NP-completeness of small conflict set generation for congruence closure

Andreas Fellner,Pascal Fontaine,Bruno Woltzenlogel Paleo

doi:10.1007/s10703-017-0283-x

Abstract

The efficiency of satisfiability modulo theories (SMT) solvers is dependent on the capability of theory reasoners to provide small conflict sets, i.e. small unsatisfiable subsets from unsatisfiable sets of literals. Decision procedures for uninterpreted symbols (i.e. congruence closure algorithms) date back from the very early days of SMT. Nevertheless, to the best of our knowledge, the complexity of generating smallest conflict sets for sets of literals with uninterpreted symbols and equalities had not yet been determined, although the corresponding decision problem was believed to be NP-complete. We provide here an NP-completeness proof, using a simple reduction from SAT.

Highlights

Satisfiability modulo theory solvers are nowadays based on a cooperation between a propositional satisfiability (SAT) solver and a theory reasoner for the combination of theories supported by the satisfiability modulo theories (SMT) solver
The propositional structure of the problem is handled by the SAT solver, whereas the theory reasoner only has to deal with conjunctions of
The complexity of the corresponding decision problem is mentioned to be NP-complete in [10]—with a reference to a private communication with Ashish Tiwari—but neither the authors of [10] nor Ashish Tiwari published a written proof of this fact.1. Our interest in this problem arose from our work on Skeptik [2], a tool for the compression of proofs generated by SAT and SMT solvers

Summary

Introduction

Satisfiability modulo theory solvers are nowadays based on a cooperation between a propositional satisfiability (SAT) solver and a theory reasoner for the combination of theories supported by the SMT solver. An optimal conflict clause, corresponding to a minimal unsatisfiable subset of literals (i.e. such that all its proper subsets are satisfiable) or even a minimum one (i.e. smallest among the minimals) is desirable This feature of the theory reasoners to generate small conflict sets (a name adopted in [1]) from their input is referred to as proof production [8,9] or explanation generation [10]. The complexity of the corresponding decision problem (i.e. of whether there exists a conflict clause with size smaller than a given k) is mentioned to be NP-complete in [10]—with a reference to a private communication with Ashish Tiwari—but neither the authors of [10] nor Ashish Tiwari published a written proof of this fact.1 Our interest in this problem arose from our work on Skeptik [2], a tool for the compression of proofs generated by SAT and SMT solvers. A preliminary version was presented at the SMT Workshop 2015 [4]

Preliminaries

Congruence closure in practice

NP-completeness of the small conflict set problem

Conclusion