**6. Conclusions**

Likewise the previous example, it is possible to establish the properties *g* <sup>∗</sup>

. Now

Let us check the naturality of *ψ*. Once again we consider the arrows *h* : *A*<sup>0</sup> ! *A*

h i *a*1, … , *an* ∈ *FA*<sup>0</sup> and compute on it the values of the arrows that form the naturality

¼ *ψ*ð Þ *Gk* ∘ *g* ∘ *h* ð Þ¼ h i *a*1, … , *an*

ð Þ *Gk* <sup>∘</sup> *<sup>g</sup>* <sup>∘</sup> *<sup>h</sup>* <sup>∗</sup> ð Þ¼ *ai*

*<sup>k</sup>* <sup>∘</sup> *<sup>g</sup>* <sup>∗</sup> ð Þ <sup>∘</sup> *<sup>h</sup>* ð Þ¼ *ai*

¼ ð Þ *k* ∘ *ψg* ∘ *Fh* ð Þ¼ h i *a*1, … , *an* ¼ *k*ð*ψg Fh a* ð Þ ð Þ h i 1, … , *an* Þ ¼ ¼ *k*ð*ψg ha* ð Þ h i ð Þ<sup>1</sup> , … , *h a*ð Þ*<sup>n</sup>* Þ ¼

*<sup>g</sup>* <sup>∗</sup> ð Þ *h a*ð Þ*<sup>i</sup>* !

*k g* <sup>∗</sup> ð Þ ð Þ *h a*ð Þ*<sup>i</sup> :*

Comparing this example with the previous one, it is seen that the first example is indeed formally simpler, since it does not require the construction of sequences of elements. The sequences in this case can be considered as representing constructions that ensure the mapping of *Mon* to *Set*, meeting the requirements formulated at the beginning of the item. Indeed, (1) the formed constructions have a functorial nature, (2) the functions that provide naming and dereferencing are explicitly constructed, and (3) due to choosing the basic categories (V*ect* or ℳ*on*, respectively), the mapping of structures are ensured in the construction of the supporting programming environment. In this case the adjoint functors can be considered as a variant of the CM's

¼

*k g* <sup>∗</sup> ð Þ ð Þ *h a*ð Þ*<sup>i</sup>*

ð Þ *k* ∘ *ψg* ∘ *Fh* ðÞ¼ *t*

<sup>¼</sup> *<sup>k</sup>* <sup>Y</sup> *i*

The naturality of *ψ* is proven by the results coincidence.

representation technique by means of a practical programming system.

<sup>¼</sup> <sup>Y</sup> *i*

ð Þ *<sup>ψ</sup><sup>g</sup>* h i *<sup>a</sup>*1, … , *an* <sup>¼</sup> *<sup>g</sup>* <sup>∗</sup> ð Þ� *<sup>a</sup>*<sup>1</sup> … � *<sup>g</sup>* <sup>∗</sup> ð Þ¼ *an*

; this is why *ψg*<sup>0</sup> : *FA*<sup>0</sup> ! *M*<sup>0</sup>

*ψ*ð Þ *Gk* ∘ *g* ∘ *h* ðÞ¼ *t*

<sup>¼</sup> <sup>Y</sup> *i*

<sup>¼</sup> <sup>Y</sup> *i*

<sup>¼</sup> <sup>Y</sup> *i*

ð Þ *<sup>g</sup>* <sup>∘</sup> *<sup>h</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>g</sup>* <sup>∗</sup> <sup>∘</sup> *<sup>h</sup>* (60)

ð Þ *Gk* <sup>∘</sup> *<sup>g</sup>* <sup>∗</sup> <sup>¼</sup> *<sup>k</sup>* <sup>∘</sup> *<sup>g</sup>* <sup>∗</sup> , (61)

Y *i*

. Let us consider the element *t* ¼

∈ *Ob*ð Þ S*et* and *B*, *B*<sup>0</sup> ∈ *Ob*ð Þ ℳ*on* . We have *g*<sup>0</sup> ¼

*<sup>g</sup>* <sup>∗</sup> ð Þ *ai :* (62)

(63)

(64)

related to the compositions

*Cognitive and Intermedial Semiotics*

where *h* : *A*<sup>0</sup> ! *A* and *k* : *M* ! *M*<sup>0</sup>

, where *A*, *A*<sup>0</sup>

and

and *k* : *M* ! *M*<sup>0</sup>

diagram:

and

**162**

*Gk* ∘ *g* ∘ *h* : *A*<sup>0</sup> ! *GM*<sup>0</sup>

The chapter considered a variant of solving the problem to store the data in a web environment and provide an access to the data based on their semantics. The semantics of data may be referred both to ensuring that the information complies with the put restrictions and to the traceability of nature of the problems that are solved by the users of different classes. The data is assumed to describe a certain subject area.

To represent the semantic nature of the data in the work, a representation in the form of a semantic network was used. The semantic network was considered as a set of marked nodes and marked links connecting them. The chapter considered the ways to access the nodes of the network, providing both the omission of irrelevant nodes and the decomposition of nodes.

The tools of describing users and their means of access to data that takes into account the specifics of the tasks to be solved must combine enough power to distinguish the relevant elements of the description and simplicity. It makes it possible to practically use the descriptions without excessive detailing, traditionally leading to an increase in the volume and complexity of the description. The work used cognitive maps to describe subjective views on the domain.

The chapter determines the CMs as hierarchically organized sets of nodes connected by unlabeled links. CMs can also contain links between nodes that are not in a hierarchical relationship. Due to:


The cognitive maps cannot be considered as semantic networks. However, it is possible to propose matching procedures that will make it possible to consider CMs as a special type of semantic networks.

To determine the language of the description of the subjects and subjective points of view on the data, the work used a variant of intensional logic language. The essential feature of the language is the possibility to construct expressions that are indexable during interpretation, which makes it possible to study and use the dependence of expressions on a parameter. A number of methods for constructing CMs are distinguished, each of which is associated with a formula of intensional logic.

The semantics of intensional logic is constructed basing on recursively defined intensions. The inclusion of lambda expressions in the language and the definition of the corresponding semantic construction provide the computational nature of semantics. The interpretation of quantifiers and operators as special types of applications (applications of functions to the argument) makes it possible to determine all constructions of the model as applicative ones and attributes a computational nature to the models.

The constructed semantics makes it possible to express constructions in the form of CMs; these constructions describe the subject area from the point of view of experts. The chapter shows the possibility of such an expression with the example of the homotopy theory of types. Ever basic construction of the theory of types is accompanied by its presentation in the form of a cognitive map. The use of dependent type theory provides a subjective description of the subject area.

Computational methods for representing CM's semantics can be promoted to the level of support for processing CMs by means of a programming system. The work develops a functor technique for this. The model constructions, naming semantic elements (CM's or their fragments), are mapped onto the constructions of the representing environment with the help of the technique of conjoint functors. In this way, the computational model can be extended to CM's support techniques. This approach ensures the correctness of the tool kits and reduces the time for their development.

**References**

2019. pp. 145-154

pp. 187-196

[2] Kosikov SV, Ismailova LY,

Wolfengagen VE. Network modeling environment for supporting families of displaced concepts. In: Proc. of the Ninth Annual Meeting of the BICA Society; 22–24 August 2018; Prague, Czech Republic. 2019.

[3] Wolfengagen VE, Ismailova LY, Kosikov SV. A computational model for refining data domains in the property

reconciliation. In: 2016 Third International Conference on Digital Information Processing, Data Mining, and Wireless Communications, Moscow, 6–8 July 2016; Moscow,

[4] Kosikov SV, Ismailova LY,

[5] Ismailova LY, Kosikov SV, Wolfengagen VE. Data domains modeling situationally determined evaluation. In: Proc. of Intelligent Systems Conference (IntelliSys); 6–7 September 2018; London, UK. 2018.

[6] Ismailova LY, Wolfengagen VE, Kosikov SV. Basic constructions of the computational model of support for access operations to the semantic

network. In: Proc. of the 8th International Conference on Biologically Inspired Cognitive Architectures (BICA 2017); 1–6 August 2017; Moscow, Russia. 2018.

Wolfengagen VE. The presentation of evolutionary concepts. Advances in Intelligent Systems and Computing.

Russia. 2016. pp. 58-63

2018;**636**:113-125

pp. 972-976

pp. 83-188

**165**

[1] Ismailova LY, Wolfengagen VE, Kosikov SV. A computational model for supporting access policies to semantic web. In: Proc. of the Ninth Annual Meeting of the BICA Society; 22–24 August 2018; Prague, Czech Republic.

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

*Computational Model for the Construction of Cognitive Maps*

[7] FreeMind. Available from: http:// freemind.sourceforge.net/ [Accessed:

[8] MindMeister. Online Mind Mapping. Mind Mapping Software. Available from: https://www.mindmeister.com/

[10] Cacoo. Mind Maps Guide. Available from: https://cacoo.com/resources/ mind-maps-guide/ [Accessed: 01 June

[11] Mindmup. Tutorials and Guides. Available from: https://www.mindmup. com/tutorials/index.html [Accessed:

[12] XMind–Mind Mapping Software. Available from: https://www.xmind. net/developer/ [Accessed: 01 June 2019]

[13] Laboratory of Cognitive Modelling and Management of Development of Situations. Institute of Control Sciences of Russian Academy of Sciences. Available from: https://www.ipu.ru/ node/11921 [Accessed: 17 May 2019]

[14] Bell DE, Raiffa H, Tversky A, editors. Decision Making: Descriptive,

Interactions. Cambridge: Cambridge University Press; 1988. pp. 9-32

[15] Chrastil ER, Warren WH. From cognitive maps to cognitive graphs. PLoS One. 2014;**9**:e112544. DOI: 10.1371/journal.pone.0112544

[16] Ferreira Opaso EV, Terelanskiy PV. Representation of cognitive maps in 3-dimensional space. In: VSPU-2014,

Normative and Prescriptive

Moscow. 2014

01 June 2019]

01 June 2019]

01 June 2019]

2019]

[Accessed: 01 June 2019]

[9] MindManager Mind Mapping Software. Available from: https://www. mindjet.com/mindmanager/ [Accessed:

In the whole the constructed computational model makes the basis for the description of subjective views on the subject area, their representation in the model, and placing in a supporting programming environment. Thus, the model can serve as the basis for the development technique and maintenance of tool kits to support the description of the domain based on CMs. The elements of the model were tested when developing the practical information systems in the field of legal regulation of the best available technology implementation.
