Skip to main content
Log in

A search-based procedure for nonlinear real arithmetic

Formal Methods in System Design Aims and scope Submit manuscript

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 on constraints generated by template-based verification and synthesis procedures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. Akbarpour B, Paulson LC (2010) MetiTarski: An automatic theorem prover for real-valued special functions. J Autom Reason 44(3):175–205

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachmair L, Ganzinger H (1994) Buchberger’s algorithm: a constraint-based completion procedure. In: Proceedings of 1st international conference constraints in computational logic, LNCS 845, Springer, pp 285–301

  3. Basu S, Pollack R, Roy MF (1996) On the combinatorial and algebraic complexity of quantifier elimination. J ACM 43(6):1002–1045

    Article  MathSciNet  MATH  Google Scholar 

  4. Bochnak J, Coste M, Roy MF (1998) Real algebraic geometry. Springer, New York

    Book  MATH  Google Scholar 

  5. Buchberger B (1983) A critical-pair completion algorithm for finitely generated ideals in rings. In: Proceedings of logic and machines: decision problems and complexity, LNCS, vol. 171, pp 137–161

  6. Cheng CH, Ruess H, Shankar N (2013) JBernstein: a validity checker for generalized polynomial constraints. In: Proceedings of 25th international conference computer aided verification, CAV, vol. LNCS 8044, Springer, pp 656–661

  7. Cheng CH, Shankar N, Ruess H, Bensalem S (2013) EFSMT: a logical framework for cyber-physical systems. CoRR abs/1306.3456

  8. Collins GE (1975) Quantifier elimination for the elementary theory of real closed fields by cylindrical algebraic decomposition. In: Proceedings of 2nd GI Conference on Automata Theory and Formal Languages, LNCS, Springer, vol. 33, pp 134–183

  9. Collins GE, Hong H (1991) Partial cylindrical algebraic decomposition for quantifier elimination. J Symb Comput 12(3):299–328

    Article  MathSciNet  MATH  Google Scholar 

  10. Colon M, Sankaranarayanan S, Sipma H (2003) Linear invariant generation using non-linear constraint solving. In: Computer-Aided Verification (CAV), LNCS, Springer-Verlag, vol. 2725, pp 420–433

  11. Davenport JH, Heintz J (1988) Real quantifier elimination is doubly exponential. J Symb Comput 5(1–2):29–35

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Demmel J, Dumitriu I, Holtz O (2007) Fast linear algebra is stable. Numer Math 108:59–91

    Article  MathSciNet  MATH  Google Scholar 

  14. Fränzle M, Herde C, Teige T, Ratschan S, Schubert T (2007) Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. JSAT 1(3–4):209–236

    MATH  Google Scholar 

  15. Ganzinger H, Hagen G, Nieuwenhuis R, Oliveras A, Tinelli C (2004) DPLL(T): fast decision procedures. In: Proceedings of 16th International Conference on Computer Aided Verification, CAV 2004, LNCS, Springer, vol 3114, pp 175–188

  16. Gao S, Avigad J, Clarke EM (2012) \(\delta \)-complete decisionprocedures for satisfiability over the reals. In: Proceedings 6th International Conference Automated Reasoning, IJCAR, LNCS 7364, pp 286–300

  17. Gao S, Kong S, Clarke EM (2013) dReal: An SMT solver for nonlinear theories over the reals. In: Proceedings of 24th International Conference on Automated Deduction, CADE, LNCS 7898, Springer, pp 208–214

  18. Grigoriev D (1988) Complexity of deciding Tarski algebra. J Symb Comput 5(1–2):65–108

    Article  MathSciNet  MATH  Google Scholar 

  19. Gulwani S, Jha S, Tiwari A, Venkatesan R (2011) Synthesis of loop-free programs. In: Proceedings of ACM Conference on Prgramming Language Designing and Implementation of PLDI, pp 62–73

  20. Gulwani S, Srivastava S, Venkatesan R (2008) Program analysis as constraint solving. In: Proceedings of ACM Conference on Prgramming Language Design and implementaion, PLDI, pp 281–292

  21. Gulwani S, Tiwari A (2008) Constraint-based approach for analysis of hybrid systems. In: Proceedings of 20th International Conference on Computer Aided Verification, CAV 2008, LNCS, vol. 5123, pp 190–203. Springer. July 7–14, 2008, Princeton, NJ

  22. Hong H (1990) An improvement of the projection operator in cylindrical algebraic decomposition. In: Proceedings of ISAAC 90, pp 261–264

  23. Hong H, Safey El Din M (2009) Variant real quantifier elimination: algorithm and application. In: ISSAC, pp 183–190. ACM. doi:10.1145/1576702.1576729

  24. Iwane H, Yanami H, Anai H (2011) An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for optimization problems. In: Proceedings of International Workshop on Symbolic Numeric Computation, SNC, pp 168–177. ACM

  25. Jovanovic D, de Moura LM (2012) Solving non-linear arithmetic. In: Proceedings of 6th International Conference Automated Reasoning, IJCAR, LNCS 7364, Springer, pp 339–354

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

  27. Leike J, Tiwari A (2014) Synthesis for polynomial lasso programs. In: Proceedings of VMCAI, LNCS 8318, pp 454–472

  28. Loup U, Scheibler K, Corzilius F, Ábrahám E, Becker B (2013) A symbiosis of interval constraint propagation and cylindrical algebraic decomposition. In: Proceedings of 24th International Conference on Automated Deduction, CADE, LNCS 7898, Springer, pp 193–207

  29. Matringe N, Moura AV, Rebiha R (2010) Generating invariants for non-linear hybrid systems by linear algebraic methods. In: Proceedings of 17th International Static Analysis Symposium, SAS, Springer, vol. LNCS 6337, pp 373–389

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

  31. Parrilo PA (2003) Semidefinite programming relaxations for semialgebraic problems. Math Program Ser B 96(2):293–320

    Article  MathSciNet  MATH  Google Scholar 

  32. Rebiha R, Matringe N, Moura AV (2012) Transcendental inductive invariants generation for non-linear differential and hybrid systems. In: Proceedings of Hybrid System: Computation and Control, HSCC, pp 25–34. ACM

  33. Renegar J (1992) On the computational complexity and geometry of the first-order theory of the reals. J Symb Comput 13(3):255–352

    Article  MathSciNet  MATH  Google Scholar 

  34. Sankaranarayanan S, Sipma H, Manna Z (2004) Constructing invariants for hybrid systems. In: R. Alur, G.J. Pappas (eds.) Hybrid systems: computation and control, HSCC 2004, Lecture Notes in Computer Science, vol. 2993, pp 539–554. Springer

  35. Sankaranarayanan S, Sipma HB, Manna Z (2008) Constructing invariants for hybrid systems. Formal Methods Syst Des 32(1):25–55

    Article  MATH  Google Scholar 

  36. SRI International: Yices: An SMT solver. http://yices.csl.sri.com/

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

    Article  Google Scholar 

  38. Stengle G (1974) A Nullstellensatz and a Positivstellensatz insemialgebraic geometry. Math Ann 207

  39. Sturm T, Tiwari A (2011) Verification and synthesis using real quantifier elimination. In: Proceedings of International Symposium on Symbolic and Algebraic Computation , ISSAC, pp 329–336. ACM

  40. Taly A, Gulwani S, Tiwari A (2009) Synthesizing switching logic using constraint solving. In: Proc. 10th Intl. Conf. on Verification, Model Checking and Abstract Interpretation, VMCAI, LNCS, Springer, vol. 5403, pp 305–319

  41. Tarski A (1948) A decision method for elementary algebra and geometry, 2nd edn. University of California Press, Berkeley

    MATH  Google Scholar 

  42. Tiwari A (2005) An algebraic approach for the unsatisfiability of nonlinear constraints. In: Computer Science Logic, 14th Annual Conference, CSL 2005, LNCS, Springer, vol. 3634, pp 248–262

  43. Tiwari A, Khanna G (2004) Nonlinear Systems: Approximating reach sets. In: Proceedings of 7th International Workshop on Hybrid Systems: Computation and Control, HSCC 2004, LNCS, Springer, vol. 2993, pp 600–614

  44. Tiwari A, Lincoln P (2014) A nonlinear real arithmetic fragment. In: Proceedings on Computer Aided Verification CAV, LNCS, Springer, vol. 8559, pp 729–736

Download references

Acknowledgments

This work was sponsored, in part, by Defense Advanced Research Projects Agency (DARPA) and Air Force Research Laboratory (AFRL), under contract FA8750-12-C-0284, and the National Science Foundation under Grant CNS-1423298. The views, opinions, and/or findings contained in this report are those of the authors and should not be interpreted as representing the official views or policies, either expressed or implied, of the funding agencies.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ashish Tiwari.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tiwari, A., Lincoln, P. A search-based procedure for nonlinear real arithmetic. Form Methods Syst Des 48, 257–273 (2016). https://doi.org/10.1007/s10703-016-0245-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10703-016-0245-8

Keywords

Navigation