Solving optimization problems with DLL

2008

Automated planning is one of the most important problems in artificial intelligence. We present a new refinement of the classical planning algorithm that formulates the planning problem as a satisfiability problem. Compared with previous techniques, the solution of the planning problem is identified using the number of truth assignments of the corresponding propositional formula and their actions' utilities. Our approach eliminates backtracking and supports efficient planners that consider additional subformulas without the need to recompute solutions for previously provided subformulas. The experimental results show that our approach can help existing SAT-based state-of-the-art planners to find the solution plan more efficiently. * This paper is an improved and unified presentation of conference papers [2] and . It contains a new section (Section 5) and complete proofs for the theoretical results (e.g., Lemma 4.1, 4.2, and Theorem 4.1), new definitions and more comprehensive experimental results.

Lecture Notes in Computer Science, 2005

We propose a new decision heuristic for DPLL-based propositional SAT solvers. Its essence is that both the initial and the conflict clauses are arranged in a list and the next decision variable is chosen from the top -most unsatisfied clause. Various methods of initially organizing the list and moving the clauses within it are studied. Our approach is an extension of one used in Berkmin, and adopted by other modern solvers, according to which only conflict clauses are organized in a list, and a literal-scoring-based secondary heuristic is used when there are no more unsatisfied conflict clauses. Our approach, implemented in the 2004 version of zChaff solver and in a generic Chaff-based SAT solver, results in a significant performance boost on hard industrial benchmarks.

Constraints, 2010

Propositional satisfiability (SAT) is a success story in Computer Science and Artificial Intelligence: SAT solvers are currently used to solve problems in many different application domains, including planning and formal verification. The main reason for this success is that modern SAT solvers can successfully deal with problems having millions of variables. All these solvers are based on the Davis-Logemann-Loveland procedure (DLL). In its original version, DLL is a decision procedure, but it can be very easily modified in order to return one or all assignments satisfying the input set of clauses, assuming at least one exists. However, in many cases it is not enough to compute assignments satisfying all the input clauses: Indeed, the returned assignments have also to be "optimal" in some sense, e.g., they have to satisfy as many other constraints -expressed as preferences-as possible. In this paper we start with qualitative preferences on literals, defined as a partially ordered set (poset) of literals. Such a poset induces a poset on total assignments and leads to the definition of optimal model for a formula ψ as a minimal element of the poset on the models of ψ. We show (i) how DLL can be extended in order to return one or all optimal models of ψ (once converted in clauses and assuming ψ is satisfiable), and (ii) how the same procedures can be used to compute optimal models wrt a qualitative preference on formulas and/or wrt a quantitative preference on literals or formulas. We implemented our ideas and we tested the resulting system on a variety of very challenging structured benchmarks. The results indicate that our implementation has comparable performances with other state-of-the-art systems, tailored for the specific problems we consider. This work extends the results presented in .

2002

The availability of decision procedures for combinations of boolean and linear mathematical propositions opens the ability to solve problems arising from real-world domains such as verification of timed systems and planning with resources. In this paper we present a general and efficient approach to the problem, based on two main ingredients. The first is a DPLL-based SAT procedure, for dealing efficiently with the propositional component of the problem. The second is a tight integration, within the DPLL architecture, of a set of mathematical deciders for theories of increasing expressive power. A preliminary experimental evaluation shows the potential of the approach.

2005

Propositional reasoning (SAT) is an essential part of many reasoning tasks. Many problems in computer science can be compiled to SAT and then effectively decided using state-of-the-art solvers. Alternatively, if reduction to SAT is not feasible, the ideas and technology of state-of-the-art SAT solvers can be useful in deciding the propositional component of the reasoning task being considered. This last approach has been used in different contexts by different authors, many times by authors of this paper. Because of the essential role played by the SAT solver, these decision procedures have been called “SAT-based”. SAT-based decision procedures have been proposed for various logics, but also in other areas such as planning. In this paper we present a unifying perspective on the various SAT-based approaches to these different reasoning tasks.

Satisfiability Problem: Theory and Applications, 1997

The satis ability (SAT) problem is a core problem in mathematical logic and computing theory. In practice, SAT is fundamental in solving many problems in automated reasoning, computer-aided design, computeraided manufacturing, machine vision, database, robotics, integrated circuit design, computer architecture design, and computer network design. Traditional methods treat SAT as a discrete, constrained decision problem. In recent years, many optimization methods, parallel algorithms, and practical techniques have been developed for solving SAT. In this survey, we present a general framework (an algorithm space) that integrates existing SAT algorithms into a uni ed perspective. We describe sequential and parallel SAT algorithms including variable splitting, resolution, local search, global optimization, mathematical programming, and practical SAT algorithms. We give performance evaluation of some existing SAT algorithms. Finally, we provide a set of practical applications of the satis ability problems.

Lecture Notes in Computer Science, 2003

Handbook of Combinatorial Optimization, 2013

The Satisfiability problem (SAT) has gained considerable attention over the past decade for two reasons. First, the performance of competitive SAT solvers has improved enormously due to the implementation of new algorithmic concepts. Second, so many real-world problems that are not naturally expressed as instances of SAT can be transformed to SAT instances and solved relatively efficiently using one of the ever improving SAT solvers or solvers based on SAT. This chapter attempts to clarify these successes by presenting the important advances in SAT algorithm design as well as the classic implementations of SAT solvers. Some applications to which SAT solvers have been successfully applied are also presented. The first section of the chapter presents some logic background and notation that will be necessary to understand and express the algorithms presented. The second section presents several representations for SAT instances in preparation for discussing the wide variety of SAT solver implementations that have been tried. The third section presents some applications of SAT. The final two sections presents algorithms for SAT, including search and heuristic algorithms plus systems that use SAT to manage the logic of complex computations.

Journal of Automated …, 2005

Progress in Artificial Intelligence, 1999

This paper studies the practical impact of the branching heuristics used in Propositional Satisfiability (SAT) algorithms, when applied to solving real-world instances of SAT. In addition, different SAT algorithms are experimentally evaluated. The main conclusion of this study is that even though branching heuristics are crucial for solving SAT, other aspects of the organization of SAT algorithms are also essential. Moreover, we provide empirical evidence that for practical instances of SAT, the search pruning techniques included in the most competitive SAT algorithms may be of more fundamental significance than branching heuristics.