Emergence, Complexity and Computation Series Editors Prof Ivan Zelinka Technical University of Ostrava Czech Republic Prof Andrew Adamatzky Unconventional Computing Centre University of the West of England Bristol United Kingdom Prof Guanrong Chen City University of Hong Kong For further volumes: http://www.springer.com/series/10624 Andrew Adamatzky Reaction-Diffusion Automata: Phenomenology, Localisations, Computation ABC Author Prof Andrew Adamatzky Unconventional Computing Centre and Department of Computer Science University of the West of England Bristol UK ISSN 2194-7287 e-ISSN 2194-7295 ISBN 978-3-642-31077-5 e-ISBN 978-3-642-31078-2 DOI 10.1007/978-3-642-31078-2 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012940855 c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer Permissions for use may be obtained through RightsLink at the Copyright Clearance Center Violations are liable to prosecution under the respective Copyright Law The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made The publisher makes no warranty, express or implied, with respect to the material contained herein Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) To those who made my life good Preface Cellular automata are regular uniform networks of locally-connected finite-state machines, or cells Cells take discrete states and update their states simultaneously in discrete time Each cell chooses its next state depending on states of its closest neighbours The cell-state transition rules are very simple and intuitive yet allows for coding a non-trivial space-time dynamics Thus the cellular automata is an ideal tool for a fast-prototyping of non-linear media models, massive-parallel computers and mathematical machines Using cellular-automaton models of reaction-diffusion and excitable systems we analyse phenomenology spatial dynamics and show how to implement computation in these automaton models of a non-linear media We complement cellular-automaton models with automata on planar proximity graphs The book consists of three parts In the first part we introduce reaction-diffusion and excitable cellular automata and automata on proximity graphs, study phenomenology of propagating patterns, and represent a ’zoo’ of travelling and stationary localizations Reaction-diffusion is represented in terms of inter-species interactions in automaton models of populations in the second part There we discuss dynamics and complexity of inter-species interactions and analyse mutualistic relationships Computation in reaction-diffusion and excitable automata is overviewed in the third part There we describe automaton networks which execute a space tessellation, we demonstrate how adders and multipliers are implemented by colliding gliders in excitable medium, we show how operate binary strings in reaction-diffusion automata and overview minimalistic models of memristive networks The book is self-consistent and does not require any special knowledge to apprehend All models are intuitive and can be implemented with minimal knowledge of programming Abundant illustrations help to appreciate expressive power of dynamical systems on cellular automata and planar automaton networks Ideas, implementations and analysis offered will attract readers from all walks of life: everyone intrigued by sophisticated behaviour of cellular automata, non-linear media and mathematical machines Andrew Adamatzky Bristol Acknowledgements I am thankful and grateful to • Genaro Martinez for contributing to Chapter by discussing phenomenological classification of binary-state reaction-diffusion automata and co-authoring paper [16] • Emmanuel Sapin for evolving cell-state transitions matrices of reaction-diffusion automata discussed in Chapter and selecting the rules rich in travelling localisations [20] • Martin Grube for adding biological flavour to Chapters and 10 • Liang Zhang for advancing my ideas on collision-based based binary arithmetics and original implementation of 2+ -medium one bit half-adder; most designs of the chapter are based on papers co-authored with Liang [281, 282] • Andy Wuensche for telling me about his beehive rule, which pushed me to discover the spiral rule automata [17], and for contributing to Chapter 13 • Leon Chua for introducing me to memristors which inspired some automaton constructs in Chapters 14 and 15 • Thomas Ditzinger, Engineering Editorial, Springer-Verlag, for being so supportive and helpful Contents Introduction 1.1 Weird Mods of Excitable Automata 1.2 Excitation on Proximity Graphs 1.3 Automated Search for Localisations 1.4 Reaction-Diffusion, Automata and Populations 1.5 Minimal Models of Population Dynamics 1.6 Towards Computing in Reaction-Diffusion Automata 1.7 Collision-Based Computing in Excitable Automata 1.8 Spiral Rule Automata 1.9 Memristors in Cellular Automata 1.10 On Colors 8 11 13 14 16 18 Part I Phenomenology and Localisations Reaction-Diffusion Binary-State Automata 2.1 Precipitating Automata 2.2 Diffusion-Association Automaton 2.3 Functional Classification 2.4 Excitable Automata without Refractory State 2.5 Summary 21 22 25 29 29 34 Retained Excitation 3.1 Rectangularly Growing Domains 3.2 Diamond-Shaped Growing Domains 3.3 Octagonally Growing Domains 3.4 Almost Disc-Shaped Growing Domains 3.5 Amoeboid Growth of Mixed-State Patterns 3.6 Not Growing Domains of Excitation 3.7 Domains with Small Number of Still Localizations 3.8 Mobile Localizations 3.9 Summary 37 38 42 48 52 53 56 61 65 65 XII Contents Mutualistic Excitation 4.1 Phenomenology of Mutualistic Excitation 4.2 Mobile Localisations 4.3 Stationary Localisations 4.4 Huge Localisations 4.5 Characterising Localisations 4.6 Excitation Rules Rich with Localisations 4.7 Summary 67 68 70 80 88 92 93 94 Dynamical Excitation Intervals: Diversity and Localisations 97 5.1 Morphological Diversity 99 5.2 Generative Diversity and Localisations 107 5.3 Summary 113 Excitable Delaunay Triangulations 115 6.1 Structural Properties of Delaunay Automata 117 6.2 Absolute Excitability 121 6.3 Relative Excitation 123 6.4 Summary 133 Excitable β -Skeletons 135 7.1 Absolutely Excitable Skeletons 137 7.2 Relatively Excitable Skeletons 141 7.3 Stability of Localised Oscillators 151 7.4 Summary 151 Evolving Localizations in Reaction-Diffusion Automata 155 8.1 Breeding Glider-Supporting Rules 156 8.2 Likehood of Gliders 156 8.3 Quasi-chemical Reaction 158 8.4 Reductions of Transitions Functions 161 8.5 Summary 162 Part II Population Dynamics Population Dynamics in Automata 165 9.1 Mutualism 166 9.2 Commensalism and Amensalism 168 9.3 Parasitism 169 9.4 Competition 177 9.5 Summary 180 10 Automaton Mechanics of Mutualism 183 10.1 Phenomenology 184 10.2 Localisations in Mutualistic Systems 185 10.3 Summary 193 Contents XIII Part III Computation with Excitation 11 Voronoi Automata 199 11.1 Voronoi Automata 199 11.2 Constructing Voronoi Diagram on Voronoi Automata 202 11.3 Arbitrary-Shaped Planar Objects and Contours 202 11.4 Summary 207 12 Adders and Multipliers in Sub-excitable Automata 209 12.1 Adders 210 12.2 Multipliers 216 12.3 Summary 228 13 Computing in Hexagonal Reaction-Diffusion Automaton 229 13.1 Input Interface 233 13.2 Memory Device 235 13.3 Routing and Tuning Signals 237 13.4 Binary Operations 240 13.5 Implementation of the Finite State Machine 242 13.6 Transformation of Two- and Four-Bit Strings 244 13.7 Six Bit Coding 250 13.8 Scylla and Charybdis: Outcomes of Passing between Two Eaters 252 13.9 Summary 255 14 Semi-memristive Automata: Retained Refractoriness 263 14.1 Methods: Experiments and Classificiation 264 14.2 Classes 264 14.3 Hierarchies 270 14.4 Travelling Localisations 277 14.5 Summary 283 15 Structural Dynamics: Memristive Excitable Automata 287 15.1 Phenomenology 289 15.2 Oscillating Localisations 296 15.3 Dynamics of Excitation on Interfaces 300 15.4 Building Conductive Pathways 302 15.5 Summary 308 15.6 Appendix 309 Epilogue 311 References 313 Index 327 Chapter Introduction Cellular automata are regular uniform networks of locally-connected finite-state machines, called cells A cell takes a finite number of states Cells are locally connected: every cell updates its state depending on states of its geographically closest neighbours All cells update their states simultaneously in discrete time steps All cells employe the same rule to calculate their states Cellular automata are discrete systems with non-trivial behaviour They are mathematical models of computation and computer models of natural systems The cellular automata forms theoretical background and, at the same time simulation tools and implementation substrates, of mathematical machines with unbounded memory, discrete theoretical structures, digital physics and modelling of spatially extended non-linear systems; massiveparallel computing, language acceptance, and computability; reversibility of computation, graph-theoretic analysis and logic; chaos and undecidability; evolution, learning and cryptography It is almost impossible to find a field of natural and technical sciences, where cellular automata are not used For those not familiar with cellular automata we recommend to have a look in few classical titles You can start with a Toffoli-Margolus’s bestseller [242] and then spoil yourself with lavishly illustrated atlas by Wuensche [266] and thought-provoking cellular automaton treatise by Wolfram [262] Plenty of interesting state transition rules and useful hints and tips on can be found in Ilachinski’s compendium of cellular-automaton universe [144] Conway’s Game of Life is the most popular cellular automaton, we recommend the collection of chapters as a treatise [25] of the Game- of Life investigations, approaches and findings Comprehensive specialised texts on modelling space-time dynamics of natural processes in cellular automata are authored by Boccara [56], Chopard and Droz [77], Weimar [258] and Deutsch and Dormann [95] Since their inception in [122], cellular automaton models of excitation became a usual tool for studying complex phenomena of excitation wave dynamics and chemical reaction-diffusionactivities in physical, chemical and biological systems [77, 144] They are now essential instruments in computational analysis of non-linear systems, and exhaustive search for non-trivial functions in cell-state transition rule spaces [9], [16] Cellular-automaton models of reaction-diffusion and excitable systems are of particular importance because by using them we can — A Adamatzky: Reaction-Diffusion Automata, ECC 1, pp 1–18 c Springer-Verlag Berlin Heidelberg 2013 springerlink.com

  Fig 14.2 Ratio of excited states to resting states vs ratio of refractory states to resting states in configurations developed by a cellular automat with retained refractoriness with D-stimulation Class C2 D-stimulation leads to propagating quasi-chaotic pattern with domination of refractory states The pattern fills the automaton array (Fig 14.1b) The pattern’s propagating front is comprised of travelling localizations, which branch periodically Excited localizations travelling along north-west-south-east and north-east-south-west axis leave a distinctive trail of refractory states (Fig 14.1b) There is a tail of gradually extinct excitations These spatially extended excitations collapse into localised oscillating localizations, most of which extinguish with time (Fig 14.2, C2 ) The class C2 has three functions M (1122), M (1211), M (1222) Class C3 Initial random stimulation leads to formation of a single circular excitation wave, followed by trains of excitation waves travelling north-west, north-east, south-west and south-east The wave-trains are followed by discoidal growing quasi-random 14.2 Classes 267 pattern of excited and refractory states (Fig 14.1c) North, south, east and west domains of cellular array lying between the growing patterns are filled entirely with refractory states These refractory domains contribute towards slight prevalence of refractory states (Fig 14.2, C3 ) The class C3 has eight functions M (1a1b), a = 3, , 6, b = 1, Class C4 The D-stimulated automata response is characterised by a single circular excitation wave, which encapsulates wave trains propagating north-west, north-east, southwest and south-east, and envelopes of wave-fronts travelling north, east, south and west (Fig 14.1d)) The wave envelopes in C4 occupying the same space domains are refractory domains in C3 This is why a cluster of C4 functions is positioned symmetrically to cluster of C3 functions with respect to the diagonal of equal ratios of excited and refractory states in Fig 14.2 The class C4 has 48 functions M (1abc), a = 3, , 6, b = 1, 2, c = 3, , Classes C5 and C6 In response D-stimulation automata from these classes generate circular waves of excitation (Fig 14.3ab) The domain D of initial perturbation is a random pattern of refractory and resting states No excitation is observed inside D in automata from class C6 and usually just few localised oscillating excitations in automata from C5 The circular wave-front leaves behind an almost uniform field of refractory cells with radial traces of resting states towards north-west, north-east, south-west and south-east directions (Fig 14.3ab) Functions from classes C5 and C6 are positioned close to class C2 in Fig 14.2 due to high ratio of refractory states in configurations they generate The class C5 has four functions M (1a22), a = 3, , and the class C6 has six functions M (1abc), a = 7, 8, b = 1, 2, b ≤ c ≤ Classes C7 and C8 Automata from these classes generate a single circular excitation wave while perturbed by D and a domain of refractory and resting states inside boundaries of D In automata from C8 a propagating excitation wave leaves a trail of refractory states along north-west-south-east and north-east-south-west axis, the rest of cells remain in the resting state Propagating wave-front leaves no traces in automata from C8 (Fig 14.3cd) Class C7 has 16 functions M (1a1b), a = 7, 8, b = 3, 4; M (1a2b), 268 14 Semi-memristive Automata: Retained Refractoriness (a) C5 (b) C6 (c) C7 (d) C8 Fig 14.3 Exemplars of configurations generated by functions from classes C5 C8 with Dstimulation Excited cells are red (light grey), refractory cells are blue (dark grey) and resting cells are white a = 7, 8, b = 3, , 8; and, class C8 has eight functions M (1a1b), a = 7, and b = 5, , Class C9 D-stimulation of an automaton from C9 produces a domain of refractory and resting states (inside boundaries of D) and localizations travelling along north-south and west-east axis (Fig 14.4a) The localizations leave traces of refractory states 14.2 Classes 269 (a) C9 (b) C10 (c) C11 Fig 14.4 Exemplars of configurations generated by functions from classes C9 C11 with D-stimulation Excited cells are red (light grey), refractory cells are blue (dark grey) and resting cells are white Configurations produced by functions from C9 has the lowest (amongst all other classes) ratio of excited states (Fig 14.2, C9 ) The class C9 has 17 functions M (22ab), a = 1, 2, a ≤ b ≤ 8; M (2a11), a = 3, Class C10 Phenomenology of automata’s behaviour is richest amongst all classes D-stimulation leads to formation of trains of excitation wave-fragments, propagating in north-west, 270 14 Semi-memristive Automata: Retained Refractoriness Class Shannon entropy Simpson’s index Space-filling ratio Expressiveness C1 7.01 1.0 0.94 7.41 5.06 0.95 0.95 5.35 C2 5.99 0.96 0.96 6.24 C3 5.55 0.98 0.96 5.8 C4 1.79 0.53 0.96 1.87 C5 2.15 0.65 0.96 2.23 C6 3.95 0.96 0.42 9.5 C7 4.56 0.99 0.15 31.21 C8 3.89 0.95 0.14 27.69 C9 5.20 0.98 0.60 8.7 C10 4.27 0.97 0.27 15.71 C11 Fig 14.5 Average values of Shannon entropy, Simpson’s index, space-filling ratio and expressiveness with D-stimulaton north-east, south-west and south-east directions; domains of localised excitations inside boundaries of D, travelling localizations with and without traces of refractory states (Fig 14.4b) Typically excited cells are in slight majority in configurations generated by automata form C10 (Fig 14.2, C10 ) The class C10 includes 61 functions M (2a1b), a, b = 3, , 8; M (2a2b), a, b = 4, , Class C11 Behaviour of automata from class C11 is somewhat similar to that of class C9 but sometimes localizations travelling along north-south and west-east axis are ’linked’ by segments from excitation wave-fronts (Fig 14.4c) The class C11 has 27 functions M (2312); M (232a), a = 2, , 8; M (2412); M (242a), a = 2, 3; M (2abc), a = 5, , 8, b = 1, 2, b ≤ c ≤ 14.3 Hierarchies Finding 42 Classes of excitable cellular automata with retained refractoriness obey the following power hierarchy: C4 C10 {C1,C11 } C7 C9 {C3,C8 } C6 C5 C2 This is a direct consequence of our joining functions into classes Finding 43 Classes of excitable cellular automata with retained refractoriness obey the following hierarchy of space-filling ratio: {C1 ,C2 ,C3 ,C5 ,C6 } C10 C7 C11 {C8 ,C9 } 14.3 Hierarchies 271    



   



 

















 


















  









 










































   














































  
Fig 14.6 Space-filling ratio values for classes of retained refractoriness rules Classes are indicated on horizontal axis Space-filling is shown on vertical axis All 225 functions are mapped to show distribution of space-filling ratio inside each class There is less discs on the chart than functions because some functions from the same class have the same space-filling ratio values Space-filling ratio values averaged amongst each class are shown in Fig 14.5 and the distribution of functions inside each class in Fig 14.6 Classes C1 to C6 , and C8 show small variance in space-filling ratio while class C11 shows largest variance Roughly the classes can be split into two space-filling groups: high space-filling ratio (C1 , ,C6 ) and low space-filling ratio (C7 , ,C11 ) Finding 44 Classes of excitable cellular automata with retained refractoriness obey the following hierarchy of diversity: C1 C3 C4 C10 C2 C8 C11 C7 C9 C6 C5 We can draw an analogy between neighbourhood states in a given configuration of cellular automaton and living species in a population Thus we can measure a diversity of a function by calculating Simpson’s index and Shannon entropy of a population of neighbourhood states in a single configuration generated by the function (Fig 14.7) Classes C5 and C6 exhibit the lowest diversity because initial random excitation pattern D extinguishes quickly and only single propagating wave-front 272 14 Semi-memristive Automata: Retained Refractoriness    
 

 
  

 











   (a)       














 
 

  ... state A Adamatzky: Reaction- Diffusion Automata, ECC 1, pp 21–35 c Springer-Verlag Berlin Heidelberg 2013 springerlink.com 22 Reaction- Diffusion Binary-State Automata • Precipitating automata: If state... Why have we chosen cellular automata to study computation in reaction- diffusion media? Because cellular automata can provide just the right fast prototypes of reaction- diffusion models The examples... quasi-chemical reactions [15] That is we not have to follow reaction- diffusion dynamic to simulate it in automata [241] but map models of cellular automata onto a space of quasi-chemical reactions