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.

scholarly journals Dependency Learning for QBF

2019 ◽  
Vol 65 ◽  
pp. 181-208 ◽  
Author(s):  
Tomáš Peitl ◽  
Friedrich Slivovsky ◽  
Stefan Szeider
Keyword(s):  
Formal Verification ◽  
New Technique ◽  
Proof System ◽  
Long Distance ◽  
Quantified Boolean Formulas ◽  
Boolean Formulas ◽  
Clause Learning ◽  
A New Technique ◽  
The Cost ◽  
Dependency Schemes

Quantified Boolean Formulas (QBFs) can be used to succinctly encode problems from domains such as formal verification, planning, and synthesis. One of the main approaches to QBF solving is Quantified Conflict Driven Clause Learning (QCDCL). By default, QCDCL assigns variables in the order of their appearance in the quantifier prefix so as to account for dependencies among variables. Dependency schemes can be used to relax this restriction and exploit independence among variables in certain cases, but only at the cost of nontrivial interferences with the proof system underlying QCDCL. We introduce dependency learning, a new technique for exploiting variable independence within QCDCL that allows solvers to learn variable dependencies on the fly. The resulting version of QCDCL enjoys improved propagation and increased flexibility in choosing variables for branching while retaining ordinary (long-distance) Q-resolution as its underlying proof system. We show that dependency learning can achieve exponential speedups over ordinary QCDCL. Experiments on standard benchmark sets demonstrate the effectiveness of this technique.

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 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.

Analysis of Incomplete Circuits Using Dependency Quantified Boolean Formulas

2017 ◽  
pp. 151-168 ◽  
Author(s):  
Ralf Wimmer ◽  
Karina Wimmer ◽  
Christoph Scholl ◽  
Bernd Becker
Keyword(s):  
Quantified Boolean Formulas ◽  
Boolean Formulas
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