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Tractable Unordered 3-CNF Games

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 12118)


The classic TQBF problem can be viewed as a game in which two players alternate turns assigning truth values to a CNF formula’s variables in a prescribed order, and the winner is determined by whether the CNF gets satisfied. The complexity of deciding which player has a winning strategy in this game is well-understood: it is \(\textsf {NL}\)-complete for 2-CNFs and \(\textsf {PSPACE}\)-complete for 3-CNFs.

We continue the study of the unordered variant of this game, in which each turn consists of picking any remaining variable and assigning it a truth value. The complexity of deciding who can win on a given CNF is less well-understood; prior work by the authors showed it is in \(\textsf {L}\) for 2-CNFs and \(\textsf {PSPACE}\)-complete for 5-CNFs. We conjecture it may be efficiently solvable on 3-CNFs, and we make progress in this direction by proving the problem is in \(\textsf {P}\), indeed in \(\textsf {L}\), for 3-CNFs with a certain restriction, namely that each width-3 clause has at least one variable that appears in no other clause. Another (incomparable) restriction of this problem was previously shown to be tractable by Kutz.


  • 3-CNF
  • Games
  • Unordered
  • Logarithmic space

This work was supported by NSF grant CCF-1657377.

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Correspondence to Md Lutfar Rahman .

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Rahman, M.L., Watson, T. (2020). Tractable Unordered 3-CNF Games. In: Kohayakawa, Y., Miyazawa, F.K. (eds) LATIN 2020: Theoretical Informatics. LATIN 2021. Lecture Notes in Computer Science(), vol 12118. Springer, Cham.

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