THF0 – The Core of the TPTP Language for Higher-Order Logic

  • Christoph Benzmüller
  • Florian Rabe
  • Geoff Sutcliffe
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5195)


One of the keys to the success of the Thousands of Problems for Theorem Provers (TPTP) problem library and related infrastructure is the consistent use of the TPTP language. This paper introduces the core of the TPTP language for higher-order logic – THF0, based on Church’s simple type theory. THF0 is a syntactically conservative extension of the untyped first-order TPTP language.


Type Theory Proof Assistant Automate Theorem Prove Type Declaration Binary Connective 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Andrews, P.B., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., Xi, H.: TPS: A Theorem-Proving System for Classical Type Theory. Journal of Automated Reasoning 16(3), 321–353 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Beeson, M.: Otter-lambda, a Theorem-prover with Untyped Lambda-unification. In: Sutcliffe, G., Schulz, S., Tammet, T. (eds.) Proceedings of the Workshop on Empirically Successful First Order Reasoning (2004)Google Scholar
  3. 3.
    Benzmüller, C., Brown, C.: A Structured Set of Higher-Order Problems. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 66–81. Springer, Heidelberg (2005)Google Scholar
  4. 4.
    Benzmüller, C., Brown, C., Kohlhase, M.: Higher-order Semantics and Extensionality. Journal of Symbolic Logic 69(4), 1027–1088 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Benzmüller, C., Kohlhase, M.: LEO - A Higher-Order Theorem Prover. In: Kirchner, C., Kirchner, H. (eds.) CADE 1998. LNCS (LNAI), vol. 1421, pp. 139–143. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  6. 6.
    Benzmüller, C., Paulson, L.: Exploring Properties of Normal Multimodal Logics in Simple Type Theory with LEO-II. In: Benzmüller, C., Brown, C., Siekmann, J., Statman, R. (eds.) Festschrift in Honour of Peter B. Andrews on his 70th Birthday. IfCoLog (to appear 2007)Google Scholar
  7. 7.
    Benzmüller, C., Sorge, V., Jamnik, M., Kerber, M.: Combined Reasoning by Automated Cooperation. Journal of Applied Logic (in print) (2008)Google Scholar
  8. 8.
    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.) Proceedings of the 4th International Joint Conference on Automated Reasoning (IJCAR 2008). LNCS (LNAI), vol. 5195. Springer, Heidelberg (2008)Google Scholar
  9. 9.
    Church, A.: A Formulation of the Simple Theory of Types. Journal of Symbolic Logic 5, 56–68 (1940)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Curry, H.B., Feys, R.: Combinatory Logic I. North Holland, Amsterdam (1958)Google Scholar
  11. 11.
    Frege, F.: Grundgesetze der Arithmetik. Jena (1893) (1903)Google Scholar
  12. 12.
    Godefroid, P.: Software Model Checking: the VeriSoft Approach. Technical Report Technical Memorandum ITD-03-44189G, Bell Labs, Lisle, USA (2003)Google Scholar
  13. 13.
    Gordon, M., Melham, T.: Introduction to HOL, a Theorem Proving Environment for Higher Order Logic. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  14. 14.
    Harper, R., Honsell, F., Plotkin, G.: A Framework for Defining Logics. Journal of the ACM 40(1), 143–184 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Harrison, J.: HOL Light: A Tutorial Introduction. In: Srivas, M., Camilleri, A. (eds.) FMCAD 1996. LNCS, vol. 1166, pp. 265–269. Springer, Heidelberg (1996)CrossRefGoogle Scholar
  16. 16.
    Henkin, L.: Completeness in the Theory of Types. Journal of Symbolic Logic 15, 81–91 (1950)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Howard, W.: The Formulas-as-types Notion of Construction. In: Seldin, J., Hindley, J. (eds.) H B Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 479–490. Academic Press, London (1980)Google Scholar
  18. 18.
    Martin-Löf, P.: An Intuitionistic Theory of Types. In: Sambin, G., Smith, J. (eds.) Twenty-Five Years of Constructive Type Theory, pp. 127–172. Oxford University Press, Oxford (1973)Google Scholar
  19. 19.
    Matuszek, C., Cabral, J., Witbrock, M., DeOliveira, J.: An Introduction to the Syntax and Content of Cyc. In: Baral, C. (ed.) Proceedings of the 2006 AAAI Spring Symposium on Formalizing and Compiling Background Knowledge and Its Applications to Knowledge Representation and Question Answering, pp. 44–49 (2006)Google Scholar
  20. 20.
    Niles, I., Pease, A.: Towards A Standard Upper Ontology. In: Welty, C., Smith, B. (eds.) Proceedings of the 2nd International Conference on Formal Ontology in Information Systems, pp. 2–9 (2001)Google Scholar
  21. 21.
    Nipkow, T., Paulson, L.C., Wenzel, M.T.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  22. 22.
    Owre, S., Rajan, S., Rushby, J.M., Shankar, N., Srivas, M.: PVS: Combining Specification, Proof Checking, and Model Checking. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 411–414. Springer, Heidelberg (1996)Google Scholar
  23. 23.
    Pfenning, F., Schürmann, C.: System Description: Twelf - A Meta-Logical Framework for Deductive Systems. In: Ganzinger, H. (ed.) CADE 1999. LNCS (LNAI), vol. 1632, pp. 202–206. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  24. 24.
    Rudnicki, P.: An Overview of the Mizar Project. In: Proceedings of the 1992 Workshop on Types for Proofs and Programs, pp. 311–332 (1992)Google Scholar
  25. 25.
    Siekmann, J., Benzmüller, C., Autexier, S.: Computer supported mathematics with omega. Journal of Applied Logic 4(4), 533–559 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Sutcliffe, G.: TPTP, TSTP, CASC, etc. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 7–23. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  27. 27.
    Sutcliffe, G., Schulz, S., Claessen, K., Gelder, A.V.: Using the TPTP Language for Writing Derivations and Finite Interpretations. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 67–81. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  28. 28.
    Sutcliffe, G., Suttner, C.: The State of CASC. AI Communications 19(1), 35–48 (2006)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Sutcliffe, G., Suttner, C.B.: The TPTP Problem Library: CNF Release v1.2.1. Journal of Automated Reasoning 21(2), 177–203 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    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, vol. 112, pp. 201–215. IOS Press, Amsterdam (2004)Google Scholar
  31. 31.
    Gelder, A.V., Sutcliffe, G.: Extending the TPTP Language to Higher-Order Logic with Automated Parser Generation. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 156–161. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  32. 32.
    Zermelo, E.: Über Grenzzahlen und Mengenbereiche. Fundamenta Mathematicae 16, 29–47 (1930)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christoph Benzmüller
    • 1
  • Florian Rabe
    • 2
  • Geoff Sutcliffe
    • 3
  1. 1.Saarland UniversityGermany
  2. 2.Jacobs University BremenGermany
  3. 3.University of MiamiUSA

Personalised recommendations