DISCRETE SYSTEMS WITH MEMORY

By: ALONSO-SANZ RAMON

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Hardcover | 8 March 2011

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478 Pages

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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.

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Non-Fiction Science Physics Mathematical Physics Mathematics Applied Mathematics Chaos Theory Algebra Calculus & Mathematical Analysis Complex Analysis Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Cellular Automata and memory	p. 1
Cellular Automata	p. 1
Memory	p. 4
Disclaimer	p. 5
Average type memory	p. 7
Average memory	p. 7
Two-dimensional lattices	p. 9
Totalistic rules	p. 9
LIFE	p. 13
One-dimensional layers	p. 17
Elementary rules	p. 17
Nearest and next-nearest neighbors	p. 33
Other memories	p. 39
Average-like memory	p. 39
Limited trailing memory	p. 45
Majority of the last three state memory	p. 47
Elementary rules as memory	p. 61
Minimal memory	p. 73
Asynchrony and probabilistic rules	p. 81
Asynchrony	p. 81
Probabilistic rules	p. 84
Cycles and random sequences	p. 95
Cycles	p. 95
Random number generation by CA	p. 98
Three state automata	p. 105
Totalistic rules	p. 105
Excitable systems	p. 113
Reversible dynamics	p. 121
Characterization	p. 121
Reversible rules with memory	p. 123
Block cellular automata	p. 143
Characterization	p. 143
Density classification task	p. 145
Structurally dynamic systems	p. 159
Introduction	p. 159
Reversible SDCA	p. 161
SDCA with memory	p. 163
Two state SDCA with memory	p. 163
Three state SDCA	p. 167
Boolean networks	p. 175
Automata on networks	p. 175
Boolean networks	p. 183
Automata on proximity graphs	p. 190
Coupled layers	p. 203
Coupled cellular automata	p. 203
Coupled Boolean networks	p. 220
Continuous state variable	p. 231
Continuous-valued automata	p. 231
Finite difference equations	p. 234
One-dimensional maps	p. 234
Two-dimensional maps	p. 246
Plane curves	p. 250
Stochastic processes	p. 268
Spatial games	p. 271
The prisoner's dilemma	p. 271
Degrees of cooperation and strategies	p. 286
The structurally dynamic PD (SDPD)	p. 295
Pavlov versus anti-Pavlov (PAP) in the PD	p. 318
Other spatial games	p. 325

Average memory starting at random	p. 343
Dynamic with short-term memory	p. 351
Heterogeneous and coupled networks	p. 371
Continuous state variable	p. 389
Spatial games	p. 403
Bibliography	p. 429
List of Figures	p. 451
List of Tables	p. 461
Index	p. 463
Table of Contents provided by Ingram. All Rights Reserved.

DISCRETE SYSTEMS WITH MEMORY

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