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On the sofic limit sets of cellular automata

Published online by Cambridge University Press:  19 September 2008

Alejandro Maass
Alejandro Maass
Affiliation:
Departamento de Ingeniería Matemática, Universidad de Chile, casilla 170/3, correo 3, Santiago, Chile(amaass@llaima.dim.uchile.cl)
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Abstract

It is not known in general whether any mixing sofic system is the limit set of some one-dimensional cellular automaton. We address two aspects of this question. We prove first that any mixing almost of finite type (AFT) sofic system with a receptive fixed point is the limit set of a cellular automaton, under which it is attained in finite time. The AFT condition is not necessary: we also give examples of non-AFT sofic systems having the same properties. Finally, we show that near Markov sofic systems (a subclass of AFT sofic systems) cannot be obtained as limit sets of cellular automata at infinity.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 4 , August 1995 , pp. 663 - 684
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[1]Adler, R. and Marcus, B.. Topological entropy and equivalence of dynamical systems. Memoirs Amer. Math. Soc. 219 (1979).Google Scholar
[2]Ashley, J.. An extension theorem for closing maps of shifts of finite type. Trans. Amer. Math. Soc. 336 (1993), 389–120.CrossRefGoogle Scholar
[3]Boyle, M.. Lower entropy factors of sofic systems. Ergod. Th. & Dynam. Sys. 4 (1984), 541–557.Google Scholar
[4]Boyle, M., Kitchens, B. and Marcus, B.. A note on minimal covers for sofic systems. Proc. Amer. Math. Soc. 95(3) (1985), 403–411.CrossRefGoogle Scholar
[5]Boyle, M. and Krieger, W.. Almost Markov and shift equivalent sofic systems. Springer Lecture Notes in Mathematics 1342, pp 33–93. Springer: Berlin, 1988.Google Scholar
[6]Coven, E. and Paul, M.. Endomorphisms of irreducible shifts of finite type. Math. Sys. Theory. 8 (1974), 167–175.CrossRefGoogle Scholar
[7]Coven, E. and Paul, M.. Sofic systems. Israel J. Math. 20 (1975), 165–177.CrossRefGoogle Scholar
[8]Culik, K. II, Pachl, J. and Yu, S.. On the limit sets of cellular automata. SIAM J. Comput. 18(4) (1989), 831–842.CrossRefGoogle Scholar
[9]Denker, M., Grillenberger, C. and Sigmund, K.. Ergodic theory on vompact spaces. Springer Lecture Notes in Mathematics 527. Springer: Berlin, 1976.Google Scholar
[10]Fischer, R.. Sofic systems and graphs. Monatshefte für Mathematik. 80 (1975), 179–186.CrossRefGoogle Scholar
[11]Goles, E., Maass, A. and Martínez, S.. On the limit set of some universal cellular automata. Theor. Comput. Sci. 110(1) (1993), 53–78.CrossRefGoogle Scholar
[12]Hedlund, G.. Endomorphisms and automorphisms of the shift dynamical systems. Math. Sys. Theory. 3 (1969), 320–375.CrossRefGoogle Scholar
[13]Hurd, L.. Formal language characterizations of cellular automaton limit sets. Complex Systems. 1 (1987), 69–80.Google Scholar
[14]Hurd, L.. The application of formal language theory to the dynamical behaviour of cellular automata. A dissertation presented to the faculty of Princeton University in Candidacy for the degree of Doctor of Philosophy (1988).Google Scholar
[15]Hurd, L.. Recursive cellular automata invariant sets. Complex Systems. 4 (1990), 119–129.Google Scholar
[16]Krieger, W.. On the subsystems of topological Markov chains. Ergod. Th. & Dynam. Sys. 2 (1982), 195–202.CrossRefGoogle Scholar
[17]Maass, A.. Some coded systems that are not unstable limit sets of CA. Cellular Automata and Cooperative Systems, NATO-ASI series 396, pp 433–449. Kluwer: Dordrecht, 1993.CrossRefGoogle Scholar
[18]Marcus, B.. Sofic systems and encoding data. IEEE-IT. 31 (1985), 366–377.CrossRefGoogle Scholar
[19]Von Neumann, J.. Theory of Self-reproducing Automata. Burks, A. W., ed. University of Illinois: Urbana, 1966.Google Scholar
[20]Weiss, B., Subshifts of finite type and sofic systems. Monatshefte Math. 11 (1973), 462–474.CrossRefGoogle Scholar
[21]Wolfram, S.. Computation theory of cellular automata. Commun. Math. Phys. 96 (1984), 15–57.CrossRefGoogle Scholar