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Space-time behavior of the REV-OR rule from different initial conditions. The time axis runs down vertically. (a) A single active cell. (b) A line of active cells which generates a periodic phase background. (c) A line of active cells with one "hole" (quiescent cell) produces two traveling dislocations.
Source publication
We present a class of reversible cellular automata which display soliton-mediated phase transitions and chemical turbulence. This class is defined as the reversible extension of simple reaction-diffusion models of excitable media such as the Schlögl and Greenberg-Hastings models. The phenomenology of these models is discussed in terms of dimensiona...
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... details of the calculation can be found in the appendix. Similar invariants have been found for the Q 2 R rule by P o m e a u [34], for some one- dimensional rules by Takesue [16] and for m a n y reversible rules with appropriate symmetries by Margolus [33]. In fig. 7 we see the space-time behavior of a single activated cell. Two outgoing solitons result which collide due to the periodic b o u n d a r y condi- tions. We can set up a periodic phase by an initial condition of a line of activated cells. W h e n the periodic phase is p e r t u r b e d two traveling dislo- cations result which go through ...
Citations
... There are numerous examples of successful explanation and simulation of real world phenomena by CAs [17,26]. For instance, they were utilized as robust tools to extensively analyze and effortlessly imitate the dynamical wave propagation in chemical media [27,38,39,18,52]. These models enabled the analysis of the dispersion and curvature of wave patterns [22], the wave dynamics associated with turbulence [27] and anisotropic media [42]. ...
... For instance, they were utilized as robust tools to extensively analyze and effortlessly imitate the dynamical wave propagation in chemical media [27,38,39,18,52]. These models enabled the analysis of the dispersion and curvature of wave patterns [22], the wave dynamics associated with turbulence [27] and anisotropic media [42]. Nonetheless, CAs are suitable tools to approximate the solution of partial differential equations (PDEs) and it has been established that they can easily represent high complexity in the initialization, constrains and anisotropies of PDEs [48,44]. ...
Cellular Automata (CAs) have been proved to be a robust tool for mimicking a plethora of biological, physical and chemical systems. CAs can be used as an alternative to partial differential equations, in order to illustrate the evolution in time of the aforementioned systems. However, CAs are preferred due to their formulation simplicity and their ability to portray the emerging of complex dynamics. Their simplicity is attributed to the fact that they are composed by simple elementary components, whereas their complexity capacities are the result of emerging behaviors from the local interactions of these elementary components. Here, the utilization of CAs on mimicking of physio-chemical reactions is presented. In specific, the implementation of chemical-based logic circuits with the use of the Belousov-Zhabotinsky (BZ) class reactions was illustrated. The BZ reaction can demonstrate non-linear oscillations that have been utilized in different scenarios as a computational substrate, whereas its photo-sensitivity have been exploited as an additional factor of manipulating the computations. A common method to mathematically represent the BZ dynamics is the Oregonator equations, which are a set of PDEs. In this work the approximation of the Oregonator equations is performed with CAs to simulate logic circuits (from classic logic gates like AND to combinatorial ones). The proposed tool has been proved to be in agreement with results produced in the lab from the actual chemical reactions. Moreover, the tool is used to design novel computing architectures in a trivial manner, without the need of specialized knowledge on chemistry, without the need to handle dangerous chemicals and alleviating unnecessary costs for equipment and consumables. The main advantage of this method can be summarized as the acceleration achieved in current implementations (serial computers), but also towards potential future implementations in massively parallel computational systems (like Field-Programmable Gate Array hardware and mainly nano-neuromorphic circuits) that have been proved to be good substrates for accelerating the implemented CA models.KeywordsCellular automataBelousov-Zabotinsky reactionUnconventional computingChemical computing
... Also, the implementation of CAs was studied in the projection and simulation of the dynamic urban expansion [26], the study of modern urban transport networks [3] and those of the past [24]. The motivation behind using CAs is based on the fact that they are proved to be efficient tools for fast prototyping and in-depth analysis of waves dynamics in chemical systems [28,35,36,60]. The CA models allow for studying a curvature and dispersion of wave patterns [27], dynamics of waves in anisotropic medium [42], turbulence [28]. ...
... The motivation behind using CAs is based on the fact that they are proved to be efficient tools for fast prototyping and in-depth analysis of waves dynamics in chemical systems [28,35,36,60]. The CA models allow for studying a curvature and dispersion of wave patterns [27], dynamics of waves in anisotropic medium [42], turbulence [28]. ...
Cellular automata (CA) have been used to simulate a variety of different chemical, biological and physical phenomena. Their ability to emulate complex dynamics, emerging from simple local interactions of their elementary cells, made them a strong candidate for mimicking these phenomena, especially when accelerated computation through parallelization is required. Belousov–Zhabotinsky (BZ) is a class of chemical reactions that due to their potential as nonlinear chemical oscillators, have inspired scientists to use them as chemical computers. The Oregonator equations, which approximate the dynamics of BZ reactions, were implemented here using CA methods. This new modelling approach (CA-based Oregonator) was tested in terms of accuracy and efficiency against previous models and laboratory-based experimental results, while the benefits of this method were outlined. It was observed that the results from the CA-based Oregonator are in good agreement with both modelled and laboratory experiments. The main advantage of this method can be summarized as the acceleration achieved in current implementations (serial computers), but also towards potential future implementations in massively parallel computational systems (like field-programmable gate array hardware and nano-neuromorphic circuits) that have been proved to be good substrates for accelerating the implemented CA models.
... It has direct correspondence with the reversibility of microscopic physical systems, implied by the laws of quantum mechanics. The reversible (or bijective) cellular automata (CAs) have been utilized in different domains, like simulation of natural phenomenon [1], cryptography [2], pattern generations [3], pseudo-random number generation [4], recognition of languages [5] etc. In this work, we are concentrating on reversibility for 1-dimensional CAs. ...
Reversibility of a one-dimensional finite cellular automaton (CA) is dependent on lattice size. A finite CA can be reversible for a set of lattice sizes. On the other hand, reversibility of an infinite CA, which is decided by exploring the rule only, is different in its kind from that of finite CA. Can we, however, link the reversibility of finite CA to that of infinite CA? In order to address this issue, we introduce a new notion, named semi-reversibility. We classify the CAs into three types with respect to reversibility property -- reversible, semi-reversible and strictly irreversible. A tool, reachability tree, has been used to decide the reversibility class of any CA. Finally, relation among the existing cases of reversibility is established.
... CA representation of reaction-diffusion and excitable systems is especially interesting because this allows us to effortlessly map already established massively-parallel architectures onto novel material base of chemical systems, and also design novel non-classical and nature-inspired computing architectures [22]. The examples of 'best practice' include cellular automata models of Belousov-Zhabotinsky reaction, [73,110], chemical systems exhibiting Turing patterns [179,176] precipitating systems [22], calcium wave dynamics [177], and chemical turbulence [84]. Reactiondiffusion modelling and simulation, particularly in a sense of chemical computation and development of wave-based chemical processors [22,21], becomes a hot topic of computer science, physics and chemistry. ...
This fascinating, colourful book offers in-depth insights and first-hand working experiences in the production of art works, using simple computational models with rich morphological behaviour, at the edge of mathematics, computer science, physics and biology. It organically combines ground breaking scientific discoveries in the theory of computation and complex systems with artistic representations of the research results. In this appealing book mathematicians, computer scientists, physicists, and engineers brought together marvelous and esoteric patterns generated by cellular automata, which are arrays of simple machines with complex behavior. Configurations produced by cellular automata uncover mechanics of dynamic patterns formation, their propagation and interaction in natural systems: heart pacemaker, bacterial membrane proteins, chemical rectors, water permeation in soil, compressed gas, cell division, population dynamics, reaction-diffusion media and self-organisation.
The book inspires artists to take on cellular automata as a tool of creativity and it persuades scientists to convert their research results into the works of art.
The book is lavishly illustrated with visually attractive examples, presented in a lively and easily accessible manner.
... This property has direct correspondence with the reversibility of microscopic physical systems, implied by the laws of quantum mechanics. The reversible (or bijective) CAs have been utilized in different domains, like simulation of natural phenomenon [16], cryptography [45,2,13], pattern generations [33,21], pseudo-random number generation [46,7], recognition of languages [22] etc. ...
This paper investigates reversibility properties of 1-dimensional
3-neighborhood d-state finite cellular automata (CAs) under periodic boundary
condition. A tool named reachability tree has been developed from de Bruijn
graph which represents all possible reachable configurations of an n-cell CA.
This tool has been used to test reversibility of CAs. We have identified a
large set of reversible CAs using this tool by following some greedy
strategies. Our conjecture is that the reversible CAs, defined over infinite
lattice, are always reversible when the CAs are finite. However, the reverse
may not be true.
... The cellular automata offers quick 'prototyping' of complex spatially extended non-linear media. The examples of 'best practice' include models of Belousov-Zhabotinsky reactions and other excitable systems [4,8], chemical systems exhibiting Turing patterns [12,9,10], precipitating systems [2], calcium wave dynamics [11], and chemical turbulence [6]. ...
Excitable cellular automata with dynamical excitation interval exhibit a
wide range of space-time dynamics based on an interplay between
propagating excitation patterns which modify excitability of the
automaton cells. Such interactions leads to formation of standing
domains of excitation, stationary waves and localized excitations. We
analyzed morphological and generative diversities of the functions
studied and characterized the functions with highest values of the
diversities. Amongst other intriguing discoveries we found that upper
boundary of excitation interval more significantly affects morphological
diversity of configurations generated than lower boundary of the
interval does and there is no match between functions which produce
configurations of excitation with highest morphological diversity and
configurations of interval boundaries with highest morphological
diversity. Potential directions of future studies of excitable media
with dynamically changing excitability may focus on relations of the
automaton model with living excitable media, e.g. neural tissue and
muscles, novel materials with memristive properties and networks of
conductive polymers.
... [7] ...
We define a cellular automaton where a resting cell excites if number of its excited neighbors belong to some specified interval and boundaries of the interval change depending on ratio of excited and refractory neighbors in the cell's neighborhood. We calculate excitability of a cell as a number of possible neighborhood configurations that excite the resting cell. We call cells with maximal values of excitability conductive. In exhaustive search of functions of excitation interval updates we select functions which lead to formation of connected configurations of conductive cells. The functions discovered are used to design conductive, wirelike, pathways in initially nonconductive arrays of cells. We demonstrate that by positioning seeds of growing conductive pathways it is possible to implement a wide range of routing operations, including reflection of wires, stopping wires, formation of conductive bridges, and generation of new wires in the result of collision. The findings presented may be applied in designing conductive circuits in excitable nonlinear media, reaction-diffusion chemical systems, neural tissue, and assemblies of conductive polymers.
... wave-fragments in a sub-excitable Belousov-Zhabotinsky (BZ) medium, and where computation is implemented in collisions of the mobile localizations [1]. Cellular automata are computationally fast prototypes of reaction-diffusion computers, the models include BZ reactions [6], chemical systems exhibiting Turing patterns [23,20,22], precipitating systems [1], calcium wave dynamics [21], and chemical turbulence [7]. ...
We present a cellular-automaton model of a reaction-diffusion excitable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their generators. We analyze a three-state totalistic cellular automaton on a two-dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specific rules support spiral glider-guns (rotating activator-inhibitor spirals emitting mobile localizations) and stationary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We describe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.
... Because CAs can provide just the right fast prototypes of reaction-diffusion models. The examples of " best practice " include models of Belousov–Zhabotinsky reactions and other excitable systems [9, 10] , chemical systems exhibiting Turing pat- terns111213, precipitating systems [1], calcium wave dynamics [14], and chemical turbulence [15]. We therefore consider it reasonable to interpret the CA rules we have discovered in terms of reaction-diffusion chemical systems. ...
A hexagonal ternary-state two-dimensional cellular automaton is de-signed which imitates an activator-inhibitor reaction-diffusion system, where the activator is self-inhibited in particular concentrations and the inhibitor dissociates in the absence of the activator. The automaton ex-hibits both stationary and mobile localizations (eaters and gliders), and generators of mobile localizations (glider-guns). A remarkable feature of the automaton is the existence of spiral glider-guns, a discrete analog of a spiral wave that splits into localized wave-fragments (gliders) at some dis-tance from the spiral tip. It is demonstrated that the rich spatio-temporal dynamics of interacting traveling localizations and their generators can be used to implement computation, namely manipulation with signals, binary logical operations, multiple-value operations, and finite-state ma-chines.
... wave-fragments in a sub-excitable Belousov-Zhabotinsky (BZ) medium, and where computation is implemented in collisions of the mobile localizations[1] . Cellular automata are computationally fast prototypes of reaction-diffusion computers, the models include BZ reactions[6], chemical systems exhibiting Turing patterns[23, 20, 22] , precipitating systems[1], calcium wave dynamics[21], and chemical turbulence[7]. Therefore we consider it reasonable to provide an interpretation of the rules we have discovered in terms of reaction-diffusion chemical systems, which we envisage will provide the basis for experimental chemical laboratory designs of reaction-diffusion computers, allowing stationary localizations to be used as memory units[2]. ...
We present a cellular-automaton model of a reaction-diusion ex- citable system with concentration dependent inhibition of the activator, and study the dynamics of mobile localizations (gliders) and their gener- ators. We analyze a three-state totalistic cellular automaton on a two- dimensional lattice with hexagonal tiling, where each cell connects with 6 others. We show that a set of specic rules support spiral glider-guns (ro- tating activator-inhibitor spirals emitting mobile localizations) and sta- tionary localizations which destroy or modify gliders, along with a rich diversity of emergent structures with computational properties. We de- scribe how structures are created and annihilated by glider collisions, and begin to explore the necessary processes that generate this kind of complex dynamics.
