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A Heuristic Prover for Real Inequalities

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Interactive Theorem Proving (ITP 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8558))

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Abstract

We describe a general method for verifying inequalities between real-valued expressions, especially the kinds of straightforward inferences that arise in interactive theorem proving. In contrast to approaches that aim to be complete with respect to a particular language or class of formulas, our method establishes claims that require heterogeneous forms of reasoning, relying on a Nelson-Oppen-style architecture in which special-purpose modules collaborate and share information. The framework is thus modular and extensible. A prototype implementation shows that the method is promising, complementing techniques that are used by contemporary interactive provers.

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References

  1. Akbarpour, B., Paulson, L.C.: MetiTarski: An Automatic Prover for the Elementary Functions. In: Autexier, S., Campbell, J., Rubio, J., Sorge, V., Suzuki, M., Wiedijk, F. (eds.) AISC/Calculemus/MKM 2008. LNCS (LNAI), vol. 5144, pp. 217–231. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  2. Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Trans. Comput. Logic 9(1), 2 (2007)

    Article  MathSciNet  Google Scholar 

  3. Avigad, J., Friedman, H.: Combining decision procedures for the reals. Log. Methods Comput. Sci. 2(4), 4:4, 42 (2006)

    Google Scholar 

  4. Avis, D.: Living with lrs. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 47–56. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  5. Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisability modulo theories. In: Biere, A., et al. (eds.) Handbook of Satisability, pp. 825–885. IOS Press (2008)

    Google Scholar 

  6. Basu, S., Pollack, R., Roy, M.: Algorithms in real algebraic geometry. Springer (2003)

    Google Scholar 

  7. Billingsley, P.: Probability and measure, 3rd edn. John Wiley & Sons Inc. (1995)

    Google Scholar 

  8. Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending Sledgehammer with SMT Solvers. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS (LNAI), vol. 6803, pp. 116–130. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  9. Boyd, S., Vandenberghe, L.: Convex optimization. Cambridge University Press (2004)

    Google Scholar 

  10. Chaieb, A., Nipkow, T.: Proof synthesis and reflection for linear arithmetic. J. Autom. Reasoning 41(1), 33–59 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  11. Contejean, E.: A Certified AC Matching Algorithm. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 70–84. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  12. de Moura, L., Bjørner, N.: Efficient E-Matching for SMT Solvers. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 183–198. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  13. de Moura, L., Bjørner, N.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  14. 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)

    Chapter  Google Scholar 

  15. Fukuda, K., Prodon, A.: Double description method revisited. In: Deza, M., Manoussakis, I., Euler, R. (eds.) CCS 1995. LNCS, vol. 1120, pp. 91–111. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  16. Garling, D.J.H.: Inequalities: a journey into linear analysis. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  17. Gao, S., Avigad, J., Clarke, E.M.: Delta-complete decision procedures for satisfiability over the reals. In: Gramlich, B., et al. (eds.) IJCAR, pp. 286–300 (2012)

    Google Scholar 

  18. Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. Cambridge University Press, Cambridge (1988), Reprint of the 1952 edition

    Google Scholar 

  19. Harrison, J.: HOL light: a tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  20. Harrison, J.: Verifying Nonlinear Real Formulas Via Sums of Squares. In: Schneider, K., Brandt, J. (eds.) TPHOLs 2007. LNCS, vol. 4732, pp. 102–118. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  21. Hunt Jr., W.A., Krug, R.B., Moore, J.: Linear and nonlinear arithmetic in ACL2. In: Geist, D., Tronci, E. (eds.) CHARME 2003. LNCS, vol. 2860, pp. 319–333. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  22. Jones, C.N., Kerrigan, E.C., Maciejowski, J.M.: Equality set projection: A new algorithm for the projection of polytopes in halfspace representation. Technical report, Department of Engineering, University of Cambridge (March 2004)

    Google Scholar 

  23. McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 295–314. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  24. Meng, J., Paulson, L.: Lightweight relevance filtering for machine-generated resolution problems. J. Applied Logic 7(1), 41–57 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Moore, R., Kearfott, R., Cloud, M.: Introduction to interval analysis. Society for Industrial and Applied Mathematics (SIAM) (2009)

    Google Scholar 

  26. Moses, J.: Algebraic simplification: A guide for the perplexed. Communications of the ACM 14, 527–537 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  27. Nelson, G., Oppen, D.: Simplification by cooperating decision procedures. ACM Transactions of Programming Languages and Systems 1, 245–257 (1979)

    Article  MATH  Google Scholar 

  28. Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  29. Polya, G.: How to solve it. Princeton University Press, Princeton (1945)

    MATH  Google Scholar 

  30. Prevosto, V., Waldmann, U.: SPASS+T. In: Sutcliffe, G., et al. (eds.) ESCoR: Empirically Successful Computerized Reasoning 2006. CEUR Workshop Proceedings, pp. 18–33 (2006)

    Google Scholar 

  31. Pugh, W.: The omega test: a fast and practical integer programming algorithm for dependence analysis. Communications of the ACM 8, 4–13 (1992)

    Google Scholar 

  32. Schrijver, A.: Theory of linear and integer programming. John Wiley & Sons (1986)

    Google Scholar 

  33. Ziegler, G.: Lectures on polytopes. Springer (1995)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Jeremy Avigad, Robert Y. Lewis & Cody Roux

Authors
  1. Jeremy Avigad
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  2. Robert Y. Lewis
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  3. Cody Roux
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Editor information

Editors and Affiliations

  1. NICTA, Sydney, NSW, Australia

    Gerwin Klein

  2. University of Wyoming, Laramie, WY, USA

    Ruben Gamboa

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Avigad, J., Lewis, R.Y., Roux, C. (2014). A Heuristic Prover for Real Inequalities. In: Klein, G., Gamboa, R. (eds) Interactive Theorem Proving. ITP 2014. Lecture Notes in Computer Science, vol 8558. Springer, Cham. https://doi.org/10.1007/978-3-319-08970-6_5

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  • DOI: https://doi.org/10.1007/978-3-319-08970-6_5

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08969-0

  • Online ISBN: 978-3-319-08970-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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