Skip to main content
SpringerLink
Log in

A Decision Procedure for Restricted Intensional Sets

  • Conference paper
  • First Online:
Automated Deduction – CADE 26 (CADE 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10395))

Included in the following conference series:

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The form of RIS terms is borrowed from the form of set comprehension expressions available in Z.

  2. 2.

    This is guaranteed by procedure remove_neq (see Sect. 3).

  3. 3.

    More precisely, each solution of \(\varPhi \) expanded to the variables occurring in \(\phi _i\) but not in \(\varPhi \), so to account for the possible fresh variables introduced into \(\phi _i\).

References

  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

    Chapter  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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. Cristiá, M., Rossi, G.: Restricted insentional sets. http://people.dmi.unipr.it/gianfranco.rossi/SETLOG/risCADEonline.pdf

  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. 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. 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. 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. Dovier, A., Piazza, C., Pontelli, E., Rossi, G.: Sets and constraint logic programming. ACM Trans. Program. Lang. Syst. 22(5), 861–931 (2000)

    Article  Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Article  MathSciNet  MATH  Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Chapter  Google Scholar 

  14. Hill, P.M., Lloyd, J.W.: The Gödel Programming Language. MIT Press, Cambridge (1994)

    MATH  Google Scholar 

  15. Jackson, D.: Software Abstractions: Logic, Language, and Analysis. The MIT Press, Cambridge (2006)

    Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Chapter  Google Scholar 

  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

    Chapter  Google Scholar 

  19. Rossi, G.: \(\{log\}\) (2008). http://people.dmi.unipr.it/gianfranco.rossi/setlog.Home.html

  20. Schneider, S.: The B-method: An Introduction. Cornerstones of Computing. Palgrave (2001). http://books.google.com.ar/books?id=Krs0OQAACAAJ

  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. 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

    Chapter  Google Scholar 

  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

    Chapter  Google Scholar 

  24. Woodcock, J., Davies, J.: Using Z: Specification, Refinement, and Proof. Prentice-Hall, Inc., Upper Saddle River (1996)

    MATH  Google Scholar 

  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

    Chapter  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Universidad Nacional de Rosario and CIFASIS, Rosario, Argentina

    Maximiliano Cristiá

  2. Università di Parma, Parma, Italy

    Gianfranco Rossi

Authors
  1. Maximiliano Cristiá
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gianfranco Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maximiliano Cristiá .

Editor information

Editors and Affiliations

  1. Microsoft Research, Redmond, Washington, USA

    Leonardo de Moura

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Cristiá, M., Rossi, G. (2017). A Decision Procedure for Restricted Intensional Sets. In: de Moura, L. (eds) Automated Deduction – CADE 26. CADE 2017. Lecture Notes in Computer Science(), vol 10395. Springer, Cham. https://doi.org/10.1007/978-3-319-63046-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-63046-5_12

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-63045-8

  • Online ISBN: 978-3-319-63046-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation