Chapter 7. Proof Complexity and SAT Solving

2021 ◽  
Author(s):  
Sam Buss ◽  
Jakob Nordström
Keyword(s):  
Linear Algebra ◽  
Cutting Planes ◽  
Proof Complexity ◽  
Sat Solving ◽  
Proof Systems ◽  
Frege Systems ◽  
Ample Supply ◽  
Clause Learning ◽  
Algebraic Approaches ◽  
Polynomial Calculus

This chapter gives an overview of proof complexity and connections to SAT solving, focusing on proof systems such as resolution, Nullstellensatz, polynomial calculus, and cutting planes (corresponding to conflict-driven clause learning, algebraic approaches using linear algebra or Gröbner bases, and pseudo-Boolean solving, respectively). There is also a discussion of extended resolution (which is closely related to DRAT proof logging) and Frege and extended Frege systems more generally. An ample supply of references for further reading is provided, including for some topics omitted in this chapter.

scholarly journals On Division Versus Saturation in Pseudo-Boolean Solving

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.

Chapter 3. Complete Algorithms

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.

Chapter 31. Quantified Boolean Formulas

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.

Minimum propositional proof length is NP-hard to linearly approximate

2001 ◽  
Vol 66 (1) ◽  
pp. 171-191 ◽  
Author(s):  
Michael Alekhnovich ◽  
Sam Buss ◽  
Shlomo Moran ◽  
Toniann Pitassi
Keyword(s):  
Assignment Problem ◽  
Sequent Calculus ◽  
Hardness Of Approximation ◽  
Np Hard ◽  
Satisfying Assignment ◽  
Proof Systems ◽  
Frege Systems ◽  
Propositional Proof ◽  
Almost All ◽  
Polynomial Calculus

AbstractWe prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within a factor of . These results are very robust in that they hold for almost all natural proof systems, including: Frege systems, extended Frege systems, resolution. Horn resolution, the polynomial calculus, the sequent calculus, the cut-free sequent calculus, as well as the polynomial calculus. Our hardness of approximation results usually apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and reduce it to the problems of approximation of the length of proofs.

scholarly journals Simulating Strong Practical Proof Systems with Extended Resolution

2020 ◽  
Vol 64 (7) ◽  
pp. 1247-1267
Author(s):  
Benjamin Kiesl ◽  
Adrián Rebola-Pardo ◽  
Marijn J. H. Heule ◽  
Armin Biere
Keyword(s):  
Propositional Logic ◽  
Proof Complexity ◽  
Proof System ◽  
Compact Representation ◽  
Sat Solving ◽  
Proof Systems ◽  
The Sixties ◽  
New Variables ◽  
Resolution Proof ◽  
Do So

Abstract Proof systems for propositional logic provide the basis for decision procedures that determine the satisfiability status of logical formulas. While the well-known proof system of extended resolution—introduced by Tseitin in the sixties—allows for the compact representation of proofs, modern SAT solvers (i.e., tools for deciding propositional logic) are based on different proof systems that capture practical solving techniques in an elegant way. The most popular of these proof systems is likely DRAT, which is considered the de-facto standard in SAT solving. Moreover, just recently, the proof system DPR has been proposed as a generalization of DRAT that allows for short proofs without the need of new variables. Since every extended-resolution proof can be regarded as a DRAT proof and since every DRAT proof is also a DPR proof, it was clear that both DRAT and DPR generalize extended resolution. In this paper, we show that—from the viewpoint of proof complexity—these two systems are no stronger than extended resolution. We do so by showing that (1) extended resolution polynomially simulates DRAT and (2) DRAT polynomially simulates DPR. We implemented our simulations as proof-transformation tools and evaluated them to observe their behavior in practice. Finally, as a side note, we show how Kullmann’s proof system based on blocked clauses (another generalization of extended resolution) is related to the other systems.

scholarly journals Computational and Proof Complexity of Partial String Avoidability

2021 ◽  
Vol 13 (1) ◽  
pp. 1-25
Author(s):  
Dmitry Itsykson ◽  
Alexander Okhotin ◽  
Vsevolod Oparin
Keyword(s):  
Lower Bounds ◽  
Circuit Complexity ◽  
Conjunctive Normal Form ◽  
Proof Complexity ◽  
Proof Systems ◽  
Finite Set ◽  
Propositional Proof Systems ◽  
Classical Proof ◽  
Exponential Size ◽  
Infinite String

The partial string avoidability problem is stated as follows: given a finite set of strings with possible “holes” (wildcard symbols), determine whether there exists a two-sided infinite string containing no substrings from this set, assuming that a hole matches every symbol. The problem is known to be NP-hard and in PSPACE, and this article establishes its PSPACE-completeness. Next, string avoidability over the binary alphabet is interpreted as a version of conjunctive normal form satisfiability problem, where each clause has infinitely many shifted variants. Non-satisfiability of these formulas can be proved using variants of classical propositional proof systems, augmented with derivation rules for shifting proof lines (such as clauses, inequalities, polynomials, etc.). First, it is proved that there is a particular formula that has a short refutation in Resolution with a shift rule but requires classical proofs of exponential size. At the same time, it is shown that exponential lower bounds for classical proof systems can be translated for their shifted versions. Finally, it is shown that superpolynomial lower bounds on the size of shifted proofs would separate NP from PSPACE; a connection to lower bounds on circuit complexity is also established.

scholarly journals Proof Complexity of Modal Resolution

2021 ◽  
Author(s):  
Sarah Sigley ◽  
Olaf Beyersdorff
Keyword(s):  
Artificial Intelligence ◽  
Automated Reasoning ◽  
Proof Complexity ◽  
Pigeonhole Principle ◽  
Analytic Tableaux ◽  
Frege Systems ◽  
International Joint ◽  
Resolution Proofs ◽  
Exponential Size ◽  
Sixth International

AbstractWe investigate the proof complexity of modal resolution systems developed by Nalon and Dixon (J Algorithms 62(3–4):117–134, 2007) and Nalon et al. (in: Automated reasoning with analytic Tableaux and related methods—24th international conference, (TABLEAUX’15), pp 185–200, 2015), which form the basis of modal theorem proving (Nalon et al., in: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI’17), pp 4919–4923, 2017). We complement these calculi by a new tighter variant and show that proofs can be efficiently translated between all these variants, meaning that the calculi are equivalent from a proof complexity perspective. We then develop the first lower bound technique for modal resolution using Prover–Delayer games, which can be used to establish “genuine” modal lower bounds for size of dag-like modal resolution proofs. We illustrate the technique by devising a new modal pigeonhole principle, which we demonstrate to require exponential-size proofs in modal resolution. Finally, we compare modal resolution to the modal Frege systems of Hrubeš (Ann Pure Appl Log 157(2–3):194–205, 2009) and obtain a “genuinely” modal separation.

scholarly journals A Theory of Satisfiability-Preserving Proofs in SAT Solving

10.29007/tc7q ◽  
2018 ◽  
Author(s):  
Adrián Rebola-Pardo ◽  
Martin Suda
Keyword(s):  
Propositional Logic ◽  
Logical Truth ◽  
Point Of View ◽  
Traditional View ◽  
Satisfiability Problem ◽  
Sat Solving ◽  
Sat Solvers ◽  
Proof Systems ◽  
Meta Level

We study the semantics of propositional interference-based proof systems such as DRAT and DPR. These are characterized by modifying a CNF formula in ways that preserve satisfiability but not necessarily logical truth. We propose an extension of propositional logic called overwrite logic with a new construct which captures the meta-level reasoning behind interferences. We analyze this new logic from the point of view of expressivity and complexity, showing that while greater expressivity is achieved, the satisfiability problem for overwrite logic is essentially as hard as SAT, and can be reduced in a way that is well-behaved for modern SAT solvers. We also show that DRAT and DPR proofs can be seen as overwrite logic proofs which preserve logical truth. This much stronger invariant than the mere satisfiability preservation maintained by the traditional view gives us better understanding on these practically important proof systems. Finally, we showcase this better understanding by finding intrinsic limitations in interference-based proof systems.

Improving SAT Solving Using Monte Carlo Tree Search-Based Clause Learning

2019 ◽  
pp. 107-133
Author(s):  
Oliver Keszocze ◽  
Kenneth Schmitz ◽  
Jens Schloeter ◽  
Rolf Drechsler
Keyword(s):  
Monte Carlo ◽  
Tree Search ◽  
Sat Solving ◽  
Monte Carlo Tree Search ◽  
Clause Learning
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