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Constraint Satisfaction with Countable Homogeneous Templates

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Computer Science Logic (CSL 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2803))

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Abstract

For a fixed countable homogeneous relational structure Γ we study the computational problem whether a given finite structure of the same signature homomorphically maps to Γ. This problem is known as the constraint satisfaction problem CSP(Γ) for Γ and was intensively studied for finite Γ. We show that – as in the case of finite Γ – the computational complexity of CSP(Γ) for countable homogeneous Γ is determinded by the clone of polymorphisms of Γ. To this end we prove the following theorem which is of independent interest: The primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are invariant under the polymorphisms of Γ.

Constraint satisfaction with countable homogeneous templates is a proper generalization of constraint satisfaction with finite templates. If the age of Γ is finitely axiomatizable, then CSP(Γ) is in NP. If Γ is a digraph we can use the classification of homogeneous digraphs by Cherlin to determine the complexity of CSP(Γ).

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Author information

Authors and Affiliations

  1. Humboldt Universität zu Berlin, Germany

    Manuel Bodirsky

  2. Institute for Theoretical Computer Science (ITI), and Department of Applied Mathematics (KAM), Charles University, Prague, Czech Republic

    Jaroslav Nešetřil

Authors
  1. Manuel Bodirsky
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  2. Jaroslav Nešetřil
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Editor information

Editors and Affiliations

  1. Institute of Discrete Mathematics and Geometry, Technical University Vienna, Wiedner Hauptstrasse 8-10, A-1040, Wien, Austria

    Matthias Baaz

  2. Members of EACSL Jury for the Ackermann Award,  

    Johann A. Makowsky

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Bodirsky, M., Nešetřil, J. (2003). Constraint Satisfaction with Countable Homogeneous Templates. In: Baaz, M., Makowsky, J.A. (eds) Computer Science Logic. CSL 2003. Lecture Notes in Computer Science, vol 2803. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45220-1_5

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  • DOI: https://doi.org/10.1007/978-3-540-45220-1_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40801-7

  • Online ISBN: 978-3-540-45220-1

  • eBook Packages: Springer Book Archive

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