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Automated proofs of the moufang identities in alternative rings

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Abstract

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.

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References

  1. Anantharaman S. and Mzali J., Unfailing Completion Modulo a set of Equations, Research Report, no. 470, LRI-Orsay (Fr.), 1989.

  2. Anantharaman, S., Hsiang J., and Mzali J., SbReve2: A Term Rewriting Laboratory with (AC-)Unfailing Completion, RTA (1989).

  3. Anantharaman, S. and Andrianarivelo, A., Heuristical Critical Pair Criteria in Automated Theorem Proving, Research Report, Université d'Orléans (Fr.), 1989.

  4. Bachmair, L. and Dershowitz, N., ‘Critical Pair Criteria for Completion’, J. Symbolic Computation, 6, 1–18 (1988).

    Google Scholar 

  5. Dershowitz, N., ‘Termination of Rewriting’, J. Symbolic Computation, 3, 59–116 (1987).

    Google Scholar 

  6. HallJr., M., The Theory of Groups, Macmillan, New York, 1959.

    Google Scholar 

  7. Hsiang, J. and Rusinowitch, M., ‘On word problems in equational theories, Proc. 14th ICALP, Springer-Verlag LNCS, Vol 267, pp. 54–71 (1987).

  8. Hsiang, J., Rusinowitch M., and Sakai, K., ‘Complete set of inference rules for the cancellation laws’, IJCAI 87, Milano, Italy, 1987.

  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. 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).

  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).

  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. 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. Rusinowitch M., ‘Démonstration Automatique: Techniques do réécriture’, Thèse d'Etat, Université de Nancy (1987).

  15. Stevens, R. L., ‘Some Experiments in Nonassociative Ring Theory with an Automated Theorem Prover’, J. Automated Reasoning, 3(2) (1987)

  16. Stevens, R. L., ‘Challenge Problems from Nonassociative Rings for Theorem Provers’, Proc. 9th CADE, Springer-Verlag, pp. 730–734 (1988).

  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).

  18. Wang, T. C., ‘Case Studies of Z-module Reasoning: Proving Benchmark Theorems from Ring Theory’, J. Automated Reasoning, 3(4) (1987).

  19. Wos, L. and McCune, W., ‘Negative Paramodulation’, Proc. 8th CADE, Springer-Verlag LNCS 230, pp. 229–239 (1986).

  20. Wos, L. and Robinson G. A., ‘Paramodulation and Set of Support’, Proc. IRIA-Symposium on Auto. Demonstration, Springer-Verlag LNCS (1968).

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Authors and Affiliations

  1. LIFO, Dépt. Math-Info., Université d'Orléans, 45067, Orléans Cedex 02, France

    Siva Anantharaman

  2. Department of Computer Science, National Taiwan University, Taipei, Taiwan, Republic of China

    Jieh Hsiang

Authors
  1. Siva Anantharaman
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  2. Jieh Hsiang
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Additional information

On leave from the Department of Computer Science, UNYY at Stony Brook, New York. Research supported in part by NSF grants CCR-8805734, INT-8715231, and CCR-8901322.

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Anantharaman, S., Hsiang, J. Automated proofs of the moufang identities in alternative rings. J Autom Reasoning 6, 79–109 (1990). https://doi.org/10.1007/BF00302643

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  • DOI: https://doi.org/10.1007/BF00302643

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