Advertisement

Synthese

, Volume 133, Issue 1–2, pp 203–335 | Cite as

Comparing Approaches To Resolution Based Higher-Order Theorem Proving

  • Christoph Benzmüller
Article

Abstract

We investigate several approaches to resolution based automated theoremproving in classical higher-order logic (based on Church's simply typedλ-calculus) and discuss their requirements with respect to Henkincompleteness and full extensionality. In particular we focus on Andrews' higher-order resolution (Andrews 1971), Huet's constrained resolution (Huet1972), higher-order E-resolution, and extensional higher-order resolution(Benzmüller and Kohlhase 1997). With the help of examples we illustratethe parallels and differences of the extensionality treatment of these approachesand demonstrate that extensional higher-order resolution is the sole approach thatcan completely avoid additional extensionality axioms.

Keywords

Theorem Prove Full Extensionality Extensionality Treatment Additional Extensionality Extensionality Axiom 
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. Andrews, P, B.: 1971, ‘Resolution in Type Theory’, Journal of Symbolic Logic 36, 414–432.Google Scholar
  2. Andrews, P. B.: 1972, ‘General Models and Extensionality’, Journal of Symbolic Logic 37, 395–397.Google Scholar
  3. Andrews, P. B.: 1973, Letter to Roger Hindley dated January 22.Google Scholar
  4. Andrews, P. B.: ‘Refutations by Matings’, IEEE Transactions on Computers C-25, 801–807.Google Scholar
  5. Andrews, P. B.: 1989, ‘On Connections and Higher Order Logic’, Journal of Automated Reasoning 5, 257–291.Google Scholar
  6. Andrews, P. B., Bishop, M., Issar, S., Nesmith, D., Pfenning, F., and Xi, H.: 1996, ‘TPS: A Theorem Proving System for Classical Type Theory’, Journal of Automated Reasoning 16, 321–353.Google Scholar
  7. Barendregt, H. P.: 1984, The Lambda Calculus — Its Syntax and Semantics, Studies in Logic and the Foundations of Mathematics 103, Amsterdam.Google Scholar
  8. Benzmüller, C.: 1997, A calculus and a System Architecture for Extensional Higher-order Resolution, Research Report 97-198, Department of Mathematical Sciences, Carnegie Mellon University.Google Scholar
  9. Benzmüller, C.: 1999a, Equality and Extensionality in Automated Higher-Order Theorem Proving, Ph.D. thesis, Technische Fakultät, Universität des Saarlandes.Google Scholar
  10. Benzmüller, C.: 1999b, in H. Ganzinger (ed.), Proceedings of the 16th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1632, pp. 399–413, Springer.Google Scholar
  11. Benzmüller, C. and Kohlhase, M.: 1997, ‘Model Existence for Higher-Order Logic’, SEKI-Report SR-97-09, Fachbereich Informatik, Universität des Saarlandes.Google Scholar
  12. Benzmüller, C. and Kohlhase, M.: 1998a, ‘Extensional Higher-order Resolution’, in Kirchner and Kirchner (eds.), Proceedings of the 15th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1421, pp.56–72, Springer.Google Scholar
  13. Benzmüller, C. and Kohlhase, M. 1998b, ‘LEO — A Higher-order Theorem Prover, in Kirchner and Kirchner (eds.), Proceedings of the 15th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 1421, pp. 139–144, Springer.Google Scholar
  14. Baader, F. and Siekmann, J.: 1994, ‘Unification Theory’, in D. M. Gabbay, C. J. Hogger, J. A. Robinson (eds.), Handbook of Logic in Artificial Intelligence and Logic Programming, Volume 2: Deduction Methodologies, Oxford, Chapter 2.2Google Scholar
  15. Church, A.: 1940, ‘A formulation of the Simple Theory of Types’, Journal of Symbolic Logic 5, 56–68.Google Scholar
  16. Dowek, G., Hardin, T., and Kirchner, C.: 1998, Theorem Proving Modulo, Rapport de Recherche 3400, Institut National de Recherche en Informatique et en Automatique.Google Scholar
  17. Dougherty, D. and Johann, P.: 1992, ‘A Combinatory Logic Approach to Higher-order E-unification’, in D. Kapur (ed.), Proceedings of the 11th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 607, pp. 79–93, Springer.Google Scholar
  18. Goldfarb, W. D. 1981, ‘The Undecidability of the Second-order Unification Problem’, Theoretical Computer Science 13, 225–230.Google Scholar
  19. Henkin, L.: 1950, ‘Completeness in the Theory of Types’, Journal of Symbolic Logic 15, 81–91.Google Scholar
  20. Huet, G. P.: 1972, Constrained Resolution: A Complete Method for Higher Order Logic, Ph.D. thesis, Case Western Reserve University.Google Scholar
  21. Huet, G. P.: 1973, ‘A Mechanization of Type Theory’, in D. E. Walker and L. Norton (eds.), Proceedings of the 3rd International Joint Conference on Artificial Intelligence, pp. 139–146.Google Scholar
  22. Huet, G. P.: 1973, ‘The Undecidability of Unification in Third Order Logic’, Information and Control 22, 257–267.Google Scholar
  23. Huet, G. P.: 1975, ‘A Unification Algorithm for Typed λ-calculus’, Theoretical Computer Science 1, 27–57.Google Scholar
  24. Jensen, D. C. and Pietrzykowski, T.: 1972, ‘A Complete Mechanization of ω-order Type Theory’, in Proceedings of the ACM annual Conference, volume 1, 89–92.Google Scholar
  25. Jensen, D. C. and Pietrzykowski, T.: 1976, ‘Mechanizing ω-order Type Theory through Unification’, Theoretical Computer Science 3, 123–171.Google Scholar
  26. Kohlhase, M.: 1994, A Mechanization of Sorted Higher-Order Logic Based on the Resolution Principle, Ph.D. thesis, Fachbereich Informatik, Universität des Saarlandes.Google Scholar
  27. Kohlhase, M.: 1995, ‘Higher-Order Tableaux’, in P. Baumgartner, R. Hähnle, and J. Posegga (eds.), Theorem Proving with Analytic Tableaux and Related Methods, Lecture Notes in Artificial Intelligence 918, pp. 294–309, Springer.Google Scholar
  28. Lucchesi, C. L.: 1972, The Undecidability of the Unification Problem for Third Order Languages, Report CSRR 2059, University of Waterloo, Waterloo, Canada.Google Scholar
  29. Miller, D.: 1983, Proofs in Higher-Order Logic, Ph.D. thesis, Carnegie Mellon University.Google Scholar
  30. Nipkow, T.: 1995, ‘Higher-order Rewrite Systems’, in J. Hsiang (ed.), Rewriting Techniques and Applications, 6th International Conference, Lecture Notes in Computer Science 914, Springer.Google Scholar
  31. Nipkow, T. and Prehofer, C.: 1998, ‘Higher-order Rewriting and Equational Reasoning’, in W. Bibel and P. Schmitt (eds.), Automated Deduction — A Basis for Applications, Dordrecht, Applied Logic Series, pp. 399–430.Google Scholar
  32. Nipkow, T. and Qian, Z,: 1991, ‘Modular Higher-order E-unification’, in R. V. Book (ed.), Proceedings of the 4th International Conference on Rewriting Techniques and Applications, Lecture Notes in Artificial Intelligence 488, pp. 200–214, Springer.Google Scholar
  33. Prehofer, C.: 1998, Solving Higher-Order Equations: From Logic to Programming, Progress in theoretical computer science, Birkhäuser.Google Scholar
  34. Qian, Z. and Wang K.: 1996, ‘Modular Higher-order Equational Preunification’, Journal of Symbolic Computation 22, 401–424.Google Scholar
  35. Robinson, J. A.: 1965, ‘A Machine-oriented Logic Based on the Resolution Principle’, Journal of the Association for Computing Machinery 12, 23–41.Google Scholar
  36. Siekmann, J. H.: 1989, ‘Unification Theory’, Journal of Symbolic Computation 7, 207–274.Google Scholar
  37. Smullyan, R. M. 1963, ‘A Unifying Principle for Quantification Theory’, Proceedings of the National Acadamy of Sciences, USA 49, pp. 828–832.Google Scholar
  38. Snyder, W.: 1990, ‘Higher Order E-unification’, in M. Stickel (ed.), Proceedings of the 10th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence 449, pp. 573–578, Springer.Google Scholar
  39. Snyder, W. and Gallier, J.: 1989, ‘Higher-order Unification Revisited: Complete Sets of Transformations’, Journal of Symbolic Computation 8, 101–140.Google Scholar
  40. Snyder, W. and Lynch, C.: 1991, ‘Goal-directed Strategies for Paramodulation’, in R. V. Book (ed.), Proceedings of the 4th International Conference on Rewriting Techniques and Applications, Lecture Notes in Artificial Intelligence 488, pp. 200–214, Springer.Google Scholar
  41. Wolfram, D. A.: 1993, The Clausal Theory of Types, Cambridge, Cambridge Tracts in Theoretical Computer Science 21.Google Scholar

Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

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

Personalised recommendations