FroCoS 2009: Frontiers of Combining Systems pp 263-278 | Cite as

Combinations of Theories for Decidable Fragments of First-Order Logic

  • Pascal Fontaine
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5749)

Abstract

The design of decision procedures for first-order theories and their combinations has been a very active research subject for thirty years; it has gained practical importance through the development of SMT (satisfiability modulo theories) solvers. Most results concentrate on combining decision procedures for data structures such as theories for arrays, bitvectors, fragments of arithmetic, and uninterpreted functions. In particular, the well-known Nelson-Oppen scheme for the combination of decision procedures requires the signatures to be disjoint and each theory to be stably infinite; every satisfiable set of literals in a stably infinite theory has an infinite model.

In this paper we consider some of the best-known decidable fragments of first-order logic with equality, including the Löwenheim class (monadic FOL with equality, but without functions), Bernays-Schönfinkel-Ramsey theories (finite sets of formulas of the form ∃ * ∀ * ϕ, where ϕ is a function-free and quantifier-free FOL formula), and the two-variable fragment of FOL. In general, these are not stably infinite, and the Nelson-Oppen scheme cannot be used to integrate them into SMT solvers. Noticing some elementary results about the cardinalities of the models of these theories, we show that they can nevertheless be combined with almost any other decidable theory.

Keywords

Decidable Theory Decision Procedure Predicate Symbol Satisfiability Modulo Theory Constant Symbol 
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 2009

Authors and Affiliations

  • Pascal Fontaine
    • 1
  1. 1.Université de Nancy, LoriaNancyFrance

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