Combinations of Theories for Decidable Fragments of First-Order Logic
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.
KeywordsDecidable Theory Decision Procedure Predicate Symbol Satisfiability Modulo Theory Constant Symbol
Unable to display preview. Download preview PDF.
- 3.Baumgartner, P., Fuchs, A., Tinelli, C.: Implementing the Model Evolution Calculus. In: Schulz, S., Sutcliffe, G., Tammet, T. (eds.) Special Issue of the International Journal of Artificial Intelligence Tools (IJAIT). International Journal of Artificial Intelligence Tools, vol. 15 (2005)Google Scholar
- 9.Fontaine, P.: Combinations of theories and the Bernays-Schönfinkel-Ramsey class. In: Beckert, B. (ed.) 4th International Verification Workshop - VERIFY 2007, Bremen (15/07/07-16/07/07) (July 2007)Google Scholar
- 10.Fontaine, P.: Combinations of theories for decidable fragments of first-order logic (2009), http://www.loria.fr/~fontaine/Fontaine12b.pdf
- 12.Fontaine, P., Gribomont, E.P.: Combining non-stably infinite, non-first order theories. In: Ahrendt, W., Baumgartner, P., de Nivelle, H., Ranise, S., Tinelli, C. (eds.) Selected Papers from the Workshops on Disproving and the Second International Workshop on Pragmatics of Decision Procedures (PDPAR 2004), July 2005. Electronic Notes in Theoretical Computer Science, vol. 125, pp. 37–51 (2005)Google Scholar
- 24.Wies, T., Piskac, R., Kuncak, V.: Combining theories with shared set operations. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS (LNAI), vol. 5749, pp. 366–382. Springer, Heidelberg (2009)Google Scholar