It is shown that conceptual mathematics comes into contact with cognitive modeling through computational thinking, which acts as an intermediary and guide on the way to computing. Computational thinking is defined as the process of defining a clear, defined, step-by-step solution to a complex problem, going through the stages of decomposition, pattern recognition, abstraction, and algorithmization. Since the subject of computing is the variety of effects caused by information processes, corresponding generated combinations, which form variable domains, are studied. In turn, they are collected from separate processes represented by individual functions. The notion of the cognitive stage is introduced, on the basis of which transition processes that occur in domains are considered, both its being and becoming are taken into account.

The notion of an equational shell is studiedto involve the objects and their environment.Appropriate methods are studied as validembeddings of refined objects. The refinementprocess determines the linkages between the varietyof possible representations, giving rise to variantsof computations. The case study is equipped withthe adjusted equational systems that validatethe initial applicative framework.This paper proposes and develops an intelligent system based on conceptual mathematics. To fully utilize the potential of symbolic computations, combinator theory is applied. Interaction occurs in stages. In the initial state, a basic representation of knowledge about the linguistic reality is formed. These are the postulates of the initial formulation of combinator theory, relative to which consistency can be ensured. Next, the system is presented with a cue in the form of an equality constructed according to the rules of the linguistic reality. An attempt is then made to reconstruct the representation of reality taking this cue into account. In effect, the response is attached to the original knowledge base. If the attempt is successful, a transition to a new state of knowledge occurs, i.e., a derivative knowledge base is formed. This process occurs cyclically. The example demonstrates how a categorical theory of combinators is generated from the basic untyped theory of combinators.​

The notion of semantic neighborhood is introduced and its properties are studied. The semantic neighborhood of a concept is understood as a commutative diagram characterizing the associated natural transformations over the representing functors. They characterize the behavior of the domains over which individual variables can range. The domains consist of generalized elements implementing the representation of an individual-as-a-process. Such a parametrization is based on the idea of the stage of knowledge that is achieved along evolvents. Another parameter is the properties of individual domains. As an example, the problem of habitability of variable domains and the corresponding problem of transmigration of individuals are considered, for which a solution is given. The life cycle of individuals and the possible elevation and spread of the effect of their entanglement are analyzed. Following the way of computational thinking, decomposition of the overall information task is performed in a standard manner. An evolvent system is chosen as a pattern, reflecting the idea of existence and becoming. When abstracting, how one class emerges from another is revealed, which is the content of some resulting mathematical assumptions. As a result, algorithmization becomes possible, involving parametrization, indexing, and functors.

The actual task of developing tools to support computational thinking is considered. Homotopy type theory is studied as a formal basis for providing computational thinking. The basic objects of the applicative environment are constructed, and considerable attention is paid to the technological means of their use. The application operation (application of a function to an argument) is considered as the main means of use, and the correctness of the application is ensured by typing the objects involved in the application. A method of parameterization of objects is considered, which allows to introduce into consideration families of objects of various types that behave similarly during application. The resulting sets of objects make it possible to define operators immersed in the basic computing environment.

The means and constructions of conceptual mathematics that are most closely related to conceptual modeling are selected. This selection is made in full accordance with the doctrine of computational thinking. An example of a construction representing the processability of individuals and concepts is given. It is shown that functors and natural transformations can serve as a basis for the subsequent construction of non-standard conceptual modeling based on variable domains. On the basis of conceptual mathematics, a conceptual framework of cognitive modeling has been laid, which allows taking into account the stages of knowledge and the transitions between them. It turns out that using conceptual mathematics, one can express cognitive maps.

The problem of object identification, considered in various contexts, has long attracted the attention of researchers. The paper proposes the use of an approach to the solution based on homotopy theory. It is assumed that various identification methods can be assembled into a single object within a suitable category, representing a set of possible methods. Thus, consideration of identification methods as objects of the first order is provided. The methods of defining functions on identification methods that provide the introduction of an algebraic structure on the corresponding objects are considered. To define functions, the recursion technique is used, which allows you to build hierarchies of functions based on reflexively defined relationships. The proposed technique can serve as a basis for building object identification systems, including in the context of security support tasks in a Web environment.

A computational approach to cognitive modeling is proposed. The computational model is a parametric construction that takes into account cognitive stages and transitions between them.The cognitive model enables the idea of information processes, from their birth and appearance in a scope, evolution and canceling out their existence and disappearing from the scope. Process habitats are Lawvere's variable domains; inter-transition is based on the notion of channeled spreading of processes.

According to modern notions, computing is not separable from cognitive modeling and activity. This paper continues the tradition of the uniform approach and proposes a small number of general mechanisms that cope with the main known effects of computing as a science — the interaction of objects-as-processes, the interaction of processes with the environment, generalized interaction. As shown, the applicative prestructure (objects-as-processes, application) generates an applicative structure (processes, application, values), which ensures the generation of the result — the value of interactions, enabling the process of evaluation. The theory of combinators is used as the main (meta)mathematical means. A diagram mechanism has been developed that implements the emerging applicative computational system of object interaction and reflects the arity of accompanying the induced information processes. The processes are bidirectional in nature, both with a decrease in arity – reduction, and with an increase in arity – expansion.

In the present work, efforts have been made to create a configuration-based approach to knowledge extraction. The notion of granularity is developed, which allows fine-tuning the expressive possibilities of the semantic network. As known, the central issues for knowledge-based systems are what’s-in-a-node and what’s-in-a-link. As shown, the answer can be obtained from the functor-as-object representation. Then the nodes are functors, and the main links are natural transformations. Such a model is applicable to represent morphing, and the object is considered as a process, which is in a harmony with current ideas on computing. It is possible to represent information channels that carry out the transformations of processes. The possibility of generating displaced concepts and the generation of families of their morphs is shown, the evolvent of stages of knowledge and properties of the process serve as parameters.

The paper shows that the use of a composition of objects representing a data structure, in fact, means the creation of a kind of information channels on their basis, along which the computation process spreads. In the case of applying the applicative computational technology, the computation process is started by means of the operation of applying the function object to the argument object. Two ways are presented to generate function objects — to use either lambda expressions or combinators to represent them. The first method uses an abstraction meta-operator and associated variables, which leads to the use of substitution systems with the potential for side effects. In the second method, only constant combinator objects participate in the construction of the function, and their application to the argument triggers a conversion based on rewriting rules, which does not cause a side effect. A practical solution to the problem of synthesizing a compositional data structure can be mixed, when both lambda terms and combinators are involved in the computations, which reduces the length of expressions. As a result, a data structure appears, which is composed of compositions of argument objects, equipped with a generated set of supporting function objects.

The paper considers the automatic construction of information systems according to the formal specification of interaction with a user. The constructed system is provided with a proof of correctness in such a way that static verification of the compliance of the constructed system with its specification and the correctness of the construction itself is possible. The specification language is an extension of linear temporal logic, which allows the user setting the specification to set a class of possible options for the interaction of an information system with the environment and/or the user. The proposed solution is based on the implementation of the interpretive function, which checks the consistency of the specification and the interaction history at each stage.

In this paper, at a qualitative level, we mean the concept as an abstraction based on the characteristics of the perceived reality. In analytic activity, a concept is considered to be some label imposed on a phenomenon, which forces us to connect individual observations and perform generalizations. For convenience, it is identified with the name given to observations and events. Each concept is associated with some kind of its properties. A variable is often understood as a measured concept as a property that is associated with the concept and changes when measured. If the property does not change, then it is considered a constant. We proceed from the fact that concepts should have a mechanism for change. Under these assumptions, a two-parameter commutative diagram is constructed that connects possible worlds and property transformations. The resulting semantic map sets the framework for their variation. Thus, the regulation of the variability of the associated structure of variable concepts is achieved, which determines the possible relationships of the displaced concepts.

In this work, an individual process, or for short, an individual is selected, and the "history" of its transformations is traced, depending on the scenario. Scenario is considered as the restriction imposed on the behavior of individual. States, or separate scenes, are achieved by possible worlds, and transitions between worlds are given by the relation of reachability, or evolvent. The evolvent restricts a transition from one stage of knowledge to later stages. In fact, in this work, the only preliminary work has been done to study the dynamics of the individual and a leading example is formulated. Its formulation gives rise to the induced commutative diagrams in a representative functor category. The mathematical apparatus involved is based on the variable domains and the functor category that represents them. Its purpose is to develop a certain intuition of semantic modeling, and then, if possible, to form a cognitive system. Some limited semantic models have been implemented and are undergoing further practical testing and improvement.

This paper considers the applicative computing technology (ACT), within the framework of which the semantic analysis of a number of natural language constructs is performed. The necessary elements of grammatical analysis are involved. Much first-order logical means is used, and predicate variables are necessarily used for analysis. At the same time, the advantages of higher-order systems, which include ACT, are extracted. The fact is that the semantics of a natural language is characterized by a multi-tiered nesting of grammatical structures, which conflicts with first-order logical systems.

Computational activity is now recognized as a natural science, and computational and information processes have been discovered in the deep structures of many areas. Computations in the natural world were present long before the invention of computers, but a remarkable shift in understanding its fundamental nature occurs, in fact, before our eyes. The present moment, in fact, is a transition from the concept of computer science as an artificial science to the understanding that information processes are abundant in nature. Computing is recognized as a natural science that studies natural and artificial information processes.

Applicative computing systems and technologies have taken a strong position in modern computing. In this paper, the basic applicative system, whether it is a lambda calculus or a system of combinators, is considered as a prototype concept system, using which it is possible to build individual systems that are practically significant for mathematics, computing, or programming. They are families of computational models that have both their own semantics and applied areas. This conceptualization/individualization technique is characteristic of the field of semantic studies. As it turns out, the applicative approach forms a metatheoretical framework that provides the basis for cognitive systems that consider abstract objects and interpret their properties and behavior in the environment of modern computing.

The task of modeling a subject whose cognitive activity is directed under the influence of the information environment continues to remain relevant. The difficulties in solving this problem are due to (1) the dependence of the truth of information on time, as a result of which the subject can make decisions based on outdated information, and (2) the heterogeneous nature of the information itself, which, in particular, can be intentionally false (fake). The paper proposes a model of interaction of subjects, including the exchange of information marked both by the time of its creation (receipt) and the quality of information. The model is based on a system of interacting processes and is presented as a set of information graphs describing the stages of interaction. A system of typing graphs and their constituent elements is proposed. The composition tools that allow one to assemble complex graphs from simpler ones are presented. The paper also presents a set of tools corresponding to the proposed model. The tools include a process language interpreter that provides computation of model constructs, as well as an information graph editor, type control tools, debugging tools, and configuration management tools for the computing environment. The proposed tools have been experimentally tested in the development of application systems from the field of implementing the best available technologies.

This paper considers the application of conceptual mathematics to construct a computational model of concepts and individuals. An applicative approach is systematically applied to build a concept-as-process design. Since modern computing considers information processes as the main objects of modeling, the developed design is indeed representative of the semantic processing of information. The nature of concepts – what concepts are – and the constraints that govern the theory of concepts have been, and continue to be, the subject of debate. It is especially interesting to discuss the nature of concepts in connection with the recently established fundamental nature of information processes that are attributable to all phenomena and events occurring in the world around us. The current trend elevates information processes to forms of computing, which can also be implemented through practices, for example, in the form of programming. The deep component is computational models, one way or another expressed by means of mathematics and metamathematics. The main meta-operations used are abstraction and application. Of greatest interest is functional abstraction and application in the form of applying a function to an argument. Despite this “conceptual minimalism”, a rich theory of concepts can be developed. Using this theory, it is possible to focus further discussion not only on the nature of concepts, but also to characterize the position on each of the five important issues that are central to many theories of concepts: (1) ontology of concepts, (2) structure of concepts, (3) empiricism, and nativism about concepts, (4) concepts and natural language, and (5) concepts and conceptual analysis.

The paper considers a model of knowledge extraction based on the conceptual modeling of user interaction with a domain-oriented virtual environment. The environment is modeled as a network of information graphs that changes its structure over time. This allows us to set the task of supporting the modeling of the cognitive activity of the subject in a changing environment. The description of changes is made on the basis of a parameterized computational model using the construction of a variable domain. The paper shows that a given set of domain variables can be considered as nested in a topos, which provides a natural construction of program structures. The paper offers a tool for working with variable domains, which provides the task of semantically stable fragments of information graphs. The tool is a specialized applicative-type evaluator that provides computations in a changing environment. The build-up of network vertices and connections can lead to system contradictions, for the resolution of which it is necessary to include special handlers, the number of which can grow excessively. The evaluator provides a solution to the problem of managing handlers. Attempts to solve the mentioned difficulties and contradictions in practice lead to the idea of a multi-layer network architecture and semantic adapters.

The given paper considers an approach to the construction of information systems, the interaction of which with the user can be described in the form of a small set of formal requirements. The means of formal specification are proposed in the form of a language allowing to express requirements compactly and close to how they are formulated by the developer. The language is an extension of the language of linear temporal logic. The language support tools ensure the construction of a supporting environment that is sufficient for the construction of the system and static verification of its correctness. Based on the previously proposed approach to the automatic construction of systems on the example of a model of asynchronous discrete interaction (question-answer) with the user, the paper demonstrates that a statically verifiable solution to the problem in the form of a function with a certain signature does exist. The specification language permits to identify a variety of interaction protocols that can be implemented regardless of user actions.