Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution

Abstract

We offer a new understanding of some aspects of practical SAT-solvers that are based on DPLL with unit-clause propagation, clause-learning, and restarts. We do so by analyzing a concrete algorithm which we claim is faithful to what practical solvers do. In particular, before making any new decision or restart, the solver repeatedly applies the unit-resolution rule until saturation, and leaves no component to the mercy of non-determinism except for some internal randomness. We prove the perhaps surprising fact that, although the solver is not explicitly designed for it, with high probability it ends up behaving as width-k resolution after no more than O(n 2k+2 ) conflicts and restarts, where n is the number of variables. In other words, width-k resolution can be thought of as O(n 2k+2 ) restarts of the unit-resolution rule with learning.

On Computing Minimal Equivalent Subformulas

“…Applications (of restricted forms) of redundancy removal can be found in [22,37,36,15,25,30]. The importance of redundant clauses in CDCL SAT solvers is addressed in [33,1]. Redundancy problems have been studied in many other settings, e.g.…”

“…Formula redundancy is often desirable. For example, modern ConflictDriven Clause Learning (CDCL) Boolean Satisfiability (SAT) solvers learn redundant clauses [33,1], which are often essential for solving practical instances of SAT. However, redundancy can also be undesirable.…”

Relating Proof Complexity Measures and Practical Hardness of SAT

“…Given a subsolver A and a unsatisfiable formula F , a (strong) backdoor set S is a subset of the variables in F such that for every truth assignment ρ over S, the subsolver A determines unsatisfiability of F ρ . 9 The algorithm A might be, for instance, unit propagation, polynomial-time restricted DPLL, or a 2-SAT algorithm.…”

“…size) and space are interesting since they in some sense model the running time and memory consumption of (optimal) CDCL solvers, while width is another measure that seems relevant for practical performance in view of e.g. [9]. An informal description of these measures is as follows (see Section 2 for the formal definitions):…”

“…Width is thus a stricter hardness measure than length. Recently, is was shown in [9] (under some theoretical assumptions) that if the minimum width of any resolution proof for a formula is w, then a CDCL solver (that does not at all care about width per se) will with high probability decide the formula in time something like n w . This seems to indicate that resolution width could be a measure that correlates well with practical hardness for CDCL solvers.…”