The identification of cellular automata

Although cellular automata have been widely studied as a class of the spatio temporal systems, very few investigators have studied how to identify the CA rules given observations of the patterns. A solution using a polynomial realization to describe the CA rule is reviewed in the present study based on the application of an orthogonal least squares algorithm. Three new neighbourhood detection methods are then reviewed as important preliminary analysis procedures to reduce the complexity of the estimation. The identification of excitable media is discussed using simulation examples and real data sets and a new method for the identification of hybrid CA is introduced

Integrated urban evolutionary modeling

Cellular automata models have proved rather popular as frameworks for simulating the physical growth of cities. Yet their brief history has been marked by a lack of application to real policy contexts, notwithstanding their obvious relevance to topical problems such as urban sprawl. Traditional urban models which emphasize transportation and demography continue to prevail despite their limitations in simulating realistic urban dynamics. To make progress, it is necessary to link CA models to these more traditional forms, focusing on the explicit simulation of the socio-economic attributes of land use activities as well as spatial interaction. There are several ways of tackling this but all are based on integration using various forms of strong and loose coupling which enable generically different models to be connected. Such integration covers many different features of urban simulation from data and software integration to internet operation, from interposing demand with the supply of urban land to enabling growth, location, and distributive mechanisms within such models to be reconciled. Here we will focus on developin

Identification of probabilistic cellular automata

The identification of probabilistic cellular automata (PCA) is studied using a new two stage neighborhood detection algorithm. It is shown that a binary probabilistic cellular automaton (BPCA) can be described by an integer-parameterized polynomial corrupted by noise. Searching for the correct neighborhood of a BPCA is then equivalent to selecting the correct terms which constitute the polynomial model of the BPCA, from a large initial term set. It is proved that the contribution values for the correct terms can be calculated independently of the contribution values for the noise terms. This allows the neighborhood detection technique developed for deterministic rules in to be applied with a larger cutoff value to discard the majority of spurious terms and to produce an initial presearch for the BPCA neighborhood. A multiobjective genetic algorithm (GA) search with integer constraints is then evolved to refine the reduced neighborhood and to identify the polynomial rule which is equivalent to the probabilistic rule with the largest probability. A probability table representing the BPCA can then be determined based on the identified neighborhood and the deterministic rule. The new algorithm is tested over a large set of one-dimensional (1D), two-dimensional (2D), and three-dimensional (3D) BPCA rules. Simulation results demonstrate the efficiency of the new method

A simple scalar coupled map lattice model for excitable media

A simple scalar coupled map lattice model for excitable media is intensively analysed in this paper. This model is used to explain the excitability of excitable media, and a Hopf-like bifurcation is employed to study the different spatio-temporal patterns produced by the model. Several basic rules for the construction of these kinds of models are proposed. Illustrative examples demonstrate that the sCML model is capable of generating complex spatiotemporal patterns

Identification of the neighborhood and CA rules from spatio-temporal CA patterns

Extracting the rules from spatio-temporal patterns generated by the evolution of cellular automata (CA) usually produces a CA rule table without providing a clear understanding of the structure of the neighborhood or the CA rule. In this paper, a new identification method based on using a modified orthogonal least squares or CA-OLS algorithm to detect the neighborhood structure and the underlying polynomial form of the CA rules is proposed. The Quine-McCluskey method is then applied to extract minimum Boolean expressions from the polynomials. Spatio-temporal patterns produced by the evolution of 1D, 2D, and higher dimensional binary CAs are used to illustrate the new algorithm, and simulation results show that the CA-OLS algorithm can quickly select both the correct neighborhood structure and the corresponding rule

Phase Transition in NK-Kauffman Networks and its Correction for Boolean Irreducibility

In a series of articles published in 1986 Derrida, and his colleagues studied two mean field treatments (the quenched and the annealed) for \textit{NK}-Kauffman Networks. Their main results lead to a phase transition curve $K_c \, 2 \, p_c \left( 1 - p_c \right) = 1$ ($0 < p_c < 1$) for the critical average connectivity $K_c$ in terms of the bias $p_c$ of extracting a "$1$" for the output of the automata. Values of $K$ bigger than $K_c$ correspond to the so-called chaotic phase; while $K < K_c$, to an ordered phase. In~[F. Zertuche, {\it On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions}. J.~Math.~Phys. {\bf 50} (2009) 043513], a new classification for the Boolean functions, called {\it Boolean irreducibility} permitted the study of new phenomena of \textit{NK}-Kauffman Networks. In the present work we study, once again the mean field treatment for \textit{NK}-Kauffman Networks, correcting it for {\it Boolean irreducibility}. A shifted phase transition curve is found. In particular, for $p_c = 1 / 2$ the predicted value $K_c = 2$ by Derrida {\it et al.} changes to $K_c = 2.62140224613 \dots$ We support our results with numerical simulations.Comment: 23 pages, 7 Figures on request. Published in Physica D: Nonlinear Phenomena: Vol.275 (2014) 35-4