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Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic

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Pursuit of the Universal (CiE 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9709))

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Abstract

We study interactions between Skolem Arithmetic and certain classes of Circuit Satisfiability and Constraint Satisfaction Problems (CSPs). We revisit results of Glaßer et al. [16] in the context of CSPs and settle the major open question from that paper, finding a certain satisfiability problem on circuits—involving complement, intersection, union and multiplication—to be decidable. This we prove using the decidability of Skolem Arithmetic. Then we solve a second question left open in [16] by proving a tight upper bound for the similar circuit satisfiability problem involving just intersection, union and multiplication. We continue by studying first-order expansions of Skolem Arithmetic without constants, \((\mathbb {N};\times )\), as CSPs. We find already here a rich landscape of problems with non-trivial instances that are in P as well as those that are NP-complete.

P. Jonsson—was partially supported by the Swedish Research Council (VR) under grant 621-2012-3239.

B. Martin—was supported by EPSRC grant EP/L005654/1.

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Notes

  1. 1.

    Weird. Thus spake Lindemann about Hilbert’s non-constructive methods in the resolution of Gordon’s problem (see [28]).

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Authors and Affiliations

  1. Theoretische Informatik, Julius-Maximilians-Universität, Würzburg, Germany

    Christian Glaßer

  2. Department of Computer and Information Science, Linköpings Universitet, 581 83, Linköping, Sweden

    Peter Jonsson

  3. School of Science and Technology, Middlesex University, The Burroughs, Hendon, London, NW4 4BT, UK

    Barnaby Martin

Authors
  1. Christian Glaßer
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  2. Peter Jonsson
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  3. Barnaby Martin
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Correspondence to Barnaby Martin .

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Editors and Affiliations

  1. Swansea University, Swansea, United Kingdom

    Arnold Beckmann

  2. Université Paris-Diderot - Paris 7, Paris, France

    Laurent Bienvenu

  3. University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

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Glaßer, C., Jonsson, P., Martin, B. (2016). Circuit Satisfiability and Constraint Satisfaction Around Skolem Arithmetic. In: Beckmann, A., Bienvenu, L., Jonoska, N. (eds) Pursuit of the Universal. CiE 2016. Lecture Notes in Computer Science(), vol 9709. Springer, Cham. https://doi.org/10.1007/978-3-319-40189-8_33

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  • DOI: https://doi.org/10.1007/978-3-319-40189-8_33

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