raSAT: An SMT Solver for Polynomial Constraints

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9706)

Abstract

This paper presents the raSAT SMT solver for polynomial constraints, which aims to handle them over both reals and integers with simple unified methodologies: (1) raSAT loop for inequalities, which extends the interval constraint propagation with testing to accelerate SAT detection, and (2) a non-constructive reasoning for equations over reals, based on the generalized intermediate value theorem.

Notes

Acknowledgements

The authors would like to thank Pascal Fontain and Nobert Müller for valuable comments and kind instructions on veriT and iRRAM. They also thank to anonymous referees for fruitful suggestions. This work is supported by JSPS KAKENHI Grant-in-Aid for Scientific Research(B) (23300005,15H02684).

References

  1. 1.
    Alliot, J.M., Gotteland, J.B., Vanaret, C., Durand, N., Gianazza, D.: Implementing an interval computation library for OCaml on x86/amd64 architectures. In: ICFP. ACM (2012)Google Scholar
  2. 2.
    Benhamou, F., Granvilliers, L.: Continuous and interval constraints. In: van Beek, P., Rossi, F., Walsh, T. (eds.) Handbook of Constraint Programming, pp. 571–604. Elsevier, Amsterdam (2006)CrossRefGoogle Scholar
  3. 3.
    Bofill, M., Nieuwenhuis, R., Oliveras, A., Rodríguez-Carbonell, E., Rubio, A.: The barcelogic SMT solver. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 294–298. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Comba, J.L.D., Stolfi, J.: Affine arithmetic and its applications to computer graphics. In: SIBGRAPI 1993, pp. 9–18 (1993)Google Scholar
  5. 5.
    Corzilius, F., Loup, U., Junges, S., Ábrahám, E.: SMT-RAT: an SMT-compliant nonlinear real arithmetic toolbox. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 442–448. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  6. 6.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  7. 7.
    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, 209–236 (2007)MATHGoogle Scholar
  8. 8.
    Ganai, M., Ivancic, F.: Efficient decision procedure for non-linear arithmetic constraints using cordic. In: FMCAD 2009, pp. 61–68, November 2009Google Scholar
  9. 9.
    Gao, S., Kong, S., Clarke, E.M.: Satisfiability modulo odes. In: FMCAD 2013, pp. 105–112, October 2013Google Scholar
  10. 10.
    Gao, S., Kong, S., Clarke, E.M.: \({\sf dReal}\): an SMT solver for nonlinear theories over the reals. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 208–214. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  11. 11.
    Granvilliers, L., Benhamou, F.: Realpaver: an interval solver using constraint satisfaction techniques. ACM Trans. Math. Softw. 32, 138–156 (2006)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Hickey, T., Ju, Q., Van Emden, M.H.: Interval arithmetic: from principles to implementation. J. ACM 48(5), 1038–1068 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Jovanović, D., de Moura, L.: Solving non-linear arithmetic. In: Gramlich, B., Miller, D., Sattler, U. (eds.) IJCAR 2012. LNCS, vol. 7364, pp. 339–354. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Khanh, T.V., Ogawa, M.: SMT for polynomial constraints on real numbers. In: TAPAS 2012. ENTCS, vol. 289, pp. 27–40 (2012)Google Scholar
  15. 15.
    Messine, F.: Extentions of affine arithmetic: application to unconstrained global optimization. J. UCS 8(11), 992–1015 (2002)MathSciNetMATHGoogle Scholar
  16. 16.
    Moore, R.: Interval Analysis. Prentice-Hall Series in Automatic Computation. Prentice-Hall, Upper Saddle River (1966)MATHGoogle Scholar
  17. 17.
    Neumaier, A.: Interval Methods for Systems of Equations. Cambridge Middle East Library. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  18. 18.
    Passmore, G.O.: Combined decision procedures for nonlinear arithmetics, real and complex. Dissertation, School of Informatics, University of Edinburgh (2011)Google Scholar
  19. 19.
    Passmore, G.O., Jackson, P.B.: Combined decision techniques for the existential theory of the reals. In: Carette, J., Dixon, L., Coen, C.S., Watt, S.M. (eds.) MKM 2009, Held as Part of CICM 2009. LNCS, vol. 5625, pp. 122–137. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  20. 20.
    Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Comput. Logic 7(4), 723–748 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Tseitin, G.: On the complexity of derivation in propositional calculus. In: Siekmann, J.H., Wrightson, G. (eds.) Automation of Reasoning. Symbolic Computation, pp. 466–483. Springer, Heidelberg (1983)CrossRefGoogle Scholar
  22. 22.
    Tung, V.X., Khanh, T.V., Ogawa, M.: raSAT: SMT for polynomial inequality. In: SMT Workshop 2014, p. 67 (2014)Google Scholar
  23. 23.
    Zankl, H., Middeldorp, A.: Satisfiability of non-linear irrational arithmetic. In: Clarke, E.M., Voronkov, A. (eds.) LPAR-16 2010. LNCS, vol. 6355, pp. 481–500. Springer, Heidelberg (2010)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Japan Advanced Institute of Science and TechnologyNomiJapan
  2. 2.University of Engineering and TechnologyVietnam National UniversityHanoiVietnam

Personalised recommendations