Journal of Automated Reasoning

, Volume 30, Issue 1, pp 99–120 | Cite as

Paramodulation and Knuth–Bendix Completion with Nontotal and Nonmonotonic Orderings

  • Miquel Bofill
  • Guillem Godoy
  • Robert Nieuwenhuis
  • Albert Rubio


Up to now, all existing completeness results for ordered paramodulation and Knuth–Bendix completion have required term ordering ≻ to be well founded, monotonic, and total(izable) on ground terms. For several applications, these requirements are too strong, and hence weakening them has been a well-known research challenge.

Here we introduce a new completeness proof technique for ordered paramodulation where the only properties required on ≻ are well-foundedness and the subterm property. The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS).

By a careful further analysis of our technique, we obtain the first Knuth–Bendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly non-totalizable) reduction ordering ≻ whenever it exists. Note that being a reduction ordering is the minimal possible requirement on ≻, since a TRS terminates if, and only if, it is contained in a reduction ordering.

term rewriting automated deduction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bachmair, L. and Dershowitz, N.: Equational inference, canonical proofs, and proof orderings, J. Assoc. Comput. Mach. 41(2) (1994), 236-276.Google Scholar
  2. Bachmair, L., Dershowitz, N. and Hsiang, J.: Orderings for equational proofs, in First IEEE Symposium on Logic in Computer Science (LICS), Cambridge, Massachusetts, USA, IEEE Computer Society Press, 1986, pp. 346-357.Google Scholar
  3. Bachmair, L. and Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification, J. Logic Comput. 4(3) (1994), 217-247.Google Scholar
  4. Bachmair, L. and Ganzinger, H.: Equational reasoning in saturation-based theorem proving, in W. Bibel and P. Schmitt (eds), Automated Deduction: A Basis for Applications, Kluwer Academic Publishers, Dordrecht, 1998.Google Scholar
  5. Bachmair, L., Ganzinger, H., Lynch, C. and Snyder, W.: Basic paramodulation, Inform. and Comput. 121(2) (1995), 172-192.CrossRefGoogle Scholar
  6. Basin, D. and Ganzinger, H.: Complexity analysis based on ordered resolution, in Eleventh Annual IEEE Symposium on Logic in Computer Science (LICS), New Brunswick, New Jersey, USA, IEEE Computer Society Press, 1996, pp. 456-465.Google Scholar
  7. Bofill, M. and Godoy, G.: On the completeness of arbitrary selection strategies for paramodulation, in Automata, Languages and Programming, 28th Int. Colloquium (ICALP), Crete, Greece, Springer-Verlag, New York, 2001.Google Scholar
  8. Comon, H. and Nieuwenhuis, R.: Induction = I-axiomatization + first-order consistency, Inform. and Comput. 159(1/2) (2000), 151-186.CrossRefGoogle Scholar
  9. Dershowitz, N.: Orderings for term-rewriting systems, Theoret. Comput. Sci. 17(3) (1982), 279-301.CrossRefGoogle Scholar
  10. Dershowitz, N. and Jouannaud, J.-P.: Rewrite systems, in J. van Leeuwen (ed.), Handbook of Theoretical Computer Science, Vol. B: Formal Models and Semantics, Elsevier Science Publishers B.V., Amsterdam, Chapt. 6, pp. 244-320.Google Scholar
  11. Dershowitz, N. and Manna, Z.: Proving termination with multiset orderings, Comm. ACM 22(8) (1979), 465-476.CrossRefGoogle Scholar
  12. Dershowitz, N., Marcus, L. and Tarlecki, A.: Existence, uniqueness, and construction of rewrite systems, SIAM J. Comput. 17(4) (1988), 629-639.Google Scholar
  13. Devie, H.: Linear completion, in S. Kaplan and M. Okada (eds), Conditional and Typed Rewriting Systems, 2nd International Workshop, Montreal, Canada, Springer-Verlag, New York, 1990, pp. 233-245.Google Scholar
  14. Devie, H.: Ph.D. thesis, Université de Paris-Sud, Orsay, France, 1992.Google Scholar
  15. Hillenbrand, T., Buch, A., Vogt, R. and Löchner, B.: WALDMEISTER - High-performance equational deduction, J. Automated Reasoning 18(2) (1997), 265-270.CrossRefGoogle Scholar
  16. Hsiang, J. and Rusinowitch, M.: Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method, J. ACM 38(3) (1991), 559-587.CrossRefGoogle Scholar
  17. Kamin, S. and Levy, J.-J.: Two generalizations of the recursive path ordering, Unpublished note, Dept. of Computer Science, Univ. of Illinois, Urbana, IL, 1980.Google Scholar
  18. Kirchner, C., Kirchner, H. and Rusinowitch, M.: Deduction with symbolic constraints, Rev. Française d’Intelligence Artificielle 4(3) (1990), 9-52.Google Scholar
  19. Lynch, C.: Oriented equational logic programming is complete, J. Symbolic Comput. 23(1) (1997), 23-46.CrossRefGoogle Scholar
  20. Marché, C.: On ground AC-completion, in R. V. Book (ed.), 4th Int. Conf. Rewriting Techniques and Applications (RTA), Como, Italy, Springer-Verlag, New York, 1991, pp. 411-422.Google Scholar
  21. Narendran, P. and Rusinowitch, M.: Any ground associative commutative theory has a finite canonical system, in 4th Int. Conf. Rewriting Techniques and Applications (RTA), Como, Italy, Springer-Verlag, New York, 1991, pp. 423-434.Google Scholar
  22. Narendran, P. and Rusinowitch, M.: The unifiability problem in ground AC theories, in Eighth Annual IEEE Symposium on Logic in Computer Science, Montreal, Canada, IEEE Computer Society Press, 1993, pp. 364-370.Google Scholar
  23. Nieuwenhuis, R.: Decidability and complexity analysis by basic paramodulation, Inform. and Comput. 147 (1998), 1-21.CrossRefGoogle Scholar
  24. Nieuwenhuis, R. and Rubio, A.: Theorem proving with ordering and equality constrained clauses, J. Symbolic Comput. 19(4) (1995), 321-351.CrossRefGoogle Scholar
  25. Nieuwenhuis, R. and Rubio, A.: Paramodulation-based theorem proving, in J. Robinson and A. Voronkov (eds), Handbook of Automated Reasoning, Elsevier Science Publishers and MIT Press, 2001.Google Scholar
  26. RTA-LOOP: Int. Conf. on Rewriting Techniques and Applications, The List of Open Problems, Maintained at∼rtaloop/ (by R. Treinen), 2001.Google Scholar
  27. Rubio, A. and Nieuwenhuis, R.: A total AC-compatible ordering based on RPO, Theoret. Comput. Sci. 142(2) (1995), 209-227.CrossRefGoogle Scholar
  28. Wechler, W.: Universal Algebra for Computer Scientists, EATCS Monogr. Theoret. Comput. Sci. 25, Springer-Verlag, Berlin, 1991.Google Scholar
  29. Weidenbach, C.: SPASS - Version 0.49, J. Automated Reasoning 18(2) (1997), 247-252.CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Miquel Bofill
    • 1
  • Guillem Godoy
    • 2
  • Robert Nieuwenhuis
    • 2
  • Albert Rubio
    • 2
  1. 1.Dept. IMA, Lluís Santaló s/nUniversitat de GironaGironaSpain
  2. 2.Dept. LSITechnical University of CataloniaBarcelonaSpain

Personalised recommendations