Satisfiability Solvers
colomba bufano
Sign up for access to the world's latest research
checkGet notified about relevant papers
checkSave papers to use in your research
checkJoin the discussion with peers
checkTrack your impact
Abstract
The past few years have seen an enormous progress in the performance of Boolean satisfiability (SAT) solvers . Despite the worst-case exponential run time of all known algorithms, satisfiability solvers are increasingly leaving their mark as a generalpurpose tool in areas as diverse as software and hardware verification , automatic test pattern generation , planning , scheduling [103], and even challenging problems from algebra . Annual SAT competitions have led to the development of dozens of clever implementations of such solvers [e.g. 13, 19, 71, 93, 109, 118, 150, 152, 161, 165, 170, 171, 173, 174, 184, 198, 211, 213, 236], an exploration of many new techniques [e.g. 15, 102, 149, 170, 174], and the creation of an extensive suite of real-world instances as well as challenging hand-crafted benchmark problems [cf. 115]. Modern SAT solvers provide a "black-box" procedure that can often solve hard structured problems with over a million variables and several million constraints.
Key takeaways
- However, large SAT encodings are no longer an obstacle for modern SAT solvers.
- It is natural to think of a CNF formula as a set of clauses and each clause as a set of literals.
- This technique has played a critical role in the success of SAT solvers, in particular those involving clause learning.
- Clause learning and random restarts can be considered to be two of the most important ideas that have lifted the scope of modern SAT solvers from experimental toy problems to large instances taken from real world challenges.
- This explains, formally, the performance gains observed empirically when clause learning is added to DPLL based solvers.
Related papers
Chaff: Engineering An Efficient SAT Solver
Design Automation …, 2001
Boolean Satisfiability is probably the most studied of combinatorial optimization/search problems. Significant effort has been devoted to trying to provide practical solutions to this problem for problem instances encountered in a range of applications in Electronic Design Automation (EDA), as well as in Artificial Intelligence (AI). This study has culminated in the development of several SAT packages, both proprietary and in the public domain (e.g. GRASP, SATO) which find significant use in both research and industry. Most existing complete solvers are variants of the Davis-Putnam (DP) search algorithm. In this paper we describe the development of a new complete solver, Chaff, which achieves significant performance gains through careful engineering of all aspects of the search-especially a particularly efficient implementation of Boolean constraint propagation (BCP) and a novel low overhead decision strategy. Chaff has been able to obtain one to two orders of magnitude performance improvement on difficult SAT benchmarks in comparison with other solvers (DP or otherwise), including GRASP and SATO.
A SAT Based Approach for Solving Formulas over Boolean and Linear Mathematical Propositions
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.
Algorithms for the Satisfiability Problem
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.
Answer Set Programming Based on Propositional Satisfiability
Journal of Automated Reasoning, 2006
Answer Set Programming (ASP) emerged in the late 1990s as a new logic programming paradigm which has been successfully applied in various application domains. Also motivated by the availability of efficient solvers for propositional satisfiability (SAT), various reductions from logic programs to SAT were introduced in the past. All these reductions either are limited to a subclass of logic programs, or introduce new variables, or may produce exponentially bigger propositional formulas. In this paper, we present a SAT-based procedure, called ASP-SAT, that (i) deals with any (non disjunctive) logic program, (ii) works on a propositional formula without additional variables (except for those possibly introduced by the clause form transformation), and (iii) is guaranteed to work in polynomial space. From a theoretical perspective, we prove soundness and completeness of ASP-SAT. From a practical perspective, we have (i) implemented ASP-SAT in Cmodels, (ii) extended the basic procedures in order to incorporate the most popular SAT reasoning strategies, and (iii) conducted an extensive comparative analysis involving also other state-of-the-art answer set solvers. The experimental analysis shows that our solver is competitive with the other solvers we considered, and that the reasoning strategies that work best on "small but hard" problems are ineffective on "big but easy" problems and vice versa.
SAT-Based Decision Procedures for Automated Reasoning: A Unifying Perspective
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.
Applying Modern SAT-solvers to Solving Hard Problems
Fundamenta Informaticae
We present nine SAT-solvers and compare their efficiency for several decision and combinatorial problems: three classical NP-complete problems of the graph theory, bounded Post correspondence problem (BPCP), extended string correction problem (ESCP), two popular chess problems, PSPACE-complete verification of UML systems, and the Towers of Hanoi (ToH) of exponential solutions. In addition to several known reductions to SAT for the problems of graph k-colouring, vertex k-cover, Hamiltonian path, and verification of UML systems, we also define new original reductions for the N-queens problem, the knight's tour problem, and ToH, SCP, and BPCP. Our extensive experimental results allow for drawing quite interesting conclusions on efficiency and applicability of SAT-solvers to different problems: they behave quite efficiently for NP-complete and harder problems but they are by far inferior to tailored algorithms for specific problems of lower complexity.
SAT-based Weighted Planning
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.
A hybrid bit-encoding for SAT planning based on clique-partitioning
AIP Conference Proceedings, 2017
Planning as satisfiability is one of the most efficient ways to solve classic automated planning problems. In SAT planning, the encoding used to convert the problem to a SAT formula is critical for the performance of the SAT solver. This paper presents a novel bit-encoding that reduces the number of bits required to represent actions in a SATbased automated planning problem. To obtain such encoding we first build a conflict graph, which represents incompatibilities of pairs of actions, and bitwise encode the subsets of actions determined by a clique partition. This reduces the number of Boolean variables and clauses of the SAT encoding, while preserving the possibility of parallel execution of compatible (non-neighbor) actions. The article also describes an appropriate algorithm for selecting the clique partition for this application and compares the new encodings obtained over some standard planning problems. CLASSICAL AUTOMATED PLANNING AS BOOLEAN SATISFIABILITY Automated Planning is a part of Artificial Intelligence that states the problem of selecting a course of actions to reach a goal. Classical automatic planning employs a planning model that on a domain description-typically an initial state and a goal state-and a set of actions that can change the current state. Classical domains have some restrictions: they are fully observable, deterministic, finite, static (that is, changes occur only when the planning agent acts), and discrete (in time, action, objects, and effects). Thus, the planning problem can be clearly defined and solved using a logical approach [1, 2]. Formally this is defined as a planning task = (, , ,) where:
CirCUs: A Satisfiability Solver Geared towards Bounded Model Checking
2004
CirCUs is a satisfiability solver that works on a combination of And-Inverter-Graph, CNF clauses, and BDDs. It has been designed to work well with bounded model checking. It takes as inputs a Boolean circuit (e.g., the model unrolled k times) and an optional set of additional constraints (for instance, requesting that a solution correspond to a simple path) in the form of CNF clauses or BDDs. The algorithms in CirCUs take advantage of the mixed representation by applying powerful BDD-based implication algorithms, and decision heuristics that are objective-driven. CirCUs supports incremental SAT solving, early termination checks, and other analyses of the model that translate into SAT. Experimental results demonstrate CirCUs’s efficiency.
Solver Scheduling via Answer Set Programming
Theory and Practice of Logic Programming
Although Boolean Constraint Technology has made tremendous progress over the last decade, the efficacy of state-of-the-art solvers is known to vary considerably across different types of problem instances and is known to depend strongly on algorithm parameters. This problem was addressed by means of a simple, yet effective approach using handmade, uniform and unordered schedules of multiple solvers in ppfolio, which showed very impressive performance in the 2011 SAT Competition. Inspired by this, we take advantage of the modeling and solving capacities of Answer Set Programming (ASP) to automatically determine more refined, that is, non-uniform and ordered solver schedules from existing benchmarking data. We begin by formulating the determination of such schedules as multi-criteria optimization problems and provide corresponding ASP encodings. The resulting encodings are easily customizable for different settings and the computation of optimum schedules can mostly be done in the bli...
Related papers
SAT-based answer set programming
PROCEEDINGS OF THE NATIONAL …, 2004
The relation between answer set programming (ASP) and propositional satisfiability (SAT) is at the center of many research papers, partly because of the tremendous performance boost of SAT solvers during last years. Various translations from ASP to SAT are known but the resulting SAT formula either includes many new variables or may have an unpractical size. There are also well known results showing a one-to-one correspondence between the answer sets of a logic program and the models of its completion. Unfortunately, these results only work for specific classes of problems. In this paper we present a SAT-based decision procedure for answer set programming that (i) deals with any (non disjunctive) logic program, (ii) works on a SAT formula without additional variables, and (iii) is guaranteed to work in polynomial space. Further, our procedure can be extended to compute all the answer sets still working in polynomial space. The experimental results of a prototypical implementation show that the approach can pay off sometimes by orders of magnitude. * We are grateful to Paolo Ferraris and Vladimir Lifschitz for their comments related to the subject of the paper; to Esra Erdem and Keijo Heljanko for providing us with the benchmarks; and to Francesco Calimeri for his support on DLV. This work is partially supported by ASI, MIUR and Texas Higher Education Coordinating Board under Grant 003658-0322-2001.
Cmodels-2: SAT-based Answer Set Solver Enhanced to Non-tight Programs
Lecture Notes in Computer Science, 2003
Generalizing Boolean Satisfiability III: Implementation
Journal of Artificial Intelligence Research
This is the third of three papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high-performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal has been to define a representation in which this structure is apparent and can be exploited to improve computational performance. The first paper surveyed existing work that (knowingly or not) exploited problem structure to improve the performance of satisfiability engines, and the second paper showed that this structure could be understood in terms of groups of permutations acting on individual clauses in any particular Boolean theory. We conclude the series by discussing the techniques needed to implement our ideas, and by reporting on their performance on a variety of problem instances.
A Compact and Efficient SAT Encoding for Planning
2008
In the planning-as-SAT paradigm there have been numerous recent developments towards improving the speed and scalability of planning at the cost of finding a step-optimal parallel plan. These developments have been towards: (1) Query strategies that efficiently yield approximately optimal plans, and (2) Having a SAT procedure compute plans from relaxed encodings of the corresponding decision problems in such a way that conflicts in a plan arising from the relaxation are resolved cheaply during a post-processing phase. In this paper we examine a third direction of tightening constraints in order to achieve a more compact, efficient, and scalable SAT-based encoding of the planning problem. For the first time, we use lifting (i.e., operator splitting) and factoring to encode the corresponding n-step decision problems with a parallel action semantics. To ensure compactness we exploit reachability and neededness analysis of the plangraph. Our encoding also captures state-dependent mutex constraints computed during that analysis. Because we adopt a lifted action representation, our encoding cannot generally support full action parallelism. Thus, our approach could be termed approximate, planning for a number of steps between that required in the optimal parallel case and the optimal linear case. We perform a detailed experimental analysis of our approach with 3 state-of-the-art SAT-based planners using benchmarks from recent international planning competitions. We find that our approach dominates optimal SAT-based planners, and is more efficient than the relaxed planners for domains where the plan existence problem is hard.
Experiments with SAT-based Answer Set Programming
2006
Answer Set Programming (ASP) emerged in the late 1990s as a new logic programming paradigm which has been successfully applied in various application domains. Propositional satisfiability (SAT) is one of the most studied problems in Computer Science. ASP and SAT are closely related: Recent works have studied their relation, and efficient SAT-based ASP solvers (like assat and Cmodels) exist. In this paper we report about (i) the extension of the basic procedures in Cmodels in order to incorporate the most popular SAT reasoning strategies, and (ii) an extensive comparative analysis involving also other state-of-the-art answer set solvers. The experimental analysis points out, besides the fact that Cmodels is highly competitive, that the reasoning strategies that work best on "small but hard" problems are ineffective on "big but easy" problems and vice-versa.
Fast SAT-based answer set solver
PROCEEDINGS OF THE NATIONAL …, 2006
Recent research shows that SAT (propositional satisfiability) techniques can be employed to build efficient systems to compute answer sets for logic programs. ASSAT and CMODELS are two well-known such systems. They find an answer set from the full models ...
A comprehensive study and analysis on SAT-solvers: advances, usages and achievements
Artificial Intelligence Review, 2018
Boolean satisfiability (SAT) has been studied for the last twenty years. Advances have been made allowing SAT solvers to be used in many applications including formal verification of digital designs. However, performance and capacity of SAT solvers are still limited. From the practical side, many of the existing applications based on SAT solvers use them as blackboxes in which the problem is translated into a monolithic conjunctive normal form instance and then throw it to the SAT solver with no interaction between the application and the SAT solver. This paper presents a comprehensive study and analysis of the latest developments in SAT-solver and new approaches that used in branching heuristics, Boolean constraint propagation and conflict analysis techniques during the last two decade. In addition, the paper provides the most effective techniques in using SAT solvers as verification techniques, mainly model checkers, to enhance the SAT solver performance, efficiency and productivity. Moreover, the paper presents the remarkable accomplishments and the main challenges facing SAT-solver techniques and contrasts between different techniques according to their efficiency, algorithms, usage and feasibility.
Generalizing Boolean Satisfiability I: Background and Survey of Existing Work
Journal of Artificial Intelligence Research
This is the first of three planned papers describing ZAP, a satisfiability engine that substantially generalizes existing tools while retaining the performance characteristics of modern high-performance solvers. The fundamental idea underlying ZAP is that many problems passed to such engines contain rich internal structure that is obscured by the Boolean representation used; our goal is to define a representation in which this structure is apparent and can easily be exploited to improve computational performance. This paper is a survey of the work underlying ZAP, and discusses previous attempts to improve the performance of the Davis-Putnam-Logemann-Loveland algorithm by exploiting the structure of the problem being solved. We examine existing ideas including extensions of the Boolean language to allow cardinality constraints, pseudo-Boolean representations, symmetry, and a limited form of quantification. While this paper is intended as a survey, our research results are contained i...
aspeed: Solver scheduling via answer set programming
Theory and Practice of Logic Programming, 2014
Although Boolean Constraint Technology has made tremendous progress over the last decade, the efficacy of state-of-the-art solvers is known to vary considerably across different types of problem instances and is known to depend strongly on algorithm parameters. This problem was addressed by means of a simple, yet effective approach using handmade, uniform and unordered schedules of multiple solvers in ppfolio, which showed very impressive performance in the 2011 SAT Competition. Inspired by this, we take advantage of the modeling and solving capacities of Answer Set Programming (ASP) to automatically determine more refined, that is, non-uniform and ordered solver schedules from existing benchmarking data. We begin by formulating the determination of such schedules as multi-criteria optimization problems and provide corresponding ASP encodings. The resulting encodings are easily customizable for different settings and the computation of optimum schedules can mostly be done in the blink of an eye, even when dealing with large runtime data sets stemming from many solvers on hundreds to thousands of instances. Also, the fact that our approach can be customized easily enabled us to swiftly adapt it to generate parallel schedules for multi-processor machines.
Algorithms for the satisfiability (SAT) problem: A survey
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.
Loading Preview
Sorry, preview is currently unavailable. You can download the paper by clicking the button above.