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FAR EARSTERN BRANCH RUSSIAN ACADEMY OF SCIENCES Alexander S. Kleshchev , Irina L. Artemjeva DOMAIN ONTOLOGIES AND KNOWLEDGE PROCESSING. Technical Report 7-99 Vladivostok 1999 Kleshchev A.S., Artemjeva I.L. DOMAIN ONTOLOGIES AND KNOWLEDGE PROCESSING. Technical Report. Vladivostok: Far Eastern Branch of the Russian Academy of Sciences, 1999. 24 p. The practice of KBS development for complex domains and tasks has shown that every domain has a structure taking an intermediate position between a domain model (a knowledge base) and the knowledge representation used in the model. At present this structure is called "domain ontology". Modern approaches to defining the notion of domain ontology are discussed in the article. A mathematical apparatus is introduced which allows us to define the notions of domain model and domain ontology model. Statements of some problems of knowledge processing are given. It is shown that the problems are closely interconnected and connecting links among them are the notion of domain and the notion of domain ontology. A progress in solving these problems is closely connected with explicit representation and studying of mathematical ontology models of practically important domains corresponding to modern ideas about these domains. Complexity of these problems is such that their solving is possible only by using mathematical methods. In the article some ways of solutions of the problems using the notions introduced are discussed. Editor-in-chief Professor M.Yu.Chernyakhovskaya Reviewer Professor B.I.Cogan C Institute for Automation & Control Processes Far Eastern Branch Russian Academy of Sciences, 1999 3 INTRODUCTION Research on ontology is becoming increasingly widespread in the computer science community, and its importance is being recognized in a multiplicity of research fields and application areas, including knowledge engineering, databases design and integration, information retrieval and extraction [1]. However, among the members of the artificial intelligence community the meaning of the term of domain ontology is understood ambiguously. There have been many articles where their authors try to define this term but no definition is satisfactory. Also, no author tries to show that his definition could be useful for solving many different problems. The goal of this article is to give a new definition of domain ontology and to show that this definition helps to solve a few problems of knowledge processing. The paper has the following structure. In section 1 modern approaches to defining the notion of domain ontology are discussed. In sections 2 a mathematical apparatus is introduced. The apparatus allows us to define the notions of domain models and domain ontology model in sections 3 and 4. In section 5 statements of some problems of knowledge processing are given and some ways of their solutions using the notions introduced are discussed. This work was carried out with financial support from the Russian Fund of Fundamental Investigations (grant 99-01-00634). 1. DIFFERENT APPROACHES TO DEFINING THE NOTION OF DOMAIN ONTOLOGY. BACKGROUND The central notion of classical technologies for knowledge processing is the notion of knowledge representation. This term means either a way of coding knowledge in a knowledge base or a formal system used for formalising knowledge. The practice of KBS development for complex domains and tasks has shown that every domain has a structure taking an intermediate position between a domain model (a knowledge base) and the knowledge representation used in the model. In some early articles the structure was described informal for certain domains [2, 3]. Since the 80's this structure has been studied in a number of Russian works [4, 5]. In the late 1980's systematic studies of this structure began in the connection with some attempts to solve some of the problems given in section 5. This structure was called "domain ontology". At present there are three main approaches to defining the notion of domain ontology. The first one, called here as humanitarian, consists in attempting to define the content of the notion of domain ontology in terms understood intuitively. Some examples of such definitions are the following (1) "A (domain) (AI) ontology is a theory of what entities can exist in the mind of a knowledgeable agent" [6], (2)"An ontology defines a taxonomy of concepts, that we feel are required for the definition of the semantics of a language for representing knowledge. [7], (3) "An ontology is an explicit knowledge-level specification of a conceptualisation, i.e. the set of 4 distinctions that are meaningful to an agent. The conceptualisation - and therefore the ontology - may be affected by the particular domain it is intended for" [8]; (4) "An ontology is an explicit, partial account of a conceptualisation" [9]; (5) "An ontology is an explicit, partial specification of a conceptualisation that is expressible as a meta-level viewpoint on a set of possible domain theories" [10]. The chief defect of the definitions above and the whole humanitarian approach is that a technical notion necessary for solving technical problems cannot be defined in such a manner. The second approach to defining the notion of domain ontology can be called the computer one. Within the framework of the approach computer languages for domain ontology representation have being developed. Everything that can be described using such a language is considered as a domain ontology. Some examples of computer languages for domain ontology representation are given in [11-14]. Unfortunately, the authors do not explain the differences between languages for describing ontologies and knowledge representation languages (for example, KRL [15]). The semantics of the both classes languages is equivalent to the semantics of predicate calculus languages. Using such a language both domain ontologies and domain models can be described. In addition, the question about meaningful interpretation of constructions of such languages in domain terms is still an open question. Therefore, any definition of the notion of domain ontology within the framework of this approach does not distinguish this notion from other notions, in particular, from the notion of domain model (knowledge base). The third approach to defining the notion of domain ontology can be called the mathematical one. Within the framework of this approach an attempt is made to define the notion of domain ontology in mathematical terms or by a mathematical construction. The definition (6) "An ontology is a logical theory that constrains the intended models of a logical language" [16] can serve as an example of a definition of the notion of domain ontology in mathematical terms. In [17] an attempt was made to define the notions of a conceptualisation, of a domain model, and of a domain ontology model strictly. Every domain conceptualisation is represented by a many-sorted algebraic system (many-sorted logic-mathematical structure) S. The signature <R, F, C> of S is a set of domain terms (relations, functions, and constants). The universe U of S is a set of domain objects. A domain model (a knowledge base) is represented a subsystem of S. Every domain ontology model is represented by another algebraic system O. Sorts of O are "constant", "concept", "relation" and others. The signature of O (<R1, F1) is the set of terms used for a description of domain models ("instant", "relation-instant" and others). The set C1 of constants of O is the signature of S. The interpretation of the symbols of the signature of O represents the meaning of terms of the signature of O. The interpretation is restricted by a set D1 of formulas. The ontology describes the vocabulary of the objects representation (C1) and their types. This formalization of conceptualisation is based on the article [18] by Genesereth and Nilsson where conceptualisation is represented by the logic-mathematical structure <D,R>, where D is a set of domain objects and R is a set of domain relations among them. Such a 5 representation of conceptualisation is possible if the world is considered as set of objects and relations among them depending on time and space (many relations belonging to R must depend on time and space). But neither Genesereth and Nilsson nor the authors of article [17] write about it explicitly. According to the definition, domain ontology is a definition of a representation of conceptualisation using problem independent terms. The notion of domain ontology defined in such a manner is not connected with its domain. In [1] the notion of conceptualisation and of ontology were defined. Conceptualisation is a (infinite) set of logic-mathematical structures {<D,Rw>wW}, where D is a set of domain objects, W is a set of relevant states of affairs also called possible worlds, every Rw is a set of relations among objects, and all logic-mathematical structures have the same type. So, a conceptualisation determines a set of intended world structures. If L is a logical language with a vocabulary V then interpretation of all the object symbols belonging to V does not depend on W but interpretation of all the predicative symbols belonging to V depends on W. An ontology O is a set of axioms represented by L. The set of models for O is an approximation of the conceptualisation. This definition has a few defects. 1. On the one hand, the set D has to be infinite one (for the set W has to be infinite one). On the other hand, it is impossible to imaging a state of affairs containing an infinite set of objects. So, the set of objects in every state of affairs should be a finite subset of the set D. 2. There are also relations among objects that do not depend on W. They are ordinary mathematical relations. Such relations in the blocks world are the relation of equality and the relation of inequality. Since different ordinary mathematical relations may be appropriated to different domain, so it is natural to include these relations into conceptualization. 3. The vocabulary of a language L can be divided into two parts V 0 and V1. The interpretation of the symbols of V0 does not depend on the W, but the interpretation of the symbols of V1 depends on W. The symbols of V0 are ordinary mathematical constants and signs of ordinary mathematical relations. An example of an object symbol from V1 in the blocks world is "the table" if in every state of affairs there is the only table, but in different state of affairs the tables may be different. 4. There are terms related to domain and being more general then terms of V. An example of such a term may be "spatial relations" that is not ordinary mathematical relation and its meaning does not depend on W. So, such terms should be a part of a conceptualization. Thus, it is possible to consider that till now no generally accepted definition of domain ontology has been suggested [19,20]. However, its is possible to select four different meaning of the term of domain ontology from this overview. 1. A domain ontology is a part of the domain knowledge that is not to be changed. The other part of the domain knowledge may be changed according to the domain ontology. 2. A domain ontology is a part of the domain knowledge that restricts the 6 meanings of the domain terms; the meanings of domain terms do not depend on the other part of the domain knowledge. 3. A domain ontology is a set of agreements about the domain; the other part of the domain knowledge is a set of empirical or other domain lows. 4. A domain ontology is an intensional approximation of a extensional conceptualisation. These meaning of the term of domain ontology supplements each other. In section 4 an attempt will be made to give another definition. But here some basic methodological principles of such a definition should be formulated. 1. On the meaningful level a domain ontology will be understood as a set of agreements (domain terms, their commentary, statements restricting a possible meaning of these terms, and also a commentary of these statements). These agreements are a result of understanding among members of the domain community. So, they cannot be disproved by any empirical observations. In their meaning they differ from empirical or other knowledge that can be disproved by empirical observations or by other way. In this regard the notion of domain ontology is analogous to the notion of paradigm by T.S. Kuhn [21]. 2. The agreements should be represented by an appropriate language of mathematical logic. 3. An explicit correspondence between elements of the mathematical construction representing every domain ontology and properties of the domain should be defined. To define the notion of domain ontology a class of mathematical constructions for domain model representation will be firstly defined. Then the notion of domain ontology will be defined by the class and its meaningful interpretation. 2. LOGICAL RELATIONSHIP SYSTEMS Logical relationship systems of the n-th order were defined in [22,23]. Here only their short description will be given. A logical relationship system of the n-th order is a tuple <, A0, 1, A2, A3,...,An >. Here is a set of formulas of a many-sorted quantor-free language of predicate calculus of the n-th order. The signature of the language is divided into nonempty subsets 0, 1, 2,..., n that are mutually disjoint. The objective symbols of 0 will be called constants, the functional symbols of 0 - signs of operations and functions, and the predicative symbols of 0 signs of relations. All the symbols are interpreted in the many-sorted algebraic system A0. The symbols of 1 will be called unknowns (objective, functional and predicative). These symbols do not have any fixed interpretation. For i from 2 to n the symbols of i will be called parameters of the i-th order (objective, functional and predicative). They are interpreted in the many-sorted algebraic system Ai. The interpretations of the signs of operations, functions and relations (in A0) are computable operations, functions and predicates. The sorts of A2 are finite ones and consist of constants and unknowns. For i from 3 to n the sorts of Ai are also finite 7 ones and consist of constants and symbols of 1, 2, ..., i-1. A solution of a logical relationship system of the n-th order is such a many-sorted algebraic system of the signature 1, that every formula from is true for any the admissible substitution instead of all the variables of the formula. Any solution has finite sorts consisting of the same objects as the sorts of A0. The set of all the solutions of a logical relationship system S will be designated as R(S). So, a logical relationship system of the n-th order uniquely defines the sets of its solutions (that can be an infinite one). Some properties of the solution sets of logical relationship systems are studied in [24-27]. Here we will only intimate the theorem about decreasing of the order of logical relationship systems important for further presentation. The theorem is: for any logical relationship system S of the k-th order (k>1) there is a logical relationship system S1 of the (k-1)-th order equivalent to S. Two logical relationship systems are equivalent if they have the same set of solutions. A logical relationship system will be called isomorphic to another one if there is an one-to-one correspondence between the sets of their solutions. A logical relationship system S2 will be called a homomorphic image of another logical relationship system S1 if there is a completely defined one-valued mapping h from the solution set of S1 into the solution set of S2. In this case we will say that there is the homomorphism h : S1 S2. A logical relationship system S will be called the product of other logical relationship systems S1, S2,..., Sm (the factors of the product) if there are homomorphisms h1 : S S1, h2 : S S2,..., hm : S Sm such that ',"R(S) ' " (h1('), h2('),..., hm(')) (h1("), h2("),..., hm(")). Further we will consider a generalisation of the notions introduced above. Let S1 = <1, A0, 1> be a logical relationship system of the first order and 2 be a set of formulas of a predicate calculus language of the first order of the signature 0 1, such that 1 2 is noncontradictory. Then a logical relationship system S2 = <1 2, A0, 1> will be considered as S1 enriched by the set 2. The system S1 will be called unenriched one, the set 2 will be called an enrichment for S1, and the system S2 will be called enriched one. A tuple О = <, A0, 1, 2, 3,..., n>, where n > 1 will be called a system with free parameters. Let S1 = <, A0,1,2,3,...,n> be a logical relationship system with free parameters, and <A2,...,An> be a tuple of algebraic systems of the signatures 2,..., n. These algebraic systems must satisfy the restrictions mentioned above and if all the symbols of i are interpreted in Ai for all i from 2 to n, then all the formulas belonging to are not false for all the admissible substitutions. In this case a logical relationship system S2 = <, A0, 1, A2, A3,...,An> will be considered as S1 enriched by the tuple <A2,..., An>. The system S1 will be called unenriched one, the tuple <A2,..., An> will be called an enrichment for S1, and the system S2 will be called enriched one. The set of all possible enrichment of S1 will be designated En(S1). If k 8 En(S1) then S2 = <S1,k> is S1 enriched by k. Thus, an unenriched system determines a set of logical relationship systems. An unenriched logical relationship system S1 will be called equivalent to another unenriched logical relationship system S2 if there is a one-to-one correspondence I between the set En(S1) and the set En(S2) such that for all kEn(S1) the systems <S1,k> and <S2, I(k)> are equivalent. The following theorem about decreasing the order of unenriched logical relationship systems takes place: if Sn = <1 2, A0, 1, 2,..., n> is an unenriched logical relationship system of the n-th order (n>1), where 1 is a set of formulas of the signature 0 1, then for the unenriched logical relationship system of the first order S1 = <1, A0, 1> there is a completely defined one-valued mapping h from En(Sn) into En(S1) such that for all k En(Sn) the system <Sn, k> is equivalent to the system <S1, h(k)>. From this theorem it follows that, in general, S n and S1 are not equivalent because {h(k) k En(Sn)} En(S1). An unenriched logical relationship system S1 will be called isomorphic to another unenriched logical relationship system S2 if there is a one-to-one correspondence I between the set En(S1) and the set En(S2) such that for all kEn(S1) the systems <S1,k> and <S2, I(k)> are isomorphic. An unenriched logical relationship system S2 will be called a homomorphic image of another unenriched logical relationship system S1 if there is a completely defined one-valued mapping h from the set En(S1) into the set En(S2) such that for all kEn(S1) the system <S2,h(k)> is a homomorphic image of the system <S1,k>. In this case we will say that there is the homomorphism h : S1 S2. An unenriched logical relationship system S will be called the product of other unenriched logical relationship systems S1, S2,..., Sm (the factors of the product) if there are homomorphisms h1 : S S1, h2 : S S2,..., hm : S Sm such that k',k"En(S) k' k" (h1(k'), h2(k'),..., hm(k')) (h1(k"), h2(k"),..., hm(k")) and for all kEn(S) the system <S,k> is the product of <S1,h1(k)>, <S2,h2(k)>,..., <Sm,hm(k)>. 3. A MATHEMATICAL MODEL OF A DOMAIN A logical relationship system of the n-th order <, A0, 1, A2,.., An> can be considered as a domain model if all its elements represent the properties of the domain mentioned below. The algebraic system A0 represents a model of mathematics (abstract algebra) used for description of knowledge and ontological agreements of the domain. The domain objects are represented by mathematical objects - elements of the sorts of A0. Such a replacement of the domain objects by mathematical objects must keep some certain properties of the objects represented by the operations, functions and relations of A0. The solution set of the logical relationship system is an approximation of the domain reality. The domain reality is the set of all the situations (states of affairs in terms of [16]) taking place in the past, 9 in present, and in the future. Every solution represents a class of situations (indistinguishable in this conceptualisation). A situation is considered here as information concerning a finite fragment of the reality that may be related to a finite part of space and a finite time period. Any situation has a finite set of objects and there is a finite set of relations among them. These relations are designated by the domain terms describing a structure of the reality. These terms are represented by the signature 1. Thus, the structure of any situation is described in the same terms, but each of these terms has its own meaning in every situation. If a model of the domain is represented by a logical relationship system of the first order then the set of formulas can be divided into two nonintersecting subsets. They are a set of formulas representing empirical or other knowledge of the domain (knowledge base) and a set of formulas representing ontological agreements. These two types of formulas are distinguished only by meaning but not by their forms. If a model of the domain is represented by a logical relationship system of a higher order then 1, the algebraic systems A2,..., An represent empirical or other knowledge of the domain on different levels of abstraction (knowledge base). This knowledge is represented by relations among terms of the domain. In this case the whole set of formulas represents ontological agreements of the domain. 4. A MATHEMATICAL MODEL OF A DOMAIN ONTOLOGY An unenriched logical relationship system can be considered as a model of a domain ontology if the set of its formulas represents all the ontological agreements of the domain and only them, and other elements of the system represent properties of the domain mentioned in the previous section. To understand such a model a meaningful commentary for all the terms and formulas being its components is necessary. Now the informal notion of domain ontology can be defined by the formal notion of domain ontology model: the part of information about a domain represented by an ontology model of the domain will be called the ontology of the domain. Some examples can be found in [22,23]. Here a predicate calculus language plays the role of a representation for models of domain ontologies, of domains and of knowledge. It determines a form (syntax) for models of domain ontologies and domains, and also it determines a basic set of symbols having semantics, i.e. propositional connectives and applications of functional and predicative symbols to arguments. This representation is extended by the model of mathematics given by the algebraic system A0. In this case syntax of the representation is the same. The set 0 of symbols interpreted in A0 is added to the set of symbols of the representation. Since a specific mathematics is used in every domain, A0 is not included in the representation, but is a part of every domain ontology model. However, it is known from practices that the same models of mathematics represented by the same algebraic system A0 may be a part of ontology models of different domains. We notice also that formulas containing at least one symbol from 1, 2,... or n can only belong to the set of formulas, because any 10 formula that does not contain such symbols can be considered only as a restriction on properties of symbols from 0. At the same time, all the symbols from 0 have already had the certain interpretation given by A0. Let a domain ontology model <1, A0, 1> be of the first order. Then the model of mathematics A0, the signature 1 and the set 1 of ontological agreements are determined by the model. The domain ontology model enriched by a set 2 of formulas (knowledge base) representing empirical or other laws of the domain is a model of the domain. Every empirical or other law can be represented only by using symbols from 01. Therefore, the empirical or other law determines properties of relations among objects of situations (properties of possible interpretations for all the symbols from 1). These relations in different situations have different contents (all the symbols from 1 have different interpretations in different models of different situations). At the same time, these relations have the same properties in different situations. Some of these properties are represented by ontological agreements and the others are represented by empirical or other laws. From the philosophical point of view, empirical laws explicitly represent the order taking place in the reality. At the same time, every empirical law must be met in every really existing situation of the domain. Therefore, a model of a domain is adequate to the domain if the set of models of all the situations possible in this domain is equal to the set of all the solutions of the logical relationship system being the model of the domain, i.e. the set of solutions is the exact approximation of the domain reality. The question of existing an adequate model for a domain is usually an open question. If it is detected that a model of a domain is not adequate to the domain then empirical laws of the domain should be modified within the restrictions imposed by the domain ontology to make the model more adequate, i.e. another model of the domain is formed by enriching the same domain ontology model by another set of empirical laws. If in the process of storing empirical or other data (a set of situations) it becomes clear that the current domain model is not always adequate to the domain, it is necessary to modify the model permanently and this process leads to constant increasing of the number of empirical or other laws and/or to constant growth of complexity of the formulas representing them then an aspiration may arise for finding another ontology (changing the paradigm) of the domain and for finding an adequate model of the domain within the restrictions of the new ontology. Let a domain ontology model <, A0, 1, 2> be of the second order. Then the model of mathematics A0, the signatures 1, 2 and the set of ontological agreements are determined by the model. In comparison with a domain ontology model of the first order a new set of terms (represented by the signature 2) is introduced in a domain ontology model of the second order. These terms are designations of relations among concepts of the domain. In this case, unlike domain ontology models of the first order, empirical or other knowledge of the domain must be represented on the higher level of abstraction, i.e. using these relations, defined on the set of terms representing objects of the domain (the set of the constants from 0), 11 and also of terms representing relations among objects in the situations (symbols from 1). The set of ontological agreements represented by formulas from can be divided into three nonintersecting subsets. They are constraints of the reality model, i.e. statements restricting the set of possible interpretations of unknowns (this set is the same in meaning and in form as the set 1 for domain ontology models of the first order), constraints of the knowledge model, i.e. statements restricting the set of possible interpretations of symbols from 2, and statements settings up a correspondence between interpretations of symbols from 1 and from 2. Every statement of the first group must contain at least one symbol from 1 and contain no symbols from 2. Every statement of the second group must contain at least one symbol from 2 and contain no symbols from 1. Every statement of the third group must contain at least one symbol both from 1 and from 2. A domain ontology model of the second order enriched by the algebraic system A2 (a knowledge base) representing empirical or other laws of the domain is a model of the domain. According to the theorem about decreasing of the order of logical relationship systems a set of formulas can be got using every formula from the third group. These formulas will represent empirical or other laws of the domain. These laws will be formulated in terms of signatures 1 and 0. In this case the formulas got in such a way must not contradict the constraints of the reality model. In addition, the theorem about decreasing of the order of logical relationship systems gives a possibility to settle the questions about meaning and adequacy of empirical or other knowledge for domain models of the second order by reducing them to equivalent models of the first order. Let a domain ontology model <, A0, 1, 2, 3> be of the third order. Then the model of mathematics A0, the signature 1, 2, 3 and the set of ontological agreements are determined by the model. Thus, in comparison with a domain ontology model of the second order a new set of terms (represented by the signature 3) is introduced in a domain ontology model of the third order. These terms are designations of relations among classes of concepts of the domain. In this case, empirical or other knowledge of the domain must be represented on two levels of abstraction by relations among concepts (as in the models of the second order) and relations among classes of concepts. These relations are defined on the set of terms representing objects of the domain (the set of the constants from 0), of terms representing relations among objects in the situations (symbols from 1), and also of terms representing relations among concepts (symbols from 2). The ontological agreements represented by formulas from can be divided into seven nonintersecting subsets. They are (1) constraints of the reality model; (2) constraints of the lower (the first) abstraction level of the knowledge model, i.e. statements restricting the set of possible interpretations of symbols from 2; (3) constrains of the upper (the second) abstraction level of the knowledge model, i.e. statements restricting the set of possible interpretations of symbols from 3; (4) constraints setting up a correspondence between the lower and the upper abstraction levels of the 12 knowledge model, i.e. statements about a correspondence between possible interpretations of symbols from 2 and 3; statements setting up a correspondence between different abstraction levels of the knowledge model and the reality model (i.e. statements setting up a correspondence between interpretations (5) of symbols from 1 and 2; (6) of symbols from 1 and 3; (7) of symbols from 1, 2 and 3). A domain ontology model of the third order enriched by the algebraic systems A2 and A3 (a knowledge base) representing empirical or other laws of the domain is a model of the domain. According to the theorem about decreasing of the order of logical relationship systems a set of formulas representing constrains of a knowledge model can be got using every formula from group (4); a set of formulas representing ontological agreements setting up a correspondence between the knowledge model and the reality model can be got using every formula from groups (6) and (7). In this case the formulas got in such a way must not contradict the formulas of groups (1), (2) and (5). Analysis of the structures of domain ontology models of a higher order can be analogically made. Below the set of all possible enrichments for a domain ontology will be called the set of knowledge bases consistent with this domain ontology model. Now we will discuss the question about facilities of domain models and domain ontology models represented by enriched and unenriched logical relationship systems of different orders. On the one hand, from the theorem about decreasing the order of logical relationship systems it follows that if there is a domain model represented by a logical relationship system of the n-th order that determines an approximation of the domain reality then there is the model of the same domain represented by a system of the (n-1)-th order that determines the same approximation of the domain reality. Therefore, any approximation of any domain reality can be determined by a model of the domain represented by a logical relationship system of the first order. In this point, domain models represented by logical relationship systems of a higher order have no advantages over models represented by a logical relationship system of the first order. On the other hand, every domain ontology model represented by an unenriched logical relationship system determines a set of intended domain models. The adequate domain model should belongs to this set. So, this set should be as small as possible. From the theorem about decreasing the order of unenriched logical relationship systems it follows that using a domain ontology model represented by an unenriched logical relationship system of the n-th (n>1) order a smaller set of intended domain models can be determined than using a domain ontology model represented by an unenriched logical relationship system of the first order. In this point, domain ontology models represented by unenriched logical relationship systems of a higher order have certain advantages over models represented by an unenriched logical relationship system of the first order. 13 5. SOME PROBLEMS OF MANUAL AND COMPUTER TECHNOLOGIES FOR KNOWLEDGE PROCESSING AND WAYS OF SOLVING THESE PROBLEMS Some problems, most important from the authors' point of view, will be considered below. The problems arise when manual and computer technologies for knowledge processing are used. The problems have not had satisfactory solution within the framework of "classical" computer technologies. The technologies worked out by the founders of expert systems and described in widely known monographs and reviews are meant by classical ones. All the problems will be divided into five groups. I. Problems related to ascertainment of correspondence between domain ontologies. 1) Within the framework of manual technologies alternative points of view on the same domain are often said to be equivalent or different domains are said to be similar. Usually such judgements are intuitive. Ascertainment of the equivalence between alternative points of view on the same domain can give a possibility to solve tasks arising within the framework of a point of view using methods worked out within the framework of another point of view. Ascertainment of the similarity between different domains can give a possibility to solve tasks arising in a domain reasoning by analogy in the case if methods for solving analogous tasks in another domain have worked out. In this connection the following problems arise: (a) how to ascertain an equivalence between two different knowledge systems or between two different ontologies of the same domain; (b) how to ascertain an equivalence between knowledge systems or between ontologies of two different domains. The problems were discussed in [8] within the framework of computer technologies. Two models of the same domain (models of two different domains) represented by logical relationship systems S1 and S2 will be called isomorphic if S1 and S2 are isomorphic. Two domain ontology models (of the same or two different domains) represented by unenriched logical relationship systems S1 and S2 will be called isomorphic if S1 and S2 are isomorphic. Thus, both problems formulated above are reduced to ascertaining an isomorphism between two different (enriched or unenriched) logical relationship systems. For ascertaining isomorphism between systems studying properties of such isomorphism can be useful. 2) Within the framework of manual technologies a cruder description of a domain can be obtained from a more detailed description of the domain. It could be necessary for specialists having different qualification. No articles are known to the authors in which a satisfactory statement of the task of "coarsening" description of domains or its solution within the framework of computer technologies was given. In this connection the following problem arises: how to provide coarsening knowledge or an ontology of a domain. If a domain has two models represented by logical relationship systems S1 and 14 S2 and the system S2 is a homomorphic image of the system S1 then the domain model S2 will be called a coarsening of the model S1. Let models of two ontologies O1 and O2 of a domain be represented by unenriched logical relationship systems S1 and S2. Then the model O2 will be called a coarsening of the model O1 if the system S2 is a homomorphic image of the system S1. Thus, this problem is reduced to studying homomorphisms of (unenriched and enriched) logical relationship systems. 3) Within the framework of manual technologies an integration of descriptions of a few domains is referred to as combination of these descriptions in a general description of the new complex domain those aspects are original domains. In this connection the following problem arises: how to provide integrating knowledge systems or ontologies of a few domains into a new knowledge system or ontology. The problem was discussed in [8, 28-31] within the framework of computer technologies. A domain model represented by a logical relationship system S will be called an integrated one from domain models represented by logical relationship systems S1, S2,...,Sk, if the system S is the product of S1, S2,..., Sk. Let domain ontology models O, O1, O2,..., Ok be represented by unenriched logical relationship systems S, S1, S2,...,Sk. The model O will be called an integrated one from O1, O2,...,Ok, if S is the product of S1, S2,..., Sk. Thus, the problem is reduced to studying products of (unenriched and enriched) logical relationship systems. . II. Problems related to using knowledge in a broad sense. 4) Within the framework of manual technologies the same knowledge is often used for different purposes. An analogous need exists within the framework of computer technologies too [8, 20]. In this connection the following problem arises: how to provide reusing the same domain knowledge or domain ontologies in different computer designs. Let S1 and S2 be models of two domains represented by logical relationship systems, S1 be the product of S and S', and S2 be the product of S and S". In this case we will say that the model of another domain represented by the system S is reused. Let O1 and O2 be ontology models of two domains represented by unenriched logical relationship systems, O1 be the product of O and O', and O2 be the product of O and O". In this case we will say that the ontology model of another domain represented by the system O and the knowledge base of this domain (any enrichment of O) is reused. Thus, the problem is reduced to studying ways of forming enriched (unenriched) logical relationship systems being homomorphic images of several enriched (unenriched) logical relationship systems and also properties of such systems and homomorphisms. 5) Within the framework of manual technologies knowledge is always shared among both different people and other bearers of knowledge (books, articles, and so on). Every bearer of knowledge possesses an integral knowledge system. The specialists concerned have more or less easy access to the knowledge system of any bearer. Within the framework of computer technologies a need to share knowledge 15 among different knowledge bases related to the same domain model and also a need to organise access of knowledge based systems to "another's" knowledge bases [8, 12, 20, 32-34] exists. In this connection the following problem arises: how to provide using shared domain models (sharing knowledge bases). A domain model that can be represented by a logical relationship system S will be called shared if, first, S is the product of logical relationship systems S 1,...,Sm that are the models of other domains, and, second, there is a way to restore S using S1,...,Sm. We consider that the systems S1,...,Sm are stored on different servers, but the system S stores nowhere and S is restored using S1,...,Sm. A domain ontology model that can be represented by an unenriched logical relationship system O and the knowledge base of this domain k (an enrichment of O) will be called shared if, first, O is the product of unenriched logical relationship systems O1,...,Om that are the ontology models of other domains, and, second, there is a way to restore O using O1,...,Om and a way to restore k using k1,k2,..,km that are enrichments of O1,...,Om. We consider that the systems O1,...,Om and their enrichments k1,k2,...,km (knowledge bases) are stored on different servers, but the system O and its enrichment k (the domain knowledge base) stores nowhere, O is restored using O1,...,Om and k is restored using k1,k2,...,km. Thus, the problem is reduced to studying ways to restore elements of S using elements of S1,...,Sm, elements of O using elements of O1,...,Om, elements of k using elements of k1,...,km. III. Problems related to translating "human" knowledge into "computer" one and back. 6) A bottleneck of using computer technologies for knowledge processing is passing knowledge from experts to computers. The process is labour-consuming because of knowledge extensiveness. When classical technologies are used it is supposed that a knowledge engineer takes part in this work besides an expert. On the one hand, the knowledge engineer is the principal motive force of this work, but on the other hand, he brings about certain difficulties in the process of passing knowledge and misrepresentation in its results. In this connection the following problem arises: how an expert can pass his knowledge into a computer without any intermediary. The article [8] is devoted to its detailed discussion. We will show how the problem can be solved if a domain ontology model of the second or a higher order is formed. In this case using the unenriched logical relationship system S = <, A0, 1, 2, 3,..., n> representing the domain ontology model, an equivalent unenriched logical relationship system S'= <', A'0, 1, '2, '3,..., 'n> representing equivalent ontology is formed in the following way. The signature '0 is an extension of 0 by set-theoretic constants, signs of operations and relations. The algebraic system A'0 is an extension of A0, giving these symbols the universally accepted set-theoretic interpretation. For i from 2 to n the signature 'i can be got from i by eliminating all the predicative symbols and by adding a set of functional symbols one-to-one corresponding to the set of all the predicative symbols of i. In the process the interpretation of a predicative symbol p i and the 16 interpretation of the functional symbol fp 'i corresponding to it are related by the equality fp(v1,...,vk) = {v p(v1,...,vk, v)}. ' is formed using so that both unenriched logical relationship systems would be equivalent (it is obviously always possible). Let a relation "<" be defined on the set of all the sorts of the signatures '2,..., 'n in the following way. If s1 and s2 are sorts and f is a functional symbol of one of the signatures '2,..., 'n, then s1 < s2, if s1 is the sort of an argument of f, and s2 is the sort of the result of f. Let the system S be such that the relation "<" is a partial order relation. In this case the process of passing knowledge from an expert to a computer consists in forming interpretations of all the sorts and all the symbols of the signatures '2,..., 'n. At first, for every terminal sort (a sort is terminal if it is the sort of the result of none of the functions) the computer asks the expert about a set of objects that is the interpretation of the sort. All the questions are asked in terms of the domain ontology. Also the computer asks the expert about values of all the objective symbols of the signatures '2,..., 'n. Next, for every functional symbol of the signatures '2,..., 'n and for every tuple of its arguments (a1,a2,...,ak) the computer asks the expert about the value of application term f(a1,a2,...,ak). In this way the interpretation of the sort of the result of f is formed. This process can be completed when every function f is completely defined within the domain of definition given by Cartesian product of the sorts of all the arguments of f. The constraints of the signatures '2,..., 'n allow us to check consistency and completeness of knowledge base generated. Thus, only a part of the domain ontology model is used to solve the problem. This part consists of the signatures 2,..., n and their constraints. In [35] a knowledge acquisition model based on this principle was described, and in [8, 36-40] several tools for developing knowledge base editors were presented. 7) Formalising knowledge using classical computer technologies makes the knowledge "clear" for a computer but unclear for domain specialists. Therefore, back translation also requires the same translator, namely, the knowledge engineer. In this connection the following problem arises: how a domain specialist can gain access to computer knowledge without any intermediary. A knowledge base editor based on domain ontology models (see the previous point) allows a domain specialist to look through a knowledge base. During a knowledge acquisition process a knowledge base editor asks an expert but during overlooking the knowledge base a domain specialist has a possibility to ask the same questions to the knowledge base editor in terms of the domain ontology. 8) When a knowledge base is formed for a real domain then after testing and debugging the knowledge constituting its content can be of interest for domain specialists. But no articles are known to the authors reporting that the contents of a real knowledge base has become available for many specialists on the appropriate domain and has had an impact on the domain. In this connection the following problem arises: how to make the content of computer knowledge well-known for 17 many domain specialists. This problem can be solved if a domain ontology model of the second or a higher order is formed. A possible way of solving the problem is transforming a knowledge base into a text. The text must have a form (structure, style) convenient for domain specialists. Such texts can be got by modern tools for generating documents using data bases [41,42]. As seen from the domain ontology model definition, a knowledge base is a relational data base in form. In addition, access to a knowledge base for a wide range of specialists can be organised by Internet using a knowledge base editor based on the domain ontology model. IV. Problems related to forming and debugging knowledge. 9) The principal way of getting empirical knowledge is their inductive forming on the basis of empirical data. Manual technologies of inductive forming knowledge are an object of study beginning with the works by F.Bacon and J.S. Mill. But it is very difficult to estimate the quality of the knowledge formed by this way. Moreover, a comparison of empirical data with knowledge inductively formed as a rule is subjective, especially in the domains where a tradition of application of mathematics is absent. This circumstance has an negative influence on the quality of the knowledge too. The task of inductive forming knowledge was considered within the framework of computer technologies. However, as D. Michie noted [43], automatically formed knowledge can be useful only if, firstly, their content is clear for domain specialists and, secondly, if the whys and wherefores of the knowledge leans upon empirical data and is made by clear ways for the specialists. No articles on inductive forming knowledge bases in which both these requirements would be met are known to the authors. In this connection the following problem arises: how to make inductively formed computer knowledge clear and convincing for domain specialists. This problem can be solved if a domain ontology model of the second or a higher order is formed. Any knowledge base consistent with the domain ontology model will be understood by a specialists of the domain (taking into account point 7) if the ontology is acceptable for him. In addition, a knowledge base is considered by domain specialist as justified, if there is a correspondence between the knowledge base and series of examples of the domain situations understandable for the specialist. With regard to these two reasons, several task specifications for inductive learning can have the following form. A domain ontology model of an order higher than 1 (unenriched logical relationship system) and a finite set of algebraic systems of the signature 1 (series of examples of the domain situation models) are given. These algebraic systems can be determined completely or partially (the interpretations of some of the sorts and/or of symbols of 1 are ambiguously given). It is necessary to find an enrichment K (a knowledge base) for the domain ontology model S such that every example of the series is a solution of S enriched by K. In addition, an explanation for the knowledge base formed inductively should be formed. The explanation represents the correspondence between the built knowledge 18 base and the series. Since, the task of inductive learning in such a specification can have many solutions then a criterion permitting to select the solution uniquely may be added to the task specification. Thus, a number of task specifications for inductive learning can be got depending on what information about examples of the series is given and what criterion for selecting solutions is chosen. It is clear that a solution of a task of inductive learning specificated in this way will meet requirements above. 10) Refining knowledge is constant modifying the knowledge to make it consistent with a constantly increasing amount of empirical data. When manual technologies are used the same difficulties arise as in the case of problem 9. Within the framework of computer technologies an analogous process is called knowledge base debugging. This process needs an expert to take part in it who is not familiar with such an activity. Besides practically insuperable difficulties related to problem 7 arise in his work. No articles are known to the authors where acceptable methods for knowledge base debugging would be suggested. In this connection the following problem arises: how to make refining computer knowledge possible and clear for experts. This problem can be solved if a domain ontology model of the second or a higher order is formed. The problem is a special case of the previous one with two restrictions on methods of its solving. Firstly, these methods must be recurrent, i.e. they must form a new version of the knowledge base "improving" the previous one using the domain ontology model, the previous version of the knowledge base, and a series of examples. Secondly, these methods must be man-machine ones, i.e. they must allow an expert to take part in forming a new version of the knowledge base. The latter gives a possibility to an expert to lop off unsatisfactory solutions using his experience, as if he had added some precedents from his previous practice to the series of examples and also as if he had determined more exactly some ambiguous examples of the series. V. Problems related to using knowledge to solve applied tasks 11) A statement of an applied task allows us to determine the properties of the output data of the task (its solution) in dependence on the input data of the task. Within the framework of manual technologies it is possible to deal with either implicit or meaningful statement of a task. In this statement knowledge determines the properties of output data of the task depending on its input data. In the domains having a tradition of using mathematics and mathematical methods for solving tasks they pass on from meaningful statement of a task to its mathematical specification. When a computer is used to solve tasks, a mathematical specification of any task is necessary, because the only property of output data of a program is they are the result of running the program. Knowledge representation means used within the framework of classical computer technologies (production systems, frames and semantic networks) have procedural semantics. It means that a knowledge base represented in such a manner is a program of solving tasks of a class. Hence, a knowledge base in such a form cannot be considered as a component of the specification of any task solved by KBS. As a result, in developing a majority of systems no attempt were 19 even made to pass from meaningful statement of tasks solved by KBS's to their mathematical specifications. In this connection the following problem arises: how to obtain mathematical specifications of applied tasks solved by KBS's. A mathematical specification of an applied task solved by a KBS can contain a domain model represented by a logical relationship system, input and output data of the task, task conditions (a set of formulas), and also criterion of selecting solutions [22,23]. All the components of the applied task specification are represented in terms of the domain model. If every value of input data is replaced by a variable (different variables correspond to different values) in the task specification then the mathematical specification of the task will be transformed into a mathematical specification of a class of applied tasks. These variables will be called variables of the class of applied tasks. There is a one-to-one correspondence between the set of tasks belonging to the class and the set of all the admissible substitutions of values instead of these variables. To get the mathematical specification of an applied task belonging to a class it is necessary to replace all the variables of the class by values of input data. Let a domain model be represented by an enriched logical relationship system of an order higher then the first. If the enriched system is replaced by the unenriched logical relationship system representing the domain ontology model and enrichments representing knowledge bases are considered as another set of input data of all the tasks of the class then the mathematical specification of the class of applied tasks will be transformed into the mathematical specification of the class of applied tasks corresponding to the domain ontology. There is a one-to-one correspondence between the set of tasks belonging to the class and the Cartesian product of the set of all the admissible substitutions of values instead of variables of the class of the tasks by the set of all the possible enrichments (knowledge bases) for the unenriched logical relationship system representing the domain ontology model. To get the mathematical specification of an applied task belonging to a class of tasks corresponding the domain ontology it is necessary to replace all the variables of the class by values of input data and to enrich the domain ontology model by an appropriate knowledge base. Finally, if domain terms in the mathematical specification of the class of applied tasks corresponding to a domain ontology are replaced by abstract designations then this mathematical specification of the class will be transformed into a mathematical task. The transformation of a mathematical specification of a class of applied tasks corresponding to a domain ontology into a mathematical task is important because different classes of applied tasks corresponding to ontologies of different domains, generally speaking, can be reduced to the same mathematical task. 12) Within the framework of manual technologies to work out a method for solving an applied task means to obtain a method using its meaningful statement or mathematical specification, both containing knowledge necessary for solving the task. There is an analogous need within the framework of computer technologies. Investigation of a method for solving a task is, firstly, obtaining some arguments for 20 the method solves the task and, secondly, estimating the labour consuming character of solving the task by the method. In the case when there is a mathematical specification of the task and the method is described sufficiently rigorously as an algorithm or calculus, then the arguments are replaced by a proof and the labour consuming character is replaced by a complexity of the method. In this connection the following problem arises: how to obtain and to investigate a method for solving a task using a mathematical specification of the task. When a method of solving a task is described it is naturally to use the terminology that was used for describing a mathematical specification of the task. Also new terms may be introduced but they must be defined by the terms used in the mathematical specification of the task. Thus, it is seen that the more abstract terminology is used for describing a mathematical specification of a task the wider the field of application of a method for solving the task. Therefore, the problem is reduced to accumulating of the set of mathematical tasks solved by KBS's and to working out methods for their solving. As is the convention of practice of software development, working out a method for solving a mathematical task should be separated from an implementation of the method. As noted in [22,23], such methods can be worked out both in the form of a calculus and in the form of an algorithm. For describing these methods appropriate generative models or computational models should be used rather then implemented knowledge representation languages or programming languages. In [22,23] problems of investigating such method were considered. 13) The same methods often can be used for solving a few tasks and subtasks. There is an analogous need within the framework of computer technologies [8]. In this connection the following problem arises: how to reuse the same methods of solving tasks for different designs. Based on point 11, this problem and also problem 12 cannot be satisfactorily posed within the framework of classical computer technologies. Abstraction of applied tasks to mathematical ones gives a possibility of reusing methods for their solving (see point 11). If different applied tasks can be reduced to the same mathematical task then a method for solving the mathematical task can be used for solving these applied tasks too. A decomposition of a mathematical task into mathematical subtasks in working out a method for solving the mathematical task gives an additional possibility for reusing methods. In this case the same mathematical subtasks can be components of decompositions of different mathematical tasks and methods for solving these subtasks can be components of methods for solving different mathematical tasks. Reusing methods for solving tasks should be distinguished from reusing programs for solving these tasks. Possibilities of reusing programs are usually given by application packages [44-45] and libraries of reusable methods [8,46,47] (production schemes [48-49]). 14) Within the framework of manual technologies applied tasks of different classes often can be solved by different methods but using the same knowledge. 21 There is an analogous need within the framework of computer technologies too [50]. However, within the framework of computer technologies every KBS, as a rule, is intended for solving the tasks of the only class. In this connection the following problem arises: how to develop KBS's intended for solving tasks of many classes using the same domain model (knowledge base). Let's consider a set of mathematical specifications of applied tasks such that every specification contains the same domain model. Such a set will be called an applied multitask. Just as an applied task was transformed into a class of applied tasks, the latter was transformed into a class of applied tasks corresponding to a domain ontology, and the latter was transformed into a mathematical task, so an applied multitask can be transformed into a class of applied multitasks, the latter can be transformed into a class of applied multitasks corresponding to a domain ontology, and the latter can be transformed into a mathematical multitask. A multitask knowledge-based system is intended for solving applied multitasks of a class of applied multitasks or for solving applied multitasks of a class of applied multitasks corresponding to a domain ontology. A shell for such knowledge-based systems is intended for solving a mathematical task. CONCLUSIONS Because knowledge plays a very important role in the life of the mankind, "manual" technologies for knowledge processing arose already in ancient times and have constantly been developing till now. At present the knowledge "industry" based on "manual" technologies is a considerable part of human activity. "Manual" technologies give a possibility to solve all the tasks facing the industry of knowledge processing. However, they are characterised by low productivity of labour and by low quality of their results in many cases. At the end of the 60's forming computer technologies for knowledge processing started within the AI framework. Main achievements of computer technologies for knowledge processing related to using knowledge for solving applied tasks. The computer technologies usually lead to considerable raising the productivity of labour and sometimes to higher quality of their results as compared with manual technologies. But for many tasks of knowledge processing solved by manual technologies no corresponding computer technologies were worked out. In addition, transition from manual technologies to computer ones and back requires special translators, i.e. special people called "knowledge engineers". It is a reason why the two classes of technologies are largely isolated one from the other. In this article an attempt is made to show that KBS's can be considered as a section of more extent area - computer processing empirical or other knowledge. It is clear from the discussion above of a few problems related to processing knowledge that these problems are closely interconnected and connecting links among them are the notion of domain and the notion of domain ontology. Therefore, a progress in solving these problems is closely connected with explicit representation and studying 22 of mathematical ontology models of practically important domains corresponding to modern ideas about these domains. In addition, complexity of these problems is such that their solving is possible only by using mathematical methods. 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IJHCS, 41: 399- 424. 48. Artemjeva I.L., Yatsnenco O.S. System of declarative production rules with generalised operations. Technical Report, Vladivostok: Institute for Automation & Control Processes of the FEBRAS, 1998. (in Russian). 49. Lenat D.B., Guha R.V., Pittman K, ets. CYC: toward programs with common sense. In Communication of ACM, 1990, vol. 33, N 8: 30-49. 50. Buchanan B.G., Bobrow D., Davis R., McDermott J., Shortliffe E. Knowledge-based systems. In Annu. Rev. Comput. Sci., 1989-1990, Palo Alto (Calif.), 1990, vol. 4: 395-416. Alexander S. Kleshchev Irina L. Artemjeva DOMAIN ONTOLOGIES AND KNOWLEDGE PROCESSING Technical Report Send to press 23.03.99 Conventional printer's sheets 1.05 Registration publishing sheets 1.4 Size 60x84/16 Circulation 150 Order Published: Institute for automation & Control Processes Far Eastern Branch of the Russian Academy of Sciences 5 Radio Street, Vladivostok, Russia