Lakhdar Sais

Boosting complete techniques thanks to local search methods

In this paper, an efficient heuristic allowing one to localize inconsistent kernels in propositional knowledge‐bases is described. Then, it is shown that local search techniques can boost the performance of logically complete methods for SAT. More precisely, local search techniques can be used to guide the branching strategy of logically complete techniques like Davis and Putnams one, giving rise to significant performance improvements, in particular when addressing locally inconsistent problems. Moreover, this approach appears very competitive in the context of consistent SAT instances, too.

Control-based clause sharing in parallel SAT solving

Conflict driven clause learning, one of the most important component of modern SAT solvers, is also recognized as very important in parallel SAT solving. Indeed, it allows clause sharing between multiple processing units working on related (sub) problems. However, without limitation, sharing clauses might lead to an exponential blow up in communication or to the sharing of irrelevant clauses. This paper, proposes two innovative policies to dynamically adjust the size of shared clauses between any pair of processing units. The first approach controls the overall number of exchanged clauses whereas the second additionally exploits the relevance quality of shared clauses. Experimental results show important improvements of the state-of the-art parallel SAT solver.

Tractability Through Symmetries in Propositional Calculus

Many propositional calculus problems — for example the Ramsey or the pigeon-hole problems — can quite naturally be represented by a small set of first-order logical clauses which becomes a very large set of propositional clauses when we substitute the variables by the constants of the domainD. In many cases the set of clauses contains several symmetries, i.e. the set of clauses remains invariant under certain permutations of variable names. We show how we can shorten the proof of such problems. We first present an algorithm which detects the symmetries and then we explain how the symmetries are introduced and used in the following methods: SLRI, Davis and Putnam and semantic evaluation. Symmetries have been used to obtain results on many known problems, such as the pigeonhole, Schurs lemma, Ramseys, the eight queen, etc. The most interesting one is that we have been able to prove for the first time the unsatisfiability of Ramseys problem (17 vertices and three colors) which has been the subject of much research.

A generalized framework for conflict analysis

This paper presents an extension of Conflict Driven Clauses Learning (CDCL). It relies on an extended notion of implication graph containing additional arcs, called inverse arcs. These are obtained by taking into account the satisfied clauses of the formula, which are usually ignored by conflict analysis. This extension captures more conveniently the whole propagation process, and opens new perspectives for CDCL-based approaches. Among other benefits, our extension leads to a new conflict analysis scheme that exploits the additional arcs to back-jump to higher levels. Experimental results show that the integration of our generalized conflict analysis scheme within two state-of-the-art solvers improves their performance.

Automatic extraction of functional dependencies

In this paper, a new polynomial time technique for extracting functional dependencies in Boolean formulas is proposed. It makes an original use of the well-known Boolean constraint propagation technique (BCP) in a new preprocessing approach that extracts more hidden Boolean functions and dependent variables than previously published approaches on many classes of instances.

Theoretical Study of Symmetries in Propositional Calculus and Applications

Many propositional calculus problems (for example the Ramsey or the pigeon hole problems) can quite naturally be represented by a small set of first order logical clauses which becomes a very large set of propositional clauses when we substitute the variables by the constants of the domain. In many cases, the set of clauses contains several symmetries i.e. the set of clauses remains invariant under a permutation of variable names. We will show how we can shorten the proofs of such problems. We present an algorithm which detects the symmetries and explain how the symmetries are introduced and used in the following methods: Slri, Davis and Putnam and Semantic Evaluation. With symmetries we have got good results on many known problems such pigeon hole, Schurs lemma, Ramsey, the eight queen etc. The most interesting one is that we have been able to prove for the first time the unsatisfiability of the Ramsey problem for 17 vertices and 3 colors.

Diversification and intensification in parallel SAT solving

In this paper, we explore the two well-known principles of diversification and intensification in portfolio-based parallel SAT solving. These dual concepts play an important role in several search algorithms including local search, and appear to be a key point in modern parallel SAT solvers. To study their trade-off, we define two roles for the computational units. Some of them classified as Masters perform an original search strategy, ensuring diversification. The remaining units, classified as Slaves are there to intensify their masters strategy. Several important questions have to be answered. The first one is what information should be given to a slave in order to intensify a given search effort? The second one is, how often, a subordinated unit has to receive such information? Finally, the question of finding the number of subordinated units and their connections with the search efforts has to be answered. Our results lead to an original intensification strategy which outperforms the best parallel SAT solver ManySAT, and solves some open SAT instances.

A SAT-based approach for discovering frequent, closed and maximal patterns in a sequence

In this paper we propose a satisfiability-based approach for enumerating all frequent, closed and maximal patterns with wildcards in a given sequence. In this context, since frequency is the most used criterion, we introduce a new polynomial inductive formulation of the cardinality constraint as a Boolean formula. A nogood-based formulation of the anti-monotonicity property is proposed and dynamically used for pruning. This declarative framework allows us to exploit the efficiency of modern SAT solvers and particularly their clause learning component. The experimental evaluation on real world data shows the feasibility of our proposed approach in practice.

On freezing and reactivating learnt clauses

In this paper, we propose a new dynamic management policy of the learnt clause database in modern SAT solvers. It is based on a dynamic freezing and activation principle of the learnt clauses. At a given search state, using a relevant selection function, it activates the most promising learnt clauses while freezing irrelevant ones. In this way, clauses learned at previous steps can be frozen at the current step and might be activated again in future steps of the search process. Our strategy tries to exploit pieces of information gathered from the past to deduce the relevance of a given clause for the remaining search steps. This policy contrasts with all the well-known deletion strategies, where a given learned clause is definitely eliminated. Experiments on SAT instances taken from the last competitions demonstrate the efficiency of our proposed technique.

Two Proof Procedures for a Cardinality Based Language in Propositional Calculus

In this paper we use the cardinality to increase the expressiveness efficiency of propositional calculus and improve the efficiency of resolution methods. Hence to express propositional problems and logical constraints we introduce the pair formulas (ρ, ℒ) which mean that “at least ρ literals among those of a list ℒ are true”. This makes a generalization of propositional clauses which express ”At least one literal is true among those of the clause”. We propose a cardinality resolution proof system for which we prove both completenesss and decidability. A linear proof for Pigeon-hole problem is given in this system showing the advantage of cardinality.