Implementation of a System of Incompletely Specified Boolean Functions in a Circuit of Two-Input Gates by Means of Bi-Decomposition

Abstract

The problem of bi-decomposition of a Boolean function is to represent a given Boolean function as a logic algebra operation over two Boolean functions. A method based on bi-decomposition of Boolean functions is suggested to implement systems of incompletely specified (partial) Boolean functions in the basis of two-input gates. This basis can be the basis of NOR gates, NAND gates or the basis of AND and OR gates with accessible input complements. The used method for bi-decomposition is reduced to the search for weighted two-block cover of a complete bipartite weighted graph with complete bipartite subgraphs (bi-cliques). The graph represents differences between the rows of Boolean matrices that specify the given system of partial Boolean functions. The system is given by two Boolean matrices, one of which represents the domain of Boolean space where the values of the given functions are specified, and the other—the values of the functions on the elements of the domain. Every bi-clique in the obtained cover is assigned in a certain way with а set of variables that are the arguments of the functions. This set is the weight of the bi-clique. Every one 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 synthesis of a combinational circuit consists in successive application of bi-decomposition to the obtained functions. The method for two-block covering the orthogonality graph of rows of ternary matrices is described.