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Complexité et automates cellulaires linéaires

Published online by Cambridge University Press:  15 April 2002

Valérie Berthé
Valérie Berthé*
Affiliation:
IML, UPR 9016, Case 907, 163 avenue de Luminy, 13288 Marseille Cedex 09, France; (berthe@iml.univ-mrs.fr)
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Abstract

The aim of this paper is to evaluate the growth order of the complexity function (in rectangles) for two-dimensional sequences generated by a linear cellular automaton with coefficients in $\mathbb{Z}/l \mathbb{Z}$, and polynomial initial condition. We prove that the complexity function is quadratic when l is a prime and that it increases with respect to the number of distinct prime factors of l.

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 34 , Issue 5 , September 2000 , pp. 403 - 423
Copyright
© EDP Sciences, 2000

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References

J.-P. Allouche, Automates finis en théorie des nombres. Exposition. Math. 5 (1987) 239-266.
Allouche, J.-P., Sur la complexité des suites infinies. Bull. Belg. Math. Soc. 1 (1994) 133-143.
Allouche, J.-P. et Berthé, V., Triangle de Pascal, complexité et automates. Bull. Belg. Math. Soc. 4 (1997) 1-23.
Allouche, J.-P. et Shallit, J., The ring of k-regular sequences. Theoret. Comput. Sci. 98 (1992) 163-197. CrossRef
J.-P. Allouche et D. Berend, Complexity of the sequence of middle-binomial coefficients (en préparation).
Allouche, J.-P., Cateland, E., Peitgen, H.-O., Shallit, J. et Skordev, G., Automatic maps on a semiring with digits. Fractals 3 (1995) 663-677. CrossRef
Allouche, J.-P., von Haeseler, F., Peitgen, H.-O. et Skordev, G., Linear cellular automata, finite automata and Pascal's triangle. Discrete Appl. Math. 66 (1996) 1-22. CrossRef
Allouche, J.-P., von Haeseler, F., Peitgen, H.-O., Petersen, A. et Skordev, G., Linear cellular automata and automatic sequences. Parallel Comput. 23 (1997) 1577-1592. CrossRef
Allouche, J.-P., von Haeseler, F., Peitgen, H.-O., Petersen, A. et Skordev, G., Automaticity of double sequences generated by one-dimensional linear cellular automata. Theoret. Comput. Sci. 88 (1997) 195-209. CrossRef
Blanchard, F., Kurka, P. et Maass, A., Topological and measure-theoretic properties of one-dimensional cellular automata. Phys. D 103 (1997) 86-99. CrossRef
J. Cassaigne, Special factors of sequences with linear subword complexity, in Developments in Language Theory II (DLT'95), Magdeburg (Allemagne). World Scientific (1996) 25-34.
Cobham, A., Uniform tag sequences. Math. Systems Theory 6 (1972) 164-192. CrossRef
Grillenberger, C., Construction of striclty ergodic systems I. Given entropy. Z. Wahrsch. Verw. Gebiete 25 (1973) 323-334. CrossRef
Litow, B. et Dumas, P., Additive cellular automata and algebraic series. Theoret. Comput. Sci. 119 (1993) 345-354. CrossRef
Manzini, G., Characterization of sensitive linear cellular automata with respect to the counting distance, in MFCS'98. Springer, Lecture Notes in Comput. Sci. 1450 (1998) 825-833. CrossRef
Manzini, G. et Margara, L., Attractors of D-dimensional linear cellular automata, in STACS 98. Springer, Lecture Notes in Comput. Sci. 1373 (1998) 128-138. CrossRef
Manzini, G. et Margara, L., Invertible cellular automata over $\mathbb{Z}_m$ : Algorithmic and dynamical aspects. J. Comput. System Sci. 56 (1998) 60-67. CrossRef
Manzini, G. et Margara, L., A complete and efficiently computable topological classification of D-dimensional linear cellular automata over $\mathbb{Z}_m$ . Theoret. Comput. Sci. 221 (1999) 157-177. CrossRef
Martin, O., Odlyzko, A. et Wolfram, S., Algebraic properties of cellular automata. Comm. Math. Phys. 93 (1984) 219-258. CrossRef
Pansiot, J.-J., Complexité des facteurs des mots infinis engendrés par morphismes itérés. Springer, Lecture Notes in Comput. Sci. 172 (1984) 380-389. CrossRef
Robinson, A.D., Fast computation of additive cellular automata. Complex Systems 1 (1987) 211-216.
Salon, O., Suites automatiques à multi-indices et algébricité. C. R. Acad. Sci. Paris Sér. I Math. 305 (1987) 501-504.
O. Salon, Suites automatiques à multi-indices, Séminaire de Théorie des Nombres de Bordeaux, Exposé 4 (1986-1987) 4-01-4-27 ; suivi par un Appendice de J. Shallit, 4-29A-4-36A.
J.W. Sander, R. Tijdeman, The complexity of functions on lattices. Theoret. Comput. Sci. (à paraître).