K.  Culik, J. Pach, and S. Yu. On the limit set of cellular automata. SIAM Journal on Computing, 18:831-842, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....an initial con guration, we represent 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 ....

K.  Culik, J. Pach, and S. Yu. On the limit set of cellular automata. SIAM Journal on Computing, 18:831-842, 1989.

....: evolution to an aperiodic or chaotic space time pattern; W 4 : evolution to complex localized structures, sometimes long lived. It is clear that this classi cation is neither complete nor well formalized. In fact, many successive works on CA were an attempt to give it a formal consistency [8, 4]. Successively, some purely topological classi cations have 1 been proposed. They are based, for instance, on the structure of attractors [16, 13] or on the equicontinuity property [16] If the predicate to have a classi cation means to have an algorithm which, given a CA as input, decides to ....

K.  Culik, J. Pachl, and S. Yu. On the limit set of cellular automata. SIAM J. on Comp., 18:167-175, 1989.

....an initial con guration, we represent 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 ....

K.  Culik, J. Pach, and S. Yu. On the limit set of cellular automata. SIAM Journal on Computing, 18:831-842, 1989.

....an initial con guration, we represent 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 ....

K.  Culik, J. Pach, and S. Yu. On the limit set of cellular automata. SIAM Journal on Computing, 18:831-842, 1989.

....cellular automata behavior in terms of computation theory, viewing cellular automata as computers whose time evolution processes the information specified by their initial configurations. CA have been studied as language recognizers [20] and as continuous functions on compact topological spaces [13, 14, 15, 16, 17, 18, 22]. For the latter characterization, let S be the finite set of cell states of a CA. S with the discrete topology is a compact space. Let d be the dimension of the CA and let Z denote the integers. Let S = S Z d be the space of functions from cells in the CA to states. Then S with the product ....

K. Culik II, J. Pachl, and S. Yu. On the limit sets of cellular automata. SIAM Journal on Computing, 18(4):831--842, 1989.

....= T 1 i=1 Omega i . We say that a CA belongs to the class NIL, and we call it nilpotent, if its limit set is a singleton. In other words, NIL = f(Q; ffi) jQj 1) j Omega Gamma Q; ffi)j = 1)g Obviously, when the limit set is a singleton it corresponds to an homogeneous configuration. In [2] it is proved that when nilpotency holds then this configuration is reached from any other one in a finite and fixed number of steps. More precisely, NIL = f(Q; ffi) jQj 1) 9s 0 2 Q; n 2 IN ) 8C 2 Q ZZ ) G n ffi (C) s 0 )g We introduce now the simplest nilpotent CA: those reaching ....

Culik II K., Pachl J., Yu S. On the limit sets of cellular automata, SIAM J. Computing, 18 (1989), 831-842.

....has been devoted to the understanding of its long time behavior (consider, for instance, the well known Wolfram s classification of [Wol84] The long time behavior of any dynamical system is described by its attractors. In this context, for the two (and higher) dimensional CA, it was proved in [CPY89] the undecidability of the nilpotency problem (which, in practice, consists to decide whether every configuration of a given CA evolves to the same fixed point in a finite number of steps) Later J. Kari proved in [Kar92] the undecidability of the nilpotency problem for the one dimensional case. ....

Culik II K., Pachl J., Yu S. On the limit sets of cellular automata, SIAM J. Computing, 18 (1989), 831-842.

....they can only be taken as initial configurations (which explains the terminology) and, consequently, only given by external input. They appeared in the context of construc8 tion universality and self reproduction, and, in fact, originated the systematic study of global functions [59] 61] and [20]. 2.4.3 Some results and open problems Let us consider a finite set S, a positive integer d, d 1. If S is endowed with the discrete topology (for which all subsets are open) the set S Z d of applications from Z d into S, canonically endowed with the product topology, is a compact metric ....

.... A) is non empty, due to the fact that the space is a compact one) allows to separate cellular automata into two classes: class 1 contains all cellular automata for which there exists i such that Omega i = Omega and class 2 the others, which is made relevant by the following result proved in [20] 3 : A belongs to class 2 if and only if there exists a countable intersection D of dense open sets such that Omega ( i2N G i A (D) Most non trivial properties of the limit sets are recursively undecidable, even when the sets are restricted to finite configurations (see [20] When ....

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Culik II K., Pachl J. and Yu S. On the limit sets of cellular automata. SIAM J. Comput. Vol. 18 no. 4: 831--842, 1989.

....localized structures, sometimes long lived. Many successive works on CA are an attempt to give a mathematical consistence to this classification scheme or to find other more satisfying ones. The standard approach to the classification problem is within the framework of dynamical systems theory [2, 6, 3]. Classifications obtained in this manner have a common drawback: they are undecidable. Moreover they take into account neither the information content nor the algorithmic complexity of the evolutions. We propose an alternative approach which is supposed to recover this gap. In order to measure ....

K. Culik, J. Pachl, and S. Yu. On the limit set of cellular automata. SIAM Journal on Computing, 18:167--175, 1989.

....as shift invariant classes of configurations in S Z . A set X S Z is said to be shift invariant if oe(X) X. ae oe ae oe ffl ffl a b L R Figure 1: An FA A Finite automata that recognize sets of bi infinite words have been defined by Nivat and Perrin [15] and studied in [2, 10, 9]. Here we use the definition from [9] An finite automaton ( FA) A is a quintuple (K; S; ffi; KL ; KR ) where ffl K is the finite set of states; ffl S is the input alphabet; ffl ffi : K Theta S [ f g 2 K is the transition function; ffl KL K is the set of left (accepting) states; ....

K. Culik II, J. Pachl and S. Yu, On the limit sets of cellular automata, SIAM J. Comput. 18, 831-842 (1989).

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