Satisfiability Modulo Transcendental Functions via Incremental Linearization

  • Alessandro Cimatti
  • Alberto Griggio
  • Ahmed Irfan
  • Marco Roveri
  • Roberto Sebastiani
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10395)

Abstract

In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted in the abstract space, which is described in terms of the combined theory of linear arithmetic on the rationals with uninterpreted functions, and are incrementally axiomatized by means of upper- and lower-bounding piecewise-linear functions. Suitable numerical techniques are used to ensure that the abstractions of the transcendental functions are sound even in presence of irrationals. Our experimental evaluation on benchmarks from verification and mathematics demonstrates the potential of our approach, showing that it compares favorably with delta-satisfiability/interval propagation and methods based on theorem proving.

References

  1. 1.
    Akbarpour, B., Paulson, L.C.: MetiTarski: an automatic theorem prover for real-valued special functions. JAR 44(3), 175–205 (2010)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Allamigeon, X., Gaubert, S., Magron, V., Werner, B.: Certification of inequalities involving transcendental functions: combining SDP and max-plus approximation. In: 2013 European Control Conference (ECC), pp. 2244–2250. IEEE (2013)Google Scholar
  3. 3.
    Bak, S., Bogomolov, S., Johnson, T.T.: HYST: a source transformation and translation tool for hybrid automaton models. In: Proceedings of the 18th International Conference on Hybrid Systems: Computation and Control, pp. 128–133. ACM (2015)Google Scholar
  4. 4.
    Cavada, R., Cimatti, A., Dorigatti, M., Griggio, A., Mariotti, A., Micheli, A., Mover, S., Roveri, M., Tonetta, S.: The nuXmv symbolic model checker. In: Biere, A., Bloem, R. (eds.) CAV 2014. LNCS, vol. 8559, pp. 334–342. Springer, Cham (2014). doi:10.1007/978-3-319-08867-9_22 Google Scholar
  5. 5.
    Cimatti, A., Griggio, A., Irfan, A., Roveri, M., Sebastiani, R.: Invariant checking of NRA transition systems via incremental reduction to LRA with EUF. In: Legay and Margaria [16], pp. 58–75. https://es-static.fbk.eu/people/griggio/papers/tacas17.pdf
  6. 6.
    Cimatti, A., Griggio, A., Mover, S., Tonetta, S.: HyComp: an SMT-Based model checker for hybrid systems. In: Baier, C., Tinelli, C. (eds.) TACAS 2015. LNCS, vol. 9035, pp. 52–67. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46681-0_4 Google Scholar
  7. 7.
    Cimatti, A., Griggio, A., Schaafsma, B.J., Sebastiani, R.: The MathSAT5 SMT solver. In: Piterman, N., Smolka, S.A. (eds.) TACAS 2013. LNCS, vol. 7795, pp. 93–107. Springer, Heidelberg (2013). doi:10.1007/978-3-642-36742-7_7 CrossRefGoogle Scholar
  8. 8.
    Cimatti, A., Mover, S., Sessa, M.: From electrical switched networks to hybrid automata. In: Fitzgerald, J., Heitmeyer, C., Gnesi, S., Philippou, A. (eds.) FM 2016. LNCS, vol. 9995, pp. 164–181. Springer, Cham (2016). doi:10.1007/978-3-319-48989-6_11 CrossRefGoogle Scholar
  9. 9.
    de Dinechin, F., Lauter, C., Melquiond, G.: Certifying the floating-point implementation of an elementary function using Gappa. IEEE Trans. Comput. 60(2), 242–253 (2011)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Eggers, A., Kruglov, E., Kupferschmid, S., Scheibler, K., Teige, T., Weidenbach, C.: Superposition modulo non-linear arithmetic. In: Tinelli, C., Sofronie-Stokkermans, V. (eds.) FroCoS 2011. LNCS (LNAI), vol. 6989, pp. 119–134. Springer, Heidelberg (2011). doi:10.1007/978-3-642-24364-6_9 CrossRefGoogle Scholar
  11. 11.
    Fränzle, M., Herde, C., Teige, T., Ratschan, S., Schubert, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT 1(3–4), 209–236 (2007)MATHGoogle Scholar
  12. 12.
    Gao, S., Avigad, J., Clarke, E.M.: \(\delta \)-Complete decision procedures for satisfiability over the reals. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 286–300. Springer, Heidelberg (2012). doi:10.1007/978-3-642-31365-3_23 CrossRefGoogle Scholar
  13. 13.
    Gao, S., Kong, S., Clarke, E.M.: dReal: An SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS (LNAI), vol. 7898, pp. 208–214. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38574-2_14 CrossRefGoogle Scholar
  14. 14.
    Gario, M., Micheli, A.: PySMT: a solver-agnostic library for fast prototyping of SMT-based algorithms. In: SMT, pp. 373–384 (2015)Google Scholar
  15. 15.
    Hazewinkel, M.: Encyclopaedia of Mathematics: Stochastic Approximation Zygmund Class of Functions. Encyclopaedia of Mathematics. Springer, Netherlands (1993). https://books.google.it/books?id=1ttmCRCerVUC
  16. 16.
    Legay, A., Margaria, T. (eds.): TACAS 2017. LNCS, vol. 10205. Springer, Heidelberg (2017)Google Scholar
  17. 17.
    Magron, V.: NLCertify: a tool for formal nonlinear optimization. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 315–320. Springer, Heidelberg (2014). doi:10.1007/978-3-662-44199-2_49 Google Scholar
  18. 18.
    Martin-Dorel, É., Melquiond, G.: Proving tight bounds on univariate expressions with elementary functions in Coq. J. Autom. Reasoning 57(3), 187–217 (2016)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Melquiond, G.: Coq-interval (2011)Google Scholar
  20. 20.
    Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley-Interscience, New York (1988)CrossRefMATHGoogle Scholar
  21. 21.
    Nieven, I.: Numbers: Rational and Irrational. Mathematical Association of America (1961)Google Scholar
  22. 22.
    Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. TOCL 7(4), 723–748 (2006)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Roohi, N., Prabhakar, P., Viswanathan, M.: HARE: A hybrid abstraction refinement engine for verifying non-linear hybrid automata. In: Legay and Margaria [16], pp. 573–588Google Scholar
  24. 24.
    Scheibler, K., Kupferschmid, S., Becker, B.: Recent improvements in the SMT solver iSAT. MBMV 13, 231–241 (2013)Google Scholar
  25. 25.
    Solovyev, A., Hales, T.C.: Formal verification of nonlinear inequalities with taylor interval approximations. In: Brat, G., Rungta, N., Venet, A. (eds.) NFM 2013. LNCS, vol. 7871, pp. 383–397. Springer, Heidelberg (2013). doi:10.1007/978-3-642-38088-4_26 CrossRefGoogle Scholar
  26. 26.
    Townsend, E.: Functions of a Complex Variable. Read Books (2007)Google Scholar
  27. 27.
    Weidenbach, C., Dimova, D., Fietzke, A., Kumar, R., Suda, M., Wischnewski, P.: SPASS version 3.5. In: Schmidt, R.A. (ed.) CADE 2009. LNCS (LNAI), vol. 5663, pp. 140–145. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02959-2_10 CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Alessandro Cimatti
    • 1
  • Alberto Griggio
    • 1
  • Ahmed Irfan
    • 1
    • 2
  • Marco Roveri
    • 1
  • Roberto Sebastiani
    • 2
  1. 1.Fondazione Bruno KesslerTrentoItaly
  2. 2.DISI, University of TrentoTrentoItaly

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