Positional Games and QBF: The Corrective Encoding

Positional games are a mathematical class of two-player games comprising Tic-tac-toe and its generalizations. We propose a novel encoding of these games into Quantified Boolean Formulas (QBF) such that a game instance admits a winning strategy for first player if and only if the corresponding formula is true. Our approach improves over previous QBF encodings of games in multiple ways. First, it is generic and lets us encode other positional games, such as Hex. Second, structural properties of positional games together with a careful treatment of illegal moves let us generate more compact instances that can be solved faster by state-of-the-art QBF solvers. We establish the latter fact through extensive experiments. Finally, the compactness of our new encoding makes it feasible to translate realistic game problems. We identify a few such problems of historical significance and put them forward to the QBF community as milestones of increasing difficulty.Comment: Accepted for publication in the 23rd International Conference on Theory and Applications of Satisfiability Testing (SAT2020

Reasons for Hardness in QBF Proof Systems

We aim to understand inherent reasons for lower bounds for QBF proof systems, and revisit and compare two previous approaches in this direction. The first of these relates size lower bounds for strong QBF Frege systems to circuit lower bounds via strategy extraction (Beyersdorff & Pich, LICS\u2716). Here we show a refined version of strategy extraction and thereby for any QBF proof system obtain a trichotomy for hardness: (1) via circuit lower bounds, (2) via propositional Resolution lower bounds, or (3) `genuine\u27 QBF lower bounds. The second approach tries to explain QBF lower bounds through quantifier alternations in a system called relaxing QU-Res (Chen, ICALP\u2716). We prove a strong lower bound for relaxing QU-Res, which also exhibits significant shortcomings of that model. Prompted by this we propose an alternative, improved version, allowing more flexible oracle queries in proofs. We show that lower bounds in our new model correspond to the trichotomy obtained via strategy extraction

Lifting QBF Resolution Calculi to DQBF

We examine the existing resolution systems for quantified Boolean formulas (QBF) and answer the question which of these calculi can be lifted to the more powerful Dependency QBFs (DQBF). An interesting picture emerges: While for QBF we have the strict chain of proof systems Q-Res < IR-calc < IRM-calc, the situation is quite different in DQBF. Q-Res and likewise universal resolution are too weak: they are not complete. IR-calc has the right strength: it is sound and complete. IRM-calc is too strong: it is not sound any more, and the same applies to long-distance resolution. Conceptually, we use the relation of DQBF to EPR and explain our new DQBF calculus based on IR-calc as a subsystem of first-order resolutio

Tackling Universal Properties of Minimal Trap Spaces of Boolean Networks

Minimal trap spaces (MTSs) capture subspaces in which the Boolean dynamics is trapped, whatever the update mode. They correspond to the attractors of the most permissive mode. Due to their versatility, the computation of MTSs has recently gained traction, essentially by focusing on their enumeration. In this paper, we address the logical reasoning on universal properties of MTSs in the scope of two problems: the reprogramming of Boolean networks for identifying the permanent freeze of Boolean variables that enforce a given property on all the MTSs, and the synthesis of Boolean networks from universal properties on their MTSs. Both problems reduce to solving the satisfiability of quantified propositional logic formula with 3 levels of quantifiers ($\exists\forall\exists$). In this paper, we introduce a Counter-Example Guided Refinement Abstraction (CEGAR) to efficiently solve these problems by coupling the resolution of two simpler formulas. We provide a prototype relying on Answer-Set Programming for each formula and show its tractability on a wide range of Boolean models of biological networks.Comment: Accepted at 21st International Conference on Computational Methods in Systems Biology (CMSB 2023

ASPQ: An ASP-Based 2QBF Solver

Answer Set Programming (ASP) is an established logic-based programming paradigm which has been successfully applied for solving complex problems. Since ASP can model problems up to the second level of the polynomial hierarchy, it can be used to model and solve the 2QBF problem. In this paper we show how to obtain a fairly effective 2QBF solver by just resorting to state-of-The-Art ASP solvers

Quantified Constraints in Twenty Seventeen

I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions

General Boolean Formula Minimization with QBF Solvers

The minimization of propositional formulae is a classical problem in logic, whose first algorithms date back at least to the 1950s with the works of Quine and Karnaugh. Most previous work in the area has focused on obtaining minimal, or quasi-minimal, formulae in conjunctive normal form (CNF) or disjunctive normal form (DNF), with applications in hardware design. In this paper, we are interested in the problem of obtaining an equivalent formula in any format, also allowing connectives that are not present in the original formula. We are primarily motivated in applying minimization algorithms to generate natural language translations of the original formula, where using shorter equivalents as input may result in better translations. Recently, Buchfuhrer and Umans have proved that the (decisional version of the) problem is $\Sigma_2^p$-complete. We analyze three possible (practical) approaches to solving the problem. First, using brute force, generating all possible formulae in increasing size and checking if they are equivalent to the original formula by testing all possible variable assignments. Second, generating the Tseitin coding of all the formulae and checking equivalence with the original using a SAT solver. Third, encoding the problem as a Quantified Boolean Formula (QBF), and using a QBF solver. Our results show that the QBF approach largely outperforms the other two

SAT Competition 2020

The SAT Competitions constitute a well-established series of yearly open international algorithm implementation competitions, focusing on the Boolean satisfiability (or propositional satisfiability, SAT) problem. In this article, we provide a detailed account on the 2020 instantiation of the SAT Competition, including the new competition tracks and benchmark selection procedures, overview of solving strategies implemented in top-performing solvers, and a detailed analysis of the empirical data obtained from running the competition

SAT Competition 2020

The SAT Competitions constitute a well-established series of yearly open international algorithm implementation competitions, focusing on the Boolean satisfiability (or propositional satisfiability, SAT) problem. In this article, we provide a detailed account on the 2020 instantiation of the SAT Competition, including the new competition tracks and benchmark selection procedures, overview of solving strategies implemented in top-performing solvers, and a detailed analysis of the empirical data obtained from running the competition. (C) 2021 The Authors. Published by Elsevier B.V.Peer reviewe