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The set of parameters of the unenriched logical relationship system being a domain ontology model will be called the structure of domain knowledge model. It follows from this definition that if an unenriched logical relationship system without parameters is a domain ontology model, then any knowledge model of this domain has no structure. If a mixed unenriched logical relationship system with parameters is a domain ontology model, then a part of any knowledge model has a structure but the other its part has no structure. If a pure unenriched logical relationship system with parameters is a domain ontology model, then all parts of any knowledge model of the domain have a structure. Let a domain ontology model be an unenriched logical relationship system with parameters. If no parameter value in its turn contains a parameter, then all domain knowledge models for this conceptualization have the same structures. If values of some parameters in their turn contain parameters, then different knowledge models of the domain can have different structures.

Using parameters whose values contain parameters makes it possible to hide some terms used for knowledge description in domain ontology model description. At the same time, the meanings of these terms are completely determined by the propositions describing the sorts of these terms.

A Comparison between Different Ontology Model Classes Now let us discuss the question about capabilities of domain models and domain ontology models, which are enriched and unenriched logical relationship systems of different classes.

Let us consider several aspects of the term domain ontology.

1. If a conceptualization contains intended situations of different structures, then any ontology representing this conceptualization cannot have a model in the class of unenriched logical relationship systems without parameters but can have a model in the class of the systems with parameters.

2. If a conceptualization contains concepts designated by terms for knowledge description, then no ontology representing this conceptualization can have a model in the class of unenriched logical relationship systems without parameters, but it can have a model in the class of the systems with parameters.

3. If a conceptualization contains concept classes and determines properties of the concepts belonging to these classes, and concepts themselves are introduced by domain knowledge, then no ontology representing this conceptualization can have a model in the class of unenriched logical relationship systems without parameters but can have a model in the class of the systems with parameters.

4. If in a conceptualization some restrictions on meanings of terms for situation description depend on the meaning of terms for knowledge description, then any ontology representing this conceptualization cannot have a model in the class of unenriched logical relationship systems without parameters, but it can have a model in the class of the systems with parameters.

XII-th International Conference "Knowledge - Dialogue - Solution" 5. The more compactly and clearly domain ontology models of a class describe agreements about domains, the better the class is. In this regard unenriched logical relationship systems without parameters require for every term for situation description to appear explicitly in these agreements. For real domains (such as medicine) the models of their ontologies turn out immense because of large number of these terms. At the same time, the systems with parameters describe agreements about domains for groups of terms, rather then only for isolated terms through using terms for knowledge description. In doing so the majority of the terms for situation description and some terms for knowledge description do not appear explicitly in agreement descriptions (they are replaced by the variables whose values are terms from appropriate groups). As a result, a model of agreements becomes compact and agreements themselves become more general.

6. The more understandable knowledge bases represented in terms of an ontology are for domain specialists, the better the class of domain ontology models is. In this respect unenriched logical relationship systems without parameters represent knowledge bases as sets of arbitrary logical formulas. The more complex these formulas are, the more difficult it is to understand them. At the same time, the systems with parameters introduce special terms for knowledge description. The meanings of these terms are determined by ontological agreements, and their connection with terms for situation description among them. In real domains these terms are commonly used to ease mutual understanding and to make communication among domain specialists economical. The meanings of these terms are, as a rule, understood equally by all domain specialists. The role of these terms is to represent domain knowledge as relation tables (sets of atomic formulas, of simple facts). It is considerably easier for domain specialists to understand the meanings of these simple facts than the meanings of arbitrary formulas.

7. The more precise approximation of a conceptualization model a class of domain ontology models assumes, the better it is.

First, let us remark that it follows from the theorem about eliminating parameters of enriched logical relationship systems [3] that if there is a domain model represented by an enriched logical relationship system with parameters which determines an approximation of the domain reality, then there is a model of the domain represented by an enriched logical relationship system without parameters which determines the same approximation of the domain reality. In this regard domain models represented by enriched logical relationship systems with parameters offer no advantages over domain models represented by enriched systems without parameters.

As for domain ontology models, every one represented by an unenriched logical relationship system determines some approximations for both the set of intended domain situation models and for the set of intended domain knowledge models. If a model O of a domain ontology represented by an unenriched logical relationship system P with parameters determines an approximation A(< OP, k >) of the set of intended domain situation kEn(OP ) models, then the unenriched logical relationship system O without parameters quasiequivalent to O determines X P the approximation A(< OX, k >) of the same set of intended situation models [1]. Let h: En(OP)En(O ) X kEn(OX ) be the map defined by the theorem about eliminating parameters of unenriched logical relationship systems and H = {h(k) k En(O )}. Then A(< OX, h(k) >) A(< OX, k >) ; but P A(< OX,k >) = kEn(OX ) kEn(OP ) kEn(OX )\H by the theorem about eliminating parameters of enriched logical relationship systems A(< OX,h(k) >) = kEn(OP ) A(< OP, k >) A(< OX, k >). Thus, the A(< OP,k >), i.e. A(< OX,k >) = kEn(OP ) kEn(OX ) kEn(OP ) kEn(OX )\H approximation of the set of intended situation models determined by the system O, is less precise than the X approximation represented by the system O.

P If a model O of a domain ontology represented by an unenriched logical relationship system with parameters P determines an approximation En(O ) of the set of intended domain knowledge models, then the unenriched P logical relationship system O without parameters determines an approximation En(O ) of the same set of X X intended knowledge models. In this case H is a subset of En(O ), i.e. the approximation of the set of intended X knowledge models determined by the system O also is less precise than the approximation determined by the X system O. In what follows we show some reasons of this fact.

P Ontologies Let us consider the case when a domain ontology model is a pure unenriched logical relationship system O with P parameters. First, the constraints of knowledge models represented by O determine the set En(O ) as a proper P P subset of the set of all possible interpretations of the system O 's parameters, whereas, if the system O without P X parameters is a domain ontology model, then this ontology model contains practically no restrictions on the set En(O ). Second, for the theorem about eliminating parameters of enriched logical relationship systems a set of X formulas representing empirical and other domain laws can be deduced from every proposition setting up a correspondence between knowledge models and situation models and from parameter values. These formulas contain no parameters. It is obvious that the forms of these formulas are restricted and determined by the forms of propositions setting up a correspondence between knowledge models and situation models. At the same time, if a domain ontology is an unenriched logical relationship system O without parameters, then this system X imposes no restrictions on the form of formulas entering its enrichments.

Let us consider the case when a domain ontology is a mixed unenriched logical relationship system O = <, P> P with parameters. In this case, if k En(O ), then h(k) = ' " where the propositions belonging to ' are P deduced from every proposition of setting up a correspondence between knowledge models and situation models and from parameter values (taking into account the parameter constraints) and " is such a set of propositions that ' " is a semantically correct applied logical theory where is the set of all the X X propositions of which contain no parameters, i.e. H En(O ).

X Domain ontology models represented by unenriched logical relationship systems with parameters are thus seen to offer certain advantages over domain ontology models represented by unenriched logical relationship systems without parameters (see also [13]).

Conclusions In the article a notion a mathematical model of a domain ontology has been introduced, the representation of different elements of a domain ontology in this model of terms for situation description and situations themselves; of knowledge and terms for knowledge description; of mathematical terms and constructions; of auxiliary terms, and ontological agreements has been considered. The structures of situations and knowledge and their properties have been considered. The notion "a domain model has been discussed. Definitions of the notions precise ontology and precise conceptualization have been presented. Some merits and demerits of different domain ontology model classes have been discussed in details.

References 1. Kleshchev A.S., Artemjeva I.L. A mathematical apparatus for domain ontology simulation. An extendable language of applied logic // Int. Journal on Inf. Theories and Appl., 2005, vol 12, 2. PP. 149-157. ISSN 1310-0513.

2. Kleshchev A.S., Artemjeva I.L. A mathematical apparatus for ontology simulation. Specialized extensions of the extendable language of applied logic // Int. Journal on Inf. Theories and Appl., 2005, vol 12, 3. PP. 265-271. ISSN 1310-0513.

3. Kleshchev A.S., Artemjeva I.L. A mathematical apparatus for domain ontology simulation. Logical relationship systems // Int. Journal on Inf. Theories and Appl., 2005, vol 12, 4. PP. 343-351. ISSN 1310-0513.

4. Guarino N. Formal Ontology and Information Systems. In Proceeding of International Conference on Formal Ontology in Information Systems (FOIS98), N. Guarino (ed.), Trento, Italy, June 6-8, 1998. Amsterdam, IOS Press, pp. 3- 15/ 5. Kleshchev A.S., Moskalenko Ph. M., Chernyakhovskaya M.Yu. Medical diagnostics domain ontology model. Part 1. An informal description and basic terms definitions. In Scientific and Technical Information, Series 2, 2005, 12. PP. 1-7.

6. Artemjeva I.L., Tsvetnikov V.A. The fragment of the physical chemistry domain ontology and its model. In Investigated in Russia, 2002, 5, pp.454-474. http://zhurnal.ape.relarn.ru/articles/2002/042.pdf 7. Artemjeva I.L., Visotsky V.A., Restanenko N.V. Domain ontology model for organic chemistry. In Scientific and technical information, 2005, 8, pp. 19-27.

8. Artemjeva I.L., Restanenko N.V. Modular ontology model for organic chemistry. In Information Science and Control Systems, 2004, 2, pp. 98-108. ISSN 1814-2400.

9. Artemjeva I.L., Miroshnichenko N.L. Ontology model for roentgen fluorescent analysis. In Information Science and Control Systems, 2005, 2, pp. 78-88. ISSN 1814-2400.

10. Artemjeva I. L., Knyazeva M.A., Kupnevich O.A. Processing of knowledge about optimization of classical optimizing transformations // International Journal on Information Theories and Applications. 2003. Vol. 10, 2. PP.126-131. ISSN 1310-0513.

XII-th International Conference "Knowledge - Dialogue - Solution" 11. Artemjeva I. L., Knyazeva M.A., Kupnevich O.A. A Model of a Domain Ontology for "Optimization of Sequential Computer Programs". The Terms for the Description of the Optimization Object. In Scientific and Technical Information, Series 2, 2002, 12, pp. 23-28.(see also http://www.iacp.dvo.ru/es/) 12. Artemjeva I. L., Knyazeva M.A., Kupnevich O.A. A Model of a Domain Ontology for "Optimization of Sequential Computer Programs". Terms for Optimization Process Description. In Scientific and Technical Information, Series 2, 2003, 1, pp. 22-29. (see also http://www.iacp.dvo.ru/es/) 13. Kleshchev A.S., Artemjeva I.L. Domain Ontologies and Knowledge Processing. Technical Report 7-99, Vladivostok:

Institute for Automation & Control Processes, Far Eastern Branch of the Russian Academy of Sciences, 1999. 25p.

(see also http://www.iacp.dvo.ru/es/).

Authors Information Alexander S. Kleshchev kleschev@iacp.dvo.ru Irene L. Artemjeva artemeva@iacp.dvo.ru Institute for Automation & Control Processes, Far Eastern Branch of the Russian Academy of Sciences 5 Radio Street, Vladivostok, Russia . , . : , , . , .

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