Logical Operations over Fuzzy Sets and Relations in Automaton Interpretation
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NEW MEANS OF CYBERNETICS, INFORMATICS, COMPUTER ENGINEERING, AND SYSTEMS ANALYSIS LOGICAL OPERATIONS OVER FUZZY SETS AND RELATIONS IN AUTOMATON INTERPRETATION S. L. Kryvyi,1 V. N. Opanasenko,2 and S. B. Zavyalov3
UDC 516.813
Abstract. This article considers logical operations over fuzzy sets and relations in automaton interpretation by means of logical networks with the possibility of adaptation. Examples of the synthesis of networks with the help of the considered automata based on structures of expressions for specified fuzzy sets and relations are given. Keywords: fuzzy set, relation, logical operation. INTRODUCTION The adaptation of hardware to the implementation of operations of partitioning vectors with integer coordinates (see [1, 2]) stipulates the possibility of implementation of operations over fuzzy sets and relations. The approach to the execution of the mentioned operations that is proposed in this article is well-known as the “technology of reconfigurable computing” [3], and its embodiment into real projects became possible thanks to the advent of programmable integrated circuits [4–6]. In particular, a method was considered in [7, 8] for solving the problem of adaptation of hardware together with a formalized substantiation of the corresponding algorithms based on the adaptive logic networks (ALNs) oriented towards the implementation of algorithms for partitioning a set of vectors with integer coordinates. Using such partition algorithms, the present article proposes algorithms for executing logical operations over fuzzy sets (FSs) and fuzzy relations (FRs). 1. OPERATIONS OVER FUZZY SETS AND FUZZY RELATIONS. PECULIARITIES OF THEIR IMPLEMENTATION Operations over fuzzy sets and fuzzy relations are subdivided into logical and algebraic. For fuzzy sets, logical operations are the operations of inclusion, equality, union, intersection, complement, difference, and disjunctive sum. Values of these operations are computed with the help of a membership function of the form m : X ® [0, 1] = M [9–11]. Let m A ( x ) and m B ( x ) be functions of membership of an element x in sets A and B , respectively. The operations of inclusion A Í B and equality A = B for fuzzy sets assumes Boolean values, i.e., the result of computing them is the value 0 or 1. This follows from the definition of the operations
A Í B Û "x Î A, B m A ( x ) £ m B ( x ) ; A = B Û "x Î A, B m A ( x ) = m B ( x ) . 1
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, [email protected]. 2V. M. Glushkov Institute of Cybernetics, National Academy of Sciences of Ukraine, Kyiv, Ukraine, [email protected]. 3LLC “Radioniks,” Kyiv, Ukraine, [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 6, November– December, 2020, pp. 175–183. Original article submitted March 12, 2020. 1012
1060-0396/20/5606-1012 ©2020 Springer Science+Business Media, LLC
The operation of union A U B of two fuzzy sets determines the smallest fuzzy set including both À and  with the membership function max ( m A ( x ), m B ( x )) . T
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