Skip to main content
SpringerLink
Log in

Combining Stable Infiniteness and (Strong) Politeness

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

Abstract

Polite theory combination is a method for obtaining a solver for a combination of two (or more) theories using the solvers of each individual theory as black boxes. Unlike the earlier Nelson–Oppen method, which is usable only when both theories are stably infinite, only one of the theories needs to be strongly polite in order to use the polite combination method. In its original presentation, politeness was required from one of the theories rather than strong politeness, which was later proven to be insufficient. The first contribution of this paper is a proof that indeed these two notions are different, obtained by presenting a polite theory that is not strongly polite. We also study several variants of this question.

The cost of the generality afforded by the polite combination method, compared to the Nelson–Oppen method, is a larger space of arrangements to consider, involving variables that are not necessarily shared between the purified parts of the input formula. The second contribution of this paper is a hybrid method (building on both polite and Nelson–Oppen combination), which aims to reduce the number of considered variables when a theory is stably infinite with respect to some of its sorts but not all of them. The time required to reason about arrangements is exponential in the worst case, so reducing the number of variables considered has the potential to improve performance significantly. We show preliminary evidence for this by demonstrating significant speed-up on a smart contract verification benchmark.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Notes

  1. A preliminary version of this work was published in the proceedings of CADE-28 [21]. The current article incorporates some updates to the text, adds detailed proofs to all claims, and is accompanied by an artifact that can be used to reproduce the case study reported in Sect. 5.

  2. In [22] this was proven more generally, for order-sorted logics which extend many-sorted logics with subsorts.

  3. For the results proven below, this full generality regarding V is not needed (e.g., it is sufficient to consider only variables in \(\phi \)). However, for the validity of other results in polite theory combination, an arbitrary (finite) V is required. For more details, see Footnote 4 of [14].

  4. In [9], the first condition is written \({\left| \sigma _{1}^{{{\mathcal {A}}}}\right| }\ge 2\). We use equality as this is equivalent and we believe it makes things clearer.

  5. The case of non-empty signatures is addressed in a recent paper by Toledo et al. [24].

  6. Notice that \({{{\mathcal {T}}}_{\textrm{Even}}^{\infty }}\) can be axiomatized using the set \(\left\{ \lnot \psi _{=2n+1}^{\sigma }\mid n\in {\mathbb {N}}\right\} \) (see Fig. 2).

  7. An artifact which includes the compiled binary of the implementation, the benchmark, the raw results, as well as reproduction instructions is available at https://doi.org/10.5281/zenodo.6538824.

References

  1. Amsden, Z., Arora, R., Bano, S., Baudet, M., Blackshear, S., Bothra, A., Cabrera, G., Catalini, C., Chalkias, K., Cheng, E., Ching, A., Chursin, A., Danezis, G., Giacomo, G.D., Dill, D.L., Ding, H., Doudchenko, N., Gao, V., Gao, Z., Garillot, F., Gorven, M., Hayes, P., Hou, J.M., Hu, Y., Hurley, K., Lewi, K., Li, C., Li, Z., Malkhi, D., Margulis, S., Maurer, B., Mohassel, P., de Naurois, L., Nikolaenko, V., Nowacki, T., Orlov, O., Perelman, D., Pott, A., Proctor, B., Qadeer, S., Rain, Russi, D., Schwab, B., Sezer, S., Sonnino, A., Venter, H., Wei, L., Wernerfelt, N., Williams, B., Wu, Q., Yan, X., Zakian, T., Zhou, R.: The Diem blockchain. https://developers.diem.com/docs/technical-papers/the-diem-blockchain-paper/ (2019)

  2. Barbosa, H., Barrett, C., Brain, M., Kremer, G., Lachnitt, H., Mann, M., Mohamed, A., Mohamed, M., Niemetz, A., Noetzli, A., Ozdemir, A., Preiner, M., Reynolds, A., Sheng, Y., Tinelli, C., Zohar, Y.: cvc5: a versatile and industrial-strength SMT solver. In: Proceedings of TACAS 2022 (2022)

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

  4. Barrett, C.W.: Checking validity of quantifier-free formulas in combinations of first-order theories. Ph.D., Stanford University (2003). http://www.cs.stanford.edu/~barrett/pubs/B03.pdf. Stanford, California

  5. Barrett, C.W., Conway, C.L., Deters, M., Hadarean, L., Jovanovic, D., King, T., Reynolds, A., Tinelli, C.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) Computer Aided Verification—23rd International Conference, CAV 2011, Snowbird, UT, USA, July 14–20, 2011. Proceedings, Lecture Notes in Computer Science, vol. 6806, pp. 171–177. Springer, New York (2011). https://doi.org/10.1007/978-3-642-22110-1_14

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

    Article  MathSciNet  MATH  Google Scholar 

  7. Blackshear, S., Cheng, E., Dill, D.L., Gao, V., Maurer, B., Nowacki, T., Pott, A., Qadeer, S., Rain, Russi, D., Sezer, S., Zakian, T., Zhou, R.: Move: a language with programmable resources. https://developers.diem.com/docs/technical-papers/move-paper/ (2019)

  8. Casal, F., Rasga, J.: Revisiting the equivalence of shininess and politeness. In: McMillan, K.L., Middeldorp, A., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning—19th International Conference, LPAR-19, Stellenbosch, South Africa, December 14–19, 2013. Proceedings, Lecture Notes in Computer Science, vol. 8312, pp. 198–212. Springer, New York (2013). https://doi.org/10.1007/978-3-642-45221-5_15

  9. Casal, F., Rasga, J.: Many-sorted equivalence of shiny and strongly polite theories. J. Autom. Reason. 60(2), 221–236 (2018). https://doi.org/10.1007/s10817-017-9411-y

    Article  MathSciNet  MATH  Google Scholar 

  10. diem: https://github.com/diem/diem

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

    MATH  Google Scholar 

  12. Fontaine, P.: Combinations of theories for decidable fragments of first-order logic. In: Ghilardi, S., Sebastiani, R. (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, New York (2009). https://doi.org/10.1007/978-3-642-04222-5_16

  13. Jovanovic, D., Barrett, C.W.: Polite theories revisited. In: Fermüller, C.G., Voronkov, A. (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, New York (2010). https://doi.org/10.1007/978-3-642-16242-8_29

  14. Jovanovic, D., Barrett, C.W.: Polite theories revisited. Tech. rep., New York University (2019). http://theory.stanford.edu/~barrett/pubs/JB10-TR.pdf. Technical Report TR2010-922

  15. Krstic, S., Goel, A., Grundy, J., Tinelli, C.: Combined satisfiability modulo parametric theories. In: Grumberg, O., Huth, M. (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, New York (2007). https://doi.org/10.1007/978-3-540-71209-1_47

  16. Leino, K.R.M.: This is Boogie 2. manuscript KRML 178(131), 9 (2008). https://www.microsoft.com/en-us/research/publication/this-is-boogie-2-2/

  17. Nelson, G.: Techniques for program verification. Tech. Rep. CSL-81-10, Xerox, Palo Alto Research Center (1981)

  18. Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. Program. Lang. Syst. 1(2), 245–257 (1979). https://doi.org/10.1145/357073.357079

    Article  MATH  Google Scholar 

  19. Ranise, S., Ringeissen, C., Zarba, C.G.: Combining data structures with nonstably infinite theories using many-sorted logic. In: Gramlich, B. (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, New York (2005). Extended technical report is available at https://hal.inria.fr/inria-00070335/

  20. Sheng, Y., Zohar, Y., Ringeissen, C., Lange, J., Fontaine, P., Barrett, C.W.: Politeness for the theory of algebraic datatypes. In: Peltier, N., Sofronie-Stokkermans, V. (eds.) Automated Reasoning—10th International Joint Conference, IJCAR 2020, Paris, France, July 1–4, 2020, Proceedings, Part I, Lecture Notes in Computer Science, vol. 12166, pp. 238–255. Springer, New York (2020). https://doi.org/10.1007/978-3-030-51074-9_14

  21. 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, New York (2021)

  22. Tinelli, C., Zarba, C.G.: Combining decision procedures for sorted theories. In: Alferes, J.J., Leite, J.A. (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, New York (2004)

  23. Tinelli, C., Zarba, C.G.: Combining nonstably infinite theories. J. Autom. Reason. 34(3), 209–238 (2005). https://doi.org/10.1007/s10817-005-5204-9

    Article  MathSciNet  MATH  Google Scholar 

  24. Toledo, G., Zohar, Y., Barrett, C.: Combining combination properties: an analysis of stable infiniteness, convexity, and politeness. In: Proceedings of CADE 2023 (2023)

  25. Zhong, J.E., Cheang, K., Qadeer, S., Grieskamp, W., Blackshear, S., Park, J., Zohar, Y., Barrett, C.W., Dill, D.L.: The Move prover. In: Lahiri, S.K., Wang, C., (eds.) Computer Aided Verification—32nd International Conference, CAV 2020, Los Angeles, CA, USA, July 21–24, 2020, Proceedings, Part I, Lecture Notes in Computer Science, vol. 12224, pp. 137–150. Springer, New York (2020). https://doi.org/10.1007/978-3-030-53288-8_7

Download references

Author information

Authors and Affiliations

  1. Stanford University, Stanford, CA, USA

    Ying Sheng & Clark Barrett

  2. Bar-Ilan University, Ramat Gan, Israel

    Yoni Zohar

  3. Université de Lorraine, CNRS, Inria, LORIA, 54000, Nancy, France

    Christophe Ringeissen

  4. The University of Iowa, Iowa City, IA, USA

    Andrew Reynolds & Cesare Tinelli

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. Andrew Reynolds
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Clark Barrett
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Cesare Tinelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed equally.

Corresponding author

Correspondence to Yoni Zohar.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Publisher's Note

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

This work was funded in part by the Stanford Center for Blockchain Research, NSF-BSF Grant numbers 2110397 (NSF) and 2020704 (BSF), ISF Grant number 619/21, and a gift from Meta Novi to the University of Iowa.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

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. Combining Stable Infiniteness and (Strong) Politeness. J Autom Reasoning 67, 34 (2023). https://doi.org/10.1007/s10817-023-09684-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10817-023-09684-0

Keywords

Search

Navigation