Advertisement

Automatic Theorem Proving With Renamable and Semantic Resolution

  • J. R. Slagle
Part of the Symbolic Computation book series (SYMBOLIC)

Abstract

The theory of J. A. Robinson’s resolution principle, an inference rule for first-order predicate calculus, is unified and extended. A theorem-proving computer program based on the new theory is proposed and the proposed semantic resolution program is compared with hyper-resolution and set-of-support resolution programs. Renamable and semantic resolution are defined and shown to be identical. Given a model M, semantic resolution is the resolution of a latent clash in which each “electron” is at least sometimes false under M; the nucleus is at least sometimes true under M.

The completeness theorem for semantic resolution and all previous completeness theorems for resolution (including ordinary, hyper-, and set-of-support resolution) can be derived from a slightly more general form of the following theorem. If U is a finite, truth-functionally un-satisfiable set of nonempty clauses and if M is a ground model, then there exists an unresolved maximal semantic clashE 1 , E 2 , ..., E q , C with nucleus C such that any set containing C and one or more of the electrons E1, E2, ..., Eq is an unresolved semantic clash in U.

Keywords

Theorem Prove Ground Model Preference Strategy Automatic Theorem Prove Empty Clause 
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. 1.
    Davis, M., and Putnam, H. A computing procedure for quantification theory. J. ACM 7 (1960), 201–215.Google Scholar
  2. 2.
    Davis, M., and Putnam, H. Eliminating the irrelevant from mechanical proofs. Proc. 15th Symp. in Appl. Math., Amer. Math. Soc., Providence, R. I., 1963, pp. 15–30.Google Scholar
  3. 3.
    Feigenbaum, E., and Feldman, J. (Eds.). Computers and Thought. McGraw-Hill, New York, 1963.MATHGoogle Scholar
  4. 4.
    Gelernter, H. Realization of a geometry theorem proving machine. Proc. Int. Conf. on Information Processing, UNESCO House, Paris, 1959, pp. 273–282. (Reprinted in [3], pp. 134–152.)Google Scholar
  5. 5.
    Gelernter, H., Hansen, J. R., and Loveland, D. W. Empirical explorations of the geometry theorem machine. Proc. 1960 Western Joint Comput. Conf., Vol. 17, pp. 143–147. (Reprinted in [3], pp. 153–163.)Google Scholar
  6. 6.
    Gilmore, P. C. A program for the production of proofs for theorems derivable within the first order predicate calculus from axioms. Proc. Int. Conf. on Information Processing, UNESCO House, Paris, 1959.Google Scholar
  7. 7.
    Mccarthy, J. Programs with common sense. Proc. Conf. on Mechanization of Thought Processes. English National Physical Laboratory, Teddington, Middlesex, England, 1959.Google Scholar
  8. 8.
    Meltzer, B. Theorem proving for computers: some results on resolution and renaming. Computer J. 8 (1966), 341–343.MATHMathSciNetGoogle Scholar
  9. 9.
    Newell, A., Shaw, J. C., and Simon, H. Empirical explorations of the logic theory machine, a case study in heuristic. Proc. 1957 Western Joint Comput. Conf., Vol. 11, pp. 218–230. (Reprinted in [3], pp. 109–133.)Google Scholar
  10. 10.
    Reynolds, J. Unpublished seminar notes. Stanford University, Palo Alto, Calif., fall 1965.Google Scholar
  11. 11.
    Robinson, J. A. A machine-oriented logic based on the resolution principle. J. ACM 12,1 (Jan. 1965), 23–41.Google Scholar
  12. 12.
    Robinson, J. A.. Automatic deduction with hyper-resolution. Int. J. Computer Math. 1 (1965), 227–234.Google Scholar
  13. 13.
    Robinson, J. A.. A review of automatic theorem proving. Proc. Symp. in Appl. Math., Amer. Math. Soc., Providence, R. I., 1967.Google Scholar
  14. 14.
    Slagle, J. R. A multipurpose, theorem proving, heuristic program that learns. Information Processing 1965, Proc. IFIP Congress 1965, Vol. 2, pp. 323–324.Google Scholar
  15. 15.
    Slagle, J. R.. A proposed preference strategy using sufficiency resolution for answering questions. UCRL-14361, Lawrence Radiation Lab., Livermore, Calif., Aug. 1965.Google Scholar
  16. 16.
    Slagle, J. R.. Experiments with a deductive, question-answering program. Comm. ACM 8 (Dec. 1965), 792–798.CrossRefGoogle Scholar
  17. 17.
    Wang, H. Formalization and automatic theorem proving. Information Processing 1965, Proc. IFIP Congress 1965, Vol. 1, pp. 51–58.Google Scholar
  18. 18.
    Wos, L., Carson, D. F., and Robinson, G. A. The unit preference strategy in theorem proving. Proc. AFIPS 1964 Fall Joint Comput. Conf., Vol. 26, pp. 616–621.Google Scholar
  19. 19.
    Wos, L., Robinson, G. A., and Carson, D. F. Efficiency and completeness of the set of support strategy in theorem proving. J. ACM 12, 4 (Oct. 1965), 536–541.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Association for Computing Machinery, Inc. 1967

Authors and Affiliations

  • J. R. Slagle

There are no affiliations available

Personalised recommendations