Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, cross-fertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a black-box approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired

Abstract. Local search is widely applied to satisfiable SAT problems, and on some classes outperforms backtrack search. An intriguing challenge posed by Selman, Kautz and McAllester in 1997 is to use it instead to prove unsatisfiability. We investigate two distinct approaches. Firstly we apply standard local search to a reformulation of the problem, such that a solution to the reformulation corresponds to a refutation of the original problem. Secondly we design a greedy randomised resolution algorithm that will eventually discover proofs of any size while using bounded memory. We show experimentally that both approaches can refute some problems more quickly than backtrack search. 1

A hybrid algorithm is devised to boost the performance of complete search on under-constrained problems. We suggest to use random variable selection in combination with restarts, augmented by a coarse-grained local search algorithm that learns favorable value heuristics over the course of several restarts. Numerical results show that this method can speedup complete search by orders of magnitude.

Local search algorithms for satisfiability testing are still the best methods for a large number of problems, despite tremendous progresses observed on complete search algorithms over the last few years. However, their intrinsic limit does not allow them to address UNSAT problems. Ten years ago, this question challenged the community without any answer: was it possible to use local search algorithm for UNSAT formulae? We propose here a first approach addressing this issue, that can beat the best resolution-based complete methods. We define the landscape of the search by approximating the number of filtered clauses by resolution proof. Furthermore, we add high-level reasoning mechanism, based on Extended Resolution and Unit Propagation Look-Ahead to make this new and challenging approach possible. Our new algorithm also tends to be the first step on two other challenging problems: obtaining short proofs for UNSAT problems and build a real local-search algorithm for QBF. 1

Abstract. We describe a new decision-making procedure for resolutionbased SAT-solvers called Decision Making with a Reference Point (DMRP). In DMRP, a complete assignment called a reference point is maintained. DMRP is aimed at finding a change of the reference point under which the number of clauses falsified by the modified point is smaller than for the original one. DMRP makes it possible for a DPLL-like algorithm to perform a ”local search strategy”. We describe a SAT-algorithm with conflict clause learning that uses DMRP. Experimental results show that even a straightforward and unoptimized implementation of this algorithm is competitive with SAT-solvers like BerkMin and Minisat on practical formulas. Interestingly, DMRP is beneficial not only for satisfiable but also for unsatisfiable formulas. 1

Recent work on Stochastic Local Search (SLS) for the SAT and CSP domains has shown the superior performance of SLS over traditional backtracking algorithms on a broad range of problem instances. In this paper, we report on a technique for

Abstract. Probabilistic inference techniques can be used to estimate variable bias, or the proportion of solutions to a given SAT problem that fix a variable positively or negatively. Methods like Belief Propagation (BP), Survey Propagation (SP), and Expectation Maximization BP (EMBP) have been used to guess solutions directly, but intuitively they should also prove useful as variable- and valueordering heuristics within full backtracking (DPLL) search. Here we report on practical design issues for realizing this intuition in the VARSAT system, which is built upon the full-featured MiniSat solver. A second, algorithmic, contribution is to present four novel inference techniques that combine BP/SP models with local/global consistency constraints via the EMBP framework. Empirically, we can also report exponential speed-up over existing complete methods, for random problems at the critically-constrained phase transition region in problem hardness. For industrial problems, VARSAT is slower that MiniSat, but comparable in the number and types problems it is able to solve.

assignments

Most successful systematic SAT-solvers are descendants of the DPLL procedure and so operate on partial assignments. Using partial assignments is explained by the “enumerative semantics ” of the DPLL procedure. Current clause learning SAT-solvers, in a sense, have outgrown this semantics. Instead of enumerating the search space as the DPLL procedure does, they explicitly build a resolution proof. In this paper, we suggest a semantics that, in our opinion, is more suitable for clause learning SAT-solvers. The idea is to consider a set of complete assignments not just as a part of the search space but as an “encryption” of a resolution proof or a part thereof. Importantly, a set of points encrypting a resolution proof can be dramatically smaller than the entire search space. We introduce a resolution based SAT-solver with clause learning called FI (short for Find point Image of a proof) that is inspired by the new semantics. FI operates on complete assignments. We compare our naive implementation of FI with Minisat and BerkMin. Experiments show that FI is competitive with Minisat and BerkMin in terms of backtracks. In terms of performance, FI is slower than Minisat and BerkMin for small CNF formulas. On the other hand, even the current primitive implementation of FI is competitive with Minisat and BerkMin on large Bounded Model Checking formulas due to its superior decision making. Keywords: SAT-solver, resolution, decision-making, local search, complete assignment