A Gentle Non-disjoint Combination of Satisfiability Procedures

  • Paula Chocron
  • Pascal Fontaine
  • Christophe Ringeissen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8562)


A satisfiability problem is often expressed in a combination of theories, and a natural approach consists in solving the problem by combining the satisfiability procedures available for the component theories. This is the purpose of the combination method introduced by Nelson and Oppen. However, in its initial presentation, the Nelson-Oppen combination method requires the theories to be signature-disjoint and stably infinite (to guarantee the existence of an infinite model). The notion of gentle theory has been introduced in the last few years as one solution to go beyond the restriction of stable infiniteness, but in the case of disjoint theories. In this paper, we adapt the notion of gentle theory to the non-disjoint combination of theories sharing only unary predicates (plus constants and the equality). Like in the disjoint case, combining two theories, one of them being gentle, requires some minor assumptions on the other one. We show that major classes of theories, i.e. Löwenheim and Bernays-Schönfinkel-Ramsey, satisfy the appropriate notion of gentleness introduced for this particular non-disjoint combination framework.


Decision Procedure Combination Method Atomic Formula Predicate Symbol Cardinality Constraint 
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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  1. 1.
    Areces, C., Fontaine, P.: Combining theories: The Ackerman and Guarded fragments. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS (LNAI), vol. 6989, pp. 40–54. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New results on rewrite-based satisfiability procedures. ACM Trans. Comput. Log. 10(1) (2009)Google Scholar
  3. 3.
    Armando, A., Ranise, S., Rusinowitch, M.: A rewriting approach to satisfiability procedures. Inf. Comput. 183(2), 140–164 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185, ch. 26, pp. 825–885. IOS Press (February 2009)Google Scholar
  5. 5.
    Bonacina, M.P., Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decidability and undecidability results for Nelson-Oppen and rewrite-based decision procedures. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 513–527. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Börger, E., Grädel, E., Gurevich, Y.: The Classical Decision Problem. In: Perspectives in Mathematical Logic. Springer, Berlin (1997)Google Scholar
  7. 7.
    Chocron, P., Fontaine, P., Ringeissen, C.: A Gentle Non-Disjoint Combination of Satisfiability Procedures (Extended Version). Research Report 8529, Inria (2014),
  8. 8.
    Dreben, B., Goldfarb, W.D.: The Decision Problem: Solvable Classes of Quantificational Formulas. Addison-Wesley, Reading (1979)zbMATHGoogle Scholar
  9. 9.
    Fontaine, P.: Combinations of theories for decidable fragments of first-order logic. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS (LNAI), vol. 5749, pp. 263–278. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Ganzinger, H., Meyer, C., Veanes, M.: The two-variable guarded fragment with transitive relations. In: Logic In Computer Science (LICS), pp. 24–34. IEEE Computer Society (1999)Google Scholar
  11. 11.
    Ganzinger, H., Nivelle, H.D.: A superposition decision procedure for the guarded fragment with equality. In: Logic In Computer Science (LICS), pp. 295–303. IEEE Computer Society Press (1999)Google Scholar
  12. 12.
    Ghilardi, S.: Model-theoretic methods in combined constraint satisfiability. Journal of Automated Reasoning 33(3-4), 221–249 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Gurevich, Y., Shelah, S.: Spectra of monadic second-order formulas with one unary function. In: Logic In Computer Science (LICS), pp. 291–300. IEEE Computer Society, Washington, DC (2003)Google Scholar
  14. 14.
    Manna, Z., Zarba, C.G.: Combining decision procedures. In: Aichernig, B.K., Maibaum, T. (eds.) Formal Methods at the Crossroads. From Panacea to Foundational Support. LNCS, vol. 2757, pp. 381–422. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  15. 15.
    Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. on Programming Languages and Systems 1(2), 245–257 (1979)CrossRefzbMATHGoogle Scholar
  16. 16.
    Nicolini, E., Ringeissen, C., Rusinowitch, M.: Combinable extensions of Abelian groups. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 51–66. Springer, Heidelberg (2009)Google Scholar
  17. 17.
    Nicolini, E., Ringeissen, C., Rusinowitch, M.: Combining satisfiability procedures for unions of theories with a shared counting operator. Fundam. Inform. 105(1-2), 163–187 (2010)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Ramsey, F.P.: On a Problem of Formal Logic. Proceedings of the London Mathematical Society 30, 264–286 (1930)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Ranise, S., Ringeissen, C., Zarba, C.G.: Combining data structures with nonstably infinite theories using many-sorted logic. In: Gramlich, B. (ed.) FroCos 2005. LNCS (LNAI), vol. 3717, pp. 48–64. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  20. 20.
    Ringeissen, C., Senni, V.: Modular termination and combinability for superposition modulo counter arithmetic. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS (LNAI), vol. 6989, pp. 211–226. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  21. 21.
    Sofronie-Stokkermans, V.: Locality results for certain extensions of theories with bridging functions. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 67–83. Springer, Heidelberg (2009)Google Scholar
  22. 22.
    Sofronie-Stokkermans, V.: On combinations of local theory extensions. In: Voronkov, A., Weidenbach, C. (eds.) Ganzinger Festschrift. LNCS, vol. 7797, pp. 392–413. Springer, Heidelberg (2013)Google Scholar
  23. 23.
    Suter, P., Dotta, M., Kuncak, V.: Decision procedures for algebraic data types with abstractions. In: Hermenegildo, M.V., Palsberg, J. (eds.) Principles of Programming Languages (POPL), pp. 199–210. ACM (2010)Google Scholar
  24. 24.
    Tinelli, C., Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures. Theoretical Computer Science 290(1), 291–353 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Tinelli, C., Zarba, C.G.: Combining non-stably infinite theories. Journal of Automated Reasoning 34(3), 209–238 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    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)CrossRefGoogle Scholar
  27. 27.
    Zarba, C.G.: Combining sets with cardinals. J. Autom. Reasoning 34(1), 1–29 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Zhang, T., Sipma, H.B., Manna, Z.: Decision procedures for term algebras with integer constraints. Inf. Comput. 204(10), 1526–1574 (2006)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Paula Chocron
    • 1
    • 3
  • Pascal Fontaine
    • 2
  • Christophe Ringeissen
    • 3
  1. 1.Universidad de Buenos AiresArgentina
  2. 2.INRIAUniversité de Lorraine & LORIANancyFrance
  3. 3.INRIA & LORIANancyFrance

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