Wolfram's Class Research Articles

Construction of phase diagram for elementary cellular automata by behavior of s-step transition function.

It is conjectured that the four classes of elementary cellular automata classified by Wolfram are related to phase transitions; however, the structure of the phase diagram is yet to be clarified. Hence, we propose two parameters that are calculated using the s-step transition function of elementary cellular automata. We calculate these two quantities based on a second-order approximation of the transition function. The phase diagram of these two parameters can separate the rules into four Wolfram classes. Based on this phase diagram, it is clear that the rules of class 4 are located on the phase boundary between the regions of classes 2 and 3.

Assessing complexity in cellular automata using information theory

ABSTRACTWe discuss two ways in which information theory can be used to assess complexity in a system of interacting agents. In the first part, we adopt a global viewpoint and propose a characterization of complexity based on successive maximum entropy estimations of the probability density describing the system, thereby quantifying the respective role played by low and high orders of interaction. In the second part we reconsider the question from a local perspective, focussing on the statistical dependencies between neighbouring agents. These tools are tried on simple cellular automata in order to put them in perspective with other notions of complexity usually employed for such systems. We show that these approaches are hardly comparable, despite some overlap in simple cases. However this allows to interpret complexity in terms of interactions at work in a system (instead of making reference to any particular realization of this dynamics), and to shed some light on the role of initial conditions in complex systems.Clustering of the 88 non-equivalent Elementary Cellular Automata according to their position in the space of information processing features. Rules are coloured according to their Wolfram class. ECA in class I are shown in black, class II in red, chaotic automata (class III) in green and automata displaying complex behaviour (class IV) in blue. In spite of some important important differences, information features and Wolfram class are seen to overlap to a certain extent.

New cellular automata-based image cryptosystem and a novel non-parametric pixel randomness test

In this paper, a novel image encryption scheme is proposed based on permutation-diffusion architecture using cellular automata. In the permutation phase, Wolfram class III radius 2 cellular automata rules are utilized for shuffling the pixels of the plain image and in the diffusion phase; the rules based on unity attractors of elementary cellular automata are employed to encrypt the pixels of the shuffled plain image. Addition to the encryption scheme, a new non-parametric statistical method for measuring the pixel randomness in the shuffled and encrypted images is also proposed in this paper. The strength of the proposed image encryption scheme is first assessed by standard methods for security analysis, and the pixel randomness is then determined by the newly proposed non-parametric method. The results of the assessment exhibit the high security of the diffusion process and its computational efficiency. Copyright © 2016 John Wiley & Sons, Ltd.

Non-uniform Cellular Automata based on Open-ended Rule Evolution

Cellular automata (CAs) are mathematical models of spatially and temporally discrete mathematical systems. Non-uniform CAs are the cellular automata in which each cell may contain a different transition rule and change it with time, while all cells share the same transition rule in regular CAs. Little is still known about the dynamics of open-ended evolution of rules in non-uniform CAs. The purpose of our study is to construct and investigate a model of non-uniform CAs capable of open-ended rule evolution exhibiting a wide variety of behavior across all Wolfram's classes. For this purpose, we construct 1-dimensional 2-state 3 neighborhood non-uniform CAs with evolving transition rules. In the model, we found an interesting dynamics that Class II (periodical behavior) and III (chaotic behavior) patterns emerged alternately, between which Class IV patterns sometimes emerged.

SOME NONROBUST BERNOULLI-SHIFT RULES

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chua's robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

Chaotic Properties of the Elementary Cellular Automaton Rule 40 in Wolfram's Class I

This paper examines the chaotic properties of the elementary cellular automaton rule 40. Rule 40 has been classified into Wolfram's class I and also into class C1 by G. Braga et al. These classifications mean that the time-space patterns generated by this cellular automaton die out in a finite time and so are not interesting. As such, we may hardly realize that rule 40 has chaotic properties. In this paper we show that the dynamical system defined by rule 40 is Devaney chaos on a class of configurations of some particular type and has every periodic point except prime period one, four, or six. In the process of the proof, it is noticed that the dynamical properties of rule 40 can be related to some interval dynamical systems. These propositions are shown in Theorems 2 and 4.

Placeholder Substructures I: The Road from NKS to Scale-Free Networks is Paved with Zero-Divisors

Zero-divisors (ZDs) derived by the Cayley--Dickson process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, and at least 4) can represent singularities and, as N goes to infinity, fractals and thereby, scale-free networks. Any integer less than 8 and not a power of 2 generates a metafractal or sky when it is interpreted as the strut constant (S) of an ensemble of octahedral vertex figures called box-kites (the fundamental ZD building blocks). Remarkably simple bit-manipulation rules or recipes provide tools for transforming one fractal genus into others within the context of Wolfram's Class 4 complexity.

Network Analysis of the State Space of Discrete Dynamical Systems

We study networks representing the dynamics of elementary 1D cellular automata (CA) on finite lattices. We analyze scaling behaviors of both local and global network properties as a function of system size. The scaling of the largest node in-degree is obtained analytically for a variety of CA including rules 22, 54, and 110. We further define the path diversity as a global network measure. The coappearance of nontrivial scaling in both the hub size and the path diversity separates simple dynamics from the more complex behaviors typically found in Wolfram's class IV and some class III CA.

Topology regulates pattern formation capacity of binary cellular automata on graphs

We study the effect of topology variation on the dynamic behavior of a system with local update rules. We implement one-dimensional binary cellular automata on graphs with various topologies by formulating two sets of degree-dependent rules, each containing a single parameter. We observe that changes in graph topology induce transitions between different dynamic domains (Wolfram classes) without a formal change in the update rule. Along with topological variations, we study the pattern formation capacities of regular, random, small-world and scale-free graphs. Pattern formation capacity is quantified in terms of two entropy measures, which for standard cellular automata allow a qualitative distinction between the four Wolfram classes. A mean-field model explains the dynamic behavior of random graphs. Implications for our understanding of information transport through complex, network-based systems are discussed.

Sub-classes and evolution stability of Wolfram's classes in the total-rule cellular automata

Abstract In this paper, we propose a concept of sub-classes and its evolution stability for the Wolfram's classes. Firstly, we obtain the sub-classes of the Wolfram's class IV, gene-piece of these sub-classes and their existing circumstance. Secondly, we introduce a new concept, the evolution stability, for the Wolfram's classes and sub-classes of Wolfram's class IV. Lastly, we find that Wolfram's classes I, II, and III have the evolution stability, but sub-classes of the Wolfram's class IV have not the evolution stability for the total rule cellular automata.

Locating Self-Organization at the Edge of Chaos

The classification of cellular automata rules by Wolfram into four universality classes is based on quantitative statistical properties of the flows produced by the rules. Of particular interest is rule class 4 whichis regarded as the class of rules capable of carrying out nontrivial computation. Langton proposed a parametric characterization of CA rules via a single tunable numerical parameter with values in the unit interval and argued that Wolfram's class 4 rules occur with parameter values near critical values corresponding to phase transitions in the chaotic properties of the rules. Evidence against Langton's thesis emerged with computational experiments and statistical analysis by Mitchell, Hraber and Crutchfield. More evidence emerged with techniques subsequently developed by Crutchfield for the detection of rules that exhibit non-chaotic complex flows and that have parameter values well inside the range of values proposed by Langton as characteristic of chaos. Nevertheless, a parametric characterization of rules based straightforwardly on Lyapunov exponents produces a plentiful supply of rules determining flows with complex nonchaotic, i.e. class 4-like behavior. Langrangian interpolation of rules yields numerical CA rules determined by several tunable parameters. A Lyapunov exponent can be associated with each point of the parameter space. The exponent measures the instability of flows produced by the corresponding rule. The resulting scalar field typically exhibits an extraordinary geometric structure that shows a tendency for stable rules to cluster into regions with apparently sharp boundaries. Sampling shows that class 4-like rules occur near these boundaries at locations corresponding to moderate local rates of change in the Lyapunov exponents. A combination of sampling and binary search can estimate the locations of these boundaries which can then be searched along in a variety of ways to find class 4-like rules. This heuristic typically finds suitable rules rapidly. We present an example of a rule found in this manner that yields self-organizing persistent noise-tolerant dynamical structures within CA trajectories that have non-periodic interactions. Although these rules are found by numerical heuristics, their stability implies a noise tolerance that in turn permits the recovery of discrete class 4-like CA-rules. Finally, Boolean-valued CA's can be recovered by using an encoding governed by the Fibonacci numbers.

Edge of Chaos in Stochastically Coupled Cellular Automata

A coupled system composed of two cellular automata in which one is driven probabilistically by the other is studied. The coupled system has a critical coupling constant P c . For P > P c , the two automata are mutually synchronized with each other exhibiting a chaotic motion called Wolfram's class 3. Class 4-type motion is generated by a class 3 rule near the critical point, and the phenomenon of edge of chaos is characterized by the Langton λ parameter and mutual information.

Traveling waves and chaotic properties in cellular automata.

Traveling wave solutions of cellular automata (CA) with two states and nearest neighbors interaction on one-dimensional (1-D) infinite lattice are computed. Space and time periods and the number of distinct waves have been computed for all representative rules, and each velocity ranging from 2 to 22. This computation shows a difference between spatially extended systems, generating only temporal chaos and those producing as well spatial complexity. In the first case wavelengths are simply related to the velocity of propagation and the dispersivity is an affine function, while in the second case (which coincides with Wolfram class 3), the dispersivity is multiform and its dependence on the velocities is highly random and discontinuous. This property is typical of space-time chaos in CA. (c) 1999 American Institute of Physics.

FPGA Implementation of one-dimensional and two-dimensional cellular automata

Abstract This report describes the hardware implementation of one-dimensional and two-dimensional cellular automata (CAs). After a general introduction to the cellular automata, we consider a one-dimensional CA used to implement pseudo-random techniques in built-in self test for VLSI. Due to the increase in digital ASIC complexity, testing is becoming one of the major costs in the VLSI production. The high electronics complexity, used in particle physics experiments, demands higher reliability than in the past time. General criterions are given to evaluate the feasibility of the circuit used for testing and some quantitative parameters are underlined to optimize the architecture of the cellular automaton. Furthermore, we propose a two-dimensional CA that performs a peak finding algorithm in a matrix of cells mapping a sub-region of a calorimeter. As in a two-dimensional filtering process, the peaks of the energy clusters are found in one evolution step. This CA belongs to Wolfram class II cellular automata. Some quantitative parameters are given to optimize the architecture of the cellular automaton implemented in a commercial field programmable gate array (FPGA).

One-dimensional cellular automata characterization by the roughness exponent

Cellular automata (CA) are discrete, spatially homogeneous, locally interacting dynamical systems of very simple construction, but which exhibit a rich intrinsic behavior. CA can, even starting from disordered initial configurations, evolve into ordered states with complex structures crystallized in its space-time patterns. In this paper we concentrate on the several Wolfram qualitative classes of CA behavior. In order to better characterize these classes we apply the roughness exponent method to the profiles generated by the spatiotemporal patterns of one-dimensional “elementary” CA rules. We find that this method can separate Wolfram class IV from other ones.

Mollusc Shell Pigmentation: Cellular Automaton Simulations and Evidence for Undecidability

We present a cellular automaton (CA) to simulate the reaction-diffusion processes which determine the pigmentation of natural mollusc shells. We obtained self-organization into stationary (Turing) structures, travelling waves and the four classes formed in Wolfram's automata. In particular, very complex “undecidable” structures (Wolfram's class IV) obtained at the transition between periodicity (class II) and chaos (class III). Agreement between the calculations and natural shells suggests evidence for class IV behaviour in the natural process.

Role of initial conditions in the classification of the rule space of cellular automata dynamics.

In the qualitative classification of cellular automata (CA) rules by Wolfram [Rev. Mod. Phys. 55, 601 (1983)], there exists a class of CA rules (called class 4) which exhibit complex pattern formation and long-lived dynamical activity (long transients). These properties of class 4 CA's has led to the conjecture that class 4 rules are universal Turing machines, i.e., they are bases for computational universality. We describe the embedding of a ``small'' universal turing machine, due to Minski [Computation: Finite and Infinite Machines (Prentice-Hall, Englewood Cliffs, NJ 1967)], into a cellular automaton rule table. This produces a collection of (k=18, r=1) cellular automata, all of which are computationally universal. However, we observe that these rules are distributed among the various Wolfram classes. More precisely, we show that the identification of the Wolfram class depends crucially on the set of initial conditions used to simulate the given CA. This work, among others, indicates that a description of complex systems and information dynamics may need a new framework for nonequilibrium statistical mechanics.

Sensitivity in cellular automata: Some examples

We study how the final pattern produced by particular one-dimensional cellular automata can depend on the starting configuration of live and dead cells. In particular, we show that slight changes in the initial state can lead to different universality classes (in the sense of Wolfram) for the final state. A peculiar anthropomorphic pattern appears in one of our experiments. Finally, for two example automata we investigate a small part of the initial condition space, a map of which initial conditions lead to which Wolfram classes.

Transformations of one-dimensional cellular automaton rules by translation-invariant local surjective mappings

If M is a noninvertible translation-invariant local surjective mapping, it is shown that some local one-dimensional deterministic cellular automaton rules F have a transform Φ by M defined by Φ ∘ M = M ∘ F. When it exists Φ is local and its Wolfram's class is the same as F. The evolution of a cellular automaton according to rule Φ is simply related to the evolution according to rule F. In the case of class-3 rules, the evolution to the attractor may often be viewed as particle-like structures evolving in a regular background. If the structure of these particles and their interactions for a rule F are known, then the structure and interactions of the transformed particles for rule Φ are also known. If M is a nontrivial invertible translation-invariant local surjective mapping, Φ always exists, but it is, in general, site- and time-dependent.

Criticality in cellular automata

Using recent results obtained for the transition to turbulence via spatiotemporal intermittency in extended dynamical systems, critical cellular automata rules are built. Thanks to a systematic procedure, the continuous phase transition observed in a coupled map lattice is translated into a sequence of cellular automata rules for which the critical properties of the original system are reproduced as precisely as desired. It is shown that criticality, as understood from the point of view of statistical mechanics, is intimately related to the various characteristics of Wolfram's class IV rules. This suggests in turn the picture of a “critical surface” in the space of rules and provides the basis for a discussion of the problem of classification schemes. We review recent cellular automata studies where questions linked to criticality arise and argue that they are unified in light of our results, leaving the relationship between computational and statistical characterizations of critical rules as the central problem for future studies on this subject.