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Some group theoretic examples with completion theorem provers

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Abstract

In this paper we investigate the performance of completion theorem provers on a number of group theoretic problems. These are of a rather different character to the usual test problems and exercise different features of the programs. Very large rewriting systems and very deeply nested terms arise, but, where the programs allow, additional mathematical information can often by used to dramatically speed the computations. We compare two general-purpose theorem provers with some more specialised tools and conclude by drawing some lessons for the design of future general-purpose provers.

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References

  1. Anick, D. J.: On the homology of associative algebra, Trans. AMS 296 (1986), 641–659.

    Google Scholar 

  2. Aschbacher, M.: Finite group theory, Cambridge Studies in Advanced Mathematics 10, Cambridge, 1986.

  3. Bachmair, L., Dershowitz, N., and Plaisted, D.: Completion without failure, in Ait-Kaci Hassan and Nivat, Maurice (eds), Proc. Coll. on Resolution of Equations in Algebraic Structures, 2. Rewriting Techniques, Academic Press, New York, 1989, pp. 1–30.

    Google Scholar 

  4. Brown, K. S.: The geometry of rewriting systems: A proof of the Anick-Groves-Squier theorem, in G. Baumslag and C. F. Miller (eds), Algorithms and Classification in Combinatorial Group Theory, Springer-Verlag, 1991, pp. 137–163.

  5. Bündgen, R.: e-mail communication, 1994.

  6. Bündgen, R.: Term Completion Versus Algebraic Completion, Technical Report WSI 91–3, Wilhelm-Schickard-Institut, Universität Tübingen, Tübingen, 1991.

    Google Scholar 

  7. Bündgen, R. and Walter, J.: The ReDuX User Guide (Version 1.3), Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany, 1993.

    Google Scholar 

  8. Campbell, C. M., Coxeter, H. S. M., and Robertson, E. F.: Some families of finite groups having two generators and two relations, Proc. R. Soc. London A357 (1977), 423–438.

    Google Scholar 

  9. Campbell, C. M. and Robertson, E. F.: Finitely presented groups of Fibonacci type II, J. Austral. Math. Soc. 28 (1979), 250–256.

    Google Scholar 

  10. Cannon, J. J., Dimino, L. A., Havas, G., and Watson, J. M.: Implementation and analysis of the Todd-Coxeter algorithm, Math. Comp. 27 (1973), 463–490.

    Google Scholar 

  11. Conway, J. H.: Advanced problem 5327, Amer. Math. Monthly 72 (1965), 91.

    Google Scholar 

  12. Conway, J. H.: Solution to advanced problem 5327, Amer. Math. Monthly 74 (1967), 91–93.

    Google Scholar 

  13. Epstein, D. B. A., Holt, D. F., and Rees, S. E.: The use of Knuth-Bendix methods to solve the word problem in automatic groups, J. Symbolic Computation 12 (1991), 397–414.

    Google Scholar 

  14. Garland, S. J. and Guttag, J. V.: A Guide to LP, The Larch Prover, Technical Report 82, DEC Systems Research Center, Palo Alto, CA., Dec. 1991.

    Google Scholar 

  15. Groves, J. R. J.: Rewriting systems and homology of groups, in L. G.Kovács (ed.), Groups — Canberra 1989, Lecture Notes in Mathematics 1456, Springer-Verlag, Heidelberg, 1990, pp. 114–141.

    Google Scholar 

  16. Groves, J. R. J. and Smith, G. C.: Soluble groups with a finite rewriting system, Proc. Edinburgh Math. Soc. 36 (1993), 283–288.

    Google Scholar 

  17. Havas, G.: Coset Enumeration Strategies, Technical Report 200, Key Centre for Software Technology, University of Queensland, 1991.

  18. Havas, G. and Newman, M. F.: Applications of computers to questions like those of Burnside, in Mennicke J. L. (ed.), Burnside Groups, Lecture Notes in Mathematics 806, Springer-Verlag, Berlin, pp. 211–230.

  19. Heineken, H.: Engelsche Elemente der Länge Drei, Illinois J. Math. 5 (1961), 681–707.

    Google Scholar 

  20. Higman, H.: A finitely generated infinite simple group, J. London Math. Soc. 26 (1951), 61–64.

    Google Scholar 

  21. Humphreys, J. E.: Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge, 1990.

  22. Jantzen, M.: A note on a special one-rule semi-Thue system, Inf. Proc. Letters 21 (1985), 135–140.

    Google Scholar 

  23. Johnson, D. L.: Presentations of Groups, Cambridge University Press, Cambridge, 1990.

    Google Scholar 

  24. Johnson, D. L., Walmsey, J. W., and Wright, D.: The Fibonacci groups, Proc. London Math. Soc. 29 (1974), 577–592.

    Google Scholar 

  25. Kapur, D. and Narendran, P.: A finite thue system with decidable word problem and without equivalent finite canonical system, Theoret. Comput. Science 35 (1985), 337–344.

    Google Scholar 

  26. Knuth, D. and Bendix, P.: Simple word problems in universal algebras, in J.Leech (ed.), Computational Problems in Abstract Algebra, Pergamon, Oxford, 1970, pp. 263–297.

    Google Scholar 

  27. Kropholler, P. H.: On groups of type (F P)∞, J. Pure Appl. Algebra 90 (1993), 55–67.

    Google Scholar 

  28. Le Chenadec, P.: Canonical forms in finitely presented algebras, in Proc. 7th Int. C.A.D.E., LNCS 170, Springer, 1984, pp. 142–165.

  29. Ledermann, W.: Introduction to Group Theory, Longman, 1973.

  30. McCune, W. M.: Otter 3.0 Reference Manual and Guide, Argonne National Laboratory, Report ANL-94/6, 1994.

  31. Madlener, K. and Otto, F.: Pseudo-natural algorithms for the word problem for finitely presented monoids and groups, J. Symbolic Computation 1 (1985), 383–418.

    Google Scholar 

  32. Madlener, K. and Otto, F.: Pseudo-natural algorithms for finitely generated presentations of monoids and groups, J. Symbolic Computation 5 (1988), 339–418.

    Google Scholar 

  33. Martin, U. and Lai, M.: Some experiments with a completion theorem prover, J. Symbolic Computation 13 (1992), 81–100.

    Google Scholar 

  34. Martin, U. and Nipkow, T.: Ordered rewriting and confluence, in Automated Deduction, CADE-10, LNAI 449, Springer-Verlag, 1990, pp. 366–380.

  35. Meinke, K. and Tucker, J. V.: Universal algebra, in S.Abramsky, D. V.Gabbay, and T. S. E.Maibaum (eds), Handbook of Logic in Computer Science, Vol. 1, Clarendon, Oxford, 1992, pp. 189–398.

    Google Scholar 

  36. Sims, C. C.: Verifying nilpotence, J. Symbolic Computation 3 (1987), 231–247.

    Google Scholar 

  37. Peterson, G. E. and Stickel, M. E.: Complete sets of reductions for some equational theories, J. ACM 28 (1981), 233–264.

    Google Scholar 

  38. Sims, C. C.: The Knuth-Bendix procedure for strings as a substitute for coset enumeration, J. Symbolic Computation 12 (1991), 439–442.

    Google Scholar 

  39. Sims, C. C.: Computation with Finitely Presented Groups, Cambridge University Press, Cambridge, 1994.

    Google Scholar 

  40. Squier, C. C.: Word problems and a homological finiteness condition for monoids, J. Pure Appl. Algebra 49 (1987), 201–217.

    Google Scholar 

  41. Squier, C. C., A finiteness condition for rewriting systems, preprint.

  42. Sutcliffe, G., Suttner, C., and Yememis, T.: The TPTP problem library, in A. Bundy (ed.), Automated Deduction, CADE-12, LNAI 814, Springer-Verlag, 1994, pp. 252–266.

  43. Thomas, R. M.: The Fibonacci groups revisited, in. C. M.Campbell and E. F.Robertson (eds), Groups St. Andrews 1989, Vol. 2, London Mathematical Society Lecture Note Series 160, Cambridge University Press, Cambridge, 1991, pp. 445–454.

    Google Scholar 

  44. Todd, J. A. and Coxeter, H. S. M.: A practical method for enumerating the cosets of a finite abstract group, Proc. Edinburgh Math. Soc. 5 (1936), 26–34.

    Google Scholar 

  45. Vaughan-Lee, M.: The Restricted Burnside Problem, Oxford, 1990.

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  1. Division of Computer Science, School of Mathematical and Computational Sciences, University of St. Andrews, KY169SS, Scotland, U.K.

    S. Linton & D. Shand

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  1. S. Linton
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  2. D. Shand
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Linton, S., Shand, D. Some group theoretic examples with completion theorem provers. J Autom Reasoning 17, 145–169 (1996). https://doi.org/10.1007/BF00244494

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