Complexity of Unordered CNF Games

Md Lutfar Rahman,Thomas Watson

doi:10.1145/3397478

Abstract

The classic TQBF problem is to determine who has a winning strategy in a game played on a given conjunctive normal form formula (CNF), where the two players alternate turns picking truth values for the variables in a given order, and the winner is determined by whether the CNF gets satisfied. We study variants of this game in which the variables may be played in any order, and each turn consists of picking a remaining variable and a truth value for it. For the version where the set of variables is partitioned into two halves and each player may only pick variables from his or her half, we prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for unbounded-width CNFs (Schaefer, STOC 1976). For the general unordered version (where each variable can be picked by either player), we also prove that the problem is PSPACE-complete for 5-CNFs and in P for 2-CNFs. Previously, it was known to be PSPACE-complete for 6-CNFs (Ahlroth and Orponen, MFCS 2012) and PSPACE-complete for positive 11-CNFs (Schaefer, STOC 1976).

Highlights

Conjunctive normal form formulas (CNFs) are among the most prevalent representations of boolean functions
A CNF is a conjunction of clauses, where each clause is a disjunction of literals; a w-CNF has at most w literals per clause
Deciding who has a winning strategy was shown to be PSPACE-complete for unbounded-width CNFs in [9, 10], where it was explicitly posed as an open problem to show PSPACE-completeness with any constant bound on the width

Summary

Introduction

Conjunctive normal form formulas (CNFs) are among the most prevalent representations of boolean functions. Deciding who has a winning strategy was shown to be PSPACE-complete for unbounded-width CNFs in [9, 10], where it was explicitly posed as an open problem to show PSPACE-completeness with any constant bound on the width. This game has been used for PSPACE-completeness reductions [3], and a variant with a matching between the two players’ variables has been studied [4]. We give the precise definitions and theorem statements

Statement of results

Intuition

Formal Proof