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A CDCL-Style Calculus for Solving Non-linear Constraints

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Frontiers of Combining Systems (FroCoS 2019)

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In this paper we propose a novel approach for checking satisfiability of non-linear constraints over the reals, called ksmt. The procedure is based on conflict resolution in CDCL-style calculus, using a composition of symbolical and numerical methods. To deal with the non-linear components in case of conflicts we use numerically constructed restricted linearisations. This approach covers a large number of computable non-linear real functions such as polynomials, rational or trigonometrical functions and beyond. A prototypical implementation has been evaluated on several non-linear SMT-LIB examples and the results have been compared with state-of-the-art SMT solvers.

The research leading to these results has received funding from the DFG grant WERA MU 1801/5-1 and the DFG/RFBR grant CAVER BE 1267/14-1 and 14-01-91334. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 731143.

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    ksmt-0.1.3, cvc4-1.6+gmp, z3-4.7.1+gmp, mathsat-5.5.2, yices-2.6+lpoly-1.7, dreal-v3.16.08.01, rasat-0.3.


  1. Benhamou, F., Granvilliers, L.: Continuous and interval constraints. In: Handbook of Constraint Programming, pp. 571–603. Elsevier (2006)

    Google Scholar 

  2. Brattka, V., Hertling, P., Weihrauch, K.: A tutorial on computable analysis. In: Cooper, S.B., Löwe, B., Sorbi, A. (eds.) New Computational Paradigms, pp. 425–491. Springer, Heidelberg (2008).

    Chapter  Google Scholar 

  3. Buchberger, B.: A theoretical basis for the reduction of polynomials to canonical forms. ACM SIGSAM Bull. 10(3), 19–29 (1976)

    Article  MathSciNet  Google Scholar 

  4. Cimatti, A., Griggio, A., Irfan, A., Roveri, M., Sebastiani, R.: Incremental linearization for satisfiability and verification modulo nonlinear arithmetic and transcendental functions. ACM Trans. Comput. Log. 19(3), 19:1–19:52 (2018)

    Article  MathSciNet  Google Scholar 

  5. Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975).

    Chapter  Google Scholar 

  6. de Moura, L., Jovanović, D.: A model-constructing satisfiability calculus. In: Giacobazzi, R., Berdine, J., Mastroeni, I. (eds.) VMCAI 2013. LNCS, vol. 7737, pp. 1–12. Springer, Heidelberg (2013).

    Chapter  Google Scholar 

  7. Dolzmann, A., Sturm, T.: REDLOG: computer algebra meets computer logic. ACM SIGSAM Bull. 31(2), 2–9 (1997)

    Article  Google Scholar 

  8. Dragan, I., Korovin, K., Kovács, L., Voronkov, A.: Bound propagation for arithmetic reasoning in Vampire. In: Proceedings SYNASC 2013, pp. 169–176. IEEE (2013)

    Google Scholar 

  9. Fontaine, P., Ogawa, M., Sturm, T., To, V.K., Vu, X.T.: Wrapping computer algebra is surprisingly successful for non-linear SMT. In: SC-Square 2018, Oxford, United Kingdom, July 2018

    Google Scholar 

  10. 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)

    MATH  Google Scholar 

  11. 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 (LNAI), vol. 7364, pp. 286–300. Springer, Heidelberg (2012).

    Chapter  Google Scholar 

  12. Hladík, M., Ratschan, S.: Efficient solution of a class of quantified constraints with quantifier prefix Exists-Forall. Math. Comput. Sci. 8(3–4), 329–340 (2014)

    Article  MathSciNet  Google Scholar 

  13. Jovanović, D., de Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS (LNAI), vol. 7364, pp. 339–354. Springer, Heidelberg (2012).

    Chapter  Google Scholar 

  14. Kapur, D., Sun, Y., Wang, D.: A new algorithm for computing comprehensive Gröbner systems. In: Proceedings ISSAC 2010, pp. 29–36. ACM, New York, USA (2010)

    Google Scholar 

  15. Korovin, K., Kos̆ta, M., Sturm, T.: Towards conflict-driven learning for virtual substitution. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 256–270. Springer, Cham (2014).

    Chapter  Google Scholar 

  16. Korovin, K., Tsiskaridze, N., Voronkov, A.: Implementing conflict resolution. In: Clarke, E., Virbitskaite, I., Voronkov, A. (eds.) PSI 2011. LNCS, vol. 7162, pp. 362–376. Springer, Heidelberg (2012).

    Chapter  Google Scholar 

  17. Korovin, K., Tsiskaridze, N., Voronkov, A.: Conflict resolution. In: Gent, I.P. (ed.) CP 2009. LNCS, vol. 5732, pp. 509–523. Springer, Heidelberg (2009).

    Chapter  Google Scholar 

  18. Korovin, K., Voronkov, A.: Solving systems of linear inequalities by bound propagation. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 369–383. Springer, Heidelberg (2011).

    Chapter  Google Scholar 

  19. Lefévre, V.: Moyens arithmetiques pour un calcul fiable. PhD thesis, École normale supérieure de Lyon (2000)

    Google Scholar 

  20. Loos, R., Weispfenning, V.: Applying linear quantifier elimination. Comput. J. 36(5), 450–462 (1993)

    Article  MathSciNet  Google Scholar 

  21. Maréchal, A., Fouilhé, A., King, T., Monniaux, D., Périn, M.: Polyhedral approximation of multivariate polynomials using Handelman’s theorem. In: Jobstmann, B., Leino, K.R.M. (eds.) VMCAI 2016. LNCS, vol. 9583, pp. 166–184. Springer, Heidelberg (2016).

    Chapter  Google Scholar 

  22. Marques-Silva, J.P., Sakallah, K.A.: GRASP: a search algorithm for propositional satisfiability. IEEE Trans. Comput. 48(5), 506–521 (1999)

    Article  MathSciNet  Google Scholar 

  23. McMillan, K.L., Kuehlmann, A., Sagiv, M.: Generalizing DPLL to richer logics. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 462–476. Springer, Heidelberg (2009).

    Chapter  Google Scholar 

  24. Müller, N.T.: The iRRAM: exact arithmetic in C++. In: Blanck, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 222–252. Springer, Heidelberg (2001).

    Chapter  Google Scholar 

  25. Nieuwenhuis, R., Oliveras, A., Tinelli, C.: Solving SAT and SAT modulo theories: from an abstract Davis-Putnam-Logemann-Loveland procedure to DPLL(T). J. ACM 53(6), 937–977 (2006)

    Article  MathSciNet  Google Scholar 

  26. Niven, I.: Irrational Numbers. Mathematical Association of America, Washington, D.C. (1956)

    MATH  Google Scholar 

  27. Passmore, G.O., Paulson, L.C., de Moura, L.: Real algebraic strategies for MetiTarski proofs. In: Jeuring, J., et al. (eds.) CICM 2012. LNCS (LNAI), vol. 7362, pp. 358–370. Springer, Heidelberg (2012).

    Chapter  Google Scholar 

  28. Reger, G., Bjorner, N., Suda, M., Voronkov, A.: AVATAR modulo theories. In: Benzmüller, C., Sutcliffe, G., Rojas, R. (eds.), 2nd Global Conference on Artificial Intelligence, EPiC Series in Computing, vol. 41, pp. 39–52. EasyChair (2016)

    Google Scholar 

  29. Reynolds, A., Tinelli, C., Jovanović, D., Barrett, C.: Designing theory solvers with extensions. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS (LNAI), vol. 10483, pp. 22–40. Springer, Cham (2017).

    Chapter  Google Scholar 

  30. Richardson, D.: Some undecidable problems involving elementary functions of a real variable. J. Symb. Log. 33(4), 514–520 (1968)

    Article  MathSciNet  Google Scholar 

  31. Tarski, A.: A decision method for elementary algebra and geometry. In: 2nd edn. University of California (1951)

    Google Scholar 

  32. Weihrauch, K.: Computable Analysis: An Introduction. Springer, Secaucus (2000).

    Book  MATH  Google Scholar 

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We thank the anonymous reviewers and Stefan Ratschan for their helpful comments.

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Correspondence to Franz Brauße .

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Brauße, F., Korovin, K., Korovina, M., Müller, N. (2019). A CDCL-Style Calculus for Solving Non-linear Constraints. In: Herzig, A., Popescu, A. (eds) Frontiers of Combining Systems. FroCoS 2019. Lecture Notes in Computer Science(), vol 11715. Springer, Cham.

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