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Extending SMT Solvers to Higher-Order Logic

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Automated Deduction – CADE 27 (CADE 2019)

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Abstract

SMT solvers have throughout the years been able to cope with increasingly expressive formulas, from ground logics to full first-order logic (FOL). In contrast, the extension of SMT solvers to higher-order logic (HOL) is mostly unexplored. We propose a pragmatic extension for SMT solvers to support HOL reasoning natively without compromising performance on FOL reasoning, thus leveraging the extensive research and implementation efforts dedicated to efficient SMT solving. We show how to generalize data structures and the ground decision procedure to support partial applications and extensionality, as well as how to reconcile quantifier instantiation techniques with higher-order variables. We also discuss a separate approach for redesigning an HOL SMT solver from the ground up via new data structures and algorithms. We apply our pragmatic extension to the CVC4 SMT solver and discuss a redesign of the veriT SMT solver. Our evaluation shows they are competitive with state-of-the-art HOL provers and often outperform the traditional encoding into FOL.

This work was partially supported by the National Science Foundation under award 1656926.

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Notes

  1. 1.

    http://matryoshka.gforge.inria.fr/pubs/hosmt/.

  2. 2.

    Since veriT does not parse TPTP, its reported results are on the equivalent benchmarks as translated by CVC4 into the HOSMT language [3].

References

  1. Andrews, P.B.: Resolution in type theory. J. Symb. Log. 36(3), 414–432 (1971)

    Article  MathSciNet  Google Scholar 

  2. Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. Log. Comput. 4(3), 217–247 (1994)

    Article  MathSciNet  Google Scholar 

  3. Barbosa, H., Blanchette, J.C., Cruanes, S., El Ouraoui, D., Fontaine, P.: Language and proofs for higher-order SMT (work in progress). In: Dubois, C., Paleo, B.W. (eds.) PXTP 2017. EPTCS, vol. 262, pp. 15–22 (2017)

    Google Scholar 

  4. Barbosa, H., Fontaine, P., Reynolds, A.: Congruence closure with free variables. In: Legay, A., Margaria, T. (eds.) TACAS 2017. LNCS, vol. 10206, pp. 214–230. Springer, Heidelberg (2017). https://doi.org/10.1007/978-3-662-54580-5_13

    Chapter  Google Scholar 

  5. Barbosa, H., Reynolds, A., El Ouraoui, D., Tinelli, C., Barrett, C.: Extending SMT solvers to higher-order logic. Technical report. The University of Iowa, May 2019

    Google Scholar 

  6. Barrett, C., et al.: CVC4. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 171–177. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-22110-1_14

    Chapter  Google Scholar 

  7. Barrett, C., Fontaine, P., Tinelli, C.: The SMT-LIB standard: version 2.6. Technical report. Department of Computer Science, The University of Iowa (2017)

    Google Scholar 

  8. Barrett, C., Sebastiani, R., Seshia, S., Tinelli, C.: Satisfiability modulo theories, Chap. 26. In: Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability. FAIA, vol. 185, pp. 825–885. IOS Press (2009)

    Google Scholar 

  9. Bentkamp, A., Blanchette, J.C., Cruanes, S., Waldmann, U.: Superposition for lambda-free higher-order logic. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS, vol. 10900, pp. 28–46. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_3

    Chapter  Google Scholar 

  10. Benzmüller, C., Miller, D.: Automation of higher-order logic. In: Siekmann, J.H. (ed.) Computational Logic. Handbook of the History of Logic, vol. 9, pp. 215–254. Elsevier (2014)

    Google Scholar 

  11. Benzmüller, C., Sultana, N., Paulson, L.C., Theiss, F.: The higher-order prover LEO-II. J. Autom. Reason. 55, 389–404 (2015)

    Article  MathSciNet  Google Scholar 

  12. Bhayat, A., Reger, G.: Set of support for higher-order reasoning. In: Konev, B., Urban, J., Rümmer, P. (eds.) PAAR-2018. CEUR Workshop Proceedings, vol. 2162, pp. 2–16. CEUR-WS.org (2018)

    Google Scholar 

  13. Blanchette, J.C.: Automatic proofs and refutations for higher-order logic. Ph.D. thesis. Technical University Munich (2012)

    Google Scholar 

  14. Blanchette, J.C., Kaliszyk, C., Paulson, L.C., Urban, J.: Hammering towards QED. J. Formaliz. Reason. 9(1), 101–148 (2016)

    MathSciNet  MATH  Google Scholar 

  15. Böhme, S., Nipkow, T.: Sledgehammer: judgement day. In: Giesl, J., Hähnle, R. (eds.) IJCAR 2010. LNCS, vol. 6173, pp. 107–121. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-14203-1_9

    Chapter  Google Scholar 

  16. Bouton, T., Caminha B. de Oliveira, D., Déharbe, D., Fontaine, P.: veriT: an open, trustable and efficient SMT-solver. In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 151–156. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-02959-2_12

    Chapter  Google Scholar 

  17. Brown, C.E.: Satallax: an automatic higher-order prover. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 111–117. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31365-3_11

    Chapter  Google Scholar 

  18. Cruanes, S.: Superposition with structural induction. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS, vol. 10483, pp. 172–188. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66167-4_10

    Chapter  Google Scholar 

  19. Czajka, Ł., Kaliszyk, C.: Hammer for Coq: automation for dependent type theory. J. Autom. Reason. 61, 423–453 (2018)

    Article  MathSciNet  Google Scholar 

  20. de Moura, L., Bjørner, N.: Efficient E-matching for SMT solvers. In: Pfenning, F. (ed.) CADE 2007. LNCS, vol. 4603, pp. 183–198. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-73595-3_13

    Chapter  Google Scholar 

  21. de Moura, L., Bjørner, N.: Generalized, efficient array decision procedures. In: FMCAD 2009, pp. 45–52. IEEE (2009)

    Google Scholar 

  22. Detlefs, D., Nelson, G., Saxe, J.B.: Simplify: a theorem prover for program checking. J. ACM 52, 365–473 (2005)

    Article  MathSciNet  Google Scholar 

  23. Dowek, G.: Higher-order unification and matching. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. II, pp. 1009–1062. Elsevier and MIT Press (2001)

    Google Scholar 

  24. Downey, P.J., Sethi, R., Tarjan, R.E.: Variations on the common subexpression problem. J. ACM 27, 758–771 (1980)

    Article  MathSciNet  Google Scholar 

  25. Färber, M., Brown, C.: Internal guidance for Satallax. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016. LNCS, vol. 9706, pp. 349–361. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-40229-1_24

    Chapter  Google Scholar 

  26. 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). https://doi.org/10.1007/978-3-642-02658-4_25

    Chapter  Google Scholar 

  27. Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15(2), 81–91 (1950)

    Article  MathSciNet  Google Scholar 

  28. Hughes, R.J.M.: Super combinators: a new implementation method for applicative languages. In: Symposium on LISP and Functional Programming, pp. 1–10 (1982)

    Google Scholar 

  29. Kohlhase, M.: Higher-order tableaux. In: Baumgartner, P., Hähnle, R., Possega, J. (eds.) TABLEAUX 1995. LNCS, vol. 918, pp. 294–309. Springer, Heidelberg (1995). https://doi.org/10.1007/3-540-59338-1_43

    Chapter  Google Scholar 

  30. Meng, J., Paulson, L.C.: Translating higher-order clauses to first-order clauses. J. Autom. Reason. 40(1), 35–60 (2008)

    Article  MathSciNet  Google Scholar 

  31. Nelson, G., Oppen, D.C.: Fast decision procedures based on congruence closure. J. ACM 27, 356–364 (1980)

    Article  MathSciNet  Google Scholar 

  32. Nieuwenhuis, R., Oliveras, A.: Fast congruence closure and extensions. Inf. Comput. IC 2005(4), 557–580 (2007)

    Article  MathSciNet  Google Scholar 

  33. Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 1, pp. 371–443. Elsevier Science (2001)

    Google Scholar 

  34. Nipkow, T., Wenzel, M., Paulson, L.C.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45949-9

    Book  MATH  Google Scholar 

  35. Noshita, K.: Translation of Turner combinators in O(n log n) space. IPL 20, 71–74 (1985)

    Article  MathSciNet  Google Scholar 

  36. Paulson, L.C., Blanchette, J.C.: Three years of experience with Sledgehammer, a practical link between automatic and interactive theorem provers. In: Sutcliffe, G., Schulz, S., Ternovska, E. (eds.) IWIL-2010. EPiC, vol. 2, pages 1–11. EasyChair (2012)

    Google Scholar 

  37. Reynolds, A., Barbosa, H., Fontaine, P.: Revisiting enumerative instantiation. In: Beyer, D., Huisman, M. (eds.) TACAS 2018. LNCS, vol. 10806, pp. 112–131. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-89963-3_7

    Chapter  Google Scholar 

  38. Reynolds, A., Tinelli, C., de Moura, L.: Finding conflicting instances of quantified formulas in SMT. In: FMCAD 2014, pp. 195–202. IEEE (2014)

    Google Scholar 

  39. 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). https://doi.org/10.1007/978-3-642-39799-8_42

    Chapter  Google Scholar 

  40. Reynolds, A., Tinelli, C., Goel, A., Krstić, S., Deters, M., Barrett, C.: Quantifier instantiation techniques for finite model finding in SMT. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 377–391. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38574-2_26

    Chapter  Google Scholar 

  41. Robinson, J.A.: Mechanizing higher order logic. Mach. Intell. 4, 151–170 (1969)

    MathSciNet  MATH  Google Scholar 

  42. Schulz, S.: E - a brainiac theorem prover. AI Commun. 15, 111–126 (2002)

    MATH  Google Scholar 

  43. Steen, A., Benzmüller, C.: The higher-order prover Leo-III. In: Galmiche, D., Schulz, S., Sebastiani, R. (eds.) IJCAR 2018. LNCS, vol. 10900, pp. 108–116. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-94205-6_8

    Chapter  Google Scholar 

  44. Stump, A., Barrett, C.W., Dill, D.L., Levitt, J.R.: A decision procedure for an extensional theory of arrays. In: LICS 2001, pp. 29–37. IEEE Computer Society (2001)

    Google Scholar 

  45. Sultana, N., Blanchette, J.C., Paulson, L.C.: LEO-II and Satallax on the Sledgehammer test bench. J. Appl. Log. 11, 91–102 (2013)

    Article  MathSciNet  Google Scholar 

  46. Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reason. 43, 337–362 (2009)

    Article  Google Scholar 

  47. Sutcliffe, G.: The CADE ATP system competition - CASC. AI Mag. 37, 99–101 (2016)

    MATH  Google Scholar 

  48. Vukmirović, P., Blanchette, J.C., Cruanes, S., Schulz, S.: Extending a brainiac prover to lambda-free higher-order logic. In: Vojnar, T., Zhang, L. (eds.) TACAS 2019. LNCS, vol. 11427, pp. 192–210. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17462-0_11

    Chapter  Google Scholar 

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Acknowledgments

We are grateful to Jasmin Blanchette and Pascal Fontaine for numerous discussions throughout the development of this work, for providing funding for research visits and for suggesting many improvements. We also thank Jasmin for generating several of the benchmarks with which we evaluate our approach; Simon Cruanes and Martin Riener for many fruitful discussions on the intricacies of HOL; Andres Nötzli for help with the table and plot scripts; Mathias Fleury, Hans-Jörg Schurr and Sophie Tourret for suggesting many improvements. This work was partially supported by the National Science Foundation under Award 1656926 and the European Research Council (ERC) under starting grant Matryoshka (713999).

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Authors and Affiliations

  1. The University of Iowa, Iowa City, USA

    Haniel Barbosa, Andrew Reynolds & Cesare Tinelli

  2. University of Lorraine, CNRS, Inria, and LORIA, Nancy, France

    Daniel El Ouraoui

  3. Stanford University, Stanford, USA

    Clark Barrett

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  1. Haniel Barbosa
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Correspondence to Daniel El Ouraoui .

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  1. University of Lorraine, Villers-lès-Nancy, France

    Pascal Fontaine

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Barbosa, H., Reynolds, A., El Ouraoui, D., Tinelli, C., Barrett, C. (2019). Extending SMT Solvers to Higher-Order Logic. In: Fontaine, P. (eds) Automated Deduction – CADE 27. CADE 2019. Lecture Notes in Computer Science(), vol 11716. Springer, Cham. https://doi.org/10.1007/978-3-030-29436-6_3

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