Combinations of Theories for Decidable Fragments of First-Order Logic

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


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.


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrial, J.-R.: The B-Book: Assigning Programs to Meanings. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  2. 2.
    Baaz, M., Egly, U., Leitsch, A.: Normal form transformations. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 5, vol. I, pp. 273–333. Elsevier Science B.V, Amsterdam (2001)CrossRefGoogle Scholar
  3. 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
  4. 4.
    Bernays, P., Schönfinkel, M.: Zum Entscheidungsproblem der mathematischen Logik. Math. Annalen 99, 342–372 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. Perspectives in Mathematical Logic. Springer, Berlin (1997)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dreben, B., Goldfarb, W.D.: The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  7. 7.
    Ebbinghaus, H.-D., Flum, J.: Finite Model Theory. Perspectives in Mathematical Logic. Springer, Berlin (1995)zbMATHGoogle Scholar
  8. 8.
    Enderton, H.B.: A Mathematical Introduction to Logic, Orlando, Florida. Academic Press Inc., London (1972)zbMATHGoogle Scholar
  9. 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. 10.
    Fontaine, P.: Combinations of theories for decidable fragments of first-order logic (2009),
  11. 11.
    Fontaine, P., Gribomont, E.P.: Decidability of invariant validation for parameterized systems. In: Garavel, H., Hatcliff, J. (eds.) TACAS 2003. LNCS, vol. 2619, pp. 97–112. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  12. 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
  13. 13.
    Grädel, E., Kolaitis, P.G., Vardi, M.Y.: On the decision problem for two-variable first-order logic. The Bulletin of Symbolic Logic 3(1), 53–69 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Gurevich, Y., Shelah, S.: Spectra of monadic second-order formulas with one unary function. In: LICS 2003: Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science, pp. 291–300. IEEE Computer Society, Washington (2003)CrossRefGoogle Scholar
  15. 15.
    Lamport, L.: Specifying Systems. Addison-Wesley, Boston (2002)zbMATHGoogle Scholar
  16. 16.
    Nelson, G., Oppen, D.C.: Simplifications by cooperating decision procedures. ACM Transactions on Programming Languages and Systems 1(2), 245–257 (1979)CrossRefzbMATHGoogle Scholar
  17. 17.
    Ramsey, F.P.: On a Problem of Formal Logic. Proceedings of the London Mathematical Society 30, 264–286 (1930)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Riazanov, A., Voronkov, A.: The design and implementation of Vampire. AI Communications 15(2), 91–110 (2002)zbMATHGoogle Scholar
  19. 19.
    Schulz, S.: System Abstract: E 0.61. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 370–375. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    Sutcliffe, G., Suttner, C.: The State of CASC. AI Communications 19(1), 35–48 (2006)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Tinelli, C., Harandi, M.T.: A new correctness proof of the Nelson–Oppen combination procedure. In: Baader, F., Schulz, K.U. (eds.) Frontiers of Combining Systems (FroCoS), Applied Logic, pp. 103–120. Kluwer Academic Publishers, Dordrecht (1996)CrossRefGoogle Scholar
  22. 22.
    Tinelli, C., Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures. Theoretical Computer Science 290(1), 291–353 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Tinelli, C., Zarba, C.G.: Combining non-stably infinite theories. Journal of Automated Reasoning 34(3), 209–238 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 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

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

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

Personalised recommendations