An ordered theory resolution calculus

  • Peter Baumgartner
Session 5: Resolution Theorem Proving
Part of the Lecture Notes in Computer Science book series (LNCS, volume 624)


In this paper we present an ordered theory resolution calculus and prove its completeness. Theory reasoning means to relieve a calculus from explicitly drawing inferences in a given theory by special purpose inference rules (e.g. E-resolution for equality reasoning). We take advantage of orderings (e.g. simplification orderings) by disallowing to resolve upon clauses which violate certain maximality constraints; stated positively, a resolvent may only be built if all the selected literals are maximal in their clauses. By this technique the search space is drastically pruned. As an instantiation for theory reasoning we show that equality can be built in by rigid E-unification.


Automated Theorem Proving Theory Resolution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [AB70]
    R. Anderson and W. Bledsoe. A linear fonnat for resolution with merging and a new technique for establishing completeness. J. of the ACM, 17:525–534, 1970.CrossRefGoogle Scholar
  2. [And81]
    P. Andrews. Theorem Proving via General Matings. J. ACM, 28(2):193–214, 1981.CrossRefGoogle Scholar
  3. [Bau91]
    P. Baumgartner. A Model Elimination Calculus with Built-in Theories. Fachbericht Informatik 7/91, Universität Koblenz, 1991.Google Scholar
  4. [BFL83]
    R. Brachmann, R. Fikes, and H. Levesque. KRYPTON: a functional approach to knowledge representation. IEEE Computer, 16(10):67–73, October 1983.Google Scholar
  5. [BG90]
    L. Bachmair and H. Ganzinger. Completion of First-Order Clauses with Equality by Strict Superposition. In Proc. Second Int. Workshop on Conditional and Typed Rewrite Systems, LNCS. Springer, 1990.Google Scholar
  6. [CL73]
    C. Chang and R. Lee. Symbolic Logic and Mechanical TheoremProving. Academic Press, 1973.Google Scholar
  7. [Der87]
    Nachum Dershowitz. Termination of Rewriting. Journal of Symbolic Computation, 3(1&2):69–116, February/April 1987.Google Scholar
  8. [GNPS90]
    J. Gallier, P. Narendran, D. Plaisted, and W. Snyder. Rigid E-unification: NP-Completeness and Applications to Equational Matings. Information and Computation, pages 129–195, 1990.Google Scholar
  9. [HR86]
    J. Hsiang and M. Rusinowitch. A New Method for Establishing Refutational Completeness in Theorem Proving. In Proc. 8th CADE, pages 141–152. Springer, 1986.Google Scholar
  10. [MR87]
    N. Murray and E. Rosenthal. Theory Links: Applications to Automated Theorem Proving. J. of Symbolic Computation, 4:173–190, 1987.Google Scholar
  11. [OS91]
    H. J. Ohlbach and J. Siekmann. The Markgraf Karl Refutation Procedure. In J.L. Lassez and G. Plotkin, editors, Computational Logic— Essays in Honor of Alan Robinson, pages 41–112. MIT Press, 1991.Google Scholar
  12. [Pet90]
    U. Petermann. Towards a connection procedure with built in theories. In JELIA 90. European Workshop on Logic in AI, Springer, LNCS, 1990.Google Scholar
  13. [Rob65]
    J.A. Robinson. A machine-oriented logic based on the resolution principle. JACM, 12(1):23–41, January 1965.CrossRefGoogle Scholar
  14. [Sti83]
    M.E. Stickel. Theory Resolution: Building in Nonequational Theories. SRI International Research Report Technical Note 286, Artificial Intelligence Center, 1983.Google Scholar
  15. [Sti85]
    M.E. Stickel. Automated deduction by theory resolution. Journal of Automated Reasoning, pages 333–356, 1985.Google Scholar
  16. [ZK88]
    H. Zhang and D. Kapur. First-Order Theorem Proving Using Conditional Rewrite Rules. In E. Lusk and R. Overbeek, editors, Lecture Notes in Computer Science: 9th International Conference on Automated Deduction, pages 1–20. Springer-Verlag, May 1988.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Peter Baumgartner
    • 1
  1. 1.Institut für InformatikUniversität KoblenzKoblenz

Personalised recommendations