Tatjana L. Plotkin

Algorithms of distributed task allocation for cooperative agents

This paper considers the problem of distributed dynamic task allocation by a set of cooperative agents. The paper describes a rather specific situation. However, its methods have wide application and, thus, it can be useful to solve general problems of computer science. One of its main ideas is to combine optimization questions with the symmetries of initial objects. There are different types of tasks that are dynamically arriving to a system. Each of the agents can satisfy only a subset of the tasks. The main goal of the agents is to maximize the overall performance of the system and to fulfill the tasks as soon as possible. The agents are modeled using a stochastic closed queueing network. The problem is divided into two subproblems: to determine a distributed policy of optimal task distribution and to find the optimal effort levels of the agents subject to certain constraints. For the first subproblem, a distributed polynomial allocation algorithm for determining an instantaneous probabilistic optimal policy for task allocation is presented. The policy is independent of the state of the system and thus does not require information exchange among the agents during the performance of the tasks. For the second subproblem, an analytical solution to find the optimal effort levels for the agents is given.

Geometrical aspect of databases and knowledge bases

Abstract. This work stands at intersection of two sciences: universal algebra on the one hand and a field we call knowledge science on the other. We view the latter as a science on languages of knowledge representation. It is strongly related to universal algebra and can be considered as an area of mathematics having motivation in computer science.¶Universal algebra deals with arbitrary algebraic operations of an arbitrary arity. They can be unary, binary, ternary, and of any finite arity. Traditionally, operations of arity over three are extremely rare in algebra. There is no such tradition in computer science and knowledge science, where operations and relations are free from restrictions. This circumstance may be one of the most significant reasons to consider operations of an arbitrary arity.¶The core point of the paper is elementary knowledge (First Order Knowledge). The main goal is to construct a model to represent some non-elementary knowledge about elementary knowledge using a universal algebraic approach. For the solution of this problem we join the methods of algebraic logic and universal algebraic geometry in logic, both defined over an arbitrary variety of algebras

Categories of elementary sets over algebras and categories of elementary algebraic knowledge

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Knowledge Bases Over Algebraic Models: Some Notes About Informational Equivalence

. We use also the Galois theory in the logic over

Relational Databases Equivalence Problem

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Isotypeness of Models and Knowledge Bases Equivalence

which generalizes the well-known theory by M. Krasner. (See also [P11]).

Symmetries of knowledge bases

For every variety of algebras θ and every algebra H in θ we consider the category of algebraic sets Kθ(H) in θ over H. We consider also the category of elementary sets LKθ(H). The latter category is associated with a geometrical approach to the First Order Logic over algebras. It is also related to the category of elementary knowledge about algebra H. Grounding on these categories we formally introduce and study the intuitive notions of coincidence of algebraic geometries over algebras H1 and H2 from θ, of coincidence of logics over H1 and H2, and of coincidence of the corresponding knowledge. This paper is a survey of ideas stimulated by this approach.

Multi-Sorted Logic, Models, and Logical Geometry

This paper discusses the informational equivalence problem for knowledge bases. The authors show that using some mathematical approach it is possible to attack this problem and end up with an implementable algorithm. An essential part of the paper is devoted to the explanation of the mathematical idea which stands behind this algorithm. The authors try to do that in common terms or, at least, in less formal terms. In the second part of the paper mathematical methods are applied to study the properties of automorphic equivalence of knowledge bases multi-models and show that this notion is much wider than the total isomorphism identity of knowledge bases. In order to make the paper self-contained, the reader is provided with the formal definition of a knowledge base. Further development of the theoretical approach presented in the paper can lead to practical applications. For example, it can be used for preventing duplication of information in knowledge bases and in other tasks of improving knowledge management.

Automorphic Equivalence of Multi-models Recognition

Two relational databases are considered as equivalent, if a reply to an arbitrary query to the first database can be obtained by the means of the second one, and vice versa. The problem is to formalize this idea and to develop an algorithm of verification of relational databases equivalence. In the paper this problem is solved for relational databases with finite data algebras.

Knowledge bases and automorphic equivalence of multi-models versus linear spaces and graphs

The aim of this work is to study the notions of elementarily equivalent and isotypic knowledge bases. We prove that isotypic knowledge bases are informationally equivalent.