CEGAR-Tableaux: Improved Modal Satisfiability via Modal Clause-Learning and SAT

The Maximum Satisfiability Problem, or MaxSAT, offers a suitable problem solving formalism for combinatorial optimization problems. Nevertheless, MaxSAT solvers implementing the Branch-and-Bound (BnB) scheme have not succeeded in solving challenging real-world optimization problems. It is widely believed that BnB MaxSAT solvers are only superior on random and some specific crafted instances. At the same time, SAT-based MaxSAT solvers perform particularly well on real-world instances. To overcome this shortcoming of BnB MaxSAT solvers, this paper proposes a new BnB MaxSAT solver called MaxCDCL. The main feature of MaxCDCL is the combination of clause learning of soft conflicts and an efficient bounding procedure. Moreover, the paper reports on an experimental investigation showing that MaxCDCL is competitive when compared with the best performing solvers of the 2020 MaxSAT Evaluation. MaxCDCL performs very well on real-world instances, and solves a number of instances that other solvers cannot solve. Furthermore, MaxCDCL, when combined with the best performing MaxSAT solvers, solves the highest number of instances of a collection from all the MaxSAT evaluations held so far.

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Clause Learning

Encyclopedia of Machine Learning and Data Mining ◽

10.1007/978-1-4899-7687-1_117 ◽

2017 ◽

pp. 225-225 ◽

Cited By ~ 1

Keyword(s):

Clause Learning

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Resolution Trees with Lemmas: Resolution Refinements that Characterize DLL Algorithms with Clause Learning

Logical Methods in Computer Science ◽

10.2168/lmcs-4(4:13)2008 ◽

2008 ◽

Vol 4 (4) ◽

Cited By ~ 15

Author(s):

Samuel Buss ◽

Jan Hoffmann ◽

Jan Johannsen

Keyword(s):

Clause Learning

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Seeking Practical CDCL Insights from Theoretical SAT Benchmarks

Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2018/181 ◽

2018 ◽

Cited By ~ 5

Author(s):

Jan Elffers ◽

Jesús Giráldez-Cru ◽

Stephan Gocht ◽

Jakob Nordström ◽

Laurent Simon

Keyword(s):

Systematic Study ◽

Boolean Satisfiability ◽

Proof System ◽

Sat Solvers ◽

Proof Search ◽

Wide Range ◽

Clause Learning ◽

Resolution Proof ◽

Shed Light

Over the last decades Boolean satisfiability (SAT) solvers based on conflict-driven clause learning (CDCL) have developed to the point where they can handle formulas with millions of variables. Yet a deeper understanding of how these solvers can be so successful has remained elusive. In this work we shed light on CDCL performance by using theoretical benchmarks, which have the attractive features of being a) scalable, b) extremal with respect to different proof search parameters, and c) theoretically easy in the sense of having short proofs in the resolution proof system underlying CDCL. This allows for a systematic study of solver heuristics and how efficiently they search for proofs. We report results from extensive experiments on a wide range of benchmarks. Our findings include several examples where theory predicts and explains CDCL behaviour, but also raise a number of intriguing questions for further study.

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On Division Versus Saturation in Pseudo-Boolean Solving

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2019/237 ◽

2019 ◽

Author(s):

Stephan Gocht ◽

Jakob Nordström ◽

Amir Yehudayoff

Keyword(s):

Linear Programming ◽

Cutting Planes ◽

The Other ◽

Sat Solving ◽

Linear Combinations ◽

Division Rule ◽

Clause Learning

The conflict-driven clause learning (CDCL) paradigm has revolutionized SAT solving over the last two decades. Extending this approach to pseudo-Boolean (PB) solvers doing 0-1 linear programming holds the promise of further exponential improvements in theory, but intriguingly such gains have not materialized in practice. Also intriguingly, most PB extensions of CDCL use not the division rule in cutting planes as defined in [Cook et al., '87] but instead the so-called saturation rule. To the best of our knowledge, there has been no study comparing the strengths of division and saturation in the context of conflict-driven PB learning, when all linear combinations of inequalities are required to cancel variables. We show that PB solvers with division instead of saturation can be exponentially stronger. In the other direction, we prove that simulating a single saturation step can require an exponential number of divisions. We also perform some experiments to see whether these phenomena can be observed in actual solvers. Our conclusion is that a careful combination of division and saturation seems to be crucial to harness more of the power of cutting planes.

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Chapter 3. Complete Algorithms

Frontiers in Artificial Intelligence and Applications - Handbook of Satisfiability ◽

10.3233/faia200986 ◽

2021 ◽

Author(s):

Adnan Darwiche ◽

Knot Pipatsrisawat

Keyword(s):

Real World ◽

Theoretical Perspective ◽

Point Of View ◽

Sat Solving ◽

Practical Point ◽

Sat Solvers ◽

Proof Systems ◽

Real World Applications ◽

Clause Learning

Complete SAT algorithms form an important part of the SAT literature. From a theoretical perspective, complete algorithms can be used as tools for studying the complexities of different proof systems. From a practical point of view, these algorithms form the basis for tackling SAT problems arising from real-world applications. The practicality of modern, complete SAT solvers undoubtedly contributes to the growing interest in the class of complete SAT algorithms. We review these algorithms in this chapter, including Davis-Putnum resolution, Stalmarck’s algorithm, symbolic SAT solving, the DPLL algorithm, and modern clause-learning SAT solvers. We also discuss the issue of certifying the answers of modern complete SAT solvers.

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Chapter 31. Quantified Boolean Formulas

Frontiers in Artificial Intelligence and Applications - Handbook of Satisfiability ◽

10.3233/faia201015 ◽

2021 ◽

Author(s):

Olaf Beyersdorff ◽

Mikoláš Janota ◽

Florian Lonsing ◽

Martina Seidl

Keyword(s):

Performance Characteristics ◽

Proof Complexity ◽

Proof Systems ◽

Quantified Boolean Formulas ◽

Boolean Formulas ◽

Clause Learning ◽

Dependency Schemes

Solvers for quantified Boolean formulas (QBF) have become powerful tools for tackling hard computational problems from various application domains, even beyond the scope of SAT. This chapter gives a description of the main algorithmic paradigms for QBF solving, including quantified conflict driven clause learning (QCDCL), expansion-based solving, dependency schemes, and QBF preprocessing. Particular emphasis is laid on the connections of these solving approaches to QBF proof systems: Q-Resolution and its variants in the case of QCDCL, expansion QBF resolution calculi for expansion-based solving, and QRAT for preprocessing. The chapter also surveys the relations between the various QBF proof systems and results on their proof complexity, thereby shedding light on the diverse performance characteristics of different solving approaches that are observed in practice.

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