Single Constant Multiplication for SAT

Hendrik Bierlee, Jip J. Dekker, Vitaly Lagoon, Peter J. Stuckey, Guido Tack

Abstract: This paper presents new methods of encoding the multiplication of a binary encoded integer variable with a constant value for Boolean Satisfiability (SAT) solvers. This problem, known as the Single Constant Multiplication (SCM) problem, is widely studied for its application in hardware design, but its techniques are currently not employed for SAT encodings. Considering the smaller and variable bit sizes and the different cost of operations in SAT, we propose further improvements to existing methods by minimizing the number of full/half adders, rather than the number of ripple carry adders. We compare our methods to simple encodings into adders, currently employed by SAT encoders, and direct Boolean encoding using logic minimization tools. We show that our methods result in improved solve-time for problems involving integer linear constraints. A library of optimal recipes for each method to encode SCM for SAT is made available as part of this publication.