Memory is a universal function of organized matter. What is the mathematics of memory? How does memory affect the space-time behaviour of spatially extended systems? Does memory increase complexity? This book provides answers to these questions. It focuses on the study of spatially extended systems, i.e., cellular automata and other related discrete complex systems. Thus, arrays of locally connected finite state machines, or cells, update their states simultaneously, in discrete time, by the same transition rule. The classical dynamics in these systems is Markovian : only the actual configuration is taken into account to generate the next one. Generalizing the conventional view on spatially extended discrete dynamical systems evolution by allowing cells (or nodes) to be featured by some trait state computed as a function of its own previous state-values, the transition maps of the classical systems are kept unaltered, so that the effect of memory can be easily traced. The book demonstrates that discrete dynamical systems with memory are not only priceless tools for modeling natural phenomena but unique mathematical and aesthetic objects. Read more... Machine generated contents note: 1. Cellular Automata and memory -- 1.1. Cellular Automata -- 1.2. Memory -- Disclaimer -- 2. Average type memory -- 2.1. Average memory -- 2.2. Two-dimensional lattices -- 2.2.1. Totalistic rules -- 2.2.2. LIFE -- 2.3. One-dimensional layers -- 2.3.1. Elementary rules -- 2.3.2. Nearest and next-nearest neighbors -- 3. Other memories -- 3.1. Average-like memory -- 3.2. Limited trailing memory -- 3.3. Majority of the last three state memory -- 3.4. Elementary rules as memory -- 3.5. Minimal memory -- 4. Asynchrony and probabilistic rules -- 4.1. Asynchrony -- 4.2. Probabilistic rules -- 5. Cycles and random sequences -- 5.1. Cycles -- 5.2. Random number generation by CA -- 6. Three state automata -- 6.1. Totalistic rules -- 6.2. Excitable systems -- 7. Reversible dynamics -- 7.1. Characterization -- 7.2. Reversible rules with memory -- 8. Block cellular automata -- 8.1. Characterization -- 8.2. Density classification task. 9. Structurally dynamic systems -- 9.1. Introduction -- 9.1.1. Reversible SDCA -- 9.2. SDCA with memory -- 9.2.1. Two state SDCA with memory -- 9.2.2. Three state SDCA -- 10. Boolean networks -- 10.1. Automata on networks -- 10.2. Boolean networks -- 10.3. Automata on proximity graphs -- 11. Coupled layers -- 11.1. Coupled cellular automata -- 11.2. Coupled Boolean networks -- 12. Continuous state variable -- 12.1. Continuous-valued automata -- 12.2. Finite difference equations -- 12.2.1. One-dimensional maps -- 12.2.2. Two-dimensional maps -- 12.3. Plane curves -- 12.4. Stochastic processes -- 13. Spatial games -- 13.1. The prisoner's dilemma -- 13.2. Degrees of cooperation and strategies -- 13.3. The structurally dynamic PD (SDPD) -- 13.4. Pavlov versus anti-Pavlov (PAP) in the PD -- 13.5. Other spatial games -- Appendices -- Appendix A Average memory starting at random -- Appendix B Dynamic with short-term memory -- Appendix C Heterogeneous and coupled networks -- Appendix D Continuous state variable -- Appendix E Spatial games
Author(s): Ramon Alonso-Sanz
Series: World Scientific Series on Nonlinear Science, Series a volume 75
Edition: 1
Publisher: World Scientific Publishing Company
Year: 2011
Language: English
Pages: 478
Tags: Математика;Дискретная математика;
Contents......Page 10
Preface......Page 8
1.1 Cellular Automata......Page 13
1.2 Memory......Page 16
Disclaimer......Page 17
2.1 Average memory......Page 19
2.2.1 Totalistic rules......Page 21
2.2.2 LIFE......Page 25
2.3.1 Elementary rules......Page 29
2.3.2 Nearest and next-nearest neighbors......Page 45
3.1 Average-like memory......Page 51
3.2 Limited trailing memory......Page 57
3.3 Majority of the last three state memory......Page 59
3.4 Elementary rules as memory......Page 73
3.5 Minimal memory......Page 85
4.1 Asynchrony......Page 93
4.2 Probabilistic rules......Page 96
5.1 Cycles......Page 107
5.2 Random number generation by CA......Page 110
6.1 Totalistic rules......Page 117
6.2 Excitable systems......Page 125
7.1 Characterization......Page 133
7.2 Reversible rules with memory......Page 135
8.1 Characterization......Page 155
8.2 Density classification task......Page 157
9.1 Introduction......Page 171
9.1.1 Reversible SDCA......Page 173
9.2.1 Two state SDCA with memory......Page 175
9.2.2 Three state SDCA......Page 179
10.1 Automata on networks......Page 187
10.2 Boolean networks......Page 195
10.3 Automata on proximity graphs......Page 202
11.1 Coupled cellular automata......Page 215
11.2 Coupled Boolean networks......Page 232
12.1 Continuous-valued automata......Page 243
12.2.1 One-dimensional maps......Page 246
12.2.2 Two-dimensional maps......Page 258
12.3 Plane curves......Page 262
12.4 Stochastic processes......Page 280
13.1 The prisoner's dilemma......Page 283
13.2 Degrees of cooperation and strategies......Page 298
13.3 The structurally dynamic PD (SDPD)......Page 307
13.4 Pavlov versus anti-Pavlov (PAP) in the PD......Page 330
13.5 Other spatial games......Page 337
Appendix A Average memory starting at random......Page 355
Appendix B Dynamic with short-term memory......Page 363
Appendix C Heterogeneous and coupled networks......Page 383
Appendix D Continuous state variable......Page 401
Appendix E Spatial games......Page 415
Bibliography......Page 441
List of Figures......Page 463
List of Tables......Page 473
Index......Page 475
