R. H. Gilman, Classes of linear automata, Ergodic Th. Dynam. Syst., 7, 105--118 (1987).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....people, with G. Cattaneo, have established a whole classi fication of the cellular autornata with two states [10] Their point of view rnay be irnproved (other topologies, rneasures on the configurations set, Lyapunov exponents, One of the refinernents leads to the Gilman s classification [25] in which the section 2.1 classification is transposed to the case where Qz is supplied with a measure. We will come back later to this type of classification (see problem 1) 4. Another classification aiming to express the Wolfram s intuition and using the dynamical systems properties was ....

....tion [37] of subsection 2.1, which presents, among others, the advantage to be rigorous and well formalized. A first question arises: Open Problem 1 : Links between known classifications Explore the relationships between known classifications, espacially compare Gilman s and Ishii s ones [25, 30] Considering the above mentioned classifications, another one appears: what configurations have to be taken into account All of them Only some of them (for example the finite ones) Introducing some measure leads to priviledge the random configurations and to reject the recursive ones into a ....

G. Gilman, Classes of linear automata. Ergodie Th. and Dynam. Sys, vol.7,105-118, 1987.

....in Z N the cellular automaton successive con gurations. Recently, a lot of articles proposed classi cations of cellular automata [18, 6] but the reference is still Wolfram s empirical classi cation [20] which has resisted numerous attempts of formalization [19] The classi cation of Gilman [10] is interesting because it is not a classi cation of CA, but a classi cation of couples (CA, measure on its con guration set) This choice, not motivated in the paper, seems interesting because we will illustrate on an example that the intuitive Wolfram s classi cation depends on a measure, that ....

....tool in CA study. To try to understand the phase space through space time diagrams which represent only one orbit, a common idea is to observe the space time diagram of a random con guration. This is one of the basic ideas of the classi cations proposed by Wolfram [20] Ishii [13] or Gilman [10]. The initial con gurations of all the space time diagrams of this paper are issued from a random process. 1.3. The Besicovitch topology The most natural topology on CA con guration sets is the product topology. The problem is that the associated distance emphasizes what is happening close to the ....

R. H. Gilman. Classes of linear automata. Ergod. Th. & Dynam. Sys., 7:105-118, 1987.

....in Z N the cellular automaton successive con gurations. Recently, a lot of articles proposed classi cations of cellular automata [12, 6] but the reference is still Wolfram s empirical classi cation [14] which has resisted numerous attempts of formalization [13] The classi cation of Gilman [7] is interesting because it is not a classi cation of CA, but a classi cation of couples (CA, measure on its con guration set) This choice, not motivated in the paper, seems interesting because we will illustrate on an example that the intuitive Wolfram s classi cation depends on a measure, that ....

R. H. Gilman. Classes of linear automata. Ergod. Th. & Dynam. Sys., 7:105-118, 1987.

....an automaton is the function which associates to each cell a state. We can thus de ne a global transition function from the set of all the con gurations to itself which associates the following con guration after one step of computation. Recently, a lot of articles proposed classi cations of CAs [5, 8, 13] but the canonical reference is still Wolfram s empirical classi cation [14] which has resisted numerous attempts of formalization. Among the latest attempts, some are based on the mathematical de nitions of chaos for dynamical systems adapted to CAs thanks to Besicovitch topology [2, 6] and [11] ....

....some are based on the mathematical de nitions of chaos for dynamical systems adapted to CAs thanks to Besicovitch topology [2, 6] and [11] introduces the almost everywhere sensitivity to initial conditions for this topology and compares this notion with information propagation formalization. As in [8], this notion does not really classify the CA but the CA for a measure. This gives, for instance, a tool to understand uid ow phenomenon modeled by CA: we can now say that the CA is not almost everywhere sensitive to initial conditions (and thus almost everywhere not sensitive) for small uid ....

R. H. Gilman, \Classes of Linear Automata," Ergodic Theory & Dynamical Systems, 7 (1987) 105-118.

....in Z N the cellular automaton successive con gurations. Recently, a lot of articles proposed classi cations of cellular automata [13, 6] but the reference is still Wolfram s empirical classi cation [15] which has resisted numerous attempts of formalization [14] The classi cation of Gilman [7] is interesting because it is not a classi cation of CAs, but a classi cation of couples (CA, measure on its con guration set) This choice, not motivated in the paper, seems interesting because we will illustrate on an example that the intuitive Wolfram s classi cation depends on a measure, that ....

R. H. Gilman. Classes of linear automata. Ergod. Th. & Dynam. Sys., 7:105-118, 1987.

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R. H. Gilman, Classes of linear automata, Ergodic Th. Dynam. Syst., 7, 105--118 (1987).

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R. H. Gilman, Classes of linear automata, Ergodic Th. & Dynam. Sys. 7 (1987), 105--118. 11

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