SAT-based Explicit LTL Reasoning

We present here a new explicit reasoning framework for linear temporal logic (LTL), which is built on top of propositional satisfiability (SAT) solving. As a proof-of-concept of this framework, we describe a new LTL satisfiability tool, Aalta\_v2.0, which is built on top of the MiniSAT SAT solver. We test the effectiveness of this approach by demonnstrating that Aalta\_v2.0 significantly outperforms all existing LTL satisfiability solvers. Furthermore, we show that the framework can be extended from propositional LTL to assertional LTL (where we allow theory atoms), by replacing MiniSAT with the Z3 SMT solver, and demonstrating that this can yield an exponential improvement in performance

Hiding Satisfying Assignments: Two are Better than One

The evaluation of incomplete satisfiability solvers depends critically on the availability of hard satisfiable instances. A plausible source of such instances consists of random k-SAT formulas whose clauses are chosen uniformly from among all clauses satisfying some randomly chosen truth assignment A. Unfortunately, instances generated in this manner tend to be relatively easy and can be solved efficiently by practical heuristics. Roughly speaking, as the formula's density increases, for a number of different algorithms, A acts as a stronger and stronger attractor. Motivated by recent results on the geometry of the space of satisfying truth assignments of random k-SAT and NAE-k-SAT formulas, we introduce a simple twist on this basic model, which appears to dramatically increase its hardness. Namely, in addition to forbidding the clauses violated by the hidden assignment A, we also forbid the clauses violated by its complement, so that both A and complement of A are satisfying. It appears that under this "symmetrization'' the effects of the two attractors largely cancel out, making it much harder for algorithms to find any truth assignment. We give theoretical and experimental evidence supporting this assertion.Comment: Preliminary version appeared in AAAI 200

Heuristic average-case analysis of the backtrack resolution of random 3-Satisfiability instances

An analysis of the average-case complexity of solving random 3-Satisfiability (SAT) instances with backtrack algorithms is presented. We first interpret previous rigorous works in a unifying framework based on the statistical physics notions of dynamical trajectories, phase diagram and growth process. It is argued that, under the action of the Davis--Putnam--Loveland--Logemann (DPLL) algorithm, 3-SAT instances are turned into 2+p-SAT instances whose characteristic parameters (ratio alpha of clauses per variable, fraction p of 3-clauses) can be followed during the operation, and define resolution trajectories. Depending on the location of trajectories in the phase diagram of the 2+p-SAT model, easy (polynomial) or hard (exponential) resolutions are generated. Three regimes are identified, depending on the ratio alpha of the 3-SAT instance to be solved. Lower sat phase: for small ratios, DPLL almost surely finds a solution in a time growing linearly with the number N of variables. Upper sat phase: for intermediate ratios, instances are almost surely satisfiable but finding a solution requires exponential time (2 ^ (N omega) with omega>0) with high probability. Unsat phase: for large ratios, there is almost always no solution and proofs of refutation are exponential. An analysis of the growth of the search tree in both upper sat and unsat regimes is presented, and allows us to estimate omega as a function of alpha. This analysis is based on an exact relationship between the average size of the search tree and the powers of the evolution operator encoding the elementary steps of the search heuristic.Comment: to appear in Theoretical Computer Scienc