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Growth Phenomena in Cellular Automata

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

In essence, analysis of growth models is an attempt to study properties of physical systems far from equilibrium (e. g., [52] and its more than 1300 references). Cellular automata (CA ) growth models, by virtue of their simplicity andamenability to computer experimentation [59,66], havebecome particularly popular in the last 30 years in many fields, such as physics [15,59,60], biology [18],chemistry [15,50], social sciences [12], and artificial life [51]. In contrast to voluminous empiricalliterature on CA in general and their growth properties in particular, precise mathematical results are rather scarce. A general CA theory is out ofthe question, since a Turing machine can be embedded in a CA, so that examples as “simple” as elementary one‐dimensionalCA [17] and Conway's Game of Life [7] are capable ofuniversal computation. Even the most basic parameterized families of CA systems exhibit a bewildering variety of phenomena: self‐organization,metastability,...

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Abbreviations

Asymptotic density :

The proportion of sites in a lattice occupied by a specified subset is called asymptotic density, or, in short, density.

Asymptotic shape :

The shape of a growing set, viewed from a sufficient distance so that the boundary fluctuations, holes, and other lower order details disappear, is called the asymptotic shape.

Cellular automaton :

A cellular automaton is a sequence of configurations on a lattice which proceeds by iterative applications of a homogeneous local update rule. A configuration attaches a state to every member (also termed a site or a cell) of the lattice. Only configurations with two states, coded 0 and 1, are considered here. Any such configuration is identified with its set of 1's.

Final set :

A site whose state changes only finitely many times is said to fixate, or attain a final state. If this happens for every site, then the sites whose final states are 1 comprise the final set.

Initial set :

A starting set for a cellular automaton evolution is called initial set, and may be deterministic or random.

Metastability :

Metastability refers to a long, but finite, time period in an evolution of a cellular automaton rule, during which the behavior of the iterates has identifiable characteristics.

Monotone cellular automaton :

A cellular automaton is monotone if addition of 1's to the initial configuration always results in more 1's in any subsequent configuration.

Nucleation :

Nucleation refers to (usually small) pockets of activity, often termed nuclei, with long range consequences.

Solidification :

A cellular automaton solidifies if any site which achieves state 1 remains forever in this state.

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Authors and Affiliations

  1. Mathematics Department, University of California, Davis, USA

    Janko Gravner

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  1. Janko Gravner
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Editors and Affiliations

  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Gravner, J. (2009). Growth Phenomena in Cellular Automata. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_266

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