Towards Understanding and Harnessing the Potential of Clause Learning

TLDR

Clause learning, integrated into DPLL, yields the fastest complete SAT solvers for many real‑world problems, yet its ultimate strengths and limitations remain largely unexplored, especially compared to stronger systems like regular and Davis‑Putnam resolution, and translating its theoretical advantages into practice is hampered by nondeterminism. This work first precisely characterizes clause learning as a proof system and relates it to resolution, then proposes exploiting high‑level problem descriptions such as graphs or PDDL to steer clause learning toward faster solutions. The authors formalize clause learning as a proof system, compare it to resolution, and introduce a novel guidance strategy that uses structural information from problem specifications to direct learning. They demonstrate that a new learning scheme yields exponentially shorter proofs than many proper refinements of general resolution, that a variant with unlimited restarts matches the power of resolution itself, and that guiding clause learning by problem structure produces exponential speed‑ups on grid and randomized pebbling problems and significant gains on certain ordering formulas.

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.