Advertisement

On Gröbner Bases in the Context of Satisfiability-Modulo-Theories Solving over the Real Numbers

  • Sebastian Junges
  • Ulrich Loup
  • Florian Corzilius
  • Erika Ábrahám
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8080)

Abstract

We address satisfiability checking for the first-order theory of the real-closed field (RCF) using satisfiability-modulo-theories (SMT) solving. SMT solvers combine a SAT solver to resolve the Boolean structure of a given formula with theory solvers to verify the consistency of sets of theory constraints.

In this paper, we report on an integration of Gröbner bases as a theory solver so that it conforms with the requirements for efficient SMT solving: (1) it allows the incremental adding and removing of polynomials from the input set and (2) it can compute an inconsistent subset of the input constraints if the Gröbner basis contains 1.

We modify Buchberger’s algorithm by implementing a new update operator to optimize the Gröbner basis and provide two methods to handle inequalities. Our implementation uses special data structures tuned to be efficient for huge sets of sparse polynomials. Besides solving, the resulting module can be used to simplify constraints before being passed to other RCF theory solvers based on, e.g., the cylindrical algebraic decomposition.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Becker, T., Weispfenning, V., Kredel, H.: Gröbner bases: a computational approach to commutative algebra. Graduate texts in mathematics. Springer (1993)Google Scholar
  2. 2.
    Biere, A., Heule, M.J.H., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)Google Scholar
  3. 3.
    Buchberger, B.: Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. PhD thesis, University of Innsbruck (1965)Google Scholar
  4. 4.
    Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975)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.
    de Moura, L., Passmore, G.O.: On locally minimal Nullstellensatz proofs. In: Proc. of SMT 2009, pp. 35–42 (2009)Google Scholar
  7. 7.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulas over ordered fields. Journal of Symbolic Computation 24, 209–231 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dutertre, B., de Moura, L.: A fast linear-arithmetic solver for DPLL(T). In: Ball, T., Jones, R.B. (eds.) CAV 2006. LNCS, vol. 4144, pp. 81–94. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Gao, S., Ganai, M.K., Ivancic, F., Gupta, A., Sankaranarayanan, S., Clarke, E.M.: Integrating ICP and LRA solvers for deciding nonlinear real arithmetic problems. In: Proc. of FMCAD 2010, pp. 81–89. IEEE (2010)Google Scholar
  10. 10.
    Junges, S., Loup, U., Corzilius, F., Ábrahám, E.: On Gröbner bases in the context of satisfiability-modulo-theories solving over the real numbers. Technical Report AIB-2013-08, RWTH Aachen University (May 2013)Google Scholar
  11. 11.
    Passmore, G.O.: Combined Decision Procedures for Nonlinear Arithmetics, Real and Complex. PhD thesis, University of Edinburgh (2011)Google Scholar
  12. 12.
    Passmore, G.O., de Moura, L., Jackson, P.B.: Gröbner basis construction algorithms based on theorem proving saturation loops. In: Decision Procedures in Software, Hardware and Bioware. Dagstuhl Seminar Proc., vol. 10161 (2010)Google Scholar
  13. 13.
    Platzer, A., Quesel, J.D., Rümmer, P.: Real world verification. In: Schmidt, R.A. (ed.) CADE-22. LNCS, vol. 5663, pp. 485–501. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Roune, B.H., Stillman, M.: Practical Gröbner basis computation. In: Proc. of ISSAC 2012, pp. 203–210. ACM (2012)Google Scholar
  15. 15.
    Weispfenning, V.: Quantifier elimination for real algebra – the quadratic case and beyond. AAECC 8(2), 85–101 (1997)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Sebastian Junges
    • 1
  • Ulrich Loup
    • 1
  • Florian Corzilius
    • 1
  • Erika Ábrahám
    • 1
  1. 1.RWTH Aachen UniversityGermany

Personalised recommendations