Richard Ostrowski

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.

Recovering and Exploiting Structural Knowledge from CNF Formulas

In this paper, a new pre-processing step is proposed in the resolution of SAT instances, that recovers and exploits structural knowledge that is hidden in the CNF. It delivers an hybrid formula made of clauses together with a set of equations of the form y = f(x1, ..., xn) where f is a standard connective operator among (?, ?, ?) and where y and xi are boolean variables of the initial SAT instance. This set of equations is then exploited to eliminate clauses and variables, while preserving satisfiability. These extraction and simplification techniques allowed us to implement a new SAT solver that proves to be the most efficient current one w.r.t. several important classes of instances.

Computing Horn Strong Backdoor Sets Thanks to Local Search

In this paper, a new approach for computing strong backdoor sets of Boolean formula in conjunctive normal form (CNF) is proposed. It makes an original use of local search techniques for finding an assignment leading to a largest renamable Horn sub-formula of a given CNF. More precisely, at each step, preference is given to variables such that when assigned to the opposite value lead to the smallest number of remaining non-Horn clauses. Consequently, if no positive or non Horn clauses remain in the formula, our approach answer the satisfiability of the original formula; otherwise, a smallest non-Horn sub-formula is used to extract the set of variables (strong backdoor) such that when assigned leads to a tractable sub-formula. Branching on the variables of the strong backdoor set leads to significant improvements of Zchaff SAT solver with respect to many real worlds SAT instances

Enhancing Clause Learning by Symmetry in SAT Solvers

The satisfiability problem (SAT) is shown to be the first decision NP-complete problem. It is central in complexity theory. A CNF formula usually contains an interesting number of symmetries. That is, the formula remains invariant under some variable permutations. Such permutations are the symmetries of the formula, their elimination can lead to make a short proof for a satisfiability proof procedure. On other hand, many improvements had been done in SAT solving, Con???ict-Driven Clause Learning (CDCL) SAT solvers are now able to solve great size and industrial SAT instances efficiently. The main theoretical key behind these modern solvers is, they use lazy data structures, a restart policy and perform clause learning at each fail end point in the search tree. Although symmetry and clause learning are shown to be powerful principles for SAT solving, but their combination, as far as we now, is not investigated. In this paper, we will show how symmetry can be used to improve clause learning in CDCL SAT solvers. We implemented the symmetry clause learning approach on the MiniSat solver and experimented it on several SAT instances. We compared both MiniSat with and without symmetry and the results obtained are very promising and show that clause learning by symmetry is profitable for CDCL SAT solvers.

Using Boolean Constraint Propagation for sub-clauses deduction

The Boolean Constraint Propagation (BCP) is a well-known helpful technique implemented in most state-of-the-art efficient satisfiability solvers. We propose in this paper a new use of the BCP to deduce sub-clauses from the associated implication graph. Our aim is to reduce the length of clauses thanks to the subsumption rule. We show how such extension can be grafted to modern SAT solvers and we provide some experimental results of the sub-clauses deduction as a pretreatment process. This work is supported by the Region Picardie under HTSC project.

Eliminating redundancies in SAT search trees

Conflict analysis is a powerful paradigm of backtrack search algorithms, in particular for solving satisfiability problems arising from practical applications. Accordingly, most recent Boolean satisfiability solvers implement forms of conflict analysis, at least to some extent. In this paper, a branching criterion initially introduced by Purdom is revisited and extended. Contrary to the authors a priori analysis, it is shown very efficient from a practical point of view in that it allows search trees in SAT solving to be pruned in a significant way while obeying an interesting time and space trade-off. More precisely, we show that redundancies during the search process can be avoided without adding new constraints explicitly. Moreover, the technique can be used not only to prune branches in the search tree, but also to derive implied literals. Extensive experimental results illustrate the feasibility and practical interest of this approach.

From XSAT to SAT by Exhibiting Equivalencies

Given a Boolean formula in conjunctive normal form (CNF), the exact satisfiability problem (XSAT), a variant of the satisfiability problem (SAT), consists in finding an assignment to the variables such that each clause contains exactly one satisfied literal. Best algorithms to solve this problem runs in O(20.2325n) (O(20.1379n) for X3SAT) [12]. Another possibility is to transform each clause in a set of equivalent clauses for the Satisfiability problem and to use modern and powerful solvers (zChaff [14], Berkmin [6], MiniSat [5], RSat [16] etc.) to find such truth assignment. In this paper we introduce two new encoding from XSAT instances to SAT instances that leads to a lot of structural information (especially equivalencies) which is naturally hidden in the pairwise transformation. Some solvers (lsat[15], march dl [7], eqsatz [10]) can take into account this kinds of structural information to make simplifications as pretreatment and speed-up the resolution. Then we show the interest of dealing with the XSAT formalism by introducing an encoding of binary CSP and graph coloring problem into XSAT instances. Preliminary results on graph coloring problem show the importance of exhibiting equivalencies for the XSAT problem.

From horn strong backdoor sets to ordered strong backdoor sets

Identifying and exploiting hidden problem structures is recognized as a fundamental way to deal with the intractability of combinatorial problems. Recently, a particular structure called (strong) backdoor has been identified in the context of the satisfiability problem. Connections has been established between backdoors and problem hardness leading to a better approximation of the worst case time complexity. Strong backdoor sets can be computed for any tractable class. In [1], a method for the approximation of strong backdoor sets for the Horn-Sat fragment was proposed. This approximation is realized in two steps. First, the best Horn renaming of the original CNF formula, in term of number of clauses, is computed. Then a Horn strong backdoor set is extracted from the non Horn part of the renamed formula. in this article, we propose computing Horn strong backdoor sets using the same scheme but minimizing the number of positive literals in the non Horn part of the renamed formula instead of minimizing the number of non Horn clauses. Then we extend this method to the class of ordered formulas [2] which is an extension of the Horn class. This method insure to obtain ordered strong backdoor sets of size less or equal than the size of Horn strong backdoor sets (never greater). Experimental results show that these new methods allow to reduce the size of strong backdoor sets on several instances and that their exploitation also allow to enhance the efficiency of satisfiability solvers.

Detecting Boolean Functions for Proving Unsatisfiability

In 1997, B. Selman and H. Kautz proposed a series of 10 challenges. One of them concerned the design of a practical stochastic local search procedure for proving unsatisfiability (Challenge 5). Today, more than 10 years later, only few attempts were led to address this challenge, in spite of the great number of incomplete methods for proving satisfiability. In this paper, we propose a two steps algorithm for proving unsatisfiability of CNF formulas\footnote{supported by ANR UNLOC project n\ensuremath{^\circ} BLAN08-1\_328904}. The first step consists in detecting

FROM XSAT TO SAT BY EXHIBITING BOOLEAN FUNCTIONS

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