**3. Intensional language for CM's description**

#### **3.1 Intensional CMs**

An essential characteristic of the semantic network of the species considered in this chapter is the possibility of linking the structures corresponding to CMs with logical formulas of a certain kind. Because the meaning of the CMs depend

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

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

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

2.Complex (poly-categorial), the number of branches in which is not limited (in

practice it is convenient to have from three to seven branches)

5.Collective (e.g., developed during the implementation of joint creative

possible to set the task of clarifying the classification of CMs both for cognitive modeling and developing the formalizations oriented to their analysis, processing,

It is easy to see that the classification is based on various reasons, which makes it

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

An essential characteristic of the semantic network of the species considered in this chapter is the possibility of linking the structures corresponding to CMs with logical formulas of a certain kind. Because the meaning of the CMs depend

3.Mnemonic, used to create an easily remembered image

of knowledge representation.

*Cognitive and Intermedial Semiotics*

1.Dyadic, containing two alternative branches

identified as follows:

4.Creative

projects)

and software generation.

interaction and computer science research.

**3.1 Intensional CMs**

**146**

**3. Intensional language for CM's description**

6.Artistic

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. Intensional logic can be chosen as such logic.

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


The solution of the problem is supposed to be obtained on the basis of a combination 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 definition is based on the adjoint functors.

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 intensional operators serve as the tools for setting contexts.

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

particular domain. Considerable interest is caused by the study of general categorical constructions, such as functors and natural transformations, in their application to CM. In particular, the adjoint functor construction can be used to describe abstract types of the data associated with CM nodes.

6. If *α*∈ *MEa*, then <sup>∧</sup>

7. If *<sup>α</sup>*<sup>∈</sup> *ME*h i *<sup>s</sup>*,*<sup>a</sup>* , then <sup>∨</sup>

than the set of CMs.

*3.2.3 Interpretation*

where

evaluation of CMs.

**149**

½ � *α* ∈ *ME*h i *<sup>s</sup>*,*<sup>a</sup>* .

*Computational Model for the Construction of Cognitive Maps*

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

½ � *α* ∈ *MEa*.

The language is the main technical tool to write formulas that are in the correspondence with CMs or their fragments. The set of formulas, however, is wider

Now let us introduce interpretation of intensional language. Let *A* and *Asg* be sets; *A* is a set of entities (or individuals), and *Asg* is a set of assignment points.

*<sup>D</sup>*h i *<sup>s</sup>*,*<sup>a</sup>* ,*A*,*Asg* <sup>¼</sup> *Da*,*A*,*AsgAsg :*

As a rule, sets *A* and *Asg* are fixed. Under these conditions, we denote *Da*,*A*,*Asg* � *Da*.

We treat A-assignment as a function *g*, whose domain is a set of variables, such that when *u* is a variable of type *a*, then *g u*ð Þ∈ *Da*,*A*,*Asg*. *G x*½ � *=u* means A-assignment:

We define the intension *α*<sup>A</sup>,*<sup>g</sup>* and the extensional *α*<sup>A</sup>,*Asg*,*<sup>g</sup>* meaningful expression *α* when using the usual recursive definition. Complete form of definition is presented in [1]. The intension is the possible value of CM, and interpretation is a tool for the

The interpretation of the constructions, composing the CM, is made up in the framework of type theory with dependent types of functions and pairs. The interpretation of CM is considered as an object, the type of which can be constructed.

*x*, if *u* � *v*, *g v*ð Þ otherwise*:*

*Da*,*A*,*Asg* ,

A ¼ h i *A*, *Asg*, *F* , (2)

(1)

(3)

8.There are no other meaningful expressions.

Define the set *Da*,*A*,*Asg* of possible denotates of type *a*:

We treat the interpretation as an ordered tuple:

2.*F* is a function whose domain is a set of constants.

*g x*½ � *=u* ð Þ¼ *v*

3. If *<sup>a</sup>*∈*<sup>Y</sup>* and *<sup>α</sup>*∈*Cona*, then *<sup>F</sup>*ð Þ *<sup>α</sup>* <sup>∈</sup> *Da*,*A*,*AsgAsg*.

**4. Problem domain description with CMs**

1.*A* and *Asg* are non-empty sets.

*De*,*A*,*Asg* ¼ *A*, *Dt*,*A*,*Asg* ¼ f g 0, 1 , *D*h i *<sup>a</sup>*,*<sup>b</sup>* ,*A*,*Asg* ¼ *Db*,*A*,*Asg*

### **3.2 Intensional language**

The intensional language contains tools for describing the nodes and links of CM. The description in the intensional language is a formalized object, matched to CM. Such an object can be used both for constructing the semantics of CM and (in practical terms) for representing CM for the purpose of storage and processing. The use of formalized objects also provides for syntactic and semantic control of objects, which makes easier their debugging and maintenance.

Types are assigned to the expressions of the intensional language; thus, the type *e* corresponds to the node of CM and the type *t* to the link between specific nodes. A set of language expressions is defined as an inductive class. This method of setting ensures the definition of CM construction operations from separate parts. The description of the language as a whole follows the paper [6]. The interpretation of language expressions is also set with the help of induction for the construction of an interpreted expression.

*3.2.1 Types*

The set of types of *ϒ* is defined inductively:


Types represent the sets of elements to interpret CMs or their fragments.

*3.2.2 Language*

We will use the enumerable set of variables and (infinite) set of constants of each type *a*. If *n* is a natural number and *a*∈*ϒ*, then *vn*,*<sup>a</sup>* is the *n*-th variable of type *a*, and *Cona* is a set of constants of type *a*.

The language includes a set of meaningful expressions *MEa* of each type *a*. It is defined recursively:

1.*vn*,*<sup>a</sup>* ∈ *MEa*; *Cona* ⊆ *MEa*.

2. If *α*∈ *MEb* and *u* is a variable of type *a*, *λuα* ∈ *ME*h i *<sup>a</sup>*,*<sup>b</sup>* .

3. If *α*∈ *ME*h i *<sup>a</sup>*,*<sup>b</sup>* and *β* ∈ *MEa*, then *α β*ð Þ∈ *MEb*.

4. If *α*, *β* ∈ *MEa*, then *α* ¼ *β* ∈ *MEt*.

5. If *φ*, *ψ* ∈ *MEt* and *u* is a variable, then ¬*φ*, ½ � *φ*∧*ψ* , ½ � *φ*∨*ψ* , ½ � *φ* ! *ψ* , ½ � *φ* \$ *ψ* , ∀*uψ*, <sup>∃</sup>*uψ*, □*ψ*, *<sup>W</sup>ψ*, and *<sup>H</sup><sup>ψ</sup>* <sup>∈</sup> *MEt*.

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

6. If *α*∈ *MEa*, then <sup>∧</sup> ½ � *α* ∈ *ME*h i *<sup>s</sup>*,*<sup>a</sup>* .

7. If *<sup>α</sup>*<sup>∈</sup> *ME*h i *<sup>s</sup>*,*<sup>a</sup>* , then <sup>∨</sup> ½ � *α* ∈ *MEa*.

8.There are no other meaningful expressions.

The language is the main technical tool to write formulas that are in the correspondence with CMs or their fragments. The set of formulas, however, is wider than the set of CMs.
