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Affine Systems of Equations and Counting Infinitary Logic

  • Albert Atserias
  • Andrei Bulatov
  • Anuj Dawar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

We study the definability of constraint satisfaction problems (CSP) in various fixed-point and infinitary logics. We show that testing the solvability of systems of equations over a finite Abelian group, a tractable CSP that was previously known not to be definable in Datalog, is not definable in an infinitary logic with counting and hence that it is not definable in least fixed point logic or its extension with counting. We relate definability of CSPs to their classification obtained from tame congruence theory of the varieties generated by the algebra of polymorphisms of the template structure. In particular, we show that if this variety admits either the unary or affine type, the corresponding CSP is not definable in the infinitary logic with counting. We also study the complexity of determining whether a CSP omits unary and affine types.

Keywords

Polynomial Time Constraint Satisfaction Problem Homomorphic Image Winning Strategy Direct Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Albert Atserias
    • 1
  • Andrei Bulatov
    • 2
  • Anuj Dawar
    • 3
  1. 1.Universitat Politécnica de Catalunya, BarcelonaSpain
  2. 2.Simon Fraser University, Burnaby BCCanada
  3. 3.University of Cambridge Computer Laboratory, CambridgeUK

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