Affine Systems of Equations and Counting Infinitary Logic

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


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.


Polynomial Time Constraint Satisfaction Problem Homomorphic Image Winning Strategy Direct Power 
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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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