular automata applications have been compiled. These contributions underline the versatility of cellular automata as models for a wide diversity of complex systems.

We hope that, after reading the outstanding contributions compiled in this book, we will have succeeded in bringing across what engineers and scientists are now doing about the application of cellular automata for solving practical problems in diverse disciplines. We also hope that this book will have been to your interest and liking.

Lastly, we would like to thank all the authors for their excellent contributions in the different topics of cellular automata covered in this book.

Emerging Applications of Cellular Automata

Emerging Applications

of Cellular Automata

*Edited by Alejandro Salcido*

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