A Decision Procedure for Restricted Intensional Sets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)


In this paper we present a decision procedure for Restricted Intensional Sets (RIS), i.e. sets given by a property rather than by enumerating their elements, similar to set comprehensions available in specification languages such as B and Z. The proposed procedure is parametric with respect to a first-order language and theory \(\mathcal {X}\), providing at least equality and a decision procedure to check for satisfiability of \(\mathcal {X}\)-formulas. We show how this framework can be applied when \(\mathcal {X}\) is the theory of hereditarily finite sets as is supported by the language CLP(\(\mathcal {SET}\)). We also present a working implementation of RIS as part of the \(\{log\}\) tool and we show how it compares with a mainstream solver and how it helps in the automatic verification of code fragments.



Part of the work of M. Cristiá is supported by ANPCyT’s grant PICT-2014-2200.


  1. 1.
    Bjørner, N., McMillan, K., Rybalchenko, A.: On solving universally quantified horn clauses. In: Logozzo, F., Fähndrich, M. (eds.) SAS 2013. LNCS, vol. 7935, pp. 105–125. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-38856-9_8 CrossRefGoogle Scholar
  2. 2.
    Cantone, D., Longo, C.: A decidable two-sorted quantified fragment of set theory with ordered pairs and some undecidable extensions. Theor. Comput. Sci. 560, 307–325 (2014). http://dx.doi.org/10.1016/j.tcs.2014.03.021 MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Claessen, K., Sörensson, N.: New techniques that improve MACE-style finite model building. In: CADE-19 Workshop: Model Computation - Principles, Algorithms, Applications, pp. 11–27 (2003)Google Scholar
  4. 4.
    Cristiá, M., Rossi, G.: Restricted insentional sets. http://people.dmi.unipr.it/gianfranco.rossi/SETLOG/risCADEonline.pdf
  5. 5.
    Cristiá, M., Rossi, G.: A decision procedure for sets, binary relations and partial functions. In: Chaudhuri, S., Farzan, A. (eds.) CAV 2016, Part I. LNCS, vol. 9779, pp. 179–198. Springer, Cham (2016). doi: 10.1007/978-3-319-41528-4_10 Google Scholar
  6. 6.
    Dal Palú, A., Dovier, A., Pontelli, E., Rossi, G.: Integrating finite domain constraints and CLP with sets. In: Proceedings of the 5th ACM SIGPLAN International Conference on Principles and Practice of Declaritive Programming, PPDP 2003, pp. 219–229. ACM, New York (2003). http://doi.acm.org/10.1145/888251.888272
  7. 7.
    Deharbe, D., Fontaine, P., Paleo, B.W.: Quantifier inference rules for SMT proofs. In: Workshop on Proof eXchange for Theorem Proving (2011)Google Scholar
  8. 8.
    Dovier, A., Omodeo, E.G., Pontelli, E., Rossi, G.: A language for programming in logic with finite sets. J. Log. Program. 28(1), 1–44 (1996). http://dx.doi.org/10.1016/0743-1066(95)00147-6
  9. 9.
    Dovier, A., Piazza, C., Pontelli, E., Rossi, G.: Sets and constraint logic programming. ACM Trans. Program. Lang. Syst. 22(5), 861–931 (2000)CrossRefGoogle Scholar
  10. 10.
    Dovier, A., Pontelli, E., Rossi, G.: Intensional sets in CLP. In: Palamidessi, C. (ed.) ICLP 2003. LNCS, vol. 2916, pp. 284–299. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-24599-5_20 CrossRefGoogle Scholar
  11. 11.
    Dovier, A., Pontelli, E., Rossi, G.: Set unification. Theor. Pract. Log. Program. 6(6), 645–701 (2006). http://dx.doi.org/10.1017/S1471068406002730 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Drăgoi, C., Henzinger, T.A., Veith, H., Widder, J., Zufferey, D.: A logic-based framework for verifying consensus algorithms. In: McMillan, K.L., Rival, X. (eds.) VMCAI 2014. LNCS, vol. 8318, pp. 161–181. Springer, Heidelberg (2014). doi: 10.1007/978-3-642-54013-4_10 CrossRefGoogle Scholar
  13. 13.
    Ge, Y., de Moura, L.: Complete instantiation for quantified formulas in satisfiabiliby modulo theories. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 306–320. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-02658-4_25 CrossRefGoogle Scholar
  14. 14.
    Hill, P.M., Lloyd, J.W.: The Gödel Programming Language. MIT Press, Cambridge (1994)MATHGoogle Scholar
  15. 15.
    Jackson, D.: Software Abstractions: Logic, Language, and Analysis. The MIT Press, Cambridge (2006)Google Scholar
  16. 16.
    Leuschel, M., Butler, M.: ProB: a model checker for B. In: Araki, K., Gnesi, S., Mandrioli, D. (eds.) FME 2003. LNCS, vol. 2805, pp. 855–874. Springer, Heidelberg (2003). doi: 10.1007/978-3-540-45236-2_46 CrossRefGoogle Scholar
  17. 17.
    Nethercote, N., Stuckey, P.J., Becket, R., Brand, S., Duck, G.J., Tack, G.: MiniZinc: towards a standard CP modelling language. In: Bessière, C. (ed.) CP 2007. LNCS, vol. 4741, pp. 529–543. Springer, Heidelberg (2007). doi: 10.1007/978-3-540-74970-7_38 CrossRefGoogle Scholar
  18. 18.
    Reynolds, A., Tinelli, C., Goel, A., Krstić, S.: Finite model finding in SMT. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 640–655. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-39799-8_42 CrossRefGoogle Scholar
  19. 19.
  20. 20.
    Schneider, S.: The B-method: An Introduction. Cornerstones of Computing. Palgrave (2001). http://books.google.com.ar/books?id=Krs0OQAACAAJ
  21. 21.
    Schwartz, J.T., Dewar, R.B.K., Dubinsky, E., Schonberg, E.: Programming with Sets - An Introduction to SETL. Texts and Monographs in Computer Science. Springer, New York (1986). http://dx.doi.org/10.1007/978-1-4613-9575-1
  22. 22.
    Veanes, M., Saabas, A.: On bounded reachability of programs with set comprehensions. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS, vol. 5330, pp. 305–317. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-89439-1_22 CrossRefGoogle Scholar
  23. 23.
    Wies, T., Piskac, R., Kuncak, V.: Combining theories with shared set operations. In: Ghilardi, S., Sebastiani, R. (eds.) FroCoS 2009. LNCS, vol. 5749, pp. 366–382. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04222-5_23 CrossRefGoogle Scholar
  24. 24.
    Woodcock, J., Davies, J.: Using Z: Specification, Refinement, and Proof. Prentice-Hall, Inc., Upper Saddle River (1996)MATHGoogle Scholar
  25. 25.
    Zhang, J., Zhang, H.: System description generating models by SEM. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 308–312. Springer, Heidelberg (1996). doi: 10.1007/3-540-61511-3_96 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Universidad Nacional de Rosario and CIFASISRosarioArgentina
  2. 2.Università di ParmaParmaItaly

Personalised recommendations