Combination of Disjoint Theories: Beyond Decidability

  • Pascal Fontaine
  • Stephan Merz
  • Christoph Weidenbach
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7364)


Combination of theories underlies the design of satisfiability modulo theories (SMT) solvers. The Nelson-Oppen framework can be used to build a decision procedure for the combination of two disjoint decidable stably infinite theories.

We here study combinations involving an arbitrary first-order theory. Decidability is lost, but refutational completeness is preserved. We consider two cases and provide complete (semi-)algorithms for them. First, we show that it is possible under minor technical conditions to combine a decidable (not necessarily stably infinite) theory and a disjoint finitely axiomatized theory, obtaining a refutationally complete procedure. Second, we provide a refutationally complete procedure for the union of two disjoint finitely axiomatized theories, that uses the assumed procedures for the underlying theories without modifying them.


Decidable Theory Decision Procedure Atomic Formula Cardinality Constraint Complete Procedure 
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 2012

Authors and Affiliations

  • Pascal Fontaine
    • 1
  • Stephan Merz
    • 2
  • Christoph Weidenbach
    • 3
  1. 1.Université de Lorraine & LORIANancyFrance
  2. 2.INRIA Nancy & LORIANancyFrance
  3. 3.Max-Planck-Institut für InformatikSaarbrückenGermany

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