MetiTarski: Past and Future

  • Lawrence C. Paulson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7406)


A brief overview is presented of MetiTarski [4], an automatic theorem prover for real-valued special functions: ln , \(\exp\), sin, cos, etc. MetiTarski operates through a unique interaction between decision procedures and resolution theorem proving. Its history is briefly outlined, along with current projects. A simple collision avoidance example is presented.


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  1. 1.
    Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Wiley (1972)Google Scholar
  2. 2.
    Akbarpour, B., Paulson, L.C.: Extending a Resolution Prover for Inequalities on Elementary Functions. In: Dershowitz, N., Voronkov, A. (eds.) LPAR 2007. LNCS (LNAI), vol. 4790, pp. 47–61. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    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 2008, Calculemus 2008, and MKM 2008. LNCS (LNAI), vol. 5144, pp. 217–231. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Akbarpour, B., Paulson, L.C.: MetiTarski: An automatic theorem prover for real-valued special functions. Journal of Automated Reasoning 44(3), 175–205 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Akbarpour, B., Paulson, L.C.: Applications of MetiTarski in the Verification of Control and Hybrid Systems. In: Majumdar, R., Tabuada, P. (eds.) HSCC 2009. LNCS, vol. 5469, pp. 1–15. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Avigad, J., Donnelly, K., Gray, D., Raff, P.: A formally verified proof of the prime number theorem. ACM Transactions on Computational Logic 9(1) (2007)Google Scholar
  8. 8.
    Beeson, M.: Automatic generation of a proof of the irrationality of e. JSC 32(4), 333–349 (2001)MathSciNetMATHGoogle Scholar
  9. 9.
    Blanchette, J.C., Böhme, S., Paulson, L.C.: Extending Sledgehammer with SMT Solvers. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 116–130. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  10. 10.
    Bledsoe, W.W.: Non-resolution theorem proving. Artificial Intelligence 9, 1–35 (1977)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Bridge, J., Paulson, L.C.: Case splitting in an automatic theorem prover for real-valued special functions. Journal of Automated Reasoning (in press, 2012),
  12. 12.
    Brown, C.W.: QEPCAD B: a program for computing with semi-algebraic sets using CADs. SIGSAM Bulletin 37(4), 97–108 (2003)MATHCrossRefGoogle Scholar
  13. 13.
    Bullen, P.S.: A Dictionary of Inequalities. Longman (1998)Google Scholar
  14. 14.
    Clarke, E., Zhao, X.: Analytica: A theorem prover for Mathematica. Mathematica Journal 3(1), 56–71 (1993)Google Scholar
  15. 15.
    Cuyt, A., Petersen, V., Verdonk, B., Waadeland, H., Jones, W.B.: Handbook of Continued Fractions for Special Functions. Springer (2008)Google Scholar
  16. 16.
    Daumas, M., Muñoz, C., Lester, D.: Verified real number calculations: A library for integer arithmetic. IEEE Trans. Computers 58(2), 226–237 (2009)CrossRefGoogle Scholar
  17. 17.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symbolic Comp. 5, 29–35 (1988)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    de Moura, L., Bjørner, N.S.: Z3: An Efficient SMT Solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Denman, W., Akbarpour, B., Tahar, S., Zaki, M., Paulson, L.C.: Formal verification of analog designs using MetiTarski. In: Biere, A., Pixley, C. (eds.) Formal Methods in Computer Aided Design, pp. 93–100. IEEE (2009)Google Scholar
  20. 20.
    van den Dries, L.: Alfred Tarski’s elimination theory for real closed fields. The Journal of Symbolic Logic 53(1), 7–19 (1988)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hardy, R.: Formal Methods for Control Engineering: A Validated Decision Procedure for Nichols Plot Analysis. PhD thesis, University of St Andrews (2006)Google Scholar
  22. 22.
    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)CrossRefGoogle Scholar
  23. 23.
    Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Di Vito, B., Muñoz, C. (eds.) Design and Application of Strategies/Tactics in Higher Order Logics, NASA/CP-2003-212448 in NASA Technical Reports, pp. 56–68 (September 2003)Google Scholar
  24. 24.
    Hurd, J.: Metis first order prover (2007), Website at,
  25. 25.
    Jovanovic̀, D., de Moura, L.: Solving Non-linear Arithmetic. Technical Report MSR-TR-2012-20, Microsoft Research (2012) (accepted to IJCAR 2012)Google Scholar
  26. 26.
    Lassez, J.-L., Maher, M.J.: On Fourier’s algorithm for linear arithmetic constraints. Journal of Automated Reasoning 9(3), 373–379 (1992)MathSciNetMATHCrossRefGoogle Scholar
  27. 27.
    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)CrossRefGoogle Scholar
  28. 28.
    Mitrinović, D.S., Vasić, P.M.: Analytic Inequalities. Springer (1970)Google Scholar
  29. 29.
    Muñoz, C., Lester, D.R.: Real Number Calculations and Theorem Proving. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 195–210. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  30. 30.
    Passmore, G.O., Paulson, L.C., de Moura, L.: Real algebraic strategies for MetiTarski proofs. In: Jeuring, J. (ed.) Conferences on Intelligent Computer Mathematics, CICM 2012. Springer (in press, 2012)Google Scholar
  31. 31.
    Platzer, A.: Differential-algebraic dynamic logic for differential-algebraic programs. J. Logic Computation 20(1), 309–352 (2010)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Platzer, A.: Logical Analysis of Hybrid Systems: Proving Theorems for Complex Dynamics. Springer (2010)Google Scholar
  33. 33.
    Platzer, A., Quesel, J.-D.: KeYmaera: A Hybrid Theorem Prover for Hybrid Systems (System Description). In: Armando, A., Baumgartner, P., Dowek, G. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 171–178. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  34. 34.
    Ratschan, S.: Efficient solving of quantified inequality constraints over the real numbers. ACM Trans. Comput. Logic 7(4), 723–748 (2006)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ratschan, S., She, Z.: Benchmarks for safety verification of hybrid systems (2008),

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Lawrence C. Paulson
    • 1
  1. 1.Computer LaboratoryUniversity of CambridgeEngland

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