Towards Understanding and Harnessing the Potential of Clause Learning

TL;DR

This paper provides a formal proof system characterization of clause learning in SAT solvers, demonstrating its potential to produce exponentially shorter proofs than resolution and proposing methods to leverage problem structure for improved performance.

Contribution

It offers the first precise proof system analysis of clause learning, relating it to resolution, and introduces strategies to exploit problem structure for faster SAT solving.

Findings

CL can produce exponentially shorter proofs than many resolution refinements.

A variant of CL with unlimited restarts matches the power of resolution.

Exploiting problem structure leads to exponential speed-ups in practice.

Abstract

Efficient implementations of DPLL with the addition of clause learning are the fastest complete Boolean satisfiability solvers and can handle many significant real-world problems, such as verification, planning and design. Despite its importance, little is known of the ultimate strengths and limitations of the technique. This paper presents the first precise characterization of clause learning as a proof system (CL), and begins the task of understanding its power by relating it to the well-studied resolution proof system. In particular, we show that with a new learning scheme, CL can provide exponentially shorter proofs than many proper refinements of general resolution (RES) satisfying a natural property. These include regular and Davis-Putnam resolution, which are already known to be much stronger than ordinary DPLL. We also show that a slight variant of CL with unlimited restarts is as powerful as RES itself. Translating these analytical results to practice, however, presents a challenge because of the nondeterministic nature of clause learning algorithms. We propose a novel way of exploiting the underlying problem structure, in the form of a high level problem description such as a graph or PDDL specification, to guide clause learning algorithms toward faster solutions. We show that this leads to exponential speed-ups on grid and randomized pebbling problems, as well as substantial improvements on certain ordering formulas.