Translating Pseudo-Boolean Constraints into SAT

Abstract

In this paper, we describe and evaluate three different techniques for translating pseudoboolean constraints (linear constraints over boolean variables) into clauses that can be handled by a standard SAT-solver. We show that by applying a proper mix of translation techniques, a SAT-solver can perform on a par with the best existing native pseudo-boolean solvers. This is particularly valuable in those cases where the constraint problem of interest is naturally expressed as a SAT problem, except for a handful of constraints. Translating those constraints to get a pure clausal problem will take full advantage of the latest improvements in SAT research. A particularly interesting result of this work is the efficiency of sorting networks to express pseudo-boolean constraints. Although tangential to this presentation, the result gives a suggestion as to how synthesis tools may be modified to produce arithmetic circuits more suitable for SAT based reasoning.

On Tackling the Limits of Resolution in SAT Solving

“…A cardinality constraint of the form x i ≤ k is referred to as an AtMostk constraint, whereas a cardinality constraint of the form x i ≥ k is referred to as an AtLeastk constraint. The study of propositional encodings of cardinality and pseudo-Boolean constraints is an area of active research [1,4,5,9,10,17,25,34,58,65,69].…”

Quantitative Verification of Neural Networks and Its Security Applications

“…Besides cardinality networks, there are many other encodings from cardinality constraints to CNF [3,4,8,31,91] that can be used as long as they are equi-witnessable. We do not formally prove here but we strongly suspect that adder networks [31] and BDDs [3] have this property. Adder networks [31] provide a compact, linear transformation resulting in a CNF with O(n) variables and clauses.…”

Low-Rank Semidefinite Programming for the MAX2SAT Problem

“…As our proposed approach in this work (which based upon continuous optimization) differs quite substantially from these approaches, we refer the reader to a recent survey (Morgado et al, 2013) for much more detailed descriptions about the current state of the art. However, broadly speaking, there have been two main classes for these discrete solvers: 1) those based upon bounding the solution via SAT method (Marques-Sila and Planes, 2011;Koshimura et al, 2012;Ansótegui, Bonet, and Levy, 2013;Fu and Malik, 2006;Le Berre and Parrain, 2010;Eén and Sorensson, 2006) which in turn exploit the heuristic developed by the SAT community such as those in the MiniSAT solver; these solvers typically are complete in that they will both produce a satisfying assignment with some number of clauses satisfied and a verification that this is the optimal solution to the problem. And 2) those based upon local search (Luo et al, 2015(Luo et al, , 2017, which maintain and locally adjust a solution to satisfy an increasing number of clauses; these solvers typically are incomplete in that they may quickly find an assignment, but often cannot prove whether or not it is optimal.…”