They are associated with the domain ontology by the fact that the name of the logical theory representing the set of logical relationships contains the names of all the extensions used for description of this theory. Using mathematical terms and constructions with this interpretation does not constrain the possibility of unenriched logical relationship systems application for representation ontologies of different domains, and mathematics is among them. In the latter case mathematical terms and constructions play the role of elements of the metalanguage with completely defined syntax and semantics, and the other terms play the role of terms of (the domain) mathematics, their semantics being defined by an ontology.

Auxiliary terms are introduced to make a domain ontology description more compact. A value of an auxiliary term is defined by the values of other domain terms: of mathematical terms, terms for situation descriptions, terms for knowledge descriptions and other auxiliary terms. The definitions of auxiliary terms are represented by a set of value descriptions for names in a domain ontology model.

Ontological Agreements Ontological agreements about a domain are represented by a set of restrictions on the interpretation of names of the unenriched logical relationship system which is an ontology model of the domain. Ontological agreements are explicitly formulated agreements about restrictions on the meanings of the terms in which the domain is described (additional restrictions on capacity of the concepts designated by these terms).

If a domain ontology model is an unenriched logical relationship system without parameters, then all the ontological agreements are only constraints of situation models. The set of ontological agreements, in this case, can be empty, too. If a domain ontology model is an unenriched logical relationship system with parameters then the set of ontological agreements can be divided into three nonintersecting groups: constraints of situation models, i.e. the agreements restricting the meanings of terms for situation description; constraints of knowledge models, i.e. the agreements restricting the meanings of terms for knowledge description; agreements setting up Ontologies a correspondence between models of knowledge and situations, i.e. the agreements setting up a correspondence between the meanings of terms for situation and knowledge description. Every proposition of the first group must contain at least one unknown or a variable whose values are unknowns and cannot contain any parameters;

every proposition of the second group must contain at least one parameter or a variable whose values are parameters and cannot contain any unknowns; every proposition of the third group must contain at least one parameter or a variable whose values are parameters and at least one unknown or a variable whose values are unknowns. In doing so, the definitions of auxiliary terms should be taken into account.

Now let us define the informal notion of domain ontology using the formal notion of a domain ontology model. The part of information about a domain, which is represented by an ontology model of the domain, will be called an ontology of the domain. It immediately follows that a domain ontology contains a set of capacity concept definitions for situations description (it cannot be empty), a set of capacity concept definitions for knowledge description (it can be empty), characteristics of mathematical apparatus for domain description, a set of auxiliary term definitions (it can be empty), a set of restrictions on the meaning of terms for situation description (it can be empty), a set of restrictions on the meaning of terms for knowledge description (it can be empty), and a set of agreements setting up a correspondence between meanings of terms for situation description and for knowledge description (it can be empty).

A Domain Model If an unenriched logical relationship system O is a domain ontology model and k En(O) is a knowledge model of the domain, then the enriched logical relationship system [3] is a model of the domain. In this case the set of solutions A() is a model of the domain reality. Thus, the domain ontology model О determines a class of the domain models {|kEn(O)}. Every domain model consists of two parts: an ontology model O that is the same for the whole class and a knowledge model k, which is specific for a particular domain model .

The set of all possible situations in a domain which have ever taken place in the past, are taking place now and will take place in the future will be called the reality of the domain. Thus, the reality has the property that the persons studying the domain, the developers of its conceptualization and its models do not know the reality completely. Only a finite subset of situations forming the reality and having taken place in the past is known (although the information forming these situations also can be not completely known). We will suggest that relative to any conceptualization of a domain the hypothesis on its adequacy is true: the reality is a subset of the set of intended situations. In view of the reality definition it is evident that this hypothesis cannot be verified.

Hence, every adequate conceptualization imposes certain limitations on the notion of the reality.

A() is an approximation of the unknown set of models of all situations which are members of the domain reality. It is apparent that the better A() approximates the reality the more adequate the domain model is. A model of a domain is adequate to the domain, if the set of models of all the situations forming the domain reality is equal to the solution set of the enriched logical relationship system which is a model of the domain, i.e.

the reality approximation is precise.

We will consider only such domain ontology models O that there is the adequate model of the domain in the class of models of the domain determined by the ontology model O (the hypothesis on existence of the adequate domain model). The hypothesis on existence of the adequate domain model is stronger than the hypothesis on conceptualization adequacy. The first hypothesis states that there is such a knowledge base (an element of the set En(O)) that A() is the same as the set of models of all the situations of the domain reality. The second one states only that the latter set is a subset of the set of models of all the intended situations.

Inasmuch as the reality is not completely known (not all the situations which took place in the past and take place at present are known, and no future situation is known either), it is unknown for any domain model how well the reality model approximates the reality. Thus, it is impermissible to hold about any domain model that it is an adequate model of the domain. At the same time, a criterion of inadequacy can be formulated: a model of a domain represented by an enriched logical relationship system is an inadequate model of the domain, if such a situation is known which took place in the reality that its model is not a solution of the logical relationship system.

If a domain ontology model O is given, and inadequacy of a domain model is revealed, then experts usually look for some other model of the domain knowledge k' En(O), so that the domain model won’t be inadequate with respect to the available data (known situations). If in the process of storing empirical data (extending the set of known situations) it becomes clear that inadequacy of the current domain model is XII-th International Conference "Knowledge - Dialogue - Solution" sufficiently often found, and that the model has to be permanently modified, and that this process leads to constant increasing of the number of empirical laws and/or to constant growth of complexity of the knowledge model, then an aspiration may arise for finding another conceptualization of the domain and an ontology representing it (changing the paradigm) and for finding an adequate model of the domain within the restrictions of the new conceptualization.

A Precise Ontology and Conceptualization A domain ontology will be called precise, if the set of situation models forming the conceptualization represented by the ontology is equal to the set A(< O,k >), where O is an unenriched logical relationship system kEn(O) that is a model of the ontology, i.e. the approximation of the conceptualization determined by the ontology is precise.

A conceptualization will be called precise, if it is the same as the domain reality. It is apparent that precise conceptualizations are impossible for the domains related to the real world. But conceptualizations are possible for theoretical (imaginary) domains (mathematics, theoretical mechanics, theoretical physics and so on) for which their precision is postulated.

If an ontology and conceptualization are precise, then the unenriched logical relationship system O being a model of this ontology must have the following property: if is the adequate model of the domain where k En(O), then A() A() for any k' En(O). If O is an unenriched logical relationship system without parameters, then the empty set of propositions is this k.

The question arises of whether in the case of precise conceptualization the empty knowledge base is always consistent with the adequate domain model. Let us discuss this question using the example of an ontology of mathematics. An ontology of mathematics (or any one of its branches) consists of definitions and axioms. Any conceptualization of mathematics is assumed to be precise. At the same time, mathematical knowledge consists of theorems (lemmas, corollaries and so on) and their proofs. Since in mathematics any theorem is a logical consequence of the ontology, the theorems impose no additional restrictions on the reality model. Thus, both the empty knowledge base and a knowledge base containing any set of theorems determine adequate (and equivalent [1]) models of mathematics. The role of theorems is to make explicit the properties implicitly given by the ontology, and the role of proofs is to make evident the truth of theorems. Some theorems can have inflexible form (identities, inequalities and so on). So a mixed unenriched logical relationship system with parameters can be a natural ontology model for mathematics where terms identities, inequalities and others describe knowledge.

The Structure of Situations and Knowledge The set of the unknowns whose values form a model of a situation will be called the structure of the situation model. We will say that models of two situations have the same structures, if the sets of the unknowns forming the structures of these situations are the same. From this point of view, the models of all the situations belonging to the reality model of any domain model have the same structures, if this domain model is an enriched logical relationship system. As for the structures of intended situation models determined by a domain ontology model that is an unenriched logical relationship system, three cases are possible.

1. A domain ontology model is an unenriched logical relationship system without parameters. In this case all intended situation models have the same structures.

2. A domain ontology model is an unenriched logical relationship system with parameters, none of parameter values being able to contain unknowns. In this case all intended situation models also have the same structures.

3. A domain ontology model is an unenriched logical relationship system with parameters, values of some parameters being able to contain unknowns. In this case the models of the situations belonging to the reality models of different models of the domain (consistent with different knowledge models) can have different structures depending on knowledge models.

The structures of all the situations determined by the ontology model of example 1 of article [3] are the same.

They are formed by the unknowns diagnosis, partition for a sign, moments of examination, blood pressure, strain of abdomen muscles, and daily diuresis. The structures of all the situations determined by the ontology model of Ontologies example 6 of article [3] also are the same. They are formed by the unknowns cubes, balls, rectangular parallelepipeds, length of an edge, volume, substance, and mass.

The parameter signs in example 2 of article [2] contains unknowns (see propositions 2.2.1 and 2.2.13 in [2]).

Thus, situation models determined by this ontology model can have different structures. In [2] an example of a knowledge model for this ontology model was given (see example 3, propositions from 3.1.1 to 3.1.9). The structure of situation models corresponding to that knowledge model is formed by the unknowns diagnosis, partition for a sign, moments of examination, strain of abdomen muscles, blood pressure and daily diuresis. If in another knowledge model of the same ontology model the parameter signs has the different value signs {pain, temperature, discharge}, and the other parameters have some proper values, then the structure of situation models corresponding to this knowledge model is formed by the unknowns diagnosis, partition for a sign, moments of examination, pain, temperature and discharge, i.e. these structures differ from one another.

Using parameters whose values contain unknowns makes it possible “to hide” some terms used for situation description in domain ontology model description. At the same time, the meanings of these unknowns are completely determined by the propositions describing the sorts of these unknowns (see proposition 2.2.13 of example 2 of [2]): models of concepts designated by these unknowns are determined, for any unknown its meaning in a situation is determined (either the unknown is a name of a role, a functional relation or an unfunctional one), for every name of relation the number of its arguments, the sorts of its arguments and the sort of its result are determined.

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