Journal of Automated Reasoning

, Volume 6, Issue 1, pp 79–109 | Cite as

Automated proofs of the moufang identities in alternative rings

  • Siva Anantharaman
  • Jieh Hsiang
Problem Corner


In this paper we present automatic proofs of the Moufang identities in alternative rings. Our approach is based on the term rewriting (Knuth-Bendix completion) method, enforced with various features. Our proofs seem to be the first computer proofs of these problems done by a general purpose theorem prover. We also present a direct proof of a certain property of alternative rings without employing any auxiliary functions. To our knowledge our computer proof seems to be the first direct proof of this property, by human or by a computer.

Key words

Automated theorem proving equational logic Knuth-Bendix completion term rewriting Moufang identities alternative rings 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Anantharaman S. and Mzali J., Unfailing Completion Modulo a set of Equations, Research Report, no. 470, LRI-Orsay (Fr.), 1989.Google Scholar
  2. 2.
    Anantharaman, S., Hsiang J., and Mzali J., SbReve2: A Term Rewriting Laboratory with (AC-)Unfailing Completion, RTA (1989).Google Scholar
  3. 3.
    Anantharaman, S. and Andrianarivelo, A., Heuristical Critical Pair Criteria in Automated Theorem Proving, Research Report, Université d'Orléans (Fr.), 1989.Google Scholar
  4. 4.
    Bachmair, L. and Dershowitz, N., ‘Critical Pair Criteria for Completion’, J. Symbolic Computation, 6, 1–18 (1988).Google Scholar
  5. 5.
    Dershowitz, N., ‘Termination of Rewriting’, J. Symbolic Computation, 3, 59–116 (1987).Google Scholar
  6. 6.
    HallJr., M., The Theory of Groups, Macmillan, New York, 1959.Google Scholar
  7. 7.
    Hsiang, J. and Rusinowitch, M., ‘On word problems in equational theories, Proc. 14th ICALP, Springer-Verlag LNCS, Vol 267, pp. 54–71 (1987).Google Scholar
  8. 8.
    Hsiang, J., Rusinowitch M., and Sakai, K., ‘Complete set of inference rules for the cancellation laws’, IJCAI 87, Milano, Italy, 1987.Google Scholar
  9. 9.
    Kapur, D., Musser, D. and Narendran, P., ‘Only Prime Superpositions Need be Considered in the Knuth-Bendix Procedure’, J. Symbolic Computation, 6, 19–36 (1988).Google Scholar
  10. 10.
    Knuth, D. E. and Bendix P. B., ‘Simple Word Problems in Universal Algebras’, Computational Problems in Abstract Algebras, Ed. J. Leech, Pergamon Press, pp 263–297 (1970).Google Scholar
  11. 11.
    Küchlin, W., ‘A Confluence criterion based on the generalised Newman Lemma’, EUROCAL'85 (ed. Caviness)2, Springer-Verlag, LNCS Vol 204, pp. 390–399 (1985).Google Scholar
  12. 12.
    Lankford, D. S. and Ballantyne, A. M., Decision Procedures for simple Equational Theories with Commutative-Associative axioms: Complete sets of commutative-associative reductions, Technical Report, Dept. of Maths., University of Texas, Austin, Texas (August 1977).Google Scholar
  13. 13.
    Peterson, G. and Stickel, M. E., ‘Complete sets of reductions for some equational theories’, J. Ass. Comp. Mach., 28(2), 233–264 (1981).Google Scholar
  14. 14.
    Rusinowitch M., ‘Démonstration Automatique: Techniques do réécriture’, Thèse d'Etat, Université de Nancy (1987).Google Scholar
  15. 15.
    Stevens, R. L., ‘Some Experiments in Nonassociative Ring Theory with an Automated Theorem Prover’, J. Automated Reasoning, 3(2) (1987)Google Scholar
  16. 16.
    Stevens, R. L., ‘Challenge Problems from Nonassociative Rings for Theorem Provers’, Proc. 9th CADE, Springer-Verlag, pp. 730–734 (1988).Google Scholar
  17. 17.
    Stickel, M. E., ‘A case study of theorem proving by the Knuth-Bendix method: Discovering that x 3=x implies ring commutativity’, Proc. 7th CADE, Springer-Verlag, LNCS Vol 170, pp. 248–258 (1984).Google Scholar
  18. 18.
    Wang, T. C., ‘Case Studies of Z-module Reasoning: Proving Benchmark Theorems from Ring Theory’, J. Automated Reasoning, 3(4) (1987).Google Scholar
  19. 19.
    Wos, L. and McCune, W., ‘Negative Paramodulation’, Proc. 8th CADE, Springer-Verlag LNCS 230, pp. 229–239 (1986).Google Scholar
  20. 20.
    Wos, L. and Robinson G. A., ‘Paramodulation and Set of Support’, Proc. IRIA-Symposium on Auto. Demonstration, Springer-Verlag LNCS (1968).Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Siva Anantharaman
    • 1
  • Jieh Hsiang
    • 2
  1. 1.LIFO, Dépt. Math-Info.Université d'OrléansOrléans Cedex 02France
  2. 2.Department of Computer ScienceNational Taiwan UniversityTaipei, TaiwanRepublic of China

Personalised recommendations