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The word problem for finitely presented monoids and finite canonical rewriting systems

Theoretical Aspects 1

Part of the Lecture Notes in Computer Science book series (LNCS,volume 256)

Abstract

The main purpose of this paper is to describe a negative answer to the following question:

Does every finitely presented monoid with a decidable word problem have a presentation (∑;R) where R is a finite canonical rewriting system?

To obtain this answer a certain homological finiteness condition for monoids is considered. If M is a monoid that can be presented by a finite canonical rewriting system, then M is an (FP)3-monoid. Since there are well-known examples of finitely presented groups that have easily decidable word problem, but that do not meet this condition, this implies that there are finitely presented monoids (and groups) with decidable word problem that cannot be presented by finite canonical rewriting systems.

References

  1. J. Avenhaus, K. Madlener; Subrekursive Komplexitaet bei Gruppen; I. Gruppen mit vorgeschriebener Komplexitaet; Acta Informatica 9 (1977), 87–104.

    Google Scholar 

  2. G. Bauer; Zur Darstellung von Monoiden durch konfluente Regelsysteme; dissertation, Fachbereich Informatik, Universitaet Kaiserslautern, 1981.

    Google Scholar 

  3. G. Bauer; N-level rewriting system; Theoret. Comput. Science 40 (1985), 85–99.

    Google Scholar 

  4. G. Bauer, F. Otto; Finite complete rewriting systems and the complexity of the word problem; Acta Informatica 21 (1984), 521–540.

    Google Scholar 

  5. R. Bieri; Homological dimension of discrete groups; Queen Mary College Mathematics Notes, London, 1976.

    Google Scholar 

  6. R. Bieri; A connection between the integral homology and the centre of a rational linear group; Math. Z. 170 (1980), 263–266.

    Google Scholar 

  7. R.V. Book; Thue systems as rewriting systems; in J.P. Jouannaud (ed.), Rewriting Techniques and Applications, Lecture Notes Comp. Science 202 (1985), 63–94.

    Google Scholar 

  8. R.V. Book, M. Jantzen, C. Wrathall; Monadic Thue systems; Theoret. Comput. Science 19 (1982), 231–251.

    Google Scholar 

  9. K.S. Brown; Finiteness properties of groups, preprint.

    Google Scholar 

  10. V. Diekert; Complete semi-Thue systems for abelian groups; Theoret. Comput. Science 44 (1986), 199–208.

    Google Scholar 

  11. A. Grzegorczyk; Some classes of recursive functions; Rozprawy Math. 4 (1953), 1–45.

    Google Scholar 

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

    Google Scholar 

  13. M. Jantzen; Thue congruences and complete string rewriting systems; Habilitationsschrift; Fachbereich Informatik, Universitaet Hamburg, 1986.

    Google Scholar 

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

    Google Scholar 

  15. S. Kemmerich; Unendliche Reduktionssysteme; dissertation, Fachbereich Mathematik, TH Aachen, 1983.

    Google Scholar 

  16. Ph. LeChenadec; Canonical Forms in Finitely Presented Algebras; Pitman, London, John Wiley & Sons,Inc., New York-Toronto, 1986.

    Google Scholar 

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

    Google Scholar 

  18. K. Madlener, F. Otto; Pseudo-natural algorithms for finitely generated presentations of monoids and groups; J. Symbolic Comput., to appear.

    Google Scholar 

  19. C. O'Dunlaing; Infinite regular Thue systems; Theoret. Comput. Science 25 (1983), 171–192.

    Google Scholar 

  20. C.C. Squier; Word problems and a homological finiteness condition for monoids; J. Pure Appl. Algebra, to appear.

    Google Scholar 

  21. J.R. Stallings; A finitely presented group whose 3-dimensional integral homology is not finitely generated; American J. of Mathematics 85 (1963), 541–543.

    Google Scholar 

  22. K. Weihrauch; Teilklassen primitiv-rekursiver Wortfunktionen; Report No. 91, GMD Bonn, 1974.

    Google Scholar 

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© 1987 Springer-Verlag Berlin Heidelberg

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Squier, C., Otto, F. (1987). The word problem for finitely presented monoids and finite canonical rewriting systems. In: Lescanne, P. (eds) Rewriting Techniques and Applications. RTA 1987. Lecture Notes in Computer Science, vol 256. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-17220-3_7

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  • DOI: https://doi.org/10.1007/3-540-17220-3_7

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