Skip to main content
SpringerLink
Log in

On the Hausdorff Measure of ω-Power Languages

  • Conference paper
Developments in Language Theory (DLT 2004)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3340))

Included in the following conference series:

  • 502 Accesses

Abstract

We use formal language theory to estimate the Hausdorff measure of sets of a certain shape in Cantor space. These sets are closely related to infinite iterated function systems in fractal geometry.

Our results are used to provide a series of simple examples for the non-coincidence of limit sets and attractors for infinite iterated function systems.

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. Autebert, J.-M., Berstel, J., Boasson, L.: Context-Free Languages and Pushdown Automata. In: [17], vol. 1, pp. 111–174.

    Google Scholar 

  2. Bandt, C.: Self-similar sets 3. Constructions with sofic systems. Monatsh. Math. 108, 89–102 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  3. Barnsley, M.F.: Fractals Everywhere, 2nd edn. 1993. Academic Press, London (1988)

    MATH  Google Scholar 

  4. Berstel, J., Perrin, D.: Theory of Codes. Academic Press, London (1985)

    MATH  Google Scholar 

  5. Čulik II, K., Dube, S.: Affine automata and related techniques for generation of complex images. Theor. Comp. Sci. 116, 373–398 (1993)

    Article  MATH  Google Scholar 

  6. Edgar, G.A.: Measure, Topology, and Fractal Geometry. Springer, Heidelberg (1990)

    MATH  Google Scholar 

  7. Falconer, K.J.: Fractal Geometry. Wiley, Chichester (1990)

    MATH  Google Scholar 

  8. Fernau, H.: Iterierte Funktionen, Sprachen und Fraktale. BI (1994)

    Google Scholar 

  9. Fernau, H.: Infinite IFS. Mathem. Nachr. 169, 79–91 (1994)

    MathSciNet  Google Scholar 

  10. Fernau, H., Staiger, L.: Iterated Function Systems and Control Languages. Inf. & Comp. 168, 125–143 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Kuich, W.: On the entropy of context-free languages. Inf. & Contr. 16, 173–200 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lindner, R., Staiger, L.: Algebraische Codierungstheorie; Theorie der sequentiellen Codierungen. Akademie-Verlag (1977)

    Google Scholar 

  13. Mauldin, R.D.: Infinite iterated function systems: Theory and applications. In: Fractal Geometry and Stochastics, vol. 37 of Progress in Probability, Birkhäuser (1995)

    Google Scholar 

  14. Mauldin, R.D., Urbański, M.: Dimensions and measures in infinite iterated function systems. Proc. Lond. Math. Soc.III. Ser. 73(1), 105–154 (1996)

    Article  MATH  Google Scholar 

  15. Mauldin, R.D., Williams, S.C.: Hausdorff dimension in graph directed constructions. Trans. AMS 309(2), 811–829 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  16. Merzenich, W., Staiger, L.: Fractals, dimension, and formal languages. RAIRO Inf. théor. Appl. 28(3–4), 361–386 (1994)

    MATH  MathSciNet  Google Scholar 

  17. Rozenberg, G., Salomaa, A. (eds.): Handbook of Formal Languages. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  18. Staiger, L.: Combinatorial properties of the Hausdorff dimension. J. Statist. Plann. Inference 23, 95–100 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Staiger, L.: Kolmogorov complexity and Hausdorff dimension. Inf. & Comp. 103, 159–194 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  20. Staiger, L.: Codes, simplifying words, and open set condition. Inf. Proc. Lett. 58, 297–301 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Staiger, L.: ω-languages. In: [17], vol. 3, pp. 339–387.

    Google Scholar 

  22. Staiger, L.: On ω-power languages. In: Păun, G., Salomaa, A. (eds.) New Trends in Formal Languages. LNCS, vol. 1218, pp. 377–393. Springer, Heidelberg (1997)

    Google Scholar 

  23. Staiger, L.: The Hausdorff measure of regular ω-languages is computable, Bull. EATCS 66, 178–182 (1998)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik, Martin-Luther-Universität Halle-Wittenberg, Halle, Germany

    Ludwig Staiger

Authors
  1. Ludwig Staiger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Auckland, New Zealand

    Cristian S. Calude

  2. Computer Science, Institute of Information and Mathematical Sciences, Massey University Albany, Auckland, New Zealand

    Elena Calude

  3. Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand

    Michael J. Dinneen

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Staiger, L. (2004). On the Hausdorff Measure of ω-Power Languages. In: Calude, C.S., Calude, E., Dinneen, M.J. (eds) Developments in Language Theory. DLT 2004. Lecture Notes in Computer Science, vol 3340. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30550-7_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-30550-7_33

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-24014-3

  • Online ISBN: 978-3-540-30550-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation