Minimization of binary decision diagrams for systems of completely defined Boolean functions using Shannon expansions and algebraic representations of cofactors

Abstract

In the systems of digital VLSI design (Very Large Integrated Circuits), the BDD (Binary Decision Diagram) is used for VLSI verification, as well as for technologically independent optimization as the first stage in the synthesis of logic circuits in various technological bases. The BDD is an acyclic graph defining a Boolean function or a system of Boolean functions. Each vertex of this graph corresponds to the complete or reduced Shannon expansion formula. When BDD representation for systems of Boolean functions is constructed, it is possible to perform additional logical optimization based on the proposed method of searching for algebraic representations of cofactors (subfunctions) of the same BDD level in the form of a disjunction, conjunction either exclusive-or of cofactors of the same level or lower levels of BDD. A directed BDD graph for a system of functions is constructed on the basis of Shannon expansion of all component functions of the system by the same permutation of variables. The method allows to reduce the number of literals by replacing the Shannon expansion formulas with simpler formulas that are disjunctions or conjunctions of cofactors, and to reduce the number of literals in specifying a system of Boolean functions. The number of literals in algebraic multilevel representations of systems of fully defined Boolean functions is the main optimization criterion in the synthesis of combinational circuits from librarian logic elements.