Well-Foundedness Is Sufficient for Completeness of Ordered Paramodulation

  • Miquel Bofill
  • Albert Rubio
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2392)


For many years all known completeness results for Knuth-Bendix completion and ordered paramodulation required the term ordering ≻ to be well-founded, monotonic and total(izable) on ground terms. Then, it was shown that well-foundedness and the subterm property were enough for ensuring completeness of ordered paramodulation.

Here we show that the subterm property is not necessary either. By using a new restricted form of rewriting we obtain a completeness proof of ordered paramodulation for Horn clauses with equality where well-foundedness of the ordering suffices. Apart from the fundamental interest of this result, some potential applications motivating the interest of dropping the subterm property are given.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [BD94]
    Leo Bachmair and Nachum Dershowitz. Equational inference, canonical proofs, and proof orderings. J. of the Association for Computing Machinery, 41(2):236–276, February 1994.Google Scholar
  2. [BDH86]
    Leo Bachmair, Nachum Dershowitz, and Jieh Hsiang. Orderings for equational proofs. In First IEEE Symposium on Logic in Computer Science (LICS), pages 346–357, Cambridge, Massachusetts, USA, June 16–18, 1986. IEEE Computer Society Press.Google Scholar
  3. [BG94]
    Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217–247, 1994.MATHCrossRefMathSciNetGoogle Scholar
  4. [BG98]
    Leo Bachmair and Harald Ganzinger. Equational reasoning in saturation-based theorem proving. In W. Bibel and P. Schmitt, editors, Automated Deduction: A Basis for Applications. Kluwer, 1998.Google Scholar
  5. [BGLS95]
    L. Bachmair, H. Ganzinger, Chr. Lynch, and W. Snyder. Basic paramod-ulation. Information and Computation, 121(2):172–192, 1995.MATHCrossRefMathSciNetGoogle Scholar
  6. [BGNR99]
    Miquel Bofill, Guillem Godoy, Robert Nieuwenhuis, and Albert Rubio. Paramodulation with non-monotonic orderings. In 14th IEEE Symposium on Logic in Computer Science (LICS), pages 225–233, Trento, Italy, July 2–5, 1999.Google Scholar
  7. [BR02]
    Miquel Bofill and Albert Rubio. Well-foundedness is sufficient for completeness of Ordered Paramodulation. Long version, 2002. Available at http://www.lsi.upc.es/~albert/papers.html.
  8. [DJ90]
    Nachum Dershowitz and Jean-Pierre Jouannaud. Rewrite systems. In Jan van Leeuwen, editor, Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, chapter 6, pages 244–320. Elsevier Science Publishers B.V., Amsterdam, New York, Oxford, Tokyo, 1990.Google Scholar
  9. [HR91]
    J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559–587, July 1991.Google Scholar
  10. [NR95]
    Robert Nieuwenhuis and Albert Rubio. Theorem Proving with Ordering and Equality Constrained Clauses. Journal of Symbolic Computation, 19(4):321–351, April 1995.Google Scholar
  11. [NR01]
    Robert Nieuwenhuis and Albert Rubio. Paramodulation-based theorem proving. In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science Publishers and MIT Press, 2001.Google Scholar
  12. [Wec91]
    W. Wechler. Universal Algebra for Computer Scientists, volume 25 of EATCS Monographs on Theoretical Computer Science. Springer-Verlag, Berlin, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Miquel Bofill
    • 1
  • Albert Rubio
    • 2
  1. 1.Dept. IMAUniversitat de GironaGironaSpain
  2. 2.Dept. LSITechnical University of CataloniaBarcelonaSpain

Personalised recommendations