# Combinations of theories for decidable fragments of first-order logic

1 MOSEL - Proof-oriented development of computer-based systems
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
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 $\exists^*\forall^* \varphi$, where $\varphi$ 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.
Type de document :
Communication dans un congrès
Silvio Ghilardi and Roberto Sebastiani. 7th International Symposium, FroCoS 2009, Sep 2009, Trento, Italy. Springer Verlag, 5749, pp.263-278, 2009, Lecture Notes in Computer Science. 〈10.1007/978-3-642-04222-5_16〉

https://hal.inria.fr/inria-00430631
Contributeur : Pascal Fontaine <>
Soumis le : lundi 9 novembre 2009 - 12:33:38
Dernière modification le : jeudi 11 janvier 2018 - 06:19:52

### Citation

Pascal Fontaine. Combinations of theories for decidable fragments of first-order logic. Silvio Ghilardi and Roberto Sebastiani. 7th International Symposium, FroCoS 2009, Sep 2009, Trento, Italy. Springer Verlag, 5749, pp.263-278, 2009, Lecture Notes in Computer Science. 〈10.1007/978-3-642-04222-5_16〉. 〈inria-00430631〉

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