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Complexity of Unordered CNF Games

Published:01 June 2020Publication History
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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).

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    • Published in

      cover image ACM Transactions on Computation Theory
      ACM Transactions on Computation Theory  Volume 12, Issue 3
      September 2020
      197 pages
      ISSN:1942-3454
      EISSN:1942-3462
      DOI:10.1145/3403647
      Issue’s Table of Contents

      Copyright © 2020 ACM

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      Association for Computing Machinery

      New York, NY, United States

      Publication History

      • Published: 1 June 2020
      • Online AM: 7 May 2020
      • Accepted: 1 April 2020
      • Revised: 1 February 2020
      • Received: 1 February 2019
      Published in toct Volume 12, Issue 3

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