Complex Systems
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Volume 32, Number 3 (2023)

Classification of Elementary Cellular Automata Based on Their Limit Cycle Lengths in Z/k Download PDF
Hans-Peter Stricker

In this paper we introduce a classification of elementary cellular automata based solely on numerical properties of the lengths of their limit cycles on finite lattices Z/k. The classification has a formal definition, and it could in principle be proved whether a given cellular automaton belongs to a given class. It will remain open if this is generally possible, that is, if the question is decidable.

Keywords: cellular automata; limit sets; limit cycles; cycle length spectra; decidability problems; classification  

Cite this publication as:
H.-P. Stricker, “Classification of Elementary Cellular Automata Based on Their Limit Cycle Lengths in Z/k,” Complex Systems, 32(3), 2023 pp. 229–251.

Turing Patterns in Networks Download PDF
Elizabeth Alejandra Ortiz Durán and Daniel Ivan Parra Verde

The first pattern formation model was proposed by the mathematician Alan M. Turing. This model consists of a system of reaction-diffusion equations that produces stationary patterns by means of the so-called “Turing instability.” In this paper, we found the conditions that the network and the parameters need to fulfill in order to achieve the Turing instability in a particular reaction-diffusion system called the Mimura–Murray model on different network topologies, including some simulations on an innovative kind of network, based on the Wolfram model, that evolves over time, generating interesting topologies that exhibit lattice-like topology. In addition, the equations are solved and simulated in Wolfram Language, and some examples of applications in biology and sociology are presented.

Keywords: Turing Patterns; Turing Instability; Mimura–Murray Model; Network Topologies; Reaction-Diffusion System; Wolfram Model

Cite this publication as:
E. A. O. Durán and D. I. P. Verde, “Turing Patterns in Networks,” Complex Systems, 32(3), 2023 pp. 253–269.

Affinity Classification Problem by Stochastic Cellular Automata Download PDF
Kamalika Bhattacharjee, Subrata Paul and Sukanta Das

This paper introduces the affinity classification problem, which is a generalization of the density classification problem. To solve this problem, we introduce temporally stochastic cellular automata where two rules are stochastically applied in each step on all cells of the automata. Our model is defined on a two-dimensional grid having affection capability. We show that this model can be used in several applications, such as modeling self-healing systems.

Keywords: Cellular Automata (CAs); Stochastic CA; Affinity Classification; Affection; Density Classification; Self-Healing

Cite this publication as:
K. Bhattacharjee, S. Paul and S. Das, “Affinity Classification Problem by Stochastic Cellular Automata,” Complex Systems, 32(3), 2023 pp. 271–288.

Evolving Multi-valued Regulatory Networks on Tunable Fitness Landscapes Download PDF
Larry Bull

Random Boolean networks have been used widely to explore aspects of gene regulatory networks. As the name implies, traditionally the model has used a binary representation scheme. This paper uses a modified form of the model to systematically explore the effects of increasing the number of gene states. These random multi-valued networks are evolved within rugged fitness landscapes to explore their behavior. Results suggest the basic properties of the original model remain, regardless of the update scheme or fitness sampling method. Changes are seen in sensitivity to high levels of connectivity, the mutation rate and the ability to vary network size.

Keywords: asynchronous; growth; mutation; NK model  

Cite this publication as:
L. Bull, “Evolving Multi-valued Regulatory Networks on Tunable Fitness Landscapes,” Complex Systems, 32(3), 2023 pp. 289–307.

Investigating Rules and Parameters of Reservoir Computing with Elementary Cellular Automata, with a Criticism of Rule 90 and the Five-Bit Memory Benchmark Download PDF
Tom Eivind Glover, Pedro Lind, Anis Yazidi, Evgeny Osipov and Stefano Nichele

Reservoir computing with cellular automata (ReCAs) is a promising concept by virtue of its potential for effective hardware implementation. In this paper, we explore elementary cellular automata rules in the context of ReCAs and the 5-bit memory benchmark. We combine elementary cellular automaton theory with our results and use them to identify and explain some of the patterns found. Furthermore, we use these findings to expose weaknesses in the 5-bit memory benchmark as it is typically applied in ReCAs, such as pointing out what features it selects for or solving it using random vectors. We look deeply into previously successful rules in ReCAs such as rule 90 and explain some of the consequences of its additive properties as well as the correlation between grid size and performance. Additionally, we present results from exhaustively exploring ReCAs on key parameters such as distractor period, iterations and grid size. The findings of this paper should motivate the ReCAs community to move away from using the 5-bit memory benchmark as it is being applied today.

Keywords: Reservoir Computing; Cellular Automata; Reservoir Computing with Cellular Automata (ReCAs); Edge of Chaos

Cite this publication as:
T. E. Glover, P. Lind, A. Yazidi, E. Osipov and S. Nichele, “Investigating Rules and Parameters of Reservoir Computing with Elementary Cellular Automata, with a Criticism of Rule 90 and the Five-Bit Memory Benchmark,” Complex Systems, 32(3), 2023 pp. 309–351.

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