Deduction with relation matching

  • Zohar Manna
  • Richard Waldinger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 206)


A new deduction rule is introduced to give streamlined treatment to relations of special importance in an automated theorem-proving system. This relation matching rule generalizes to an arbitrary binary relation the E-resolution and RUE-resolution rules for equality, and may operate within a nonclausal or clausal system. The new rule depends on an extension of the notion of polarity to apply to subterms as well as to subsentences, with respect to a given binary relation. It allows the system to draw a conclusion even if the unification algorithm fails to find a complete match, provided the polarities of the mismatched terms are auspicious. The rule allows us to eliminate troublesome axioms, such as transitivity and monotonicity, from the system; proofs are shorter and more comprehensible, and the search space is correspondingly deflated.


Binary Relation Deduction Rule Relation Match Resolution 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

7. References

  1. Anderson [70]
    R. Anderson, Completeness results for E-resolution, AFIPS Spring Joint Computer Conference, 1970, pp. 652–656.Google Scholar
  2. Boyer and Moore [79]
    R. S. Boyer and J. S. Moore, A Computational Logic, Academic Press, New York, N.Y., 1979.Google Scholar
  3. Brand [75]
    D. Brand, Proving theorems with the modification method, SIAM Journal of Computing, Vol. 4, No. 2, 1975, pp. 412–430.CrossRefGoogle Scholar
  4. Digricoli [83]
    V. Digricoli, Resolution By Unification and Equality, Ph.D. thesis, New York University, New York, N.Y., 1983.Google Scholar
  5. Kowalski [79]
    R. Kowalski, Logic for Problem Solving, North Holland, New York, N.Y., 1979.Google Scholar
  6. Manna and Waldinger [80]
    Z. Manna and R. Waldinger, A deductive approach to program synthesis, ACM Transactions on Programming Languages and Systems, Vol. 2, No. 1, January 1980, pp. 90–121.CrossRefGoogle Scholar
  7. Manna and Waldinger [81]
    Z. Manna and R. Waldinger, Deductive synthesis of the unification algorithm, Science of Computer Programming, Vol. 1, 1981, pp. 5–48.CrossRefGoogle Scholar
  8. Manna and Waldinger [82]
    Z. Manna and R. Waldinger, Special relations in program-synthetic deduction, Technical Report, Computer Science Department, Stanford University, Stanford, Calif., March 1982.Google Scholar
  9. Manna and Waldinger [85a]
    Z. Manna and R. Waldinger, Special relations in automated deduction, Journal of the ACM (to appear).Google Scholar
  10. Manna and Waldinger [85b]
    Z. Manna and R. Waldinger, Origin of the binary search paradigm, in the Proceedings of IJCAI-85, Los Angeles, August 1985.Google Scholar
  11. Morris [69]
    J. B. Morris, E-resolution: extension of resolution to include the equality relation, International Joint Conference on Artificial Intelligence, Washington, D.C., May 1969, pp. 287–294.Google Scholar
  12. Murray [82]
    N. V. Murray, Completely nonclausal theorem proving, Artificial Intelligence, Vol. 18, No. 1, 1982, pp. 67–85.CrossRefGoogle Scholar
  13. Robinson [65]
    J. A. Robinson, A machine-oriented logic based on the resolution principle, Journal of the ACM, Vol. 12, No. 1, January 1965, pp. 23–41.CrossRefGoogle Scholar
  14. Robinson [79]
    J. A. Robinson, Logic: Form and Function, North-Holland, New York, N.Y., 1979.Google Scholar
  15. Stickel [82]
    M. E. Stickel, A nonclausal connection-graph resolution theorem-proving program. National Conference on AI, Pittsburgh, Pa., 1982, pp. 229–233.Google Scholar
  16. Traugott [85]
    J. Traugott, Deductive synthesis of sorting algorithms, Technical Report, Computer Science Department, Stanford University, Stanford, Calif. (forthcoming).Google Scholar
  17. Wos and Robinson [69]
    L. Wos and G. Robinson, Paramodulation and theorem proving in first order theories with equality, in Machine Intelligence 4 (B. Meltzer and D. Michie, editors) American Elsevier, New York, N.Y., 1969, pp. 135–150.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Zohar Manna
    • 1
  • Richard Waldinger
    • 2
  1. 1.Computer Science DepartmentStanford UniversityUSA
  2. 2.Artificial Intelligence CenterSRI InternationalUSA

Personalised recommendations