Skip to main content
SpringerLink
Log in

Typed Monoids – An Eilenberg-Like Theorem for Non Regular Languages

  • Conference paper
Book cover Algebraic Informatics (CAI 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6742))

Included in the following conference series:

Abstract

Based on different concepts to obtain a finer notion of language recognition via finite monoids we develop an algebraic structure called typed monoid. This leads to an algebraic description of regular and non regular languages.

We obtain for each language a unique minimal recognizing typed monoid, the typed syntactic monoid. We prove an Eilenberg-like theorem for varieties of typed monoids as well as a similar correspondence for classes of languages with weaker closure properties than varieties.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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. Ajtai, M.: First-order definability on finite structures. Ann. Pure Appl. Logic 45(3), 211–225 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Almeida, J.: Finite Semigroups and Universal Algebra. World Scientific, Singapore (1995)

    Book  MATH  Google Scholar 

  3. Barrington, D.A.M., Compton, K.J., Straubing, H., Thérien, D.: Regular languages in NC1. J. Comput. Syst. Sci. 44(3), 478–499 (1992)

    Article  MATH  Google Scholar 

  4. Barrington, D.A.M., Immerman, N., Straubing, H.: On Uniformity within NC1. J. Comput. Syst. Sci. 41(3), 274–306 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barrington, D.A.M., Straubing, H., Thérien, D.: Non-uniform automata over groups. Inf. Comput. 89(2), 109–132 (1990)

    Article  MATH  Google Scholar 

  6. Behle, C., Krebs, A., Mercer, M.: Linear circuits, two-variable logic and weakly blocked monoids. In: Kučera, L., Kučera, A. (eds.) MFCS 2007. LNCS, vol. 4708, pp. 147–158. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Eilenberg, S.: Automata, Languages and Machines, vol. A+B. Academic Press, London (1976)

    MATH  Google Scholar 

  8. Ésik, Z., Larsen, K.G.: Regular languages definable by Lindström quantifiers. ITA 37(3), 179–241 (2003)

    MATH  Google Scholar 

  9. Furst, M.L., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. In: FOCS, pp. 260–270 (1981)

    Google Scholar 

  10. Gurevich, Y., Lewis, H.R.: A logic for constant-depth circuits. Information and Control 61(1), 65–74 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Immerman, N.: Languages that capture complexity classes. SIAM J. Comput. 16(4), 760–778 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  12. Krebs, A.: Typed Semigroups, Majority Logic, and Threshold Circuits. Ph.D. thesis, Universität Tübingen (2008)

    Google Scholar 

  13. Krebs, A., Lange, K.J., Reifferscheid, S.: Characterizing TC0 in terms of infinite groups. Theory Comput. Syst. 40(4), 303–325 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. McNaughton, R., Papert, S.: Counter-free automata. With an appendix by William Henneman. Research Monograph, vol. 65, XIX, 163 p. The M.I.T. Press, Cambridge (1971)

    MATH  Google Scholar 

  15. Pin, J.E.: Varieties of formal languages. Plenum, London (1986)

    Book  MATH  Google Scholar 

  16. Pin, J.E.: A variety theorem without complementation. Izvestiya VUZ Matematika 39, 80–90 (1995); english version: Russian Mathem. (Iz. VUZ) 39, 74–83 (1995)

    MathSciNet  Google Scholar 

  17. Pin, J.E., Straubing, H.: Some results on C-varieties. ITA 39(1), 239–262 (2005)

    MathSciNet  MATH  Google Scholar 

  18. Rhodes, J., Weil, P.: Algebraic and topological theory of languages. ITA 29(1), 1–44 (1995)

    MathSciNet  MATH  Google Scholar 

  19. Rhodes, J.L., Tilson, B.: The kernel of monoid morphisms. J. Pure Applied Alg. 62, 27–268 (1989)

    MathSciNet  MATH  Google Scholar 

  20. Sakarovitch, J.: An algebraic framework for the study of the syntactic monoids application to the group languages. In: Mazurkiewicz, A. (ed.) MFCS 1976. LNCS, vol. 45, pp. 510–516. Springer, Heidelberg (1976)

    Chapter  Google Scholar 

  21. Schützenberger, M.P.: On finite monoids having only trivial subgroups. Information and Control 8(2), 190–194 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  22. Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994)

    Book  MATH  Google Scholar 

  23. Tesson, P., Thérien, D.: Logic meets algebra: the case of regular languages. Logical Methods in Computer Science 3(1) (2007)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Wilhelm-Schickard-Institut, Universität Tübingen, Germany

    Christoph Behle, Andreas Krebs & Stephanie Reifferscheid

Authors
  1. Christoph Behle
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas Krebs
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Stephanie Reifferscheid
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria

    Franz Winkler

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Behle, C., Krebs, A., Reifferscheid, S. (2011). Typed Monoids – An Eilenberg-Like Theorem for Non Regular Languages. In: Winkler, F. (eds) Algebraic Informatics. CAI 2011. Lecture Notes in Computer Science, vol 6742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21493-6_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-21493-6_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21492-9

  • Online ISBN: 978-3-642-21493-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation