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The complexity of propositional implication

Beyersdorff, O, Meier, A, Thomas, M and Vollmer, H (2009) The complexity of propositional implication. Information Processing Letters, 109 (18). 1071 - 1077 . ISSN 0020-0190


The question whether a set of formulae Γ implies a formula φ is fundamental. The present paper studies the complexity of the above implication problem for propositional formulae that are built from a systematically restricted set of Boolean connectives. We give a complete complexity-theoretic classification for all sets of Boolean functions in the meaning of Post's lattice and show that the implication problem is efficiently solvable only if the connectives are definable using the constants {0,1} and only one of {∧,∨,⊕}. The problem remains coNP-complete in all other cases. We also consider the restriction of Γ to singletons which makes the problem strictly easier in some cases.

Item Type: Article
Keywords: Computational complexity, Propositional implication, Post's lattice
Institution: The University of Leeds
Academic Units: The University of Leeds > Faculty of Engineering (Leeds) > School of Computing (Leeds)
Depositing User: Symplectic Publications
Date Deposited: 07 Aug 2012 12:40
Last Modified: 15 Sep 2014 03:38
Published Version: http://dx.doi.org/10.1016/j.ipl.2009.06.015
Status: Published
Publisher: Elsevier
Identification Number: 10.1016/j.ipl.2009.06.015
URI: http://eprints.whiterose.ac.uk/id/eprint/74442

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