Skip to main content
SpringerLink
Log in

Coxeter Groups and Asynchronous Cellular Automata

  • Conference paper
Cellular Automata (ACRI 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6350))

Included in the following conference series:

Abstract

The dynamics group of an asynchronous cellular automaton (ACA) relates properties of its long term dynamics to the structure of Coxeter groups. The key mathematical feature connecting these diverse fields is involutions. Group-theoretic results in the latter domain may lead to insight about the dynamics in the former, and vice-versa. In this article, we highlight some central themes and common structures, and discuss novel approaches to some open and open-ended problems. We introduce the state automaton of an ACA, and show how the root automaton of a Coxeter group is essentially part of the state automaton of a related ACA.

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. Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  2. Brink, B., Howlett, R.B.: A finiteness property and an automatic structure for Coxeter groups. Math. Ann. 296, 179–190 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Eriksson, H.: Computational and combinatorial aspects of Coxeter groups. PhD thesis, KTH Stockholm (1994)

    Google Scholar 

  4. Eriksson, H., Eriksson, K.: Conjugacy of Coxeter elements. Elect. J. Comb. 16, #R4 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Geck, M., Pfeiffer, G.: Characters of finite Coxeter groups and Iwahori-Hecke algebras. Oxford Science Press (2000)

    Google Scholar 

  6. Hansson, A.Å., Mortveit, H.S., Reidys, C.M.: On asynchronous cellular automata. Adv. Comp. Sys. 8, 521–538 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  8. Macauley, M., McCammond, J., Mortveit, H.S.: Order independence in asynchronous cellular automata. J. Cell. Autom. 3, 37–56 (2008)

    MathSciNet  MATH  Google Scholar 

  9. Macauley, M., McCammond, J., Mortveit, H.S.: Dynamics groups of asynchronous cellular automata. J. Algebraic Combin. (2010) (in press)

    Google Scholar 

  10. Macauley, M., Mortveit, H.S.: On enumeration of conjugacy classes of Coxeter elements. Proc. Amer. Math. Soc. 136, 4157–4165 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Macauley, M., Mortveit, H.S.: Posets from admissible Coxeter sequences (2010) (submitted)

    Google Scholar 

  12. Macauley, M., Mortveit, H.S.: Cycle equivalence of graph dynamical systems. Nonlinearity 22, 421–436 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Macauley, M., Mortveit, H.S.: Update sequence stability in graph dynamical systems. Discrete Cont. Dyn. Sys. Ser. S (2010) (in press)

    Google Scholar 

  14. Mortveit, H.S., Reidys, C.M.: An Introduction to Sequential Dynamical Systems. Springer, New York (2007)

    MATH  Google Scholar 

  15. Tutte, W.T.: A contribution to the theory of chromatic polynomials. Canad. J. Math. 6, 80–91 (1954)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Clemson University, USA

    Matthew Macauley

  2. Department of Mathematics and NDSSL/VBI, Virginia Tech., USA

    Henning S. Mortveit

Authors
  1. Matthew Macauley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Henning S. Mortveit
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Complex Systems and Artificial Intelligence Research Center, University of Milano-Bicocca, Viale Sarca 336, Milano, Italy

    Stefania Bandini , Sara Manzoni  & Giuseppe Vizzari ,  & 

  2. University of Osaka Electro-Communication, 572-8530, Neyagawa, Osaka, Japan

    Hiroshi Umeo

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Macauley, M., Mortveit, H.S. (2010). Coxeter Groups and Asynchronous Cellular Automata. In: Bandini, S., Manzoni, S., Umeo, H., Vizzari, G. (eds) Cellular Automata. ACRI 2010. Lecture Notes in Computer Science, vol 6350. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15979-4_43

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-15979-4_43

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-15978-7

  • Online ISBN: 978-3-642-15979-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation