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Journal of Automated Reasoning

, Volume 44, Issue 3, pp 175–205 | Cite as

MetiTarski: An Automatic Theorem Prover for Real-Valued Special Functions

  • Behzad Akbarpour
  • Lawrence Charles PaulsonEmail author
Article
First Online:

Abstract

Many theorems involving special functions such as ln, exp and sin can be proved automatically by MetiTarski: a resolution theorem prover modified to call a decision procedure for the theory of real closed fields. Special functions are approximated by upper and lower bounds, which are typically rational functions derived from Taylor or continued fraction expansions. The decision procedure simplifies clauses by deleting literals that are inconsistent with other algebraic facts. MetiTarski simplifies arithmetic expressions by conversion to a recursive representation, followed by flattening of nested quotients. Applications include verifying hybrid and control systems.

Keywords

Special Function Decision Procedure Continue Fraction Theorem Prover Axiom System 
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.
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References

  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Wiley, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Akbarpour, B., Paulson, L.: Extending a resolution prover for inequalities on elementary functions. In: Logic for Programming, Artificial Intelligence, and Reasoning, pp. 47–61 (2007)Google Scholar
  3. 3.
    Akbarpour, B., Paulson, L.: MetiTarski: an automatic prover for the elementary functions. In: Autexier, S., et al. (eds.) Intelligent Computer Mathematics. LNCS, vol. 5144, pp. 217–231. Springer, New York (2008)CrossRefGoogle Scholar
  4. 4.
    Akbarpour, B., Paulson, L.C.: Towards automatic proofs of inequalities involving elementary functions. In: Cook, B., Sebastiani, R. (eds.): PDPAR: Pragmatics of Decision Procedures in Automated Reasoning, pp. 27–37 (2006)Google Scholar
  5. 5.
    Akbarpour, B., Paulson, L.C.: Applications of MetiTarski in the verification of control and hybrid systems. In: Majumdar, R., Tabuada, P. (eds.) Hybrid Systems: Computation and Control. LNCS, vol. 5469, pp. 1–15. Springer, New York (2009)CrossRefGoogle Scholar
  6. 6.
    Baader, F., Nipkow, T.: Term Rewriting and All That. Cambridge University Press, Cambridge (1998)Google Scholar
  7. 7.
    Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, chap. 2, pp. 19–99. Elsevier, Amsterdam (2001)CrossRefGoogle Scholar
  8. 8.
    Backeljauw, F., Becuwe, S., Colman, M., Cuyt, A., Docx, T.: Special functions: continued fraction and series representations. http://www.cfhblive.ua.ac.be/ (2008)
  9. 9.
    Beeson., M.: Automatic generation of a proof of the irrationality of e. J. Symb. Comput. 32(4), 333–349 (2001)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Bergstra, J.A., Tucker, J.V.: The rational numbers as an abstract data type. J. ACM 54(2), Article No. 7 (2007)Google Scholar
  11. 11.
    Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. SIGSAM Bull. 37(4), 97–108 (2003)zbMATHCrossRefGoogle Scholar
  12. 12.
    Bullen, P.S.: A Dictionary of Inequalities. Longman, New York (1998)zbMATHGoogle Scholar
  13. 13.
    Clarke, E., Zhao, X.: Analytica: a theorem prover for mathematica. Math. J. 3(1), 56–71 (1993)Google Scholar
  14. 14.
    Cuyt, A., Petersen, V., Verdonk, B., Waadeland, H., Jones, W.: Handbook of Continued Fractions for Special Functions. Springer, New York (2008)zbMATHGoogle Scholar
  15. 15.
    Cuyt, A.A.M.: Upper/lower bounds (2008). E-mail dated 7 September 2008Google Scholar
  16. 16.
    Cuyt, A.A.M., Verdonk, B., Waadeland, H.: Efficient and reliable multiprecision implementation of elementary and special functions. SIAM J. Sci. Comput. 28(4), 1437–1462 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Daumas, M., Muñoz, C., Lester, D.: Verified real number calculations: a library for integer arithmetic. IEEE Trans. Comput. 58(2), 226–237 (2009)CrossRefGoogle Scholar
  18. 18.
    Denman, W., Akbarpour, B., Tahar, S., Zaki, M., Paulson, L.C.: Automated formal verification of analog designs using MetiTarski. In: Biere, A., Pixley, C. (eds.) Formal Methods in Computer Aided Design (2009)Google Scholar
  19. 19.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. Technical report MIP-9720, Universität Passau, D-94030, Germany (1997)Google Scholar
  20. 20.
    Fränzle, M., Herde, C., Ratschan, S., Schubert, T., Teige, T.: Efficient solving of large non-linear arithmetic constraint systems with complex boolean structure. Journal on Satisfiability, Boolean Modeling and Computation 1, 209–236 (2007)Google Scholar
  21. 21.
    Gottliebsen, H., Hardy, R., Lightfoot, O., Martin, U.: Applications of real number theorem proving in PVS. Preprint (2007)Google Scholar
  22. 22.
    Grégoire, B., Mahboubi, A.: Proving equalities in a commutative ring done right in Coq. In: Hurd, J., Melham, T. (eds.) Theorem Proving in Higher Order Logics: TPHOLs 2005. LNCS, vol. 3603, pp. 98–113. Springer, New York (2005)Google Scholar
  23. 23.
    Hardy, R.: Formal methods for control engineering: a validated decision procedure for Nichols plot analysis. Ph.D. thesis, University of St Andrews (2006)Google Scholar
  24. 24.
    Harrison, J.: Handbook of Practical Logic and Automated Reasoning. Cambridge University Press, Cambridge (2009)zbMATHCrossRefGoogle Scholar
  25. 25.
    Herde, C.: HySAT Quick Start Guide. University of Oldenburg. http://hysat.informatik.uni-oldenburg.de/user_guide/hysat-user-guide.pdf (2008)
  26. 26.
    Hong, H.: QEPCAD—quantifier elimination by partial cylindrical algebraic decomposition. http://www.cs.usna.edu/~qepcad/B/QEPCAD.html (2008)
  27. 27.
    Hörmander, L.: The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficient. Springer, New York (2006) (First published in 1983; cited by McLaughlin and Harrison [34])Google Scholar
  28. 28.
    Hurd, J.: Metis first order prover. http://gilith.com/software/metis/ (2007)
  29. 29.
    Kaczor, W.J., Nowak, M.T.: Problems in Mathematical Analysis II: Continuity and Differentiation. American Mathematical Society, Providence (2001)Google Scholar
  30. 30.
    Korovkin, P.P.: Inequalities. Pergamon, Oxford (1961)zbMATHGoogle Scholar
  31. 31.
    Lafferriere, G., Pappas, G.J., Yovine, S.: Symbolic reachability computation for families of linear vector fields. J. Symb. Comput. 32(3), 231–253 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Ludwig, M., Waldmann, U.: An extension of the Knuth-Bendix ordering with LPO-like properties. In: Dershowitz, N., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2007. Lecture Notes in Computer Science, vol. 4790, pp. 348–362. Springer, New York (2007)CrossRefGoogle Scholar
  33. 33.
    McCune, W., Wos, L.: Otter: the CADE-13 competition incarnations. J. Autom. Reason. 18(2), 211–220 (1997)CrossRefGoogle Scholar
  34. 34.
    McLaughlin, S., Harrison, J.: A proof-producing decision procedure for real arithmetic. In: Nieuwenhuis, R. (ed.) Automated Deduction—CADE-20 International Conference. LNAI, vol. 3632, pp. 295–314. Springer, New York (2005)Google Scholar
  35. 35.
    Mitrinović, D.S., Vasić, P.M.: Analytic Inequalities. Springer, New York (1970)zbMATHGoogle Scholar
  36. 36.
    Muller, J.M.: Elementary Functions: Algorithms and Implementation, 2nd edn. Birkhäuser, Boston (2006)zbMATHGoogle Scholar
  37. 37.
    Parrilo, P.A.: Semidefinite programming relaxations for semialgebraic problems. Math. Program. 96(2), 293–320 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    Prevosto, V., Waldmann, U.: SPASS+T. In: Sutcliffe, G., Schmidt, R., Schulz, S. (eds.) FLoC’06 Workshop on Empirically Successful Computerized Reasoning, CEUR Workshop Proceedings, vol. 192, pp. 18–33 (2006)Google Scholar
  39. 39.
    Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Comput. Log. 7(4), 723–748 (2006)MathSciNetGoogle Scholar
  40. 40.
    Ratschan, S.: RSolver user manual. Academy of Sciences of the Czech Republic. http://rsolver.sourceforge.net/documentation/manual.pdf (2007)
  41. 41.
    Ratschan, S., She, Z.: Safety verification of hybrid systems by constraint propagation based abstraction refinement ACM Trans. Embed. Comput. Syst. 6(1) (2007)Google Scholar
  42. 42.
    Ratschan, S., She, Z.: Benchmarks for safety verification of hybrid systems. http://hsolver.sourceforge.net/benchmarks/ (2008)
  43. 43.
    Sutcliffe, G., Zimmer, J., Schulz, S.: TSTP data-exchange formats for automated theorem proving tools. In: Zhang, W., Sorge, V. (eds.) Distributed Constraint Problem Solving and Reasoning in Multi-Agent Systems, Frontiers in Artificial Intelligence and Applications, number 112, pp. 201–215. IOS, Amsterdam (2004)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Computer LaboratoryUniversity of CambridgeCambridgeUK

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