Abstract
The problem of decomposition of a Boolean function is to represent a given Boolean function in the form of a superposition of some Boolean functions whose number of arguments are less than the number of given function. The bi-decomposition represents a given function as a logic algebra operation, which is also given, over two Boolean functions. The task is reduced to specification of those two functions. A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0. The complete graph of orthogonality of Boolean vectors that constitute the definitional domain of the given function is considered. In the graph, the edges are picked out, any of which has its ends corresponding the elements of Boolean space where the given function has different values. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs (bicliques). Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is a pair of certain parameters of assigned DNF. According to each biclique of obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF. A technique for constructing the mentioned cover for two kinds of output function is described.
Highlights
The given Boolean function is specified by two sets, one of which is the part of the Boolean space of the arguments of the function where its value is 1, and the other set is the part of the space where the function has the value 0
The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the set of picked out edges of considered graph by its complete bipartite subgraphs
Every biclique is assigned with a disjunctive normal form (DNF) in definite way
Summary
A method for bi-decomposition of incompletely specified (partial) Boolean function is suggested. A heuristic method for bi-decomposition of partial Boolean functions. Задача алгебраической декомпозиции при функции , выражаемой операцией сложения по модулю 2, при заданном разбиении (Z1, Z2) рассматривается в работе [5], где для ее решения предлагается использовать логические уравнения. В предлагаемой статье, как уже отмечалось, множества Z1 и Z2 определяются в процессе решения задачи: либо выдается решение, либо делается заключение, что декомпозиции не существует.
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