A method for bi-decomposition of partial Boolean functions

• Main Author: Yu. V. Pottosin
• Format: Article
• Language: Russian
• Published: The United Institute of Informatics Problems of the National Academy of Sciences of Belarus 2019-12-01
• Series: Informatika
• Subjects: partial boolean function; boolean function bi-decomposition; superposition of functions; logic algebra operations; matrix representation of boolean functions; covering problem; complete bipartite subgraph
• Online Access: https://inf.grid.by/jour/article/view/881

The problem of bi-decomposition of a Boolean function is to represent a given Boolean function in the form of a given logic algebra operation over two Boolean functions and so is reduced to specification of these functions. Any of the required functions must have fewer arguments than the given function. A method of bi-decomposition for an incompletely specified (partial) Boolean function is suggested, this method uses the approach applied in solving the general problem of parallel decomposition of partial Boolean functions. The specification of the given function must be in the form of a pair of matrices. One of them, argument matrix, can be ternary or binary and represents the definitional domain of the given function. The other one, value matrix, is a binary column-vector and represents the function values on the intervals or elements of the Boolean space of the arguments. The graph of orthogonality of the argument matrix rows and the graph of orthogonality of one-element rows of the value matrix are considered. The problem of bi-decomposition is reduced to the problem of a weighted two-block covering the edge set of the orthogonality graph of the value matrix rows by complete bipartite subgraphs (bicliques) of the orthogonality graph of the argument matrix rows. Every biclique is assigned with a disjunctive normal form (DNF) in definite way. The weight of a biclique is the minimum rank of a term of the assigned DNF. According to each biclique of the obtained cover, a Boolean function is constructed whose arguments are the variables from the term of minimal rank on the DNF.