**6. References**


URL: *http://dx.doi.org/10.1111/j.0014-3820.2006.tb01159.x*

22 Will-be-set-by-IN-TECH

We will analyze one experiment using Homologous crossover and tournament selection to see generic behavior of GP problems given similar operators and selection pressure.

Figure 11 shows various quantitative genetic metrics similar to previous experiments

• The results of the experiment as seen in Figure 11 is comparative to the results on the ant problem in figure 5. We can see several trends that we saw before, for example, the curve for **D** follows the same type of path, converging until a fixed level of variation is

• Also, the perturbation curve and the population heritability curve show the same trend

In this chapter we have provided a detailed tutorial on quantitative genetics and some high level design methods to define phenotypic traits needed by quantitative genetics. Using these methods we performed various experiments changing the selection and breeding operator in GP to analyze different evolutionary behaviors of the problem. Evolutionary forces like exploration and exploitation were quantified using quantitative genetics tool set and some interesting correlation with other forces like bloat, diversity, convergence and fitness were made. Many observations and correlations made were generalized across different

In future we would like to perform more experiments to further understand the balance of bloat, selection and breeding operators, as well as designing new operators for resolving

[1] Altenberg, L. [1994]. The evolution of evolvability in genetic programming, *in* K. E. Kinnear (ed.), *Advances in Genetic Programming*, MIT Press, Cambridge, MA, pp. 47–74. [2] Altenberg, L. [1995]. The schema theorem and Price's theorem, *in* L. D. Whitley & M. D. Vose (eds), *Foundations of Genetic Algorithms III*, Morgan Kaufmann, San Francisco, CA,

[3] Bassett, J. K. & De Jong, K. [2011]. Using multivariate quantitative genetics theory to assist in ea customization, *Foundations of Genetic Algorithms 7*, Morgan Kaufmann, San

[4] Bassett, J. K., Kamath, U. & De Jong, K. A. [2012]. A new methodology for the GP theory toolbox, *Proceedings of the Genetic and Evolutionary Computation Conference (GECCO-2012)*,

• **Homologous Crossover with Tournament Selection size 2**

2. **Regression GP Experiment**

for quintic regression problem. 3. **Symbolic Regression Observations**

reached, and then staying there.

**5. Conclusions and future work**

benchmark GP problems.

**Author details**

**6. References**

pp. 23–49.

Francisco.

ACM.

issues in a given problem domain.

of continual increase over generations.

Uday Kamath, Jeffrey K. Bassett and Kenneth A. De Jong

*Computer Science Department, George Mason University, Fairfax, USA*

	- [20] Mühlenbein, H., Bendisch, J. & Voigt, H.-M. [1996]. From recombination of genes to the estimation of distributions: II. continuous parameters, *in* H.-M. Voigt, W. Ebeling, I. Rechenberg & H.-P. Schwefel (eds), *Parallel Problem Solving from Nature – PPSN IV*, Springer, Berlin, pp. 188–197.

**Continuous Schemes for Program Evolution**

**Chapter 2**

Genetic Programming (GP) is a technique aiming at the automatic generation of programs. It was successfully used to solve a wide variety of problems, and it can be now viewed as a mature method as even patents for old and new discovery have been filled, see e.g. [1, 2]. GP is used in fields as different as bio-informatics [3], quantum computing [4] or robotics [5],

The most widely used scheme in GP was proposed by Koza, where programs are represented as Lisp-like trees and evolved by a genetic algorithm. Many other paradigms were devised these last years to automatically evolve programs. For instance, linear genetic programming (LGP) [6] is based on an interesting feature: instead of creating program trees, LGP directly evolves programs represented as linear sequences of imperative computer instructions. LGP is successful enough to have given birth to a derived commercial product named *discipulus*. The representation (or genotype) of programs in LGP is a bounded-length list of integers. These integers are mapped into imperative instructions of a simple imperative language (a

While the previous schemes are mainly based on discrete optimization, a few other evolutionary schemes for automatic programming have been proposed that rely on some sort of continuous representation. These include notably Ant Colony Optimization in AntTAG [7, 8], or the use of probabilistic models like Probabilistic Incremental Program

In 1997, Storn and Price proposed a new evolutionary algorithm for continuous optimization, called Differential Evolution (DE) [11]. Another popular continuous evolution scheme is the Covariance Matrix Adaptation Evolution Strategy (CMA-ES) that was proposed by Hansen and Ostermeier [12] in 1996. Differential Evolution differs from Evolution Strategies in the way it uses information from the current population to determine the perturbation brought to

In this chapter, we propose to evolve programs with continuous representation, using these two continuous evolution engines, Differential Evolution and CMA Evolution Strategy. A

> ©2012 Fonlupt 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

©2012 Fonlupt 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.

Cyril Fonlupt, Denis Robilliard and Virginie Marion-Poty

Additional information is available at the end of the chapter

Evolution [9] or Bayesian Automatic Programming [10].

cited.

solutions (this can be seen as determining the direction of the search).

http://dx.doi.org/10.5772/50023

**1. Introduction**

among others.

subset of C for instance).

