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International Joint Conference on Automated Reasoning

IJCAR 2018: Automated Reasoning pp 263–278Cite as

Complexity of Combinations of Qualitative Constraint Satisfaction Problems

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10900))

Abstract

The CSP of a first-order theory T is the problem of deciding for a given finite set S of atomic formulas whether \(T \cup S\) is satisfiable. Let \(T_1\) and \(T_2\) be two theories with countably infinite models and disjoint signatures. Nelson and Oppen presented conditions that imply decidability (or polynomial-time decidability) of \({{\mathrm{CSP}}}(T_1 \cup T_2)\) under the assumption that \({{\mathrm{CSP}}}(T_1)\) and \({{\mathrm{CSP}}}(T_2)\) are decidable (or polynomial-time decidable). We show that for a large class of \(\omega \)-categorical theories \(T_1, T_2\) the Nelson-Oppen conditions are not only sufficient, but also necessary for polynomial-time tractability of \({{\mathrm{CSP}}}(T_1 \cup T_2)\) (unless P = NP).

M. Bodirsky and J. Greiner have received funding from the European Research Council (ERC Grant Agreement no. 681988), the German Research Foundation (DFG, project number 622397), and the DFG Graduiertenkolleg 1763 (QuantLA).

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Notes

  1. 1.

    In other words: for all sets S of atomic \((\tau _1 \cup \tau _2)\)-formulas, we have that \(S \cup T\) is satisfiable if and only if \(S \cup (T_1 \cup T_2)\) is satisfiable.

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Authors and Affiliations

  1. Institut für Algebra, TU Dresden, Dresden, Germany

    Manuel Bodirsky & Johannes Greiner

Authors
  1. Manuel Bodirsky
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  2. Johannes Greiner
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Correspondence to Manuel Bodirsky or Johannes Greiner .

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Editors and Affiliations

  1. Université de Lorraine, Vandoeuvre-lès-Nancy, France

    Didier Galmiche

  2. Baden-Wuerttemberg Cooperative State University, Stuttgart, Germany

    Stephan Schulz

  3. University of Trento, Trento, Italy

    Roberto Sebastiani

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Bodirsky, M., Greiner, J. (2018). Complexity of Combinations of Qualitative Constraint Satisfaction Problems. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds) Automated Reasoning. IJCAR 2018. Lecture Notes in Computer Science(), vol 10900. Springer, Cham. https://doi.org/10.1007/978-3-319-94205-6_18

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  • DOI: https://doi.org/10.1007/978-3-319-94205-6_18

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  • Online ISBN: 978-3-319-94205-6

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