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Algebraic Combinatorics
coxeter group
Discrete Dynamical System
fibonacci number
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Cellular Automata
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Dynamics groups of asynchronous cellular automata
Dynamics groups of asynchronous cellular automata,10.1007/s108010100231y,Journal of Algebraic Combinatorics,Matthew Macauley,Jon McCammond,Henning
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Dynamics groups of asynchronous cellular automata
(
Citations: 4
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Matthew Macauley
,
Jon McCammond
,
Henning S. Mortveit
We say that a finite asynchronous
cellular automaton
(or more generally, any sequential dynamical system) is πindependent if its set of periodic points are independent of the order that the local functions are applied. In this case, the local functions permute the periodic points, and these permutations generate the dynamics group. We have previously shown that exactly 104 of the possible
cellular automaton
rules are πindependent. In the article, we classify the periodic states of these systems and describe their dynamics groups, which are quotients of Coxeter groups. The dynamics groups provide information about permissible dynamics as a function of update sequence and, as such, connect discrete dynamical systems, group theory, and
algebraic combinatorics
in a new and interesting way. We conclude with a discussion of numerous open problems and directions for future research.
Journal:
Journal of Algebraic Combinatorics  J ALGEBR COMB
, vol. 33, no. 1, pp. 1135, 2011
DOI:
10.1007/s108010100231y
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References
(5)
Order Independence in Asynchronous Cellular Automata
(
Citations: 6
)
Matthew Macauley
,
Jon McCammond
,
Henning S. Mortveit
Published in 2007.
Algebraic properties of cellular automata
(
Citations: 125
)
Olivier Martin
,
Andrew M. Odlyzko
,
Stephen Wolfram
Journal:
Communications in Mathematical Physics  COMMUN MATH PHYS
, vol. 93, no. 2, pp. 219258, 1984
On Asynchronous Cellular Automata
(
Citations: 7
)
Anders Å. Hansson
,
Henning S. Mortveit
,
Christian M. Reidys
Journal:
Advances in Complex Systems  ACS
, vol. 8, no. 4, pp. 521538, 2005
Sylow pSubgroups of the General Linear Group Over Finite Fields of Characteristic p
(
Citations: 27
)
A. J. Weir
Journal:
Proceedings of The American Mathematical Society  PROC AMER MATH SOC
, vol. 6, no. 3, 1955
The OnLine Encyclopedia of Integer Sequences
(
Citations: 1122
)
N. J. A. Sloane
Journal:
The Electronic Journal of Combinatorics  Electr. J. Comb.
, vol. 1, 1994
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Citations
(4)
Coxeter Groups and Asynchronous Cellular Automata
Matthew Macauley
,
Henning S. Mortveit
Conference:
International Conference on Cellular Automata for Research and Industry  ACRI
, vol. abs/1010.1, pp. 409418, 2010
Update Sequence Stability in Graph Dynamical Systems
(
Citations: 2
)
Matthew Macauley
,
Henning S. Mortveit
Published in 2009.
Instability of Graph Dynamical Systems and Edge Shattering
Matthew Macauley
What do Coxeter Groups and Boolean Networks have in Common?
Matthew Macauley