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A Polite Non-Disjoint Combination Method: Theories with Bridging Functions Revisited

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Automated Deduction - CADE-25 (CADE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9195))

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Abstract

The Nelson-Oppen combination method is ubiquitous in Satisfiability Modulo Theories solvers. However, one of its major drawbacks is to be restricted to disjoint unions of theories. We investigate the problem of extending this combination method to particular non-disjoint unions of theories connected via bridging functions. The motivation is, e.g., to solve verification problems expressed in a combination of data structures connected to arithmetic with bridging functions such as the length of lists and the size of trees. We present a sound and complete combination procedure à la Nelson-Oppen for the theory of absolutely free data structures, including lists and trees. This combination procedure is then refined for standard interpretations. The resulting theory has a nice politeness property, enabling combinations with arbitrary decidable theories of elements.

This work has been partially supported by the project ANR-13-IS02-0001 of the Agence Nationale de la Recherche, by the European Union Seventh Framework Programme under grant agreement no. 295261 (MEALS), and by the STIC AmSud MISMT.

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References

  1. Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New results on rewrite-based satisfiability procedures. ACM Trans. Comput. Log. 10(1), 4 (2009)

    Article  MathSciNet  Google Scholar 

  2. Baader, F., Ghilardi, S.: Connecting many-sorted theories. J. Symb. Log. 72(2), 535–583 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baumgartner, P., Waldmann, U.: Hierarchic superposition with weak abstraction. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 39–57. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  4. Chocron, P., Fontaine, P., Ringeissen, C.: A gentle non-disjoint combination of satisfiability procedures. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014. LNCS, vol. 8562, pp. 122–136. Springer, Heidelberg (2014). http://hal.inria.fr/hal-00985135

    Google Scholar 

  5. Chocron, P., Fontaine, P., Ringeissen, C.: A Polite Non-Disjoint Combination Method: Theories with Bridging Functions Revisited (Extended Version) (2015). http://hal.inria.fr

  6. Fontaine, P., Ranise, S., Zarba, C.G.: Combining lists with non-stably infinite theories. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 51–66. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  7. Ghilardi, S.: Model-theoretic methods in combined constraint satisfiability. J. Autom. Reasoning 33(3–4), 221–249 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Jovanović, D., Barrett, C.: Polite theories revisited. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 402–416. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  9. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979)

    Article  MATH  Google Scholar 

  10. Nicolini, E., Ringeissen, C., Rusinowitch, M.: Combinable extensions of abelian groups. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 51–66. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  11. Nicolini, E., Ringeissen, C., Rusinowitch, M.: Combining satisfiability procedures for unions of theories with a shared counting operator. Fundam. Inf. 105(1–2), 163–187 (2010)

    MathSciNet  MATH  Google Scholar 

  12. Pham, T.-H., Whalen, M.W.: An improved unrolling-based decision procedure for algebraic data types. In: Cohen, E., Rybalchenko, A. (eds.) VSTTE 2013. LNCS, vol. 8164, pp. 129–148. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  13. 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)

    Chapter  Google Scholar 

  14. Sofronie-Stokkermans, V.: Locality results for certain extensions of theories with bridging functions. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 67–83. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  15. 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, New York (2010)

    Google Scholar 

  16. Suter, P., Köksal, A.S., Kuncak, V.: Satisfiability modulo recursive programs. In: Yahav, E. (ed.) Static Analysis. LNCS, vol. 6887, pp. 298–315. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  17. 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 (1996)

    Google Scholar 

  18. Tinelli, C., Ringeissen, C.: Unions of non-disjoint theories and combinations of satisfiability procedures. Theoret. Comput. Sci. 290(1), 291–353 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Chapter  Google Scholar 

  20. Zarba, C.G.: Combining lists with integers. In: International Joint Conference on Automated Reasoning (Short Papers), Technical report DII 11/01, pp. 170–179. University of Siena (2001)

    Google Scholar 

  21. Zarba, C.G.: Combining multisets with integers. In: Voronkov, A. (ed.) CADE 2002. LNCS (LNAI), vol. 2392, pp. 363–376. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  22. Zarba, C.G.: Combining sets with cardinals. J. Autom. Reasoning 34(1), 1–29 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, T., Sipma, H.B., Manna, Z.: Decision procedures for term algebras with integer constraints. Inf. Comput. 204(10), 1526–1574 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

We are grateful to Jasmin Blanchette and to the anonymous reviewers for many constructive remarks.

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Authors and Affiliations

  1. IIIA-CSIC, Bellaterra, Catalonia, Spain

    Paula Chocron

  2. INRIA, Université de Lorraine and LORIA, Nancy, France

    Pascal Fontaine & Christophe Ringeissen

Authors
  1. Paula Chocron
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  2. Pascal Fontaine
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  3. Christophe Ringeissen
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Corresponding author

Correspondence to Christophe Ringeissen .

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Editors and Affiliations

  1. University of Ottawa, Ottawa, Canada

    Amy P. Felty

  2. University of Innsbruck, Innsbruck, Austria

    Aart Middeldorp

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Chocron, P., Fontaine, P., Ringeissen, C. (2015). A Polite Non-Disjoint Combination Method: Theories with Bridging Functions Revisited. In: Felty, A., Middeldorp, A. (eds) Automated Deduction - CADE-25. CADE 2015. Lecture Notes in Computer Science(), vol 9195. Springer, Cham. https://doi.org/10.1007/978-3-319-21401-6_29

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  • DOI: https://doi.org/10.1007/978-3-319-21401-6_29

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21400-9

  • Online ISBN: 978-3-319-21401-6

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