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[EN]In the last decades, we have witnessed a remarkable success of algorithms solving the Boolean Satisfiability problem (SAT) on instances encoding application or real-world problems arising from a very diverse number of domains, such as hardware and software verification, planning or cryptography. These algorithms are the so known Conflict-Driven Clause Learning (CDCL) SAT solvers. Interestingly enough, the reasons for the success of these solvers on this diverse range of problems are not completely understood yet. A common issue when facing this open challenge is the heterogeneity of this set of benchmarks. Another problem is the limited number of existing instances. In this context, random models of SAT formulas capturing features shared by the majority of these application benchmarks become crucial, for both theoretical and practical purposes. On the one hand, it is undoubtedly necessary to have random models where theoretical properties, like hardness, can be studied. Therefore, realistic random SAT models may contribute to explain the success of these solvers on these industrial problems. On the other hand, the limited number of benchmarks and their hardness in practice makes the evaluation of new solving techniques a costly task. Therefore, these realistic random SAT generators can provide an unlimited number of pseudo-industrial random SAT instances with some desired properties. In this work, we present a random SAT instances generator based on the notion of locality. This notion is complementary to the popularity of variables, which is present in the scale-free structure, observable in actual application problems and achievable by previous generators. Our random SAT model combines both locality and popularity, and we show that they are two decisive dimensions of attractiveness among the variables of a formula, and how CDCL SAT solvers take advantage of them. Locality is closely related to the community structure, another important feature of application SA

Nowadays, modern SAT solvers are able to efficiently solve many industrial, or real-world, SAT instances. However, the process of development and testing of new SAT solving techniques is conditioned to the finite and reduced number of known industrial benchmarks. Therefore, new models of random SAT instances generation that capture realistically the features of real-world problems can be beneficial to the SAT community. In many works, the structure of industrial instances has been analyzed representing them as graphs and studying some of their properties, like modularity. In this work, we use the notion of modularity to define a new model of generation of random SAT instances with community structure, called Community Attachment. For high values of modularity (i.e., clear community structure), we realistically model pseudo-industrial random SAT formulas. This model also generates SAT instances very similar to classical random formulas using a low value of modularity. We also prove that the phase transition point, if exists, is independent on the modularity. We evaluate the adequacy of this model to real industrial SAT problems in terms of SAT solvers performance, and show that modern solvers do actually exploit this community structure. Finally, we use this generator to observe the connections between the modularity of the instance and some components of the solver, such as the variable branching heuristics or the clause learning mechanism.

Signed clausal forms offer a suitable logical framework for automated reasoning in multiple-valued logics. It turns out that the satisfiability problem of any finitely-valued propositional logic, as well as of certain infinitely-valued logics, can be easily reduced, in polynomial time, to the satisfiability problem of signed clausal forms. On the other hand, signed clausal forms are a powerful knowledge representation language for constraint programming, and have shown to be a practical and competitive approach to solving combinatorial decision problems. Motivated by the existing theoretical and practical results for the satisfiability problem of signed clausal forms, as well as by the recent logical and algorithmic results on the Boolean maximum satisfiability problem, in this paper we investigate the maximum satisfiability problem of propositional signed clausal forms from the logical and practical points of view. From the logical perspective, our aim is to define complete inference systems taking as a starting point the resolution-style calculi defined for the Boolean CNF case. The result is the definition of two sound and complete resolution-style rules, called signed binary resolution and signed parallel resolution for maximum satisfiability. From the practical perspective, our main motivation is to use the language of signed clausal forms along with the newly defined inference systems as a generic approach to solve combinatorial optimization problems, and not just for solving decision problems as so far. The result is the establishment of a link between signed logic and constraint programming that provides a concise and elegant logical framework for weighted constraint programming. © 2012 Elsevier Inc. All rights reserved.

Many industrial optimization problems can be translated to MaxSAT. Although the general problem is NP hard, like SAT, many practical problems may be solved using modern MaxSAT solvers. In this paper we present several algorithms specially designed to deal with industrial or real problems. All of them are based on the idea of solving MaxSAT through successive calls to a SAT solver. We show that this SAT-based technique is efficient in solving industrial problems. In fact, all state-of-the-art MaxSAT solvers that perform well in industrial instances are based on this technique. In particular, our solvers won the 2009 partial MaxSAT and the 2011 weighted partial MaxSAT industrial categories of the MaxSAT evaluation. We prove the correctness of all our algorithms. We also present a complete experimental study comparing the performance of our algorithms with latest MaxSAT solvers. © 2013 Elsevier B.V.

Both Sequence and Context Unification generalize the same problem: Word Unification. Besides that, Sequence Unification solves equations between unranked terms involving sequence variables, and seems to be appealing for information extraction in XML documents, program transformation, knowledge representation, and rule-based programming. It is decidable. Context Unification deals with the same problem for ranked terms involving context variables, and has applications in computational linguistics and program transformation. Its decidability is a long-standing open question. In this work we study a relation between these two problems. We introduce a variant (restriction) of Context Unification, called Left-Hole Context Unification (LHCU), to which Sequence Unification is P-reduced: We define a partial currying procedure to translate Sequence Unification problems into Left-Hole Context Unification problems, and prove the soundness of the translation. Furthermore, a precise characterization of the shape of the unifiers allows us to easily reduce Left-Hole Context Unification to (the decidable problem of) Word Unification with Regular Constraints, obtaining then a new decidability proof for Sequence Unification. Finally, we define an extension of Sequence Unification (ESU) and, closing the circle, prove the inter-P-reducibility of LHCU and ESU. © 2009 Elsevier Ltd. All rights reserved.

Max-SAT is the problem of finding an assignment minimizing the number of unsatisfied clauses in a CNF formula. We propose a resolution-like calculus for Max-SAT and prove its soundness and completeness. We also prove the completeness of some refinements of this calculus. From the completeness proof we derive an exact algorithm for Max-SAT and a time upper bound. We also define a weighted Max-SAT resolution-like rule, and show how to adapt the soundness and completeness proofs of the Max-SAT rule to the weighted Max-SAT rule. Finally, we give several particular Max-SAT problems that require an exponential number of steps of our Max-SAT rule to obtain the minimal number of unsatisfied clauses of the combinatorial principle. These results are based on the corresponding resolution lower bounds for those particular problems. © 2007 Elsevier B.V. All rights reserved.

There is a close relationship between word unification and second-order unification. This similarity has been exploited, for instance, in order to prove decidability of monadic second-order unification and decidability of linear second-order unification when no second-order variable occurs more than twice. The attempt to prove the second result for (nonlinear) second-order unification failed and led instead to a natural reduction from simultaneous rigid E-unification to this second-order unification. This reduction is the first main result of this paper, and it is the starting point for proving some novel results about the undecidability of second-order unification presented in the rest of the paper. We prove that second-order unification is undecidable in the following three cases: (1) each second-order variable occurs at most twice and there are only two second-order variables; (2) there is only one second-order variable and it is unary; (3) the following conditions (i)-(iv) hold for some fixed integer n: (i) the arguments of all second-order variables are ground terms of size