L. Hurd. Formal language characterizations of cellular automata limit sets. Complex Systems, 1(1):69-80, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that for automata in Wolfram s classes III and IV , the sequence grows exponentially. For the lower classes the sequence appears to be bounded (and therefore eventually constant) or grow polynomially. The limit languages may be undecidable, though they are trivially co recursively enumerable [10]. Surprisingly, there are examples of computationally universal cellular automata whose limit languages are regular [7] so one should not expect a simple answer to Wolfram s questions. Recent work in computational mechanics also focuses on nite state machines as a tool to describe interesting ....

L. Hurd. Formal language characterizations of cellular automata limit sets. Complex Systems, 1(1):69-80, 1987.

....that arise after one application of the global map of the cellular automaton. Discussions of the language theoretic aspects of linear cellular automata and sofic systems, in particular with respect to their relation to the topology of the space of configurations, can be found in [8] 10] and [7]. In this paper, we will study two measures of complexity associated with L(ae) that are based on minimal finite state machines of a certain type. The first is simply the size of the minimal automaton for L(ae) or, equivalently, the number of left quotients of this language. For the second ....

L. Hurd. Formal language characterizations of cellular automata limit sets. Complex Systems, 1(1):69--80, 1987.

....that for automata in Wolfram s classes III and IV , the sequence grows exponentially. For the lower classes the sequence appears to be bounded (and therefore eventually constant) or grow polynomially. The limit languages may be undecidable, though they are trivially co recursively enumerable [10]. Surprisingly, there are examples of computationally universal cellular automata whose limit languages are regular [7] so one should not expect a simple answer to Wolfram s questions. Recent work in computational mechanics also focuses on nite state machines as a tool to describe interesting ....

L. Hurd. Formal language characterizations of cellular automata limit sets. Complex Systems, 1(1):69-80, 1987.

.... the plane can correspond to universal Turing machines [43] Similarly, the image of a cellular automaton after a nite number of timesteps is regular [60] as is the set of xed points; but limit sets can be contextfree, context sensitive, or the complement of the halting set of a Turing machine [24]. The purpose of this paper is to introduce the reader to an analogous hierarchy of two dimensional languages or patterns of symbols. This hierarchy turns out to be much richer than in one dimension, in that several equivalent de nitions of regular languages generalize in subtle ways to become ....

....get mapped to di erent states by F t , and we can choose to ll each block of width 2rt 1 with one or the other, giving the stated result. Thus the entropy of t cannot decrease faster than t d unless the CA converges to a single homogeneous xed point in nite time. 3. 4 Limit sets In [24], Hurd uses travelling particles to enforce context free and context sensitive structures in the limit sets of one dimensional CA s. We can use a similar strategy to construct limit sets in two dimensions which are DFA, NFA, or h(LLL) The CA rule sketched in gure 9, for instance, is designed to ....

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L.P. Hurd, \Formal language characterization of cellular automaton limit sets." Complex Systems 1 (1987) 69-80.

....of the elements of Omega Gamma A) for 1 CA s A, make up a language, the limit language of A, that has been intensively studied. It was proved that there exist 1 cellular automata with regular, context free, context sensitive, recursive enumerable and nonrecursive enumerable limit languages ([39], 20] 35] all results that contribute to answer a question set up by Wolfram (Problem 13 in [84] Let us remark that reversible cellular automata have rational limit sets: the whole set of words. As there exist universal reversible cellular automata ( 58] 26] there exist universal ....

Hurd L. Formal language characterizations of cellular automata limit sets. Complex Systems Vol. 1 no. 1: 69--80, 1987.

....systems and automata, languages and grammars. Moreover, cellular automata [6] can be Supported by the National Funds for Scientific Research (Belgium) Hierarchy of Dynamical Systems 2 studied from both viewpoints, which gives important results in physics, mathematics and computer science [7, 8, 9, 10, 11, 12, 13, 14]. There are also many papers and works about universality of these different kinds of systems [15, 2, 16, 17] To compare automata and dynamical systems more precisely, we need to unify their respective frameworks. As we have said before, several authors have proposed new bridges between them but ....

L.P. Hurd. Formal language characterizations of cellular automaton limit sets. Complex Systems, 1:69--80, 1987.

....as distinct dynamical systems. Some people have seen interesting relationships between dynamical systems and automata, languages and grammars [24, 23, 26, 44] Cellular automata [42, 12] can be studied from both viewpoints, which gives important results in physics, mathematics and computer science [7, 11, 13, 14, 21, 23, 27, 40]. There are also many papers and works about universality of these different kinds of systems [4, 20, 22] Recently, some papers have shown models strictly more powerful than Turing machines. This does not contradict the Church Turing Thesis because these models are based on analog or real ....

L.P. Hurd. Formal language characterizations of cellular automaton limit sets. Complex Systems, 1:69--80, 1987.

.... plane can correspond to universal Turing machines [38] Similarly, the image of a cellular automaton after a finite number of time2 steps is regular [55] as is the set of fixed points; but limit sets can be contextfree, context sensitive, or the complement of the halting set of a Turing machine [21]. The purpose of this paper is to introduce the reader to an analogous hierarchy of two dimensional languages or patterns of symbols. This hierarchy turns out to be much richer than in one dimension, in that several equivalent definitions of regular languages generalize in subtle ways to become ....

....to different states by F t , and we can choose to fill each block of width 2rt 1 with one or the other, giving the stated result. Thus the entropy of Omega t cannot decrease faster than t Gammad unless the CA converges to a single homogeneous fixed point in finite time. 3. 4 Limit sets In [21], Hurd uses travelling particles to enforce context free and context sensitive structures in the limit sets of one dimensional CA s. We can use a similar strategy to construct limit sets in two dimensions which are DFA, NFA, or h(LLL) The CA rule sketched in figure 9, for instance, is designed to ....

[Article contains additional citation context not shown here]

L.P. Hurd, "Formal language characterization of cellular automaton limit sets." Complex Systems 1 (1987) 69--80.

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L. Hurd. Formal language characterizations of cellular automata limit sets. Complex Systems, 1(1):69-80, 1987.

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L. Hurd. Formal language characterizations of cellular automata limit sets. Complex Systems, 1(1):69-80, 1987.

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L. Hurd. Formal language characterizations of cellular automaton limit sets. Complex Systems, 1:69, 1987.

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