Skip to main content
Book cover

International Conference on Computer Aided Verification

CAV 2014: Computer Aided Verification pp 729–736Cite as

A Nonlinear Real Arithmetic Fragment

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 8559)

Abstract

We present a new procedure for testing satisfiability (over the reals) of a conjunction of polynomial equations. There are three possible return values for our procedure: it either returns a model for the input formula, or it says that the input is unsatisfiable, or it fails because the applicability condition for the procedure, called the eigen-condition, is violated. For the class of constraints where the eigen-condition holds, our procedure is a decision procedure. We describe satisfiability-preserving transformations that can potentially convert problems into a form where eigen-condition holds. We experimentally evaluate the procedure and discuss applicability.

Keywords

  • Inference Rule
  • Decision Procedure
  • Real Eigenvalue
  • Input Formula
  • Cylindrical Algebraic Decomposition

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Supported in part by DARPA under contract FA8750-12-C-0284 and by the NSF grant SHF:CSR-1017483. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the funding agencies.

Download conference paper PDF

References

  1. Akbarpour, B., Paulson, L.C.: Metitarski: An automatic theorem prover for real-valued special functions. J. Autom. Reasoning 44(3), 175–205 (2010)

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Bachmair, L., Ganzinger, H.: Buchberger’s algorithm: A constraint-based completion procedure. In: Jouannaud, J.-P. (ed.) CCL 1994. LNCS, vol. 845, pp. 285–301. Springer, Heidelberg (1994)

    CrossRef  Google Scholar 

  3. Buchberger, B.: A critical-pair completion algorithm for finitely generated ideals in rings. In: Börger, E., Hasenjaeger, G., Rödding, D. (eds.) Rekursive Kombinatorik 1983. LNCS, vol. 171, pp. 137–161. Springer, Heidelberg (1983)

    Google Scholar 

  4. Cheng, C.-H., Ruess, H., Shankar, N.: Jbernstein: A validity checker for generalized polynomial constraints. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 656–661. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  5. Cheng, C.-H., Shankar, N., Ruess, H., Bensalem, S.: Efsmt: A logical framework for cyber-physical systems. CoRR, abs/1306.3456 (2013)

    Google Scholar 

  6. Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. CACM 5(7), 394–397 (1962)

    CrossRef  MATH  MathSciNet  Google Scholar 

  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(3-4), 209–236 (2007)

    Google Scholar 

  8. Ganzinger, H., Hagen, G., Nieuwenhuis, R., Oliveras, A., Tinelli, C.: DPLL(T): Fast decision procedures. In: Alur, R., Peled, D.A. (eds.) CAV 2004. LNCS, vol. 3114, pp. 175–188. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  9. Gao, S., Avigad, J., Clarke, E.M.: δ-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)

    CrossRef  Google Scholar 

  10. 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, vol. 7898, pp. 208–214. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  11. Gulwani, S., Tiwari, A.: Constraint-based approach for analysis of hybrid systems. In: Gupta, A., Malik, S. (eds.) CAV 2008. LNCS, vol. 5123, pp. 190–203. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  12. Iwane, H., Yanami, H., Anai, H.: An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for optimization problems. In: Proc. Intl. Workshop on Symb. Numeric Comp., SNC, pp. 168–177. ACM (2011)

    Google Scholar 

  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)

    CrossRef  Google Scholar 

  14. Leike, J.: Software and benchmarks for synthesis for polynomial lasso programs (2014), http://www.csl.sri.com/~tiwari/softwares/synthesis_for_polynomial_lasso_programs_source.zip

  15. Leike, J., Tiwari, A.: Synthesis for polynomial lasso programs. In: McMillan, K.L., Rival, X. (eds.) VMCAI 2014. LNCS, vol. 8318, pp. 434–452. Springer, Heidelberg (2014)

    CrossRef  Google Scholar 

  16. Loup, U., Scheibler, K., Corzilius, F., Ábrahám, E., Becker, B.: A symbiosis of interval constraint propagation and cylindrical algebraic decomposition. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 193–207. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  17. Matringe, N., Moura, A.V., Rebiha, R.: Generating invariants for non-linear hybrid systems by linear algebraic methods. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 373–389. Springer, Heidelberg (2010)

    CrossRef  Google Scholar 

  18. Microsoft Research. Z3: An efficient SMT solver, http://research.microsoft.com/projects/z3/

  19. Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Mathematical Programming Ser. B 96(2), 293–320 (2003)

    CrossRef  MATH  MathSciNet  Google Scholar 

  20. Rebiha, R., Matringe, N., Moura, A.V.: Transcendental inductive invariants generation for non-linear differential and hybrid systems. In: Proc. Hybrid Syst.: Comp. and Cntrl., HSCC, pp. 25–34. ACM (2012)

    Google Scholar 

  21. Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 539–554. Springer, Heidelberg (2004)

    Google Scholar 

  22. Sankaranarayanan, S., Sipma, H.B., Manna, Z.: Constructing invariants for hybrid systems. Formal Methods in System Design 32(1), 25–55 (2008)

    CrossRef  MATH  Google Scholar 

  23. Srivastava, S., Gulwani, S., Foster, J.S.: Template-based program verification and program synthesis. STTT 15(5-6), 497–518 (2013)

    CrossRef  Google Scholar 

  24. Stengle, G.: A Nullstellensatz and a Positivstellensatz in semialgebraic geometry. Math. Ann. 207 (1974)

    Google Scholar 

  25. Sturm, T., Tiwari, A.: Verification and synthesis using real quantifier elimination. In: Proc. Intl. Symp. on Symb. and Alg. Comp., ISSAC, pp. 329–336. ACM (2011)

    Google Scholar 

  26. Tiwari, A.: An algebraic approach for the unsatisfiability of nonlinear constraints. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 248–262. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  27. Tiwari, A., Khanna, G.: Nonlinear Systems: Approximating reach sets. In: Alur, R., Pappas, G.J. (eds.) HSCC 2004. LNCS, vol. 2993, pp. 600–614. Springer, Heidelberg (2004)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SRI International, Menlo Park, CA, USA

    Ashish Tiwari & Patrick Lincoln

Authors
  1. Ashish Tiwari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Patrick Lincoln
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Johannes Kepler University Linz, Altenberger Str. 69, 4040, Linz, Austria

    Armin Biere

  2. IAIK, Graz University of Technology, Inffeldgasse 16a, 8010, Graz, Austria

    Roderick Bloem

Rights and permissions

Reprints and Permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Tiwari, A., Lincoln, P. (2014). A Nonlinear Real Arithmetic Fragment. In: Biere, A., Bloem, R. (eds) Computer Aided Verification. CAV 2014. Lecture Notes in Computer Science, vol 8559. Springer, Cham. https://doi.org/10.1007/978-3-319-08867-9_48

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08867-9_48

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08866-2

  • Online ISBN: 978-3-319-08867-9

  • eBook Packages: Computer ScienceComputer Science (R0)