**2. Related work**

The use of CMs for setting management methods of access to the distributed semantic network assumes the study of their expressive potential in the whole. In this connection approaches to formal definition of CM's semantics cause special interest. The diversity of approaches is reasoned not least by the fact that different researchers use the term "cognitive map" in various meanings.

The CM's applications cover different areas: sociology, economics, medicine, international relations, etc. Among the problems solved by CMs, the following ones can be singled out: (i) problems of conceptual modeling, especially in the context of initial understanding of problems in weakly structured subject areas; (ii) problems of further modeling of subject areas, especially if it is necessary to describe the dynamics; and (iii) management problems in the subject area. Some lines of research and application of CMs are shown in **Figure 1**.

Nevertheless, along with all the diversity of tools for creating CMs, including those widely disseminated ones, FreeMind [7], MindMeister [8], MindManager [9], Cacoo [10], MindMup [11], XMind [12] etc., only a small part of them can be considered as tool kits for cognitive modeling based on CMs or a ready-to-use tool for supporting cognitive architectures.

## *Computational Model for the Construction of Cognitive Maps DOI: http://dx.doi.org/10.5772/intechopen.90173*

**Figure 1.** *Lines of research and application of CMs.*

correctness of data interpretation and the construction of appropriate computational procedures. For this purpose this chapter uses the formalism developed on the basis of intensional logic [6]. The computational aspect is ensured, in particular, by the possibility to include the means of a typical lambda calculus into the logical

the entire set of assignment points or on some of its subsets.

The necessity to take into account the subjective view on the semantic network requires modeling the dependence of the interpretation of the system's structures on the subject. This requirement is considered in the intensional logic by defining an interpretation structure using a parameter, the assignment point. The value of each construction corresponds to specified parameter value. In this case the constructions of the language of intensional logic are divided into extensional and intensional ones. The value of the composite extensional construction at the specified assignment point is a function of the values of its constituent structures at the same assignment point. The determination of the value of the intensional composite constructions requires determining the values of its constituent constructions on

To take into account the interpretation of various entries of an information object requires the construction of models of interpretation dependence on the context. The context determination can also be performed using intensional constructions. In this case, it is possible to use intensional operators or constants—

The applied method of parameterization allows to take into account the seman-

This chapter is structured as follows. Section 2 describes some approaches to the definition and construction of cognitive maps; special attention is paid to the degree of use of semantic information. Section 3 contains a statement of the problem of supporting the language of description of cognitive maps and means of its interpretation and offers a solution as a variant of the language of intensional logic. Section 4 describes the use of cognitive maps in the description of the problem area on the example of dependent types. In Section 5 we propose an approach to build a support system of cognitive maps on the basis of adjoint functors. In conclusion, the results

The use of CMs for setting management methods of access to the distributed semantic network assumes the study of their expressive potential in the whole. In this connection approaches to formal definition of CM's semantics cause special interest. The diversity of approaches is reasoned not least by the fact that different

The CM's applications cover different areas: sociology, economics, medicine, international relations, etc. Among the problems solved by CMs, the following ones can be singled out: (i) problems of conceptual modeling, especially in the context of initial understanding of problems in weakly structured subject areas; (ii) problems of further modeling of subject areas, especially if it is necessary to describe the dynamics; and (iii) management problems in the subject area. Some lines of

Nevertheless, along with all the diversity of tools for creating CMs, including those widely disseminated ones, FreeMind [7], MindMeister [8], MindManager [9], Cacoo [10], MindMup [11], XMind [12] etc., only a small part of them can be considered as tool kits for cognitive modeling based on CMs or a ready-to-use tool

researchers use the term "cognitive map" in various meanings.

research and application of CMs are shown in **Figure 1**.

for supporting cognitive architectures.

system under consideration.

*Cognitive and Intermedial Semiotics*

intensions of higher orders.

are summarized briefly.

**2. Related work**

**142**

tic characteristics of users of various classes.

Regarding this some lines of research in the field of cognitive modeling should be mentioned. The work [13] proposes the most common approach. According to the approach, the cognitive modeling is "a line based on a knowledge-intensive interdisciplinary methodology for solving applied problems through cognitive maps with more or less support to special information technologies." In this case the cognitive map is understood as a formalized model of the situation that reflects the knowledge and/or beliefs of the subject, individual or collective, about the cause– effect impacts between the important factors of the situation.

Within the considered line, the formal models of CMs relate to the questions of reducing the risk, introduced by the human factor, when solving problems in various subject areas using CMs. The work [13] proposes an option of describing the approaches to the formal definition of the methods of interpreting CMs. This work distinguishes two approaches to the interpretation of CMs: descriptive (pinning methods of understanding the notions of the subject area) and normative (fixing the methods of solving problems in the subject area), which trace back to the approach accepted in the work [14]. The first approach aims to use CMs for developing an internal model of a man's knowledge about a certain situation. The second approach suggests CMs of different types as normative models (schemes or rules) for the external presentation of knowledge about situations.

In general, depending on the objective of the study, the details of the CM's definition differ from each other; in particular, CMs may have a different structure. In the whole within the formal approach, the CM definition is often extended to a cognitive graph. For example, the work [15] considers the structure of spatial knowledge that arises from the study of a new spatial environment and gives grounds for generalizing CMs up to cognitive graphs. The studies of the optimization of CM's representation adjoin the works of this type. Thus, the work [16] proposes a three-dimensional representation of CMs. The representation is based on the selection of the node kernels and daughter nodes, the nodes being located in three-dimensional space and being represented by balls of different radius. The proposed representation, as stated, enhances the cognitive clarity of the representation, which is interpreted as the ease of its intuitive understanding.

A cognitive map or, with a graph-based approach, a cognitive graph can represent parts of systems with a cognitive architecture and in this way be put in one or another cognitive architecture. The work [17] describes some cognitive architectures, the method of description giving an opportunity to think about the

compatibility of the presented architectures with the formalism of cognitive maps. The abovementioned work understands the cognitive architectures as software systems that might think about problems in different areas, develop ideas, adapt to new situations, and reflect upon themselves. To this end, the cognitive architectures are trying to provide evidence of which specific mechanisms successfully reproduce intellectual behavior and thereby contribute to cognitive science.

A somewhat different approach is adopted in the work [23], where cause–effect relationships are modeled based on interactive cognitive maps. A cognitive map is considered as a family of cognitive models. The models can be computed in parallel by exchanging data between themselves. In such conditions, the network implementation becomes natural, which also allows to hide data that a particular component "does not want" to make it visible to other components. The paper takes up the position that the adoption of the CM's network model leads to the construction of the CM's ecosystem, the development of which is managed by cognitive agents—

Apart are fuzzy cognitive maps. This rapidly advancing branch, develops the formalisms of cognitive maps. In the general case, a fuzzy cognitive map is defined as a set of nodes and links, the nodes being associated with the concepts of the domain, and links to causal relationships between concepts. Each node is associated with the degree of the presence of a concept in a situation—a number or an element of a qualitative scale with which a number is associated. The nodes of the graph also associate with numbers that determine the degree of influence of one concept to another. A positive number corresponds to an increase in the presence of the

The specified fuzzy cognitive map can serve to model the dynamics of a situation. To do this the initial degree of the concept's presence in the situation is set. Then the changes of the degrees of presence are determined in accordance with the links of the graph as the sum of the corresponding degrees of influence. The given process is repeated iteratively until it reaches the specified time limit. The experiments demonstrate that three main types of behavior are possible: (1) stabilization, i.e., convergence at a given point; (2) way out to the cyclic mode; and (3) chaotic

An approach based on fuzzy cognitive maps is exampled in the work [24]. This paper shows the use of cognitive maps for making the decision, which is understood as the choice of a single decision or a group from the given set of alternatives. The cognitive maps are used thanks to their ability to explain the applied process of thinking. The work studies the process of convergence of cognitive maps and their

The fuzzy cognitive maps can be used in different domains, including optimizing the learning process. One of the optimization techniques is to analyze data from learning management system logs and to identify patterns of users' behavior related to the content. The work [25] proposes the use of fuzzy cognitive maps to model the behavior of users of learning management systems. The proposed model describes the user's interaction with the content of the system and can be used to forecast the

The relational approach to the construction of CM's semantics is gradually getting more of dissemination. So, besides the already cited work [21], the relational approach is also accepted in the work [26], in which dynamic models of fuzzy relational cognitive maps are analyzed. A frame-based approach, accepted, for example, in [27], can be considered as a generalization of the relational approach. In this case the frames are considered as stereotypical structures that provide orientation in the physical or conceptual space. In addition to the orientation, the choice of path can be provided, which corresponds to the solution of the planning problem. The frame approach is a synthesis of graph representations and cognitive maps and solves problems connected with explaining orientation-based behavior on graphs or

The frame approach can be successfully applied both in systems with common objectives and in systems oriented to specific applications. Thus, the work [28] solves the problem of presenting historical knowledge on the basis of CMs,

corresponding concept and a negative one to its decrease.

*Computational Model for the Construction of Cognitive Maps*

*DOI: http://dx.doi.org/10.5772/intechopen.90173*

behavior, characterized by the absence of limit modes.

reaction of users to its training, test, and practical elements.

application for decision-making.

maps or when they are used in parallel.

**145**

system components.

The considered paper emphasizes three large classes of cognitive architecture's character-coded, emergent, and hybrid ones. The character-coded systems represent concepts using characters that can be manipulated using a given set of operations. The emergent architecture assumes the use of multi-node parallel models, in which the flow of information is represented as the propagation of signals between the nodes. The hybrid architecture combines both approaches, but this combination can be made in different proportions. It is obvious that the system based on cognitive maps in this classification must be referred to the character-coded architecture.

The flexible way to represent data with different degrees of abstraction using CMs grounds the possibility of data using to represent ontological information. The work [18] considers how CMs can be used for the work in the situation when the information is missing or is unreliable in e-commerce. The paper presents a knowledge management system based on CMs and ontology and also proposes a framework solution for joint use of information along with the use of a common repository based on CMs. Using CMs provides modeling of a virtual environment by generating and checking the sequence of events that take place in the environment when modeling.

An interesting use of the CM's capabilities to represent dynamic information is the modeling of cause–effect (causal) relationships. Thus, the work [19] identifies the cognitive nature of a business model designated for a cognitive representation that describes business development activities. Attention is also drawn to the cause– effect structure of the business model, that is, the model of cause–effect relationships which, according to top managers or entrepreneurs' view, connects the creation of value and activities for its creation. The conceptualization and analysis of business models as cognitive maps can shed light on four important properties of the causal structure of a business model: levels of complexity, focusing and clustering, which characterize the causal structure, as well as the mechanisms underlying causal relationships shown in this structure.

There were some attempts to model CMs with the help of more general modallogical contexts. The work [20] proposes an interpretation of cognitive maps, correlated with elements of large-scale spatial environment, for constructing geometrically impossible environments. Then the constructed CMs are proposed for joint interpretation with geometrically possible maps. Such an interpretation logically corresponds to the possibility of considering the interpreted cognitive map from different points of view, and the case of geometrically impossible interpretation is not excluded in advance.

The CM's use in the network environments (in particular, in WWW) is based on the CM's capabilities to represent information in a form that allows storing some nodes of map on separate nodes of the computational network, as well as parallel processing of stored information. Thus, for example, the work [21] is a description of the CM's use for working in a multi-agent environment. The strength of this work is the exact semantics of CMs, based on relational algebra according to [22]. Unfortunately, the constructed semantics has a very special form due to the chosen ad hoc three-valued logic system. Nevertheless, within the framework of the chosen semantics, it is possible to construct the forms for representation (when describing the subject area) of the agents' point of view on cognitive maps, as well as to determine decision-making procedures for such agents.

compatibility of the presented architectures with the formalism of cognitive maps. The abovementioned work understands the cognitive architectures as software systems that might think about problems in different areas, develop ideas, adapt to new situations, and reflect upon themselves. To this end, the cognitive architectures are trying to provide evidence of which specific mechanisms successfully reproduce

The considered paper emphasizes three large classes of cognitive architecture's character-coded, emergent, and hybrid ones. The character-coded systems represent concepts using characters that can be manipulated using a given set of operations. The emergent architecture assumes the use of multi-node parallel models, in which the flow of information is represented as the propagation of signals between the nodes. The hybrid architecture combines both approaches, but this combination can be made in different proportions. It is obvious that the system based on cognitive maps in this classification must be referred to the character-coded architecture. The flexible way to represent data with different degrees of abstraction using CMs grounds the possibility of data using to represent ontological information. The work [18] considers how CMs can be used for the work in the situation when the information is missing or is unreliable in e-commerce. The paper presents a knowledge management system based on CMs and ontology and also proposes a framework solution for joint use of information along with the use of a common repository based on CMs. Using CMs provides modeling of a virtual environment by generating and checking the sequence of events that take place in the environ-

An interesting use of the CM's capabilities to represent dynamic information is the modeling of cause–effect (causal) relationships. Thus, the work [19] identifies the cognitive nature of a business model designated for a cognitive representation that describes business development activities. Attention is also drawn to the cause– effect structure of the business model, that is, the model of cause–effect relationships which, according to top managers or entrepreneurs' view, connects the creation of value and activities for its creation. The conceptualization and analysis of business models as cognitive maps can shed light on four important properties of the causal structure of a business model: levels of complexity, focusing and clustering, which characterize the causal structure, as well as the mechanisms underlying

There were some attempts to model CMs with the help of more general modallogical contexts. The work [20] proposes an interpretation of cognitive maps, correlated with elements of large-scale spatial environment, for constructing geometrically impossible environments. Then the constructed CMs are proposed for joint interpretation with geometrically possible maps. Such an interpretation logically corresponds to the possibility of considering the interpreted cognitive map from different points of view, and the case of geometrically impossible interpretation is

The CM's use in the network environments (in particular, in WWW) is based on the CM's capabilities to represent information in a form that allows storing some nodes of map on separate nodes of the computational network, as well as parallel processing of stored information. Thus, for example, the work [21] is a description of the CM's use for working in a multi-agent environment. The strength of this work is the exact semantics of CMs, based on relational algebra according to [22]. Unfortunately, the constructed semantics has a very special form due to the chosen ad hoc three-valued logic system. Nevertheless, within the framework of the chosen semantics, it is possible to construct the forms for representation (when describing the subject area) of the agents' point of view on cognitive maps, as well as to

intellectual behavior and thereby contribute to cognitive science.

ment when modeling.

*Cognitive and Intermedial Semiotics*

not excluded in advance.

**144**

causal relationships shown in this structure.

determine decision-making procedures for such agents.

A somewhat different approach is adopted in the work [23], where cause–effect relationships are modeled based on interactive cognitive maps. A cognitive map is considered as a family of cognitive models. The models can be computed in parallel by exchanging data between themselves. In such conditions, the network implementation becomes natural, which also allows to hide data that a particular component "does not want" to make it visible to other components. The paper takes up the position that the adoption of the CM's network model leads to the construction of the CM's ecosystem, the development of which is managed by cognitive agents system components.

Apart are fuzzy cognitive maps. This rapidly advancing branch, develops the formalisms of cognitive maps. In the general case, a fuzzy cognitive map is defined as a set of nodes and links, the nodes being associated with the concepts of the domain, and links to causal relationships between concepts. Each node is associated with the degree of the presence of a concept in a situation—a number or an element of a qualitative scale with which a number is associated. The nodes of the graph also associate with numbers that determine the degree of influence of one concept to another. A positive number corresponds to an increase in the presence of the corresponding concept and a negative one to its decrease.

The specified fuzzy cognitive map can serve to model the dynamics of a situation. To do this the initial degree of the concept's presence in the situation is set. Then the changes of the degrees of presence are determined in accordance with the links of the graph as the sum of the corresponding degrees of influence. The given process is repeated iteratively until it reaches the specified time limit. The experiments demonstrate that three main types of behavior are possible: (1) stabilization, i.e., convergence at a given point; (2) way out to the cyclic mode; and (3) chaotic behavior, characterized by the absence of limit modes.

An approach based on fuzzy cognitive maps is exampled in the work [24]. This paper shows the use of cognitive maps for making the decision, which is understood as the choice of a single decision or a group from the given set of alternatives. The cognitive maps are used thanks to their ability to explain the applied process of thinking. The work studies the process of convergence of cognitive maps and their application for decision-making.

The fuzzy cognitive maps can be used in different domains, including optimizing the learning process. One of the optimization techniques is to analyze data from learning management system logs and to identify patterns of users' behavior related to the content. The work [25] proposes the use of fuzzy cognitive maps to model the behavior of users of learning management systems. The proposed model describes the user's interaction with the content of the system and can be used to forecast the reaction of users to its training, test, and practical elements.

The relational approach to the construction of CM's semantics is gradually getting more of dissemination. So, besides the already cited work [21], the relational approach is also accepted in the work [26], in which dynamic models of fuzzy relational cognitive maps are analyzed. A frame-based approach, accepted, for example, in [27], can be considered as a generalization of the relational approach. In this case the frames are considered as stereotypical structures that provide orientation in the physical or conceptual space. In addition to the orientation, the choice of path can be provided, which corresponds to the solution of the planning problem. The frame approach is a synthesis of graph representations and cognitive maps and solves problems connected with explaining orientation-based behavior on graphs or maps or when they are used in parallel.

The frame approach can be successfully applied both in systems with common objectives and in systems oriented to specific applications. Thus, the work [28] solves the problem of presenting historical knowledge on the basis of CMs,

practically, on the basis of the frame approach. Actually, the CM's models are characterized as a specific type of dialectic interaction of logical and graphic forms of knowledge representation.

essentially on time, subject, etc., the logic appropriate for the basis for the interpretation of CMs must be explicitly focused on the consideration of semantic factors.

The intensional logic allows to operate with the formulas containing functional abstraction and application of function to arguments. Thus, it is possible to obtain the value of CM's structures using the evaluation. The result of the computation can also be represented as a CM's construction. In this case, the value depends on the parameter—the assignment point—which gives the CMs an intensional character. The need for an intensional description of CMs leads to the problem of determining the language means of parameterized computation of semantic network structures as the task of developing methods to support a specialized language for describing the

1.The definition of means of interpretation of CM's structures on the basis of

2.The definition of interpretation methods as specialized CMs, which can be

3.The definition of general limitations on interpretation methods, as well as procedures for the harmonization of interpretations that ensure the

The solution of the problem is supposed to be obtained on the basis of a combi-

The research method centers on the systematic use of the formalization of CM with the further determination of the semantics of the constructed formal objects. The object formalization is carried out using methods of intensional logic by constructing an intensional language to describe the objects that compose the CM. The intensional nature of the language makes it possible to take into account the contexts of objects used. The means of intensional logic provide for both the definition of objects, the interpretation of which is independent of the context (extensional objects) and objects of a different kind, and the interpretation of which requires consideration of one or more contexts (intensional objects). The inten-

The semantics of objects is determined by the means of category theory. The use of category theory ensures a sufficient general definition of semantics, on the basis of which types of changes in the domain can be taken into account. Changes, in particular, can affect the domains of change of the variables of the CM description language, forming the so-called variable domains. Taking into account the changes allows describing the dynamic subject areas of the same CM, which in practical terms saves the efforts spent on developing and debugging the descriptions of CM use. The analysis of methods of CM use to describe the subject areas consists of systematic consideration of the applied formalized methods and the identification of stereotypical structures used to describe objects and situations specific to a

nation of methods of intensional logic to describe the language and applicative methods of interpretation to compute the values of CMs. At the same time, it is possible to describe some constructions of the domain model in the form of CMs. The chapter presents a description technique on the example of dependent types. Support to the implementation of intensional descriptions CMs requires the use of methods which agree with the methods of the description of the CMs. In this chapter, a functor technique is used for this purpose. The specialized functors are determined to represent CMs in supporting the programming environment. The

semantic network and means of its interpretation, which should provide:

embedded in objects that parameterize the interpretation

Intensional logic can be chosen as such logic.

*DOI: http://dx.doi.org/10.5772/intechopen.90173*

*Computational Model for the Construction of Cognitive Maps*

their assigned semantic characteristics

implementation of the imposed restrictions

definition is based on the adjoint functors.

**147**

sional operators serve as the tools for setting contexts.

The considered work contains a detailed classification of cognitive maps. Thus, depending on the construction technology, they distinguish (1) associative maps or mind maps based on associations and (2) conceptual maps that serve to represent the connections of concepts between them. Among the mind maps are the maps identified as follows:


It is easy to see that the classification is based on various reasons, which makes it possible to set the task of clarifying the classification of CMs both for cognitive modeling and developing the formalizations oriented to their analysis, processing, and software generation.

All described applications may be characterized by one common feature—they are either not based on the use of formal semantics and use CMs as a convenient representation of knowledge about the subject area for informal analysis or, at best, use CMs as a tool for determining a finite state machine of a special type. However, such an approach seems to unreasonably narrow the scope of CM's application. It seems more reasonable to consider cognitive maps as the formalism, providing, on the one hand, pinning informal considerations about the described subject area and, on the other hand, obtaining more or less formalized descriptions that are compatible with descriptions in modeling languages or even programming languages.

An important sphere of application of solutions based on cognitive maps is information support for legal applications. For example, the work [29] analyzes the findings and contributions of existing research in the field of decision-making about the confidentiality, and it proposes to fill up the gaps in the modern understanding by applying a cognitive architecture to model confidential decision-making. In order to solve the issues related to confidentiality, it is necessary to consider aspects of human cognition, using, for example, the methods used in human-computer interaction and computer science research.
