Skip to main content
SpringerLink
Log in

Undecidability Results for Monoids with Linear-Time Decidable Word Problems

  • Conference paper
  • First Online:
Algorithms and Computation (ISAAC 2000)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1969))

Included in the following conference series:

  • 705 Accesses

Abstract

Using a particularly adopted simulation of Turing machines by finite string-rewriting systems we show that all strong Markov properties are undecidable for the class C lin of finitely presented monoids with linear-time decidable word problems. Expanding on this construction it is then shown that also many other properties are undecidable for Clin, among them the property of having a context-free (or a regular) cross-section, the existence of a finite convergent presentation, and the homological and homotopical finiteness conditions left- and right-FPn (n≥3), left- and right-FP∞, FDT and FHT.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. Avenhaus and K. Madlener. Subrekursive Komplexität bei Gruppen: I. Gruppen mit vorgeschriebener Komplexität. Acta Inform., 9:87–104, 1977.

    Article  MATH  MathSciNet  Google Scholar 

  2. D. J. Anick. On the homology of associative algebra. Trans. Amer. Math. Soc., 296:641–659, 1987.

    Article  MathSciNet  Google Scholar 

  3. R. V. Book and F. Otto. String-Rewriting Systems. Springer-Verlag, New York, 1993.

    MATH  Google Scholar 

  4. R. Cremanns and F. Otto. FP∞ is undecidable for finitely presented monoids with word problems decidable in polynomial time. Mathematische Schriften Kassel 11/98, Universität Kassel, September 1998.

    Google Scholar 

  5. V. S. Guba and M. Sapir. Diagram groups. Memoirs Amer. Math. Soc., 130:1–117, 1997.

    MathSciNet  Google Scholar 

  6. D. Knuth and P. Bendix. Simple word problems in universal algebras. In J. Leech, editor, Computational Problems in Abstract Algebra, pages 263–297. Pergamon Press, New York, 1970.

    Google Scholar 

  7. M. Katsura, Y. Kobayashi, and F. Otto. Undecidable properties of monoids with word problems solvable in linear time-II. Cross-sections and homological and homotopical finiteness conditions. Mathematische Schriften Kassel, Universität Kassel, 2000. In preparation.

    Google Scholar 

  8. D. Kapur and P. Narendran. The Knuth-Bendix completion procedure and Thue systems. SIAM J. Comput., 14:1052–1072, 1985.

    Article  MATH  MathSciNet  Google Scholar 

  9. Y. Kobayashi and F. Otto. On homotopical and homological finiteness conditions for finitely presented monoids. Mathematische Schriften Kassel 1/00, Universität Kassel, January 2000.

    Google Scholar 

  10. Y. Kobayashi. Complete rewriting systems and homology of monoid algebras. J. Pure Applied Algebra, 65:263–275, 1990.

    Article  MATH  Google Scholar 

  11. A. Markov. Impossibility of algorithms for recognizing some properties of associative systems. Doklady Adakemii Nauk SSSR, 77:953–956, 1951.

    MATH  Google Scholar 

  12. F. Otto, M. Katsura, and Y. Kobayashi. In finite convergent string-rewriting systems and cross-sections for finitely presented monoids. J. Symb. Comput., 26:621–648, 1998.

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Otto and A. Sattler-Klein. The property FDT is undecidable for finitely presented monoids that have polynomial-time decidable word problems. Intern. J. Algebra Comput., 10:285–307, 2000.

    MATH  MathSciNet  Google Scholar 

  14. S. J. Pride. Low-dimensional homotopy theory for monoids. Intern. J. Algebra Comput., 5:631–649, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. J. Pride and J. Wang. Rewriting systems, finiteness conditions, and associated functions. In J. C. Birget, S. Margolis, J. Meakin, and M. Sapir, editors, Algorithmic Problems in Groups and Semigroups, pages 195–216. Birkhäuser, Boston, 2000.

    Google Scholar 

  16. A. Sattler-Klein. New undecidability results for finitely presented monoids. In H. Comon, editor, Rewriting Techniques and Applications, Lecture Notes in Computer Science 1232, pages 68–82. Springer-Verlag, Berlin, 1997.

    Google Scholar 

  17. C. C. Squier, F. Otto, and Y. Kobayashi. A finiteness condition for rewriting systems. Theoret. Comput. Sci., 131:271–294, 1994.

    Article  MATH  MathSciNet  Google Scholar 

  18. C. C. Squier. Word problems and a homological finiteness condition for monoids. J. Pure Applied Algebra, 49:201–217, 1987.

    Article  MATH  MathSciNet  Google Scholar 

  19. X. Wang and S. J. Pride. Second order Dehn functions of groups and monoids. Preprint, 1998.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kyoto-Sangyo University, 603, Kyoto, Japan

    Masashi Katsura

  2. Department of Information Science, Toho University, 274, Funabashi, Japan

    Yuji Kobayashi

  3. Fachbereich Mathematik/Informatik, Universität Kassel, 34109, Kassel, Germany

    Friedrich Otto

Authors
  1. Masashi Katsura
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuji Kobayashi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Friedrich Otto
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Karlsruhe University, Germany

    Gerhard Goos

  2. Cornell University, NY, USA

    Juris Hartmanis

  3. Utrecht University, The Netherlands

    Jan van Leeuwen

  4. Academia Sinica, Institute of Information Science, 128 Academia Road, Section 2, Nankang, 115, Taipei, Taiwan, R.O.C.

    D. T. Lee

  5. Department of Computer Science and Akamai Technologies, University of Illinois at Urbana Champaign, 500 Technology Square, 02139, Cambridge, MA, USA

    Shang-Hua Teng

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Katsura, M., Kobayashi, Y., Otto, F. (2000). Undecidability Results for Monoids with Linear-Time Decidable Word Problems. In: Goos, G., Hartmanis, J., van Leeuwen, J., Lee, D.T., Teng, SH. (eds) Algorithms and Computation. ISAAC 2000. Lecture Notes in Computer Science, vol 1969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-40996-3_24

Download citation

  • DOI: https://doi.org/10.1007/3-540-40996-3_24

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41255-7

  • Online ISBN: 978-3-540-40996-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation