Skip to main content
SpringerLink
Log in

Polite Combination of Algebraic Datatypes

  • Published:
Journal of Automated Reasoning Aims and scope Submit manuscript

Abstract

Algebraic datatypes, and among them lists and trees, have attracted a lot of interest in automated reasoning and Satisfiability Modulo Theories (SMT). Since its latest stable version, the SMT-LIB standard defines a theory of algebraic datatypes, which is currently supported by several mainstream SMT solvers. In this paper, we study this particular theory of datatypes and prove that it is strongly polite, showing how it can be combined with other arbitrary disjoint theories using polite combination. The combination method uses a new, simple, and natural notion of additivity that enables deducing strong politeness from (weak) politeness.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

Notes

  1. A preliminary version of this work was published in the proceedings of IJCAR 2020 [26]. The current article extends the original versions with complete proofs, as well as a discussion and results regarding existential theories (see Proposition 2). Additionally, Sect. 5 is extended to provide a more comprehensive treatment of axiomatizations for trees.

  2. We note that in practice, the new variables in \( wtn (\phi )\) are assumed to be fresh not only with respect to \(\phi \), but also with respect to the formula from the second theory being combined.

References

  1. Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: New results on rewrite-based satisfiability procedures. ACM Trans. Comput. Log. 10(1), 4:1-4:51 (2009)

    Article  MathSciNet  Google Scholar 

  2. Baader, F., Snyder, W., Narendran, P., Schmidt-Schauß, M., Schulz, K.U.: Unification theory. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 Volumes), pp. 445–532. Elsevier/MIT Press, New York (2001)

    Chapter  Google Scholar 

  3. Barrett, C.W., Dill, D.L., Stump, A.: A generalization of shostak’s method for combining decision procedures. In: A. Armando (ed.) Frontiers of Combining Systems, 4th International Workshop, FroCoS 2002, Santa Margherita Ligure, Italy, April 8–10, 2002, Proceedings, Lecture Notes in Computer Science, vol. 2309, pp. 132–146. Springer (2002)

  4. Barrett, C.W., Shikanian, I., Tinelli, C.: An abstract decision procedure for a theory of inductive data types. J. Satisfiab. Boolean Model. Comput. 3(1–2), 21–46 (2007)

    Article  MathSciNet  Google Scholar 

  5. Barrett, C., Conway, C.L., Deters, M., Hadarean, L., Jovanović, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Proceedings of the 23rd International Conference on Computer Aided Verification, CAV’11, pp. 171–177. Springer (2011). http://dl.acm.org/citation.cfm?id=2032305.2032319

  6. Barrett, C., Fontaine, P., Tinelli, C.: The SMT-LIB Standard: Version 2.6. Tech. rep., Department of Computer Science, The University of Iowa (2017). Available at www.SMT-LIB.org

  7. Berthon, R., Ringeissen, C.: Satisfiability modulo free data structures combined with bridging functions. In: T. King, R. Piskac (eds.) Proceedings of SMT@IJCAR 2016, CEUR Workshop Proceedings, vol. 1617, pp. 71–80. CEUR-WS.org (2016)

  8. Bonacina, M.P., Echenim, M.: Rewrite-based satisfiability procedures for recursive data structures. Electron. Notes Theor. Comput. Sci. 174(8), 55–70 (2007)

    Article  Google Scholar 

  9. Bonacina, M.P., Fontaine, P., Ringeissen, C., Tinelli, C.: Theory combination: Beyond equality sharing. In: C. Lutz, U. Sattler, C. Tinelli, A. Turhan, F. Wolter (eds.) Description Logic, Theory Combination, and All That - Essays Dedicated to Franz Baader on the Occasion of His 60th Birthday, Lecture Notes in Computer Science, vol. 11560, pp. 57–89. Springer (2019)

  10. Casal, F., Rasga, J.: Many-sorted equivalence of shiny and strongly polite theories. J. Autom. Reason. 60(2), 221–236 (2018)

    Article  MathSciNet  Google Scholar 

  11. Chocron, P., Fontaine, P., Ringeissen, C.: Politeness and combination methods for theories with bridging functions. J. Autom. Reason. 64(1), 97–134 (2020)

    Article  MathSciNet  Google Scholar 

  12. Enderton, H.B.: A Mathematical Introduction to Logic. Academic Press, New York (2001)

    MATH  Google Scholar 

  13. Fontaine, P.: Combinations of theories for decidable fragments of first-order logic. In: S. Ghilardi, R. Sebastiani (eds.) Frontiers of Combining Systems, 7th International Symposium, FroCoS 2009, Trento, Italy, September 16–18, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5749, pp. 263–278. Springer (2009)

  14. Gutiérrez, R., Meseguer, J.: Variant-based decidable satisfiability in initial algebras with predicates. In: F. Fioravanti, J.P. Gallagher (eds.) Logic-Based Program Synthesis and Transformation—27th International Symposium, LOPSTR 2017, Namur, Belgium, October 10–12, 2017, Revised Selected Papers, Lecture Notes in Computer Science, vol. 10855, pp. 306–322. Springer (2017)

  15. Hojjat, H., Rümmer, P.: Deciding and interpolating algebraic data types by reduction. In: T. Jebelean, V. Negru, D. Petcu, D. Zaharie, T. Ida, S.M. Watt (eds.) 19th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing, SYNASC 2017, Timisoara, Romania, September 21–24, 2017, pp. 145–152. IEEE Computer Society (2017)

  16. Jovanovic, D., Barrett, C.W.: Polite theories revisited. In: C.G. Fermüller, A. Voronkov (eds.) Logic for Programming, Artificial Intelligence, and Reasoning—17th International Conference, LPAR-17, Yogyakarta, Indonesia, October 10–15, 2010. Proceedings, Lecture Notes in Computer Science, vol. 6397, pp. 402–416. Springer (2010). Extended technical report is available at http://theory.stanford.edu/~barrett/pubs/JB10-TR.pdf

  17. Kovács, L., Robillard, S., Voronkov, A.: Coming to terms with quantified reasoning. In: G. Castagna, A.D. Gordon (eds.) Proceedings of the 44th ACM SIGPLAN Symposium on Principles of Programming Languages, POPL 2017, Paris, France, January 18–20, 2017, pp. 260–270. ACM (2017)

  18. Krstic, S., Conchon, S.: Canonization for disjoint unions of theories. Inf. Comput. 199(1–2), 87–106 (2005)

    Article  MathSciNet  Google Scholar 

  19. Krstic, S., Goel, A., Grundy, J., Tinelli, C.: Combined satisfiability modulo parametric theories. In: O. Grumberg, M. Huth (eds.) Tools and Algorithms for the Construction and Analysis of Systems, 13th International Conference, TACAS 2007, Held as Part of the Joint European Conferences on Theory and Practice of Software, ETAPS 2007 Braga, Portugal, March 24–April 1, 2007, Proceedings, Lecture Notes in Computer Science, vol. 4424, pp. 602–617. Springer (2007)

  20. Manna, Z., Zarba, C.G.: Combining decision procedures. In: B.K. Aichernig, T.S.E. Maibaum (eds.) Formal Methods at the Crossroads. From Panacea to Foundational Support, 10th Anniversary Colloquium of UNU/IIST, the International Institute for Software Technology of The United Nations University, Lisbon, Portugal, March 18–20, 2002, Revised Papers, Lecture Notes in Computer Science, vol. 2757, pp. 381–422. Springer (2002)

  21. Meseguer, J.: Variant-based satisfiability in initial algebras. Sci. Comput. Program. 154, 3–41 (2018)

    Article  Google Scholar 

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

    Article  Google Scholar 

  23. Ranise, S., Ringeissen, C., Zarba, C.G.: Combining data structures with nonstably infinite theories using many-sorted logic. In: B. Gramlich (ed.) Frontiers of Combining Systems, 5th International Workshop, FroCoS 2005, Vienna, Austria, September 19–21, 2005, Proceedings, Lecture Notes in Computer Science, vol. 3717, pp. 48–64. Springer (2005). Extended technical report is available at https://hal.inria.fr/inria-00070335/

  24. Reynolds, A., Blanchette, J.C.: A decision procedure for (co)datatypes in SMT solvers. J. Autom. Reason. 58(3), 341–362 (2017)

    Article  MathSciNet  Google Scholar 

  25. Reynolds, A., Viswanathan, A., Barbosa, H., Tinelli, C., Barrett, C.W.: Datatypes with shared selectors. In: D. Galmiche, S. Schulz, R. Sebastiani (eds.) Automated Reasoning - 9th International Joint Conference, IJCAR 2018, Held as Part of the Federated Logic Conference, FloC 2018, Oxford, UK, July 14–17, 2018, Proceedings, Lecture Notes in Computer Science, vol. 10900, pp. 591–608. Springer (2018)

  26. Sheng, Y., Zohar, Y., Ringeissen, C., Lange, J., Fontaine, P., Barrett, C.W.: Politeness for the theory of algebraic datatypes. In: IJCAR (1), Lecture Notes in Computer Science, vol. 12166, pp. 238–255. Springer (2020)

  27. Sheng, Y., Zohar, Y., Ringeissen, C., Reynolds, A., Barrett, C.W., Tinelli, C.: Politeness and stable infiniteness: Stronger together. In: CADE, Lecture Notes in Computer Science, vol. 12699, pp. 148–165. Springer (2021)

  28. Shostak, R.E.: A practical decision procedure for arithmetic with function symbols. J. ACM 26(2), 351–360 (1979)

    Article  MathSciNet  Google Scholar 

  29. Sofronie-Stokkermans, V.: Locality results for certain extensions of theories with bridging functions. In: R.A. Schmidt (ed.) Automated Deduction - CADE-22, 22nd International Conference on Automated Deduction, Montreal, Canada, August 2–7, 2009. Proceedings, Lecture Notes in Computer Science, vol. 5663, pp. 67–83. Springer (2009)

  30. Tinelli, C.: Cooperation of background reasoners in theory reasoning by residue sharing. J. Autom. Reason. 30(1), 1–31 (2003)

    Article  MathSciNet  Google Scholar 

  31. Tinelli, C., Zarba, C.G.: Combining decision procedures for sorted theories. In: J.J. Alferes, J.A. Leite (eds.) Logics in Artificial Intelligence, 9th European Conference, JELIA 2004, Lisbon, Portugal, September 27–30, 2004, Proceedings, Lecture Notes in Computer Science, vol. 3229, pp. 641–653. Springer (2004)

  32. Tinelli, C., Zarba, C.G.: Combining nonstably infinite theories. J. Autom. Reason. 34(3), 209–238 (2005)

    Article  MathSciNet  Google Scholar 

  33. Tran, D., Ringeissen, C., Ranise, S., Kirchner, H.: Combination of convex theories: modularity, deduction completeness, and explanation. J. Symb. Comput. 45(2), 261–286 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Stanford University, Stanford, USA

    Ying Sheng & Clark Barrett

  2. Bar Ilan University, Ramat Gan, Israel

    Yoni Zohar

  3. Université de Lorraine, CNRS, Inria, LORIA, Nancy, France

    Christophe Ringeissen

  4. MIT, Cambridge, USA

    Jane Lange

  5. Université de Liège, Liège, Belgium

    Pascal Fontaine

Authors
  1. Ying Sheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yoni Zohar
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christophe Ringeissen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Jane Lange
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Pascal Fontaine
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Clark Barrett
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ying Sheng.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This project was partially supported by a Grant from the Defense Advanced Research Projects Agency (N66001-18-C-4012), the Stanford CURIS program, and Jasmin Blanchette’s European Research Council (ERC) starting Grant Matryoshka (713999)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sheng, Y., Zohar, Y., Ringeissen, C. et al. Polite Combination of Algebraic Datatypes. J Autom Reasoning 66, 331–355 (2022). https://doi.org/10.1007/s10817-022-09625-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10817-022-09625-3

Keywords

Search

Navigation