Logical Design of Discrete Automata

Synthesis of combinational circuits by means of bi-decomposition of Boolean functions

Yu. V. Pottosin<sup>ab

a</sup> United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk, Belarus
^b Belarusian State University of Informatics and Radioelectronics, Minsk, Belarus

Abstract: The problem of combinational circuits synthesis in the basis of two-input gates is considered. Those gates are AND, OR, NAND and NOR. A method for solving this problem by means of Boolean functions bi-decomposition is suggested. The method reduces the problem to the search for a weighted two-block cover of the orthogonality graph of ternary matrice rows representing the given Boolean function by complete bipartite subgraphs (bi-cliques). Each bi-clique in the obtained cover is assigned in a certain way with a set of variables that are the arguments of the function. This set is the weight of the bi-clique. Each of those bi-cliques defines a Boolean function whose arguments are the variables assigned to it. The functions obtained in such a way constitute the required decomposition. The process of combinational circuit synthesis consists in successively applying bi-decomposition to the functions obtained. The method for two-block covering the orthogonality graph of ternary matrice rows is described.

Keywords: synthesis of combinational circuits, Boolean function, decomposition of Boolean functions, ternary matrix, complete bipartite subgraph, two-block cover.

UDC: 519.711

Language: English

DOI: 10.17223/20710410/60/8