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In this paper, we construct a cellular automaton on the pentagrid which is planar, weakly universal and which have five states only. This result much improves the best result which was with nine states
Boolean variables are such that they take only values on $ \mathbb{Z}_2 \cong \left\{0, 1 \right\} $. \textit{NK}-Kauffman networks are dynamical deterministic systems of $ N $ Boolean functions that depend only on $ K \leq N $ Boolean variables. They were proposed by Kauffman as a first step to understand cellular behaviour [Kauffman, S.A.; {\rm The Large Scale Structure and Dynamics of Gene Control Circuits: An Ensemble Approach}. {\it J. Theoret. Biol.} {\bf 44} (1974) 167.] with great success. Among the problems that still have been not well understood in Kauffman networks, is the mechanism that regulates the phase transition of the system from an ordered phase; where small changes of the initial state decay, to a chaotic, where they grow exponentially. We show, that this mechanism is regulated through \textsf{the irreducible decomposition} of Boolean functions proposed in [ Zertuche, F. {\rm On the robustness of NK-Kauffman networks against changes in their connections and Boolean functions}, {\it J. Math. Phys.} {\bf 50} (2009) 043513]. This is in contrast to previous knowledge that attributed it to \textsf{canalization}. We also review other statistical properties of Kauffman networks that have been shown that \textsf{Boolean irreducibility} explains.
Building self-adaptive and self-organizing (SASO) systems is a challenging problem, in part because SASO principles are not yet well understood and few platforms exist for exploring them. Cellular automata (CA) are a well-studied approach to exploring the principles underlying self-organization. A CA comprises a lattice of cells whose states change over time based on a discrete update function. One challenge to developing CA is that the relationship of an update function, which describes the local behavior of each cell, to the global behavior of the entire CA is often unclear. As a result, many researchers have used stochastic search techniques, such as evolutionary algorithms, to automatically discover update functions that produce a desired global behavior. However, these update functions are typically defined in a way that does not provide for self-adaptation. Here we describe an approach to discovering CA update functions that are both self-adaptive and self-organizing. Specifically, we use a novel evolutionary algorithm-based approach to discover finite state machines (FSMs) that implement update functions for CA. We show how this approach is able to evolve FSM-based update functions that perform well on the density classification task for 1-, 2-, and 3-dimensional CA. Moreover, we show that these FSMs are self-adaptive, self-organizing, and highly scalable, often performing well on CA that are orders of magnitude larger than those used to evaluate performance during the evolutionary search. These results demonstrate that CA are a viable platform for studying the integration of self-adaptation and self-organization, and strengthen the case for using evolutionary algorithms as a component of SASO systems.
In contrast to entropy, which increases monotonically, the "complexity" or "interestingness" of closed systems seems intuitively to increase at first and then decrease as equilibrium is approached. For example, our universe lacked complex structures at the Big Bang and will also lack them after black holes evaporate and particles are dispersed. This paper makes an initial attempt to quantify this pattern. As a model system, we use a simple, two-dimensional cellular automaton that simulates the mixing of two liquids ("coffee" and "cream"). A plausible complexity measure is then the Kolmogorov complexity of a coarse-grained approximation of the automaton's state, which we dub the "apparent complexity." We study this complexity measure, and show analytically that it never becomes large when the liquid particles are non-interacting. By contrast, when the particles do interact, we give numerical evidence that the complexity reaches a maximum comparable to the "coffee cup's" horizontal dimension. We raise the problem of proving this behavior analytically.
Research on asynchronous cellular automata has received a great amount of attention these last years and has turned to a thriving field. We survey the recent research that has been carried out on this topic and present a wide state of the art where computing and modelling issues are both represented.
We show that a large number of elementary cellular automata are computationally simple. This work is the first systematic classification of elementary cellular automata based on a formal notion of computational complexity. Thanks to the generality of communication complexity, the perspectives of our method include its application to other natural systems such as neural networks and gene regulatory networks.
Recursive equations for the number of cells with nonzero values at $n$-th step for some two-dimensional reversible second-order cellular automata are proved in this work. Initial configuration is a single cell with the value one and all others zero.
We discuss a characterization of complexity based on successive approximations of the probability density describing a system by means of maximum entropy methods, thereby quantifying the respective role played by different orders of interaction. This characterization is applied on simple cellular automata in order to put it in perspective with the usual notion of complexity for such systems based on Wolfram classes. The overlap is shown to be good, but not perfect. This suggests that complexity in the sense of Wolfram emerges as an intermediate regime of maximum entropy-based complexity, but also gives insights regarding the role of initial conditions in complexity-related issues.
The path-preference traffic flow cellular automaton is suggested to model the dynamics of transcription. The main difference from the simple traffic flow model is that it contains another preferential paths at some sites. In this paper, we propose an exact analysis for the simplest version of this model. We find that the number of particles is dominant to the dynamics of this cellular automaton and observed that there are not only expected phase shift but also several gaps as the number of particles increases. By considering the behavior of periodic orbits, we also determine the point where such gaps in the flow appear and the exact value of the flow.
The density classification task is a famous problem in the theory of cellular automata. It is unsolvable for deterministic automata, but recently solutions for stochastic cellular automata have been found. One of them is a set of stochastic transition rules depending on a parameter $η$, the traffic-majority rules. Here I derive a simplified model for these cellular automata. It is valid for a subset of the initial configurations and uses random walks and generating functions. I compare its prediction with computer simulations and show that it expresses recognition quality and time correctly for a large range of $η$ values.
In this paper, we construct a cellular automaton on the heptagrid which is planar, weakly universal and which have three states only. This result improves the best result which was with four states.
The dynamics of rule 54 one-dimensional two-state cellular automaton (CA) are a discrete analog of a space-time dynamics of excitations in nonlinear active medium with mutual inhibition. A cell switches its state 0 to state 1 if one of its two neighbors is in state 1 (propagation of a perturbation) and a cell remains in state 1 only if its two neighbors are in state 0. A lateral inhibition is because a 1-state neighbor causes a 1-state cell to switch to state 0. The rule produces a rich spectrum of space-time dynamics, including gliders and glider guns just from four primitive gliders. We construct a catalogue of gliders and describe them by tiles. We calculate a subset of regular expressions $Ψ_{R54}$ to encode gliders. The regular expressions are derived from de Bruijn diagrams, tile-based representation of gliders, and cycle diagrams sometimes. We construct an abstract machine that recognizes regular expressions of gliders in rule 54 and validate $Ψ_{R54}$. We also propose a way to code initial configurations of gliders to depict any type of collision between the gliders and explore self-organization of gliders, formation of larger tiles, and soliton-like interactions of gliders and computable devices.
We study the emergence of information integration in cellular automata (CA) with respect to states in the long run. Information integration is in this case quantified by applying the information-theoretic measure known as total correlation to the long-run distribution of CA states. Total correlation is the amount by which the total uncertainty associated with cell states surpasses the uncertainty of the CA state taken as a whole. It is an emergent property, in the sense that it can only be ascribed to how the cells interact with one another, and has been linked to the rise of consciousness in the brain. We investigate total correlation in the evolution of elementary CA for all update rules that are unique with respect to negation or reflection. For each rule we consider the usual, deterministic CA behavior, assuming that the initial state is chosen uniformly at random, and also the probabilistic variant in which every cell, at all time steps and independently of all others, disobeys the rule's prescription with a fixed probability. We have found rules that generate as much total correlation as possible, or nearly so, particularly in Wolfram classes 2 and 3. We conjecture that some of these rules can be used as CA models of information integration.
In previous works, hexagonal cellular automata (CA) have been studied as a variation of the famous Game of Life CA, mainly for spiral phenomena simulations; where the most interesting constructions are related to the Belousov-Zhabotinsky reaction. In this paper, we analyse a special kind of hexagonal CA, {\it Spiral rule}. Such automaton shows a non-trivial complex behaviour related to discrete models of reaction-diffusion chemical media, dominated by spiral guns which easily emerge from random initial conditions. The computing capabilities of this automaton are shown by means of logic gates. These are defined by collisions between mobile localizations. Also, an extended classification of complex self-localisation patterns is presented, including some self-organised patterns.
The principle of Generalized Natural Selection (GNS) states that in nature, computational processes of high computational sophistication are more likely to maintain/abide than processes of lower computational sophistication provided that sufficiently many resources are around to sustain the processes. In this paper we give a concrete set-up how to test GNS in a weak sense. In particular, we work in the setting of Cellular Automata and see how GNS can manifest itself in this setting.
We propose the characterization of binary cellular automata using a set of behavioral metrics that are applied to the minimal Boolean form of a cellular automaton's transition function. These behavioral metrics are formulated to satisfy heuristic criteria derived from elementary cellular automata. Behaviors characterized through these metrics are growth, decrease, chaoticity, and stability. From these metrics, two measures of global behavior are calculated: 1) a static measure that considers all possible input patterns and counts the occurrence of the proposed metrics in the truth table of the minimal Boolean form of the automaton; 2) a dynamic measure, corresponding to the mean of the behavioral metrics in $n$ executions of the automaton, starting from $n$ random initial states. We use these measures to characterize a cellular automaton and guide a genetic search algorithm, which selects cellular automata similar to the Game of Life. Using this method, we found an extensive set of complex binary cellular automata with interesting properties, including self-replication.
A simple algorithm for constructing an effective traffic model is presented. The algorithm uses statistically well-defined quantities extracted from the flow-density plot, and the resulting effective model naturally captures and predicts many quantitative and qualitative empirical features of the highway traffic, especially with the presence of an on-ramp bottleneck. The simplicity of the effective model provides strong evidence that stochasticity, diversity of vehicle types and modeling of complicated driving behaviors are \emph{not} fundamental to many observations in the complex real traffic dynamics. We also propose the nature of the congested phase can be well characterized by the long lasting transient states of the effective model, from which the wide moving jams evolve.
A general family of $D$-dimensional, $K$-state cellular automata is proposed where the update rule is sequentially applied in each dimension. This includes the Biham--Middleton--Levine traffic model, which is a 2D cellular automaton with 3 states. Using computer simulations, we discover new properties of intermediate states for the BML model. We present some new 2D, 3-state cellular automata belonging to this family with application to percolation, annealing, biological membranes, and more. Many of these models exhibit sharp phase transitions, self organization, and interesting patterns.