Advertisement

Extensional Higher-Order Paramodulation and RUE-Resolution

  • Christoph Benzmüller
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1632)

Abstract

This paper presents two approaches to primitive equality treatment in higher-order (HO) automated theorem proving: a calculus EP adapting traditional first-order (FO) paramodulation [RW69], and a calculus εRUε adapting FO RUE-Resolution [Dig79] to classical type theory, i.e., HO logic based on Church’s simply typed λ-calculus. εP and εRUε extend the extensional HO resolution approach εR [BK98a]. In order to reach Henkin completeness without the need for additional extensionality axioms both calculi employ new, positive extensionality rules analogously to the respective negative ones provided by εR that operate on unification constraints. As the extensionality rules have an intrinsic and unavoidable difference-reducing character the HO paramodulation approach loses its pure term-rewriting character. On the other hand examples demonstrate that the extensionality rules harmonise quite well with the difference-reducing HO RUE-resolution idea.

Keywords

Primitive Equality Rule Para Extensionality Principle Extensionality Rule Substitutivity Property 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. And71.
    P. B. Andrews. Resolution in type theory. JSL, 36(3):414–432, 1971.zbMATHCrossRefGoogle Scholar
  2. And86.
    P. B. Andrews. An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof. Academic Press, 1986.Google Scholar
  3. Bar84.
    H. P. Barendregt. The Lambda-Calculus: Its Syntax and Semantics. North-Holland, 2nd edition 1984.Google Scholar
  4. Ben99.
    C. Benzmüller. Equality and Extensionality in Automated Higher-Order Theorem Proving. PhD thesis, FB 14, Universität des Saarlandes, 1999.Google Scholar
  5. BK97.
    C. Benzmüller and M. Kohlhase. Model existence for higher-order logic. Seki-Report SR-97-09, FB 14, Universität des Saarlandes, 1997, submitted to JSL.Google Scholar
  6. BK98a.
    C. Benzmüller and M. Kohlhase. Extensional higher-order resolution. In Kirchner and Kirchner [KK98], pages 56–72.Google Scholar
  7. BK98b.
    C. Benzmüller and M. Kohlhase. LEO — a higher-order theorem prover. In Kirchner and Kirchner [KK98], pages 139–144.Google Scholar
  8. Chu40.
    A. Church. A formulation of the simple theory of types. JSL, 5:56–68, 1940.zbMATHCrossRefMathSciNetGoogle Scholar
  9. Dar68.
    J. L. Darlington. Automatic theorem proving with equality substitutions and mathematical induction. Machine Intelligence, 3:113–130, 1968.zbMATHGoogle Scholar
  10. Dig79.
    V. J. Digricoli. Resolution by unification and equality. In W. H. Joyner, editor, Proc. of the 4th Workshop on Automated Deduction, Austin, 1979.Google Scholar
  11. Hen50.
    L. Henkin. Completeness in the theory of types. JSL, 15(2):81–91, 1950.zbMATHCrossRefMathSciNetGoogle Scholar
  12. Hue72.
    G. P. Huet. Constrained Resolution: A Complete Method for Higher Order Logic. PhD thesis, Case Western Reserve University, 1972.Google Scholar
  13. KK98.
    C. Kirchner and H. Kirchner, editors. Proc. of the 15th Conference on Automated Deduction, number 1421 in LNAI, Springer, 1998.Google Scholar
  14. Koh94.
    M. Kohlhase. A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle. PhD thesis, Universität des Saarlandes, 1994.Google Scholar
  15. Mil83.
    D. Miller. Proofs in Higher-Order Logic. PhD thesis, Carnegie-Mellon University, 1983.Google Scholar
  16. RW69.
    G. A. Robinson and L. Wos. Paramodulation and TP in first order theories with equality. Machine Intelligence, 4:135–150, 1969.MathSciNetzbMATHGoogle Scholar
  17. SG89.
    W. Snyder and J. H. Gallier. Higher-Order Unification Revisited: Complete Sets of Transformations. Journal of Symbolic Computation, 8:101–140, 1989.MathSciNetCrossRefzbMATHGoogle Scholar
  18. Smu63.
    Raymond M. Smullyan. A unifying principle for quantification theory. Proc. Nat. Acad Sciences, 49:828–832, 1963.zbMATHCrossRefMathSciNetGoogle Scholar
  19. Wol93.
    D. Wolfram. The Clausal Theory of Types. Cambridge University Press, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Christoph Benzmüller
    • 1
  1. 1.Fachbereich InformatikUniversität des SaarlandesGermany

Personalised recommendations