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

Abstract

O b j e c t i v e s . The problem of synthesis of combinational circuits in the basis of two-input gates is considered. Those gates are AND, OR, NAND and NOR. The objective of the paper is to investigate the possibilities of application of bi-decomposition of Boolean functions to the synthesis of combinational circuits.M e t h o d s . The method for bi-decomposition is reduced to the search in a graph for a weighted two-block cover with complete bipartite subgraphs (bi-cliques).R e s u l t s . The initial Boolean function is given as two ternary matrices, one of which represents the domain of Boolean space where the function has the value 1, and the other is the domain of Boolean space where the function has the value 0. The orthogonality graph of rows of ternary matrices representing the given function is considered.  The method for two-bi-clique covering the orthogonality graph of rows of ternary matrices is described. Every bi-clique in the obtained cover is assigned in a certain way with а 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.Co n c l u s i o n .  The  process  of  synthesis  of  a  combinational  circuit  consists  of  a  successive  application  of bi-decomposition to obtained functions. The suggested method allows obtaining the circuits with short delay.