Knowledge bases over algebraic models. Some notes about informational equivalence

Knowledge bases over algebraic models. Some notes about informational equivalence Abstract The recent advances in knowledge base research and the growing importance of effective knowledge management raised an important question of knowledge base equivalence verification. This problem has not been stated earlier, at least in a way that allows speaking about algorithms for verification of informational equivalence, because the informal definition of knowledge bases makes formal solution of this problem impossible. The goal of this paper is to provide an implementable formal algorithm for knowledge base equivalence verification based on the formal definition of knowledge base given in [24, 26, 28, 29] and to study some important properties of automorphic equivalence of models. We will describe the concept of equivalence and formulate the criterion for the equivalence of knowledge bases defined over finite models. Further we will define multi-models and automorphic equivalence of models and multi-models that are generalization of automorphic equivalence of algebras. 1 Introduction and Motivation The paper is inspired by a natural question:

When two knowledge bases are equivalent? This question contains some uncertainty, namely it operates with the terms “knowledge base” and “equivalence of knowledge bases”. Let us dwell briefly on these notions. 1.1 Knowledge bases. Descriptive definitions. As a rule knowledge bases are defined in a various descriptive ways. The definitions reflect a common sense intuition how a knowledge base should look like. They are informal and well known for the specialists in computer science. For the sake of completeness and for the needs of mathematicians looking for applications we provide the reader with some of them. 1 A knowledge base is defined as a special kind of database for knowledge management. It provides the means for the computerized collection, organization, and retrieval of knowledge. In its turn, knowledge management comprises a range of practices used to identify, create, represent, and distribute knowledge. The definition of “knowledge” is equally a philosophical and a practical task. There is no single agreed definition of knowledge presently, and there remain numerous competing theories. In any case knowledge is some essence which requires representation of knowledge. Various artificial languages and notations have been proposed for representing knowledge. They are typically based on logic and mathematics, and have easily parsed grammars to ease machine processing [11, 20, 21, etc.]. Knowledge bases store knowledge in a computer-readable form, usually for the purpose of having automated deductive reasoning applied to them. They contain a set of data, often in the form of rules that describe the knowledge in a logically consistent manner. Logical operators, such as conjunction and disjunction, may be used to build knowledge up from the atomic data. Consequently, classical deduction can be used to reason about the knowledge in the knowledge base. In general, a knowledge base is not a static collection of information (like a database), but a dynamic resource that may itself have the capacity to learn, as part of an artificial intelligence component. These kinds of knowledge bases can suggest solutions to problems sometimes based on feedback provided by the user, and are capable of learning from experience (like an expert system). Knowledge representation, automated reasoning, argumentation and other areas of artificial intelligence are tightly connected with knowledge bases. 1.2 Equivalence problem One can ask, for example, whether google and yahoo are equivalent? Obviously, we need to restrict concept of equivalence to some special meaning. For example, they are equivalent if they answer in the same time, or they are accessible in the same way, or using fees of these systems are the same, etc., etc., depending on equivalence criterion. 2 We study an equivalence of knowledge bases in respect to their informational abilities. In other words, we would like to discuss informational equivalence of knowledge bases. If we ask google and yahoo the same question we expect to get the equivalent answers. It means, we expect to get the same information but may be in different formats. Thus, we can specify the main question stated in the beginning of the paper in a more precise form: When two knowledge bases are informationally equivalent? The principal task here is to find out whether the problem of informational equivalence verification is algorithmically solvable. If we concentrate on finite objects then the reasonable answer is yes, we can build the step-by-step procedure used to solve the problem. But when we consider infinite objects it may be problematic. Evidently, knowledge bases are the example of this case (for more details see subsections 2.2 and 4.2). On other side, if we could

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