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Analytic Tableaux for Higher-Order Logic with Choice

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Automated Reasoning (IJCAR 2010)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 6173))

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Abstract

While many higher-order interactive theorem provers include a choice operator, higher-order automated theorem provers currently do not. As a step towards supporting 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. We prove completeness of the tableau calculus relative to Henkin models.

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References

  1. Andrews, P.B.: Resolution in type theory. J. Symb. Log. 36, 414–432 (1971)

    Article  MathSciNet  Google Scholar 

  2. Andrews, P.B.: General models and extensionality. J. Symb. Log. 37, 395–397 (1972)

    Article  MathSciNet  Google Scholar 

  3. Andrews, P.B., Brown, C.E.: TPS: A hybrid automatic-interactive system for developing proofs. Journal of Applied Logic 4(4), 367–395 (2006)

    Article  MathSciNet  Google Scholar 

  4. Backes, J.: Tableaux for higher-order logic with if-then-else, description and choice. Master’s thesis, Universität des Saarlandes (2010)

    Google Scholar 

  5. Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. Technical report, Programming Systems Lab, Saarland University, Saarbrücken, Germany (January 2010)

    Google Scholar 

  6. Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher-order semantics and extensionality. J. Symb. Log. 69, 1027–1088 (2004)

    Article  MathSciNet  Google Scholar 

  7. 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.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162–170. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  8. Brown, C.E., Smolka, G.: Analytic tableaux for simple type theory and its first-order fragment. Technical report, Programming Systems Lab, Saarland University, Saarbrücken, Germany (December 2009) (accepted for publication by Logical Methods in Computer Science)

    Google Scholar 

  9. Brown, C.E., Smolka, G.: Extended first-order logic. In: Urban, C. (ed.) TPHOLs 2009. LNCS, vol. 5674, pp. 164–179. Springer, Heidelberg (2009)

    Google Scholar 

  10. Brown, C.E., Smolka, G.: Terminating tableaux for the basic fragment of simple type theory. In: Giese, M., Waaler, A. (eds.) TABLEAUX 2009. LNCS (LNAI), vol. 5607, pp. 138–151. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  11. Church, A.: A formulation of the simple theory of types. J. Symb. Log. 5, 56–68 (1940)

    Article  MathSciNet  Google Scholar 

  12. Gordon, M.J., Melham, T.F.: Introduction to HOL: A Theorem-Proving Environment for Higher-Order Logic. Cambridge University Press, Cambridge (1993)

    MATH  Google Scholar 

  13. Henkin, L.: Completeness in the theory of types. J. Symb. Log. 15, 81–91 (1950)

    Article  MathSciNet  Google Scholar 

  14. Huet, G.P.: Constrained Resolution: A Complete Method for Higher Order Logic. PhD thesis, Case Western Reserve University (1972)

    Google Scholar 

  15. Mints, G.: Cut-elimination for simple type theory with an axiom of choice. J. Symb. Log. 64(2), 479–485 (1999)

    Article  MathSciNet  Google Scholar 

  16. Nipkow, T., Paulson, L.C., Wenzel, M.T. (eds.): Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    MATH  Google Scholar 

  17. Smullyan, R.M.: First-Order Logic. Springer, Heidelberg (1968)

    Book  Google Scholar 

  18. 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.) CADE 2009. LNCS, vol. 5663, pp. 116–130. Springer, Heidelberg (2009)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Saarland University, Saarbrücken, Germany

    Julian Backes & Chad E. Brown

Authors
  1. Julian Backes
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  2. Chad E. Brown
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Editors and Affiliations

  1. RWTH Aachen, LuFG Informatik, 2, Ahornstr. 55, 52074, Aachen, Germany

    Jürgen Giesl

  2. Department of Computer Science, Chalmers University of Technology, 41296, Gothenburg, Sweden

    Reiner Hähnle

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Backes, J., Brown, C.E. (2010). Analytic Tableaux for Higher-Order Logic with Choice. In: Giesl, J., Hähnle, R. (eds) Automated Reasoning. IJCAR 2010. Lecture Notes in Computer Science(), vol 6173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14203-1_7

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  • DOI: https://doi.org/10.1007/978-3-642-14203-1_7

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-14202-4

  • Online ISBN: 978-3-642-14203-1

  • eBook Packages: Computer ScienceComputer Science (R0)

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