Advertisement

Journal of Automated Reasoning

, Volume 40, Issue 1, pp 35–60 | Cite as

Translating Higher-Order Clauses to First-Order Clauses

  • Jia Meng
  • Lawrence C. Paulson
Article

Abstract

Interactive provers typically use higher-order logic, while automatic provers typically use first-order logic. To integrate interactive provers with automatic ones, one must translate higher-order formulas to first-order form. The translation should ideally be both sound and practical. We have investigated several methods of translating function applications, types, and λ-abstractions. Omitting some type information improves the success rate but can be unsound, so the interactive prover must verify the proofs. This paper presents experimental data that compares the translations in respect of their success rates for three automatic provers.

Keywords

Interactive theorem provers Higher-order logic First-order logic Clause translation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beeson, M.: Mathematical induction in Otter-lambda. J. Autom. Reason. 36(4), 311–344 (2006)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Benzmüller, C., Sorge, V., Jamnik, M., Kerber, M.: Can a higher-order and a first-order theorem prover cooperate? In: Baader, F., Voronkov, A. (eds.) Logic for Programming, Artificial Intelligence, and Reasoning – 11th International Workshop, LPAR 2004, LNAI, vol. 3452, pp. 415–431. Springer (2005)Google Scholar
  3. 3.
    Bouillaguet, C., Kuncak, V., Wies, T., Zee, K., Rinard, M.: Using first-order theorem provers in the Jahob data structure verification system. In: Cook, B., Podelski, A. (eds.) Verification, Model Checking, and Abstract Interpretation, LNCS, vol. 4349, pp. 74–88. Springer (2007)Google Scholar
  4. 4.
    Gordon, M.J.C., Melham, T.F.: Introduction to HOL: a Theorem Proving Environment for Higher Order Logic. Cambridge Univ. Press (1993)Google Scholar
  5. 5.
    Hughes, R.J.M.: Supercombinators: a new implementation method for applicative languages. In: LISP and Func. Prog. ACM Press (1982)Google Scholar
  6. 6.
    Hurd, J.: An LCF-style interface between HOL and first-order logic. In: Voronkov, A. (ed.) Automated Deduction – CADE-18 International Conference, LNAI, vol. 2392, pp. 134–138. Springer (2002)Google Scholar
  7. 7.
    Hurd, J.: First-order proof tactics in higher-order logic theorem provers. In: Archer, M., Vito, B.D., Muñoz, C. (eds.) Design and Application of Strategies/Tactics in Higher Order Logics, Number NASA/CP-2003-212448 in NASA Technical Reports, pp. 56–68, September (2003)Google Scholar
  8. 8.
    Kennaway, R., Sleep, R.: Director strings as combinators. ACM Trans. Program. Lang. Syst. 10(4), 602–626 (1988)MATHCrossRefGoogle Scholar
  9. 9.
    Meng, J., Paulson, L.C.: Translating higher-order problems to first-order clauses. In: Sutcliffe, G., Schmidt, R., Schulz, S. (eds.) FLoC’06 Workshop on Empirically Successful Computerized Reasoning, CEUR Workshop Proceedings, vol. 192, pp. 70–80 (2006)Google Scholar
  10. 10.
    Meng, J., Paulson, L.C.: Lightweight relevance filtering for machine-generated resolution problems. Journal of Applied Logic (2007) (in press)Google Scholar
  11. 11.
    Meng, J., Quigley, C., Paulson, L.C.: Automation for interactive proof: first prototype. Inf. Comput. 204(10), 1575–1596 (2006)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Miller, D.: Unification under a mixed prefix. J. Symb. Comput. 14(4), 321–358 (1992)MATHCrossRefGoogle Scholar
  13. 13.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: a Proof Assistant for Higher-Order Logic. LNCS Tutorial, vol. 2283. Springer (2002)Google Scholar
  14. 14.
    Owre, S., Rajan, S., Rushby, J.M., Shankar, N., Srivas, M.K.: PVS: Combining specification, proof checking, and model checking. In: Alur, R., Henzinger, T.A. (eds.) Computer Aided Verification: 8th International Conference, CAV ’96, LNCS, vol. 1102, pp. 411–414. Springer (1996)Google Scholar
  15. 15.
    Paulson, L.C., Susanto, K.W.: Source-level proof reconstruction for interactive theorem proving. In: Klaus, S., Brandt, J. (eds.) Theorem Proving in Higher Order Logics, LNCS, vol. 4732, pp. 232–245. Springer (2007)Google Scholar
  16. 16.
    Peyton Jones, S.L.: The Implementation of Functional Programming Languages. Prentice Hall (1987)Google Scholar
  17. 17.
    Pierce, B.C.: Types and Programming Languages. MIT Press (2002)Google Scholar
  18. 18.
    Riazanov, A., Voronkov, A.: Vampire 1.1 (system description). In: Goré, R., Leitsch, A., Nipkow, T. (eds.) Automated Reasoning – First International Joint Conference, IJCAR 2001, LNAI, vol. 2083, pp. 376–380. Springer (2001)Google Scholar
  19. 19.
    Schulz, S.: System description: E 0.81. In: Basin, D., Rusinowitch, M. (eds.) Automated Reasoning – Second International Joint Conference, IJCAR 2004, LNAI, vol. 3097, pp. 223–228. Springer (2004)Google Scholar
  20. 20.
    Sutcliffe, G., Suttner, C.: The TPTP problem library for automated theorem proving. On the internet at http://www.cs.miami.edu/~tptp/ (2004)
  21. 21.
    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, Number 112 in Frontiers in Artificial Intelligence and Applications, pp. 201–215. IOS Press (2004)Google Scholar
  22. 22.
    Turner, D.A.: Another algorithm for bracket abstraction. J. Symb. Log. 44(2), 267–270, June 1979MATHCrossRefGoogle Scholar
  23. 23.
    Turner, D.A.: A new implementation technique for applicative languages. Softw. Pract. Exp. 9, 31–49 (1979)MATHCrossRefGoogle Scholar
  24. 24.
    Weidenbach, C.: Combining superposition, sorts and splitting. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. II, chapter 27, pp. 1965–2013. Elsevier Science (2001)Google Scholar
  25. 25.
    Zimmer, J., Meier, A., Sutcliffe, G., Zhang, Y.: Integrated proof transformation services. In: Benzmüller, C., Windsteiger, W. (eds.) Workshop on Computer-Supported Mathematical Theory Development, 2nd International Joint Conference on Automated Reasoning, Electronic Notes in Theoretical Computer Science (2004)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.National ICTCanberraAustralia
  2. 2.Computer LaboratoryUniversity of CambridgeCambridgeUK

Personalised recommendations