Abstract
Cellular automata are often used to model the “real” world in a physical or a biological context. Then, global properties such as surjectivity or reversibility correspond to physical properties of the modeled world namely the reachability of all states or the macroscopic reversibility of the phenomenon.
Boundary conditions are often used — although not always. The problem is that they affect the global behavior of cellular automata. Furthermore, these influences depend on the dimension of the space that is used.
In this paper, we present a review of global properties of cellular automata. We focus on injectivity, surjectivity and reversibility and thus present the state of the art (at least as far as we know) on the relationships between these properties when the cellular automata are used on infinite, finite, or periodic configurations. We explain the difference between the 1D case and higher dimensions. We then discuss the representation problem of transition tables.
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References
Amoroso, S. & Y.N. Patt, Decision Procedures for Surjectivity and Injectivity of Parallel Maps for Tesselation Structures, J. Comp. Syst. Sci. 6, 448–464 (1972).
Berger, R., The Undecidability of the Domino Problem, Memoirs of the American Mathematical Society 66 (1966).
Börger, E., E. Grädel & Y. Gurevich, The Classical Decision Problem, chapter “Tiling problems” by C. Allauzen and B. Durand, Springer-Verlag (1996).
Dubacq, J-C., Etude des Automates Cellulaires Inversibles, Technical report, Rapport de lère année, Ecole Normale Supérieure de Lyon (1993).
Durand, B., Automates Cellulaires: Réversibilité et Complexité, PhD thesis, Ecole Normale Supérieure de Lyon (1994).
Durand, B., Inversion of 2d Cellular Automata: Some Complexity Results, Theoretical Computer Science 134, 387–401 (1994).
Durand, B., The Surjectivity Problem for 2D Cellular Automata, Journal of Computer and Systems Science 49 (3), 718–725 (1994).
Durand, B., A Random NP-Complete Problem for Inversion of 2D Cellular Automata, Theoretical Computer Science 148 (1), 19–32 (1995).
Gurevich, Y. & I. Koriakov, A Remark on Berger’s Paper on the Domino Problem, Siberian Journal of Mathematics 13, 459–463 (1972), (in Russian).
Head, T., One-Dimensional Cellular Automata: Injectivity from Unambiguity, Complex Systems 3, 343–348 (1989).
Kari, J., Reversibility and Surjectivity Problems of Cellular Automata, Journal of Computer and System Sciences 48, 149–182 (1994).
Maruoka, A. & M. Kimura, Conditions for Injectivity of Global Maps for Tessallation Automata, Information and Control 32, 158–162 (1976).
Moore, E.F., Machine Models of Self—Reproduction, Proc. Symp. Apl. Math. 14, 13–33 (1962).
Myhill, J., The Converse to Moore’s Garden-of-Eden Theorem, Proc. Am. Math. Soc. 14, 685–686 (1963).
Richardson, D., Tesselations with Local Transformations, Journal of Computer and System Sciences 6, 373–388 (1972).
Robinson, R.M., Undecidability and Nonperiodicity for Tilings of the Plane, Inventiones Mathematicae 12, 177–209 (1971).
Sutner, K., De Bruijn Graphs and Linear Cellular Automata, Complex Systems 5, 19–30 (1991).
Toffoli, T. & N. Margolus, Invertible Cellular Automata: A Review, Physica D 45, 229–253 (1990).
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© 1999 Springer Science+Business Media Dordrecht
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Durand, B. (1999). Global Properties of Cellular Automata. In: Goles, E., Martínez, S. (eds) Cellular Automata and Complex Systems. Nonlinear Phenomena and Complex Systems, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-9223-9_1
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DOI: https://doi.org/10.1007/978-94-015-9223-9_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-5154-7
Online ISBN: 978-94-015-9223-9
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