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Trichotomies in the Complexity of Minimal Inference

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Abstract

We study the complexity of the propositional minimal inference problem. Although the complexity of this problem has been already extensively studied before because of its fundamental importance in nonmonotonic logics and commonsense reasoning, no complete classification of its complexity was found. We classify the complexity of four different and well-studied formalizations of the problem in the version with unbounded queries, proving that the complexity of the minimal inference problem for each of them has a trichotomy (between P, coNP-complete, and Π2P-complete). One of these results finally settles with a positive answer the trichotomy conjecture of Kirousis and Kolaitis (Theory Comput. Syst. 37(6):659–715, 2004). In the process we also strengthen and give a much simplified proof of the main result from Durand and Hermann (Proceedings 20th Symposium on Theoretical Aspects of Computer Science (STACS 2003), pp. 451–462, 2003).

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Authors and Affiliations

  1. Equipe de Logique Mathématique, Institut de Mathématiques de Jussieu (CNRS UMR 7586), 75251, Paris cedex 05, France

    Arnaud Durand

  2. LIX (CNRS UMR 7161), École Polytechnique, 91128, Palaiseau cedex, France

    Miki Hermann

  3. Department of Computer and Information Sciences, Linköpings Universitet, 581 83, Linköping, Sweden

    Gustav Nordh

Authors
  1. Arnaud Durand
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  2. Miki Hermann
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  3. Gustav Nordh
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Corresponding author

Correspondence to Miki Hermann.

Additional information

This research was partially supported by the grant ANR-07-BLAN-0327. A preliminary version of the results in Sect. 5 has been published in [24] and of those in Sect. 9 has been published in [12].

G. Nordh was partially supported by the Swedish Research Council (VR) under grant 2008-4675 and the Swedish-French Foundation. Part of this work has been completed during a post-doctoral stay of this author at LIX, Ecole Polytechnique.

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Durand, A., Hermann, M. & Nordh, G. Trichotomies in the Complexity of Minimal Inference. Theory Comput Syst 50, 446–491 (2012). https://doi.org/10.1007/s00224-011-9320-0

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