Theorem proving with ordering constrained clauses

  • Robert Nieuwenhuis
  • Albert Rubio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 607)


We use clauses with ordering constraints to reduce the search space in ordered inference systems for clauses with or without equality, such as ordered resolution or superposition. In our completion procedure for ordering constrained clauses redundant inferences can be ignored and redundant clauses can be deleted without loosing refutational completeness. Two new results needed for fast ordering constraint solving and incrementality of the set of function symbols are given. We discuss the use of our methods for reasoning about infinite sets of clauses defined by a finite number of ordering constrained ones.


Inference Rule Ground Term Ground Instance Empty Clause Completion Procedure 
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. [BG 90]
    L. Bachmair, H. Ganzinger: On restrictions of ordered paramodulation with simplification. In Proc. 10th CADE, Kaiserslautern, LNCS 449, 1990.Google Scholar
  2. [BG 91]
    L. Bachmair, H. Ganzinger: Rewrite-based equational theorem proving with selection and simplification, Report 208 Max-Planck Inst. Saarbrücken, 1991.Google Scholar
  3. [Com 90]
    H. Comon: Solving Symbolic Ordering Constraints. In proc. 5th IEEE Symp. Logic in Comp. Sc. Philadelphia. (1990).Google Scholar
  4. [DJ 90]
    N. Dershowitz, J-P. Jouannaud: Rewrite systems, in Handbook of Theoretical Comp. Sc. vol. B: Formal Methods and Semantics. North Holland, 1990.Google Scholar
  5. [HR 89]
    J. Hsiang, M. Rusinowitch: Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method. Submitted. (1989).Google Scholar
  6. [Hue 72]
    G. Huet: Constrained Resolution: A complete method for higher-order logic. PhD. Thesis. Case Western Reserve University. 1972.Google Scholar
  7. [Hul 80]
    J.M. Hullot: Compilation de Formes Canoniques dans les Teories Equationnelles, These de 3eme Cycle, Universite de Paris Sud, 1980.Google Scholar
  8. [JO 91]
    J-P. Jouannaud, M. Okada: Satisfiability of systems of ordinal notations with the subterm property is decidable, ICALP 1991, LNCS 510, Madrid, 1991.Google Scholar
  9. [KB 70]
    D.E. Knuth, P.B. Bendix: Simple word problems in universal algebras. J. Leech, editor, Computational Problems in Abstract Algebra, 263–297, Pergamon Press, Oxford, 1970.Google Scholar
  10. [KKR 90]
    C. Kirchner and H. Kirchner, M. Rusinowitch: Deduction with Symbolic Constraints, Revue Francaise d'Intelligence Artificielle, Vol 4, no. 3, pp. 9–52, 1990.Google Scholar
  11. [NN 91]
    R. Nieuwenhuis, P. Nivela: Efficient deduction in equality Horn logic by Horn-completion, Information Processing Letters, no. 39, pp. 1–6, July 1991.Google Scholar
  12. [NO 91]
    R. Nieuwenhuis, F. Orejas: Clausal Rewriting, 2nd Intl. Workshop on Conditional and Typed Term Rewriting, Montreal LNCS 516, pp. 246–261, (1991).Google Scholar
  13. [NOR 90]
    R. Nieuwenhuis, F. Orejas, A. Rubio: TRIP: an implementation of clausal rewriting, Proc. 10th Int. Conf. on Automated Deduction, Kaiserslautern, LNCS 449, pp. 667–668 (1990).Google Scholar
  14. [NR 92]
    R. Nieuwenhuis, A. Rubio: Basic Superposition is Complete, Proc. European Symposium On Programming, Rennes, France, LNCS, February 1992.Google Scholar
  15. [Pet 90]
    G.E. Peterson: Complete Sets of Reductions with Constraints, Proc. 10th Int. Conf. on Automated Deduction, Kaiserslautern, LNCS 449, pp. 381–395 (1990).Google Scholar
  16. [RN 91]
    A. Rubio, R. Nieuwenhuis: Handling ordering constraints of clauses in theorem proving, Research Report LSI-UPC, 1991.Google Scholar
  17. [Zha 88]
    H. Zhang: Reduction, Superposition, and Induction: Automated Reasoning in an Equational Logic, PhD. Thesis, Renselaer Polytechnic Institute, (1988).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Robert Nieuwenhuis
    • 1
  • Albert Rubio
    • 1
  1. 1.Dept. Lenguajes y Sistemas InformáticosTechnical University of CataloniaBarcelonaSpain

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