Definition and Application of a Five-Parameter Characterization of One-Dimensional Cellular Automata Rule Space

Abstract

Cellular automata (CA) are important as prototypical, spatially extended, discrete dynamical systems. Because the problem of forecasting dynamic behavior of CA is undecidable, various parameter-based approximations have been developed to address the problem. Out of the analysis of the most important parameters available to this end we proposed some guidelines that should be followed when defining a parameter of that kind. Based upon the guidelines, new parameters were proposed and a set of five parameters was selected; two of them were drawn from the literature and three are new ones, defined here. This article presents all of them and makes their qualities evident. Then, two results are described, related to the use of the parameter set in the Elementary Rule Space: a phase transition diagram, and some general heuristics for forecasting the dynamics of one-dimensional CA. Finally, as an example of the application of the selected parameters in high cardinality spaces, results are presented from experiments involving the evolution of radius-3 CA in the Density Classification Task, and radius-2 CA in the Synchronization Task.

Deeper Investigating Adequate Secret Key Specifications for a Variable Length Cryptographic Cellular Automata Based Model

“…It verifies whether the new value of the central cell "follows" the state that appears the most in the neighborhood (Oliveira et al, 2001). For example, in transition 100 → 1 there is no neighborhood dominance, although the central cell changes.…”

“…The goal of Absolute Activity parameter was to improve the measure of activity verifying the transitions that lead to a different state of the central cell's state or it's neighbors (Oliveira et al, 2001). It quantifies how much change is entailed by the rule, in the state of the central cell, in relation to the current state of the central cell, and the states of the pair of cells which are equally apart from the centre (Oliveira et al, 2001). …”

“…Besides, a relative weight is associated to each neighborhood in such way that greater is the homogeneity, greater is the weight of dominance associated to this neigborhood. For example, in elementary space, the neighborhoods 000 and 111 have weigth 3, and the others, 1 (Oliveira et al, 2001). …”

“…It was employed an analysis based on several CA static parameters, trying to capture a pattern associating such parameters to underperforming rules. Static parameters like Z, Sensitivity, Absolute Activity, Activity Propagation and Neighborhood Dominance (Oliveira et al, 2001) were investigated to capture the pattern associated to underperforming rules. A database was generated associating rules performance in VLE ciphering with their parameters.…”

“…A detailed explanation about this parameter is given in (Wuensche, 1998). In (Oliveira et al, 2001) an analysis of Z parameter is presented and the main conclusions are: (i) it is good discriminators of the chaotic behavior (ii) rules with a Z value close to 1 have high probability to be chaotic. The components Z le ft and Z right have shown very important in the specification of the rule transitions used as secret keys in the method discussed in .…”