A new method for establishing refutational completeness in theorem proving

  • Jieh Hsiang
  • Michael Rusinowitch
Term Rewriting Systems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 230)


We present here a new technique for establishing completeness of refutational theorem proving strategies. This method employs semantic trees and, in contrast to most of the semantic tree methods, is based on proof by refutation instead of proof by induction. Thus, it works well on transfinite semantic trees as well as on finite ones. This method is particularly useful for proving the completeness of strategies with the presence of the equality predicate. We have used the method to prove the completeness of the following strategies (without the need of the functional reflexive axioms), where the precise definition of oriented paramodulation will be given later.

• Resolution + oriented paramodulation

• P1-resolution + oriented paramodulation

• Resolution with ordered predicates + oriented paramodulation using clauses only containing the equality predicate

• unfailing Knuth-Bendix-Huet algorithm

• The EN-Strategy ([Hsi85])


Inference Rule Theorem Prove Failure Node Predicate Symbol Ground Term 
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.


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1. References

  1. [Bra75]
    D. Brand, “Proving Theorems with the Modification Method”, SIAM J. of Computing, 4, (1975), 412–430.CrossRefGoogle Scholar
  2. [ChL73]
    C. L. Chang and C. T. Lee, Symbolic Logic and Mechanical Theorem Proving, Academic Press, 1973.Google Scholar
  3. [Der82]
    N. Dershowitz, “Orderings for Term Rewriting Systems”, J.TCS, 17, 3 (1982), 279–301.Google Scholar
  4. [Hsi85]
    J. Hsiang, “Two Results in Term Rewriting Theorem Proving”, Proc. of 1st International Conference in Rewrite Techniques and Applications, May, 1985.Google Scholar
  5. [JLR82]
    J. P. Jouannaud, P. Lescanne and F. Reinig, “Recursive Decomposition Ordering”, Conf. on Formal Description of Programming Concepts II, 1982, 331–346.Google Scholar
  6. [KnB70]
    D. E. Knuth and P. B. Bendix, “Simple Word Problems in Universal Algebras”, in Computational Algebra, J. Leach, (ed.), Pergamon Press, 1970, 263–297.Google Scholar
  7. [KoH69]
    R. A. Kowalski and P. Hayes, “Semantic Trees in Automatic Theorem Proving”, in Machine Intelligence, vol. 5, B. Meltzer and D. Michie, (eds.), American Elsevier, 1969, 181–201.Google Scholar
  8. [LaB79]
    D. S. Lankford and A. M. Ballantyne, “The Refutation Completeness of Blocked Permutative Narrowing and Resolution”, 4th Conf. on Automated Deduction, Austin, TX, 1979.Google Scholar
  9. [Pet83]
    G. E. Peterson, “A Technique for Establishing Completeness Results in Theorem Proving with Equality”, SIAM J. of Computing, 12, 1 (1983), 82–100.CrossRefGoogle Scholar
  10. [Pla78]
    D. A. Plaisted, “A Recursively Defined Ordering for Proving Termination of Term Rewriting Systems”, UIUCDCS-R-78-943, Univ. of Illinois, Urbana, IL, 1978.Google Scholar
  11. [Rob65]
    J. A. Robinson, “A Machine Oriented Logic based on the Resolution Principle”, J. ACM, 12, 1 (January 1965), 23–41.CrossRefGoogle Scholar
  12. [Sla74]
    J. Slagle, “Automated Theorem Proving with Simplifiers, Commutativity, Associativity”, J. ACM, 21, (1974), 622–642.CrossRefGoogle Scholar
  13. [WoR70]
    L. Wos and G. A. Robinson, “Paramodulation and Set of Support”, Lecture Notes in Math. No 125, Springer-Verlag, 1970.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1986

Authors and Affiliations

  • Jieh Hsiang
    • 1
  • Michael Rusinowitch
    • 2
  1. 1.Department of Computer ScienceSUNY at Stony BrookStony BrookUSA
  2. 2.CRINVandoeuvre-les-NancyFrance

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