Journal of Automated Reasoning

, Volume 47, Issue 4, pp 451–479 | Cite as

Analytic Tableaux for Higher-Order Logic with Choice

  • Julian Backes
  • Chad Edward BrownEmail author


While many higher-order interactive theorem provers include a choice operator, higher-order automated theorem provers so far have not. In order to support automated reasoning in the presence of a choice operator, we present a cut-free ground tableau calculus for Church’s simple type theory with choice. The tableau calculus is designed with automated search in mind. In particular, the rules only operate on the top level structure of formulas. Additionally, we restrict the instantiation terms for quantifiers to a universe that depends on the current branch. At base types the universe of instantiations is finite. Both of these restrictions are intended to minimize the number of rules a corresponding search procedure is obligated to consider. We prove completeness of the tableau calculus relative to Henkin models.


Higher-order logic Simple type theory Tableaux Completeness Axiom of choice Choice operators Henkin models 


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  1. 1.
    Altenkirch, T., Uustalu, T.: Normalization by evaluation for λ →2. In: Kameyama, Y., Stuckey, P.J. (eds.) Functional and Logic Programming, 7th International Symposium, FLOPS 2004, Proceedings. LNCS, vol. 2998, pp. 260–275. Springer (2004)Google Scholar
  2. 2.
    Andrews, P.B.: Resolution in type theory. J. Symb. Log. 36, 414–432 (1971)CrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews, P.B.: General models and extensionality. J. Symb. Log. 37, 395–397 (1972)CrossRefzbMATHGoogle Scholar
  4. 4.
    Andrews, P.B., Brown, C.E.: TPS: a hybrid automatic-interactive system for developing proofs. J. Appl. Logic 4(4), 367–395 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Backes, J.: Tableaux for higher-order logic with if-then-else, description and choice. Master’s thesis, Universität des Saarlandes (2010)Google Scholar
  6. 6.
    Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. In: Giesl, J., Hähnle, R. (eds.) Automated Reasoning: 5th International Joint Conference, IJCAR 2010, Proceedings. LNCS/LNAI, vol. 6173, pp. 76–90. Springer (2010)Google Scholar
  7. 7.
    Beeson, M.: Unification in lambda-calculi with if-then-else. In: Kirchner, C., Kirchner, H. (eds.) Proceedings of the 15th International Conference on Automated Deduction. LNAI, vol. 1421, pp. 103–118. Springer, Lindau, Germany (1998)Google Scholar
  8. 8.
    Benzmueller, C., Brown, C.E., Kohlhase, M.: Cut-simulation and impredicativity. LMCS 5(1:6), 1–21 (2009)Google Scholar
  9. 9.
    Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher-order semantics and extensionality. J. Symb. Log. 69, 1027–1088 (2004)CrossRefzbMATHGoogle Scholar
  10. 10.
    Benzmüller, C., Theiss, F., Paulson, L., Fietzke, A.: LEO-II — a cooperative automatic theorem prover for higher-order logic. In: Armando, A., Baumgartner, P., Dowek, G. (eds.) Fourth International Joint Conference on Automated Reasoning, IJCAR’08. LNAI, vol. 5195. Springer (2008)Google Scholar
  11. 11.
    Brown, C.E.: Automated Reasoning in Higher-Order Logic: Set Comprehension and Extensionality in Church’s Type Theory. College Publications (2007)Google Scholar
  12. 12.
    Brown, C.E., Smolka, G.: Extended first-order logic. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) Theorem Proving in Higher Order Logics, 22nd International Conference, TPHOLs 2009, Proceedings. LNCS, vol. 5674, pp. 164–179. Springer (2009)Google Scholar
  13. 13.
    Brown, C.E., Smolka, G.: Terminating tableaux for the basic fragment of simple type theory. In: Giese, M., Waaler, A. (eds.) Automated Reasoning with Analytic Tableaux and Related Methods: 18th International Conference, TABLEAUX 2009, Proceedings. LNCS/LNAI, vol. 5607, pp. 138–151. Springer (2009)Google Scholar
  14. 14.
    Brown, C.E., Smolka, G.: Analytic tableaux for simple type theory and its first-order fragment. LMCS 6(2), 1–33 (2010)MathSciNetGoogle Scholar
  15. 15.
    Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5, 56–68 (1940)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) Theory and Applications of Satisfiability Testing, LNCS, vol. 2919, pp. 333–336. Springer, Berlin/Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Gordon, M., Melham, T.: Introduction to HOL: A Theorem-Proving Environment for Higher-Order Logic. Cambridge University Press (1993)Google Scholar
  18. 18.
    Harrison, J.: HOL light: a tutorial introduction. In: Srivas, M., Camilleri, A. (eds.) Proceedings of the First International Conference on Formal Methods in Computer-Aided Design (FMCAD’96), LNCS, vol. 1166, pp. 265–269. Springer (1996)Google Scholar
  19. 19.
    Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15, 81–91 (1950)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Hintikka, K.J.J.: Form and content in quantification theory. Two papers on symbolic logic. Acta Philos. Fenn. 8, 7–55 (1955)MathSciNetGoogle Scholar
  21. 21.
    Huet, G.P.: Constrained resolution: a complete method for higher order logic. PhD thesis, Case Western Reserve University (1972)Google Scholar
  22. 22.
    King, D.J., Arthan, R.D.: Development of practical verification tools. ICL Systems J. 11(1) (1996)Google Scholar
  23. 23.
    Miller, D.A.: A compact representation of proofs. Stud. Log. 46(4), 347–370 (1987)CrossRefzbMATHGoogle Scholar
  24. 24.
    Mints, G.: Cut-elimination for simple type theory with an axiom of choice. J. Symb. Log. 64(2), 479–485 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Mitchell, J.C., Hoang, M., Howard, B.T.: Labeling techniques and typed fixed-point operators. In: Higher Order Operational Techniques in Semantics, pp. 137–174. Cambridge University Press, New York (1998)Google Scholar
  26. 26.
    Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL — a proof assistant for higher-order logic. LNCS, vol. 2283. Springer (2002)Google Scholar
  27. 27.
    Prawitz, D.: Hauptsatz for higher order logic. J. Symb. Log. 33, 452–457 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Slind, K., Norrish, M.: A brief overview of HOL4. In: Proceedings of the 21st International Conference on Theorem Proving in Higher Order Logics. LNCS, vol. 5170, pp. 28–32. Springer, Berlin, Heidelberg (2008)CrossRefGoogle Scholar
  29. 29.
    Smullyan, R.M.: A unifying principle in quantification theory. Proc. Natl. Acad. Sci. U.S.A. 49, 828–832 (1963)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Smullyan, R.M.: First-Order Logic. Springer (1968)Google Scholar
  31. 31.
    Sutcliffe, G.: The 5th IJCAR automated theorem proving system competition - CASC-J5. AI Commun. 24(1), 75–89 (2011)Google Scholar
  32. 32.
    Sutcliffe, G., Benzmüller, C., Brown, C.E., Theiss, F.: Progress in the development of automated theorem proving for higher-order logic. In: Schmidt, R.A. (ed.) Automated Deduction - CADE-22. 22nd International Conference on Automated Deduction, Proceedings. LNCS, vol. 5663, pp. 116–130. Springer (2009)Google Scholar
  33. 33.
    Takahashi, M.: Simple type theory of gentzen style with the inference of extensionality. Proc. Jpn. Acad. 44(2), 43–45 (1968)CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Saarland UniversitySaarbrückenGermany

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