A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science, Vol. 4

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE Volume IV

7835 tp.indd 1

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WORLD SCIENTIFIC SERIES ON

NONLINEAR SCIENCE

Series A

Vol. 76

Series Editor: Leon O. Chua

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE Volume IV

Leon O Chua University of California at Berkeley, USA

World Scientific NEW JERSEY

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WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua

University of California, Berkeley

Series A.

MONOGRAPHS AND TREATISES'

Volume 60:

Smooth and I\lonsmooth High Dimensional Chaos and the Melnikov-Type Methods J Awrejcewicz & M. M. h'olicke

Volume 61:

A Gallery of Chua Attractors (with CD-ROM) E Bilotta & F Pantano

Volume 62:

Numerical Simulation of Waves and Fronts in Inhomogeneous Solids A BerezQvski, J. Engefbrecf?t & G. A Maugin

Volume 63:

Advanced Topics on Cellular Self-Organizing Nets and Chaotic Nonlinear Dynamics to Model and Control Complex Systems R. Caponetlo, L Fortuna & M. Frasca

Volume 64:

Control of Chaos in Nonlinear Circuits and Systems B. W -K. Ling, H. H. -C Lu & H K. Lam

Volume 65'

Chua's Circuit Implementations: Yesterday, Today and Tomorrow L. Fortuna, M. Frasca & M. G. Xibi/ia

Volume 66:

Differential Geometry Applied to Dynamical Systems

J.-M. Ginoux Volume 67:

Determining Thresholds of Complete Synchronization, and Application A Stefanski

Volume 68:

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science (Volume III) L O. Chua

Volume 69:

Modeling by Nonlinear Differential Equations p. E Phi/Jipson & P Schuster

Volume 70:

Bifurcations in Piecewise-Smooth Continuous Systems D. J \I\!arvvick Simpson

Volume 71:

A Practical Guide for Studying Chua's Circuits R. K.iliq

Volume 72:

Fractional Order Systems: Modeling and Control Applications

R. Caponetto, G. Dongola, L. Fortuna & I. Petras Volume 73:

2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach E. Zeraou/ia & J. C. Sprott

Volume 74:

Physarum Machines: Computers from Slime Mould A Adamatzky

Volume 75:

Discrete Systems with Memory R. Alonso-Sanz

Volume 76:

A Nonlinear Dynamics Perspective of Wolfram's New Kind of Science (Volume IV) L 0 Chua

¥To vlevv tile complete list of' the pulJHshed volumes :in the series, please visit: http://Vi'VvVi',worldscibo (I ks, co mls eries/wss nsa series, sh trnl

Published by

World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, N] 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE Volume IV Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4317-30-6 ISBN-lO 981-4317-30-6

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Printed in Singapore.

February 26, 2011

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contents

To

CYNTHIA

who thrives on games, from ergodic to quasi-ergodic

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♦

CONTENTS ♦

Ë An Ode to the Unknowable

xi

Volume IV Chapter 1. Quasi-Ergodicity

1

1. Remembrance of Things Past

2

1.1. Boolean cube representation

3

1.2. Index of complexity

3

1.3. One formula specifies all 256 rules

11

1.4. Space-time pattern and time-τ return maps

12

1.5. We only need to study 88 rules!

29

1.6. The “Magic” rule spaces

29

1.7. Symmetries among Boolean cubes

32

1.7.1. Local complementation T c 1.7.2. Three equivalence transformations

32 T†

, T , and T

∗

32

1.7.3. Perfect complementary rules

36

1.7.4. Permutive rules

36

1.7.5. Superposition of local rules

37

1.7.6. Rules with explicit period-1 and/or period-2 orbits

37

1.7.7. Most rules harbor at least one Isle of Eden

38

2. Quasi-Ergodicity

59

2.1. Only complex and hyper Bernoulli- shift rules are quasi-ergodic

59

2.2. Quasi-ergodicity and Gardens of Eden

76

vii

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contents

D D Automata Fractals in 1D Cellular

79

3.1. All time-1 characteristic functions are fractals

79

3.2. Fractals in CA additive rules

83

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

3.

o

o

3.2.1. Rule 60

84

3.2.2. Rule 90

86

3.2.3. Rule 105

88

3.2.4. Rule 150

90

D function 0 to the rule number 3.3. From the time-1 characteristic

93

3.4. Number of fractal patterns

95

4. New Results about Isles of Eden

97

4.1. Definitions and basic lemmas

98

4.2. Alternate proof that rules 45 and 154 are conservative for odd lengths

98

4.3. There are exactly 28 strictly-dissipative local rules

99

4.4. Isles of Eden for rules of group 1 5. How to Find Analytically the Basin-Tree Diagrams for Bernoulli Attractors

105 106

5.1. Bernoulli-στ basin-tree generation formula

106

5.2. A practical application of the Bernoulli στ -shift basin tree generation formula

109

6. Old Theorems and New Results for Additive Cellular Automata

111

6.1. Theorems on the maximum period of attractors and Isles of Eden

113

6.2. Scale-free property for additive rules

116

7. Concluding Remarks

152

Appendix A

152

Appendix B

153

Appendix C

154

Chapter 2. Period-1 Rules 1. The Second Time Around 1.1. Recap of Period-1 Rules 2. Basin Tree Diagrams

157 157 163 163

2.1. Each bit string has a decimal and a fractional code

285

2.2. Attractor, basin of attraction, and basin tree

285

2.3. Explicit formula for generating isomorphic basin trees

302

2.4. Robustness coefficient ρ

303

2.5. Genotype and phenotype

303

3. Omega-Limit Orbits 3.1. General observations on ω-limit orbits from Table 6

304 304

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contents

Contents

ix

3.2. Portraits of ω-limit orbits from the basin tree diagrams exhibited in Table 6

304

3.3. Concatenated ω-limit orbit generation algorithm

367

4. Rules of Group 1 have Robust Period-1 ω-Limit Orbits

371

5. Concluding Remarks

385

References

387

Index

389

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♦

AN ODE TO THE UNKNOWABLE ♦

Ë Mirroring Dicken’s A Tale of Two Cities, this volume depicts the extreme contrast between the simplest, and the unknowable, local rules of one-dimensional cellular automata. The simplest class (Group 1) consists of 67, out of 256, local rules where almost all initial configurations tend to period-1 attractors. The unknowable class (Group 5 and Group 6) embodies 40 local rules where almost all initial configurations tend to unpredictable, dubbed quasi-ergodic, attractors bearing the telltale fingerprints of G¨ odel’s incompleteness theorem. While despairing over the human frailty to decipher God’s forbidden secrets, CA affecionados can rejoice at the prospect of never running out of challenges of teasing out partial truths, however meager, hidden within the 18 globally-equivalent unknowable local rules, including rule 137, the prototypic universal Turing machine.

xi

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♦

Chapter 1 QUASI-ERGODICITY ♦

Ë

Our scientific odyssey through the theory of 1-D cellular automata is enriched by the definition of quasi-ergodicity, a new empirical property discovered by analyzing the time-1 return maps of local rules. Quasi-ergodicity plays a key role in the classification of rules into six groups: in fact, it is an exclusive characteristic of complex and hyper Bernoulli-shift rules. Besides introducing quasi-ergodicity, this paper answers several questions posed in the previous chapters of our quest. To start with, we offer a rigorous explanation of the fractal behavior of the time-1 characteristic functions, finding the equations that describe this phenomenon. Then, we propose a classification of rules according to the presence of Isles of Eden, and prove that only 28 local rules out of 256 do not have any of them; this result sheds light on the importance of Isles of Eden. A section of this paper is devoted to the characterization of Bernoulli basin-tree diagrams through modular arithmetic; the formulas obtained allow us to shorten drastically the number of cases to take into consideration during numerical simulations. Last but not least, we present some theorems about additive rules, including an analytical explanation of their scale-free property. Keywords: Cellular automata; quasi-ergodicity; ergodicity; nonlinear dynamics; attractors; Isles of Eden; Bernoulli shift; shift maps; basin tree diagram; Bernoulli velocity; Bernoulli return time; complex Bernoulli shifts; hyper Bernoulli shifts; Binomial series; scale-free phenomena; Rule 45; Rule 60; Rule 90; Rule 105; Rule 150; Rule 154; additive rules; permutive rules; dissipative rules; conservative rules; fractals; basin-tree generation formula.

Contents 1. Remembrance of Things Past . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Boolean cube representation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. Index of complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

2 3 3

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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

1.3. 1.4. 1.5. 1.6. 1.7.

One formula specifies all 256 rules . . . . . . . . . . . . . . . . . . . . . Space-time pattern and time-τ return maps . . . . . . . . . . . . . . . . We only need to study 88 rules! . . . . . . . . . . . . . . . . . . . . . . The “Magic” rule spaces . . . . . . . . . . . . . . . . . . . . . . . . . Symmetries among Boolean cubes . . . . . . . . . . . . . . . . . . . . . 1.7.1. Local complementation T c . . . . . . . . . . . . . . . . . . . . . 1.7.2. Three equivalence transformations T † , T , and T ∗ . . . . . . . . . . 1.7.3. Perfect complementary rules . . . . . . . . . . . . . . . . . . . . 1.7.4. Permutive rules . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.5. Superposition of local rules . . . . . . . . . . . . . . . . . . . . . 1.7.6. Rules with explicit period-1 and/or period-2 orbits . . . . . . . . . 1.7.7. Most rules harbor at least one Isle of Eden . . . . . . . . . . . . . 2. Quasi-Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D 2.1. Only complex and hyper Bernoulli- shift rules are quasi-ergodic . . . . . . D 2.2. Quasi-ergodicity and Gardens of Eden . . . . . . . . . . . . . . . . . . . D 3. Fractals in 1D Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . D 3.1. All time-1 characteristic functions are fractals . . . . . . . . . . . . . . . 3.2. Fractals in CA additive rules . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Rule 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Rule 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Rule 105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D D 3.2.4. Rule 150 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. From the time-1 characteristic function to the rule number . . . . . . . . . 3.4. Number of fractal patterns . . . . . . . . . . . . . . . . . . . . . . . . 4. New Results about Isles of Eden . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Definitions and basic lemmas . . . . . . . . . . . . . . . . . . . . . . . 4.2. Alternate proof that rules 45 and 154 are conservative for odd lengths . . 4.3. There are exactly 28 strictly-dissipative local rules . . . . . . . . . . . . . 4.4. Isles of Eden for rules of group 1 . . . . . . . . . . . . . . . . . . . . . 5. How to Find Analytically the Basin-Tree Diagrams for Bernoulli Attractors . . . 5.1. Bernoulli-στ basin-tree generation formula . . . . . . . . . . . . . . . . . 5.2. A practical application of the Bernoulli στ -shift basin tree generation formula 6. Old Theorems and New Results for Additive Cellular Automata . . . . . . . . . 6.1. Theorems on the maximum period of attractors and Isles of Eden . . . . . 6.2. Scale-free property for additive rules . . . . . . . . . . . . . . . . . . . . 7. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. Remembrance of Things Past This exposition continues our saga on a nonlinear dynamics perspective of the 256 elementary cellular automata rules, as featured in eight tutorial-review papers: Part I [Chua et al., 2002], Part II [Chua et al., 2003], Part III [Chua et al., 2004], Part IV 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11 12 29 29 32 32 32 36 36 37 37 38 59 59 76 79 79 83 84 86 88 90 93 95 97 98 98 99 105 106 106 109 111 113 116 152 152 153 154

DDD

[Chua et al., 2005a], Part V [Chua et al., 2005b], Part VI [Chua et al., 2006], Part VII [Chua et al., 2007a], and Part VIII [Chua et al., 2007b].1 In this paper, we examine the 18 yet untamed rules listed in Tables 11 and 12 of [Chua et al., 2007a]; namely,
the ten complex Bernoulli-shift rules 18 , 22 , 54 ,

Parts I to VI have been collected into two recent edited volumes [Chua, 2006] and [Chua, 2007], respectively. Part VII and VIII will appear in a future edited volume III.

February 14, 2011

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0000000 DOD 00000 Chapter 1: Quasi-Ergodicity

 73 , 90 , 105 , 122 , 126 , 146 ,
150 and the eight hyper Bernoulli-shift rules 26 , 30 , 41 , 45 , 60 , 106 , 110 , 154 . Remarkably, we have observed empirically that all complex and hyper Bernoulli-shift rules exhibit an Ergodic-like dynamics, which we christened quasi-ergodicity. Our main goal of this paper is to describe and characterize this unifying empirical phenomenon. We will revisit our fabled Isles of Eden from Parts VII and VIII and offer an alternate perspective of such rare gems. We will show that all local rules harbored a few precious Isles of Eden, except for 28 rules, which we will prove analytically to be devoid of Isles of Eden; these are the God forsaken rules! DOD 0 We will also revisit the scale-free phenomenon reported in Parts VII and VIII for additive rules, and prove these empirical observations are in fact fundamental properties possessed by such rules. In particular, we will present and prove several analytical theorems for rules 60 , 90 , 105 and 150 . For the reader’s convenience, let us briefly review some highlights from our earlier adventures, henceforth referred to collectively as Age-1 Episodes, a la Tolkien’s “The Lord of the Rings”. We are concerned exclusively with our tiny universe of 256 one-dimensional binary cellular automata, with a periodic boundary condition, as depicted in Fig. 1(a). Each “ring” has L
I + 1 cells, labeled consecutively from i = 0 to i = I. Each cell “i” has two states xi ∈ {0, 1}, where we usually code the states “0” and “1” by the color “blue” and “red ”, respectively. A clock sets the pace in discrete times, dubbed “iterations” by the mathematical community, or “generations” by the life science of all “i” at time t + 1 community. The state xt+1 i (i.e. the next generation) is determined by the state of its nearest neighbors xti−1 , and xti+1 , and itself xti , at time t [Fig. 1(c)], in accordance with a prescribed Boolean truth table of eight distinct 3-input patterns [Fig. 1(d)].2

1.1. Boolean cube representation We have found it extremely useful to map these eight 3-input patterns into the eight vertices of 2

3

the “cube” shown in Fig. 1(b), henceforth called a Boolean cube. The rationale for identifying which vertex corresponds to which pattern was presented in [Chua et al., 2002], in order to provide the genesis of the truth tables from a nonlinear physical system perspective, namely, cellular neural networks (CNN) [Chua, 1996], thereby providing a bridge between nonlinear dynamics and cellular automata. For readers who have not been exposed to the age1 episodes alluded to above, it is not necessary to read the cited literature. Simply map the output of each prescribed Boolean function (i.e. the 8-bit, yet unspecified, binary string in Fig. 1(d)) onto the corresponding colors (red for 1, blue for 0) at the vertices of the Boolean cube. Since there are 28 = 256 distinct combinations of eight bits, there are exactly 256 Boolean cubes with distinct vertex color combinations, one for each Boolean function, as displayed in Table 1.

1.2. Index of complexity DO 0

D

A careful examination of these 256 Boolean cubes shows that it is possible to separate, and segregate, all red vertices of each Boolean cube from the blue vertices by κ = 1, 2, or 3 parallel planes. An example illustrating this separation is shown in Fig. 2 for rules 170 , 110 and 184 , respectively. The integer κ is called the index of complexity of rule N . We will always use the color red for κ = 1, blue for κ = 2, and green for κ = 3 to code the rule number N of each of the 256 Boolean cubes, as printed at the bottom of each Boolean cube in Table 1. It is natural to associate the 8-bit pattern of each Boolean function with a decimal number N representing the corresponding 8-bit word; namely,

D

N = β7 • 27 + β6 • 26 + β5 • 25 + β4 • 24 + β3 • 23 + β2 • 22 + β1 • 21 + β0 • 20 ,

β ∈ {0, 1}.

Observe that since βi = 0 for each blue vertex in Fig. 1(b), N is simply obtained by adding the weights (indicated next to each pattern in Fig. 1(b)) associated with all red vertices. For example, for the Boolean cube shown in Fig. 3(b), we have N = 0 • 27 + 1 • 26 + 1 • 25 + 0 • 24 + 1 • 23 + 1 • 22 + 1 • 21 + 1 • 20 = 26 + 25 + 23 + 22 + 21 = 110

Throughout the paper we intentionally use both t and n to indicate the time (or iterations) to stress the equivalence between a discrete Cellular Automaton and a continuous nonlinear system.

t

(

Final

bit~

Cell Cell Cell 0 1 2

L

X~1

...,.

t

... ,.

X i+I Cell Cell C~II (i+1) (i-I)

(c)

I

Local Rule

•••

~

t+I X.1

=

N(

t t X Xi-I' i'

INPUT at time

~23=8

INPUT CODE

I

X~

1+1

)

Xi_I

I

(b) ••• ~24 =16 (d)

g

Fig. 1.

7

Xl

Xi

..... '"

0 '"·0 ..... '" 0 ..... '"' 0 ..... '"..... 0 '"·0 ..... '"..... 0 '"·0 .....

(f) Heaviside "step" Function

[C6 + Csl (C + c3x1-1 + C x1 + C, x1+, ) 4

1+1

X i +1

1

mmm IT] m m D rn m D m rn m D D rn D m m rn D m D WD n m rn D D D

(e) Universal Formula for all 256 Rules

{c + C

t

OUTPUT at time t+ 1

I

[ill

4

!J

x1+ = 4

~

[KJ

{a, I} HI X.1

E

Initial bit

••• ~22=4

1

output

2

Notations, symbols, and universal formula for local rule N .

4(W)

I]I}

_~11 D

~w

ch01

a)

Cell (1-2) Cell Cell (I-I) 1

,.

15:12

.1111

X'1-I

February 14, 2011

{a, I}

input E

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity codes for a 3-bit input pattern. Boolean cubes defining 256 CA rules. Each vertex km

Table 1.

2 6

1

0

4

1

1

1

7

4

1

1

4

1 5

4

1 5

1

4

3 1 5

4

4

1

57

6

1

0 4

5

58

4

1 5

6

6 4

7 1 5

59

2

1

6 4

7 1

0 5

60

1 5

2

1

0 4

5

61

1

2

4

1 5

62

3 7

6

1

0 5

2

3

0

5

55

7

6

1

0

4

5

2

7

6

54 3

3

47

7

4

7

6

2

3

0

5

4

5

6

53 3

1

2

1

0

46 3

0 4

5

2

4

7

6

52 3

0

7

4

5

7

6

3

0

3

39

7

6

5

4

5

2

45 3

0

1

0

1

38 3

1

2

3

0

7

31

7

6

3

0 4

5

4

7

6 4

5

2

5

2

44

7

2

3

1

0

3

0

7

6

1

1 5

6

30

37 3

1

2

7 0

2

3

0

3

23

7

6

3

0

6 4

5

4

7

6 4

5

2

51

7

6

5

4

5

50

7 5

1

2

1

2

3

0

3

0

1

36

43

7

6

0

5

2

1

2

29 3

7

6 4

7

6 4

5

2

3

0

1

42

7

6

7

1

5

2

4

5

1

0

2

22 3

5

15

7

4

7

6

28

35 3

0

7

1

2

5

1

0

3

0

3 7

6 4

5

6

5

2

14 1

2

1

2

1

0

3

0

3

0

6

7

7

6

21

7

4

2

3 7

4

5

4

7

6

3

0

5

4

5

6

3

0 4

5

6

5

6

34

7

2

1

2

1

2

1

20

7

1

2

1

6

13

7

2

0

3

0

3

0

4

3

0

3

0

6

6

2

3 7

4

7

6 4

5

2

27

7

6

1 5

6

1

12

7 0

4

5

2

3

0

1

26 3

4

5

2

3

0

1

2

5

7

6

3

2

3

0

2

19

7

6

49

7

56

2

5

4

5

4

7

6

3

5

5

2

6

18 1

0

1

0 4

7

6

7

6

1

2

0 4

5

3 7

6

4

11 3

1

0

3

0

2

3 7

6 4

7

6 4

5

2

3

2

48

0

1 5

4

5

6

3

0

1

10

7

6

3

2

4

2

3

0

5

2

3

7

6

1

0 4

5

2

41

7

4

1

2

40

0

3

0

1

3 7

6

2

5

4

5

6

0

33

1

2

6

2

3 7

4

7

4

7

4

5

6

3

0

1

2

32 6

7

2

25

5

2

3

0

3

0

1 1 1

6

24 6

7

4

5

2

1 1 0

17

7

4

6

2

3

0

1 0 1

0

16 6

5

4

5

2

1 0 0

9

7

4

4

6

3

0

0 1 1

2

8 6

3

4

5

2

0 1 0

1

7

4

2

0

3

0

0 0 1

6

0 6

1

4

5

2

0 0 0

2

3 7

0

6 4

3 7 1

0 5

63

5

February 14, 2011

6

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 1. 2 7

6

1

0 4

2

3

4

2

3 1

0

6

1

0 4

1

0 4

6

7

6

1

0 4

4

2

3 1

4

1

4

2

3 7 1

0 5

120

5

4

1 5

121

6

7

6

1

0 4

5

122

1 5

2

4

3 1

0 4

5

123

1

6

4

1

124

5

2

3

5

1

0 4

6

3 1

0 4

5

125

1

2

1

4

126

1

0 4

5

119

1 5

3 7

6

2

3

0

5

111

7

6

1

0 4

5

2

3 7

6

118

7

5

2

3 7

6

1

0

103

5

2

3 7

6

110

117

7 0

0

3

0

2

3 7

6

5

4

5

4

7

4

5

6

5

2

3

2

1

0

116

7

6

6

1

2

1

0

102 3

7

6

3

0

3

95

7

6

109

7 0

2

1 5

4

5

4

7

4

5

6

5

2

108

115 3

1

2

1

0

1

94 3

7 0

2

3

0

3

87

7

6 4

7

6

3

0

3

0

2

6 4

5

2

101

7

6

1 5

4

5

4

7

4

5

2

5

2

114 3

0

1

0 4

7

6

7

6

113

112 6

1

1

107 3

1

2

3

0

1

86

7

6

100 3

0

2

3

0

4

5

2

3

0

1

0

79

7

6

3 7

6 4

5

2

93

7

6

1 5

4

5

4

7

4

5

2

5

6

106 3

0

1

0

1

2

3

1

2

3

0

1

78

7

6

92 3

0

2

3

0

4

2

3

0

5

71

7

6

85

7

6

5

4

5

4

7

6

1

2

99

7

6 4

7

6 4

5

2

5

2

3

0

1

5

4

5

2

105

7

1

98 3

0

104 6

4

7

6 4

5

2

5

1

2

7 0

1

2

1

0 4

5

3 7

6

70

77 3

1

0

3

0

84 3

0

3

0

6 4

7

6

6

2

3 7

6 4

7

4

5

2

91

7

6

1 5

4

5

2

97

7 0

1

0

1

90 3

7 0

1

76 3

2

3

0

4

2

3

0

5

2

69

7

6

83

7

6 4

7

6

96 6

5

4

5

2

1

2

3

6 4

5

2

5

2

82

89

88

1

0

3

0

7

6 4

7

4

5

2

5

2

3 7

6

1

1

2

1

0 4

5

3 7

6

68

75 3

1

0

3

0

2

3 7

6 4

7

6 4

5

2

81

80

1

74 3

0

2

3

0

5

2

67

7

6 4

7

6 4

5

2

5

2

3 7

6`

1

2

1

0 4

5

3 7

6

66

73

72

1

0

3

0

2

3 7

6 4

7

4

5

2

5

2

65

7

6

1

0

64 2

7

6 4

5

3

(Continued )

3 7

6

1

0 4

5

127

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity Table 1. 2 7

6

1

0 4

2

3

4

2

3 1

0

6

1

0 4

1

0 4

6

7

6

1

0 4

4

2

3 1

4

1

4

2

3 7 1

0 5

184

5

4

1 5

185

6

7

6

1

0 4

5

186

1 5

2

4

3 1

0 4

5

187

1

6

4

1

188

5

2

3

5

1

0 4

6

3 1

0 4

5

189

1

2

1

4

190

1

0 4

5

183

1 5

3 7

6

2

3

0

5

175

7

6

1

0 4

5

2

3 7

6

182

7

5

2

3 7

6

1

0

167

5

2

3 7

6

174

181

7 0

0

3

0

2

3 7

6

5

4

5

4

7

4

5

6

5

2

3

2

1

0

180

7

6

6

1

2

1

0

166 3

7

6

3

0

3

159

7

6

173

7 0

2

1 5

4

5

4

7

4

5

6

5

2

172

179 3

1

2

1

0

1

158 3

7 0

2

3

0

3

151

7

6 4

7

6

3

0

3

0

2

6 4

5

2

165

7

6

1 5

4

5

4

7

4

5

2

5

2

178 3

0

1

0 4

7

6

7

6

177

176 6

1

1

171 3

1

2

3

0

1

150

7

6

164 3

0

2

3

0

4

5

2

3

0

1

0

143

7

6

3 7

6 4

5

2

157

7

6

1 5

4

5

4

7

4

5

2

5

6

170 3

0

1

0

1

2

3

1

2

3

0

1

142

7

6

156 3

0

2

3

0

4

2

3

0

5

135

7

6

149

7

6

5

4

5

4

7

6

1

2

163

7

6 4

7

6 4

5

2

5

2

3

0

1

5

4

5

2

169

7

1

162 3

0

168 6

4

7

6 4

5

2

5

1

2

7 0

1

2

1

0 4

5

3 7

6

134

141 3

1

0

3

0

148 3

0

3

0

6 4

7

6

6

2

3 7

6 4

7

4

5

2

155

7

6

1 5

4

5

2

161

7 0

1

0

1

154 3

7 0

1

140 3

2

3

0

4

2

3

0

5

2

133

7

6

147

7

6 4

7

6

160 6

5

4

5

2

1

2

3

6 4

5

2

5

2

146

153

152

1

0

3

0

7

6 4

7

4

5

2

5

2

3 7

6

1

1

2

1

0 4

5

3 7

6

132

139 3

1

0

3

0

2

3 7

6 4

7

6 4

5

2

145

144

1

138 3

0

2

3

0

5

2

131

7

6 4

7

6 4

5

2

5

2

3 7

6

1

2

1

0 4

5

3 7

6

130

137

136

1

0

3

0

2

3 7

6 4

7

4

5

2

5

2

129

7

6

1

0

128 2

7

6 4

5

3

(Continued )

3 7

6

1

0 4

5

191

7

February 14, 2011

8

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 1. 2 7

6

1

0 4

2

3

4

2

3 1

0

6

1

0 4

1

0 4

6

7

6

1

0 4

4

2

3 1

4

1

4

2

3 7 1

0 5

248

5

4

1 5

249

6

7

6

1

0 4

5

250

1 5

2

4

3 1

0 4

5

251

1

6

4

1

252

5

2

3

5

1

0 4

6

3 1

0 4

5

253

1

2

1

4

254

1

0 4

5

247

1 5

3 7

6

2

3

0

5

239

7

6

1

0 4

5

2

3 7

6

246

7

5

2

3 7

6

1

0

231

5

2

3 7

6

238

245

7 0

0

3

0

2

3 7

6

5

4

5

4

7

4

5

6

5

2

3

2

1

0

244

7

6

6

1

2

1

0

230 3

7

6

3

0

3

223

7

6

237

7 0

2

1 5

4

5

4

7

4

5

6

5

2

236

243 3

1

2

1

0

1

222 3

7 0

2

3

0

3

215

7

6 4

7

6

3

0

3

0

2

6 4

5

2

229

7

6

1 5

4

5

4

7

4

5

2

5

2

242 3

0

1

0 4

7

6

7

6

241

240 6

1

1

235 3

1

2

3

0

1

214

7

6

228 3

0

2

3

0

4

5

2

3

0

1

0

207

7

6

3 7

6 4

5

2

221

7

6

1 5

4

5

4

7

4

5

2

5

6

234 3

0

1

0

1

2

3

1

2

3

0

1

206

7

6

220 3

0

2

3

0

4

2

3

0

5

199

7

6

213

7

6

5

4

5

4

7

6

1

2

227

7

6 4

7

6 4

5

2

5

2

3

0

1

5

4

5

2

233

7

1

226 3

0

232 6

4

7

6 4

5

2

5

1

2

7 0

1

2

1

0 4

5

3 7

6

198

205 3

1

0

3

0

212 3

0

3

0

6 4

7

6

6

2

3 7

6 4

7

4

5

2

219

7

6

1 5

4

5

2

225

7 0

1

0

1

218 3

7 0

1

204 3

2

3

0

4

2

3

0

5

2

197

7

6

211

7

6 4

7

6

224 6

5

4

5

2

1

2

3

6 4

5

2

5

2

210

217

216

1

0

3

0

7

6 4

7

4

5

2

5

2

3 7

6

1

1

2

1

0 4

5

3 7

6

196

203 3

1

0

3

0

2

3 7

6 4

7

6 4

5

2

209

208

1

202 3

0

2

3

0

5

2

195

7

6 4

7

6 4

5

2

5

2

3 7

6

1

2

1

0 4

5

3 7

6

194

201

200

1

0

3

0

2

3 7

6 4

7

4

5

2

5

2

193

7

6

1

0

192 2

7

6 4

5

3

(Continued )

3 7

6

1

0 4

5

255

February 14, 2011

15:12

ch01

Boolean Cube Chapter 1: Quasi-Ergodicity

9

170 Boolean Cube

110 Boolean Cube

184 K~3 D

Fig. 2. The number κ of parallel planes which separate all vertices having one color from those having a different color on the other side is called the index of complexity of rule N .

February 14, 2011

t= 0

15:12

t= 1 (

Cell (/-2) Cell Cell (/-1) /

Final bit-.J

L

(i-I)

I

Initial bit

• • • --.+2

2

=4

t= 2

~I

3

• • • --.+2 =8

E

.~

t= 3 t=4

ch01

a)

Cell Cell C~II (i+1)

Cell Cell Cell 0 1 2

I t= 5 t= 6

10

(b)

N

bit string length L = 30

Sum of weights of red vertices:

=

{O,8,e,O,(t} (d) Formula for Rule

=2+4 +8+32+64=110

III

X:+ I = 4[-1+ I(X:-1 + 2x: -3x:+1 Fig. 3.

(e) Heaviside "step" Function 4(W)

~) I]

` ´ Dnotations, symbols and universal formula from Fig. 1. An example Rule 110 illustrating the

___I}

~w

February 14, 2011

15:12

ch01

o

o Chapter 1: Quasi-Ergodicity

Consequently, the Boolean function defined by the Boolean cube in Fig. 3(b) is identified as N = 110 .

1.3. One formula specifies all 256 rules While the usual procedure for specifying a Boolean function is to give the truth table, as God-given laws, we have discovered the following nonlinear difference equation, with eight parameters {c1 , c2 , . . . , c8 }, which is capable of generating any of the 256 Boolean cubes in Table 1, by merely assigning eight real numbers to these eight parameters: = xt+1 i

{c8 + c7 |[c6 + c5 |(c4 + c3 xti−1

+ c2 xti

+

c1 xti+1 )|]|}

(1)

The difference equation (1) is extremely robust in the sense that a very large set of real numbers can be chosen to generate each Boolean cube, as depicted in the parameter space R8 in Fig. 4. One such set of numbers is given for each of the 256 rules in Table A-3 of [Chua et al., 2006]. All points inside give the same rule

~

(

11

For example, for rule 110 , we read off the following values from page 1363 of [Chua et al., 2006]:

oc

c1 = −3,

c2 = 2,

3

c5 = 1,

c6 = −1,

c7 = 0,

= 1,

1 c4 = − , 2

(2)

c8 = 0

Substituting the eight real numbers from Eq. (2) into Eq. (1), we obtain the following difference equation for executing rule 110 :     t   x + 2xt − 3xt − 1  −1 + xt+1 = (3) i i+1 i  i−1 2 

o

To verify that Eq. (3) can indeed generate the truth table for rule 110 , let us substitute the eight input patterns listed on the left side of Fig. 1(d): Input code 0m: (xti−1 , xti , xti+1 ) = (0, 0, 0)     1  t+1  −1 +  0 + 2 • 0 − 3 • 0 − xi = 2  =

[−1 + 0.5]

=0

(4a)

Input code 1m: (xti−1 , xti , xti+1 ) = (0, 0, 1)     1  t+1  −1 +  0 + 2 • 0 − 3 • 1 − xi = 2  =

[−1 + 3.5]

=1

(4b)

Input code 2m: (xti−1 , xti , xti+1 ) = (0, 1, 0)     1  t+1  xi = −1 +  0 + 2 • 1 − 3 • 0 − 2  =

[−1 + 1.5]

=1

(4c)

Input code 3m: (xti−1 , xti , xti+1 ) = (0, 1, 1)     1  t+1 xi = −1 +  0 + 2 • 1 − 3 • 1 − 2  =

[−1 + 1.5]

=1

Fig. 4. An abstraction showing a curve meandering through 8-dimensional parameter space, showing all 256 local rules, not necessary in consecutive order. The robustness of the uni`versal formula is depicted by an open set of parameter´points surrounding a typical parameter vector for rule N2 all of which would generate the same truth table as N2 .

(4d)

Input code 4m: (xti−1 , xti , xti+1 ) = (1, 0, 0)      1 t+1  −1 +  1 + 2 • 0 − 3 • 0 − xi = 2  = =0

[−1 + 0.5] (4e)

February 14, 2011

12

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

Input code 5m: (xti−1 , xti , xti+1 ) = (1, 0, 1)     1  t+1  −1 +  1 + 2 • 0 − 3 • 1 − xi = 2 

o

[−1 + 4.5]

= =1

(4f)

Input code 6m: (xti−1 , xti , xti+1 ) = (1, 1, 0)     1  t+1  xi = −1 +  1 + 2 • 1 − 3 • 0 − 2 

D

[−1 + 2.5]

= =1

(4g)

Input code 7m: (xti−1 , xti , xti+1 ) = (1, 1, 1)     1  t+1  xi = −1 +  1 + 2 • 1 − 3 • 1 − 2  =

DO =0

[−1 + 0.5] (4h)

Mapping each vertex kmred if xt+1 = 1, and i t+1 = 0, onto the “blank ” vertices in the blue if xi Boolean cube in Fig. 1(b), we obtain the Boolean cube for 110 in Fig. 3(b), which indeed defines rule 110 , as expected. For ease of future reference, we have enshrined the “shot-gun” Eq. (1) in Fig. 1(e). Observe that in the most general case, this universal formula has eight non-zero parameters and two “nested ” absolute value functions. The analytical derivation of this formula in [Chua et al., 2002] shows that the number of absolute value functions required for D each rule N is precisely equal to κ − 1, where κ is the index of complexity of N , which in turn is defined to be the number of parallel planes needed to separate the “red” vertices from the “blue” vertices. In the case of rule 110 , we recall from Fig. 3 that κ 110 = 2 because we only need two parallel planes. Consequently, we expect the two parameters c7 = c8 = 0 so that only one absolute value function is needed for 110 .3 For the readers convenience, Table 2 gives the explicit formula for each of the 256 local rules defined by the Boolean cubes displayed in Table 1.

D

D

We end this subsection by emphasizing that the significance of the universal D difference O Dequation D enshrined in Fig. 1(e) should not be construed merely as an elegant mathematical formula, but rather as a mathematical bridge essential for deriving and proving analytical results and theorems, as demonstrated in the rigorous derivation of the Bernoulli shift formulas for rules 170 , 240 , 15 , 85 , and 184 for finite L in [Chua et al., 2005a]. Such a feat would not have been possible without exploiting this universal formula in an essential way.

1.4. Space-time pattern and time-τ return maps Given any initial binary bit-string configuration at time t = 0, the local rule N is used to update the of each cell “i” at time t + 1, using the state xt+1 i states xti−1 , xti , and xti+1 of the three neighboring cells i − 1, i, and i + 1, centered at location “i”, respectively. The space-time pattern for the initial state shown in Fig. 3(a) is shown in Fig. 3(c) for t = 0, 1, 2, . . . , 11. For simplicity, space-time patterns are generally plotted by displaying a line of L cells, with the implicit understanding that the leftmost bit is “glued ” to the rightmost bit. Such space-time patterns are useful if they are T-periodic with a small period T and a relatively short transient regime. For large period T , such as the spacetime patterns generated from the 18 complex and hyper Bernoulli rules to be studied in this paper, it is much more revealing to recast space-time patterns into a time-τ return map [Chua et al., 2005a] ρτ [N ]: φn−τ → φn

o

o

3

(5)

where φ


I 

2−(i+1) xi

(6)

i=0

is the decimal equivalent of the binary bit string xn = (xn0

xn1

xn2

···

xnL−1 )

(7)

and τ is an integer. In most cases, such as the quasi-ergodic space-time patterns to be presented in Sec. 2, we choose τ = 1 and will be concerned usually with the time-1 map ρ[N ]
ρ1 [N ], where we drop the subscript “1” to avoid clutter. Observe

We caution the reader that to avoid ambiguity, Eq. (1) should actually be written as three separate “telescoping” equations, as in Eq. (A.3) of [Chua et al., 2006]. Hence, c6 = c7 = 0 in Eq. (1) should be interpreted to mean the deletion of the outer absolute value function associated with c6 and c7 .

Formula for Local Rule X;+1 =

1+1

Xi

neurons => I All quenched

.l[-Xi- 1 -

=

Q,

=

d-

I

[

I

-Xi - 1

_

I

I

Xi - X i+1

I

Xi

+ X iI+1 _ _21]

13

+.!.] 2

I XIHI = d-[-XI~ -Xl I

HI

Xi

H1

Xi

=

=

X;+I =

HI

Xi

=

+ 21 ]

+x.II -x.HI I _.!.] 2

.l[-X~

-

I

I-I

Q,

X~

HI

d-[.!.-I 2 (2x;.l[

Q,

-

2X I-

I

I

-X; - X;+I

I

X;+I

=

H1

=

Xi

X

+.!.] 2

i 1 -Xi -Xi+1

XiHI =

XiHI =

d-[-x.

I-I

Formula for Local Rule

+1) I]

+ 23 ]

H1 I

d-[

1

.l [

Q,

23 ]

X;+I)

I]

I 1- ] -x.1-1 +X.HI 1 -2

d-[- 2

X iI- I -XiI

+ X iI+1 + 21 ]

I = d-[-XI~ +Xl _.!.] 2 I

1]

.l[

-

2X iI- I + XiI -XiI+1 + 2

.l[

-

2X iI- I + XiI + X iI+1 - 21 ]

Q,

XiHI =

Q,

HI

+ XiI + X iI+1 -

d-[.!.-I 2 (2x;- +xI -

XiHI =

Xi

-XiI- 1

ch01

HI

Xi

[ 1] = 0

d- - 2

N

=d-[-x +.!.] 2 =""7" ~ ICO~Pleme~ted t

1-1

15:12

N

February 14, 2011

Formula defining the truth table of all 256 local rules.

Table 2.

1-1 ----T

RIght shIft

N

Formula for Local Rule = 4[X'

I-I

-~] 2

ID

Formula for Local Rule Xt+1 = I

111----I

4[-X' I

-x'

HI

+~] 2

I

III III

x:+

1

=

4[-1+1 (x'

I-I

-Xl_X' I HI

+~)I] 2

~[~ -1(xf-I- 2xf +4x:+, -l)l]

11----1-------------

Xf+l

=

~[~ -I (X:_ + 2xJ + Xf+l -1) I] 1

x: +1

=

~[~ -I (2xJ-l -X: +4x:+ -3)!] 1

14

Xit+l =

4[ ,

2' 3] +1 2" Xi -Xi'+

~+1 = 4[1-1 (-~+I (-2xf-l -4~ +~+1 +1)I) I]

Xit+1 =

4['

+ 23 ]

~+l = 4[1-1(32 -1(-2~- +~ -4~+1 +1)1)1]

Xf+l =

4[~ -I (Xf-l +xf +Xf+l- 1)1]

-Xi _1 -

-Xi _1 -Xi'-2X'i+1

~[, Xit+l = g, -Xi _1 -Xi, -Xi+1

'+ 2"3 ]

1

,

~[3' '+ 2"7 ] Xi-I-Xi-Xi+l

Xi,+1 =g, -

ch01

I Xt+l =

1

-x'I -x'H 1

N

15:12

X i'+1

1

(Continued )

February 14, 2011

Table 2.

N

Formula for Local Rule

-l[Xi-II - XiI + XiI+1--3 ]

Xi1+1 =

g,

x:+1 = 4.[~ -I (-2x:_ 1 +x: +2x:+.) 1]

__ I IIIIiiIIIII •

15

• • •

x:+'

=

4.[-1+ 1(x:_. -x: +x:+,- ~)I]

x:+!

=

4.[~ -I (2X:-l - x: -4x:+. + 1) I]

XJ+I = I X:+ =

4[2-1 (-xJ- + 2xJ +4xJ+I - 3) I] 2 I

4[1-1(-~+I(-4X: 1-2x; +X:+I +1)1)1] 2

-

Xt+1 = I

4[~-I(-x' 2

I-I

I-I

-2x'I +X't+ I

I -XlI +Xt+1

+1)1]

)1]

1]

-l[-Xi-II - 2' 1] Xi + XiI+1+ 2

g,

4[~-I(-2xI 2

I - XiI + 2' Xit+1 = g,-l[-Xi-I Xi+1-2

Xi'+I -- g-1, [ -x·II +xt+I 1 -2-1 ] Xit+1 =

Xt+1 = I

Xit+1 =

m r.t

III

-1 g,

[-Xi-I I - Xi'+' 1] Xi+1+ 2

XJ+I =

4[2-1 2 (-4xJ- [-2xJ +XJ+I +2)1]

X:+!

4.[H(-2X:-I-X: +x:+. + ~)I]

=

x:+ 1 = 4[1-1(~ -1(x:_ 1 -2x: -3x:+ 1 +4)1)1] Xit+1 =

4[- 3Xi-II -XiI + Xi'+1+ 25 ]

ch01

2

Formula for Local Rule

15:12

N

(Continued )

February 14, 2011

Table 2.

N

(Continued )

N

Formula for Local Rule 1

I

1

Xit+1

=

4[Xi-It - 2Xit- Xti+1+2" l ]

Xit+1

=

g,

11-----1-------------

•

,l[Xi-It - 2Xit + Xit+1-2"1]

16

Xf+1 = 4 [ -xf +2"1] = tXi =>

IComplement Self

Xf+1 = 4[~ -I (-2xf_1 -4xf - Xf+1 + 3) I] xf+1 =4[1-1 (-~+I (4xf_1 +xf - 3xf+1 -2) I) Xf+1

=

4[1-1 (-xf_1 - 2xf -xf+1 + ~) I]

Xt+1

=

4[-Xt -3xt -xt +2] 2

I

I-I

I

HI

x:+1=4[~ -I (2x:_1+4x, -x,+1-2)1]

•

I]

• •

x:+1=4[1-!(-x:_1-2x: +x:+1+ ~)I] xf+1

=4[1-1(~-I(-xf_1 +2xf -4xf+1 +3)1)1]

Xit+1

=g,

X:+

I

,l[-Xi-It - 3Xit+1+ Xit+1+ -5 ] 2

=4[~ -I (X:_1+X: -1)1]

X:+ I

=4[~ -I (-2X,_1 -2x, +X:+1+1)1]

Xf+1

=4[~ -I (-2xJ_I - 2xJ - Xf+l + 2) 1]

Xt+1 =4[-Xt -xt +~] 2 1

I-I

1

ch01

I

1-

15:12

III

Xt+1= 4[Xt [-xt +-~] 2

Formula for Local Rule

February 14, 2011

Table 2.

N

Formula for Local Rule Xl+l I

=

4[XI

+XII -Xl1+1

1-1

-~] 2

Xi1+1

= g,-l[ XiI -XiI+1 -2"1 ]

Xi1+1 -_

-l[-Xi-

g,

I

I 1 +Xi

-

1]

2X iI+1 +2"

2xI 4[!-IC2

I

l -2x1+1 +1)1]

•

Xl+l I

=

•

Xf+1

=ti[~-I(Xf-1-Xf+Xf+1)1]

•

x,+1 = ti[~ -I (2xf-l +xf +4x,+1- 4)1]

•

xt1=ti[H(2xf-1-xf+xf+1-~)I]

I. ' II_I

Xi1+1

1-1

+XI

= g,-l[-XiI- 1 + 2XiI -XiI+I -2"1 ]

Xi1+1 -_

-l[-Xi-

g,

I

I 1 +Xi

1]

-XiI+1 +2"

x,+1 = tiH-1 (-Xf-1 +4xf +2xf+1- 3)1]

III : I

Left Shift

(+2 t + 1 ]

= a~[t -Xi- 1 - Xi

X i+1

-

2

=4[1-1 (~ -I (-3xf-l -4xf +2xJ+l +3)1)1]

=4[~-le-2x:-l-X: +2x:+1) 1] 2

t+l =

Xi

t+l =

Xi

[

4[

4[

(+ Xi(+ 2X i(+1-21 ]

-Xi- 1

-Xit- 1

+ X it+1 +21 ]

ch01

1 X:+ =

Xi

t

-4 [ Xi-1+Xi+1

N

15:12

N

(Continued )

February 14, 2011

Table 2.

N

(Continued )

Formula for Local Rule I

=

&'[2x'1-1 -x' +x'HI -~] 2 I

1

1

24

ImiJ 11II 1m III

1

Xi-1- Xi'+' Xi+1-2"1 ]

Xi'+1

=

-1 [ , U-

XiHI

=

&.['Xi-1- 3'Xi + Xi'++ 2"3 ] 1

tL[l-1 (-2+ 2 I(x:-

X:+1 =

4[~ -I (-x:- 2x: + 2x:+ 1]

Xi'+1

=

&.['Xi-1-Xi,+ 2'Xi+ -2"1 ]

XHI

=

&.[-x' +x'HI +~] 2

r,+l

=

4[% -I (2r,_1 + 2x: - x:+

HI

=

&'[1-1 (x'

I

X:+1 = 4[H(-2x:_1 -x: +X:+1 + ~)I] X:+ 1= &.[~-1(2X: 2 - 1 +X: -2x:+1-1)1]

X:+1 =

4[- ~+ 1(-r,-I +X: - X:+ I]

X:+ I

&.[- ~ + 1(x:_ 1- 2x: + x:+ + 1) I]

=

x:+1 =

1)

1

1m

1-

2x: +3x:+ 1 - 3)

1-

DI]

1)

I

I

1-

2) 1]

+ x' - x'HI -~) 2 I]

III!ftI

XI

IfIII'III I.Ii6II

XHI = &.[-~+ (-2x'1-1 + 2x' +x'HI ) I] 2 1

. .

1-1

I

XiHI

I

I

-1[' -Xi-1 - Xi'+' Xi+1 + 2"3 ]

= u-

ch01

x:+ = tL[l-1 (-%+1(4x:_ -2X: -x:+ -2)DI]

Formula for Local Rule

15:12

X'+1

N

February 14, 2011

Table 2.

Formula for Local Rule

=

0-

=

4[~-1(4xf 2 -

Xf+l

=

4[1-I(xf_I-2xf +Xf+l + ~)I]

=

4[~ -I (-2xf_1 +4xf +xf+1 - 2) 1]

=

4[1-1(~ -1(3xf_I- 4x: + 2xf+l)DI]

Xf+l

=

4[~ -I (-2xf_l +2xf -Xf+l) 1]

Xf+1

=4[xf - ~]= x;

Xit+l

=

4[-Xi- + 2Xi - Xi+ +'21 ]

Xit+l

=

0-

Xt+1

=

xt l

I

1-1

I

1-

25

Xf+1 = 4[~ -I (-xLI =

2xf -Xf+l-1)1]

+xf)l]

-l[X.t I + 2X·t -x·t I - -23 ]

0-

1-

I

H

Xf+l = 4[1-1(~-I(-3x: 2 - 1 +4x: +2x:+1 -2)DI] - 2x' - XlHI +~) I] 2

II1I"I I.MI

XI+1 =

4[1-1 (Xl

l1l':I I.rM

Xt+l

4[~2 -I (2x t

I

=

1-1

I

I-I

-

2xt - XlHI. + 1) I] I

x:+

I

1

I

I

I

I

I

1

=> I

IIdentity I

1

-l[-Xi-I + 2Xit + Xi+1 -'21 ] I

I

4[-X' +XI +~] 2 I-I

I

ch01

Xf+l

I

-l[Xi_1+2Xi + Xi+1 -'25]

Xit+l

=

X·It+l

Formula for Local Rule

4[X' +X _~] 2

Xt+1 I

1m 1m III

N

15:12

N

(Continued )

February 14, 2011

Table 2.

N

(Continued )

Formula for Local Rule

N I Formula for Local Rule

4[

= 2Xi-II + XiI -XiI+1 (2xf_J

26

Xf+1

1m

1+1

Xi

+ xf -4xf+l -1)

2

- I +2xf -Xf+l

2

4[

X i _1

4[

X i- l

=

23 ]

I

1]

+ XiI -XiI+1 - 2

I

mJ

•

1m III Xt+l =ti[-.!.+I(x' +x'-x' )1] mJ l1li Xf+l =ti[- ~ +1 (2xf_, +2.x, +.x,+1-3)1] 11m 1+1

Xi

I

=

I

2

+ XiI - 3X iI+1 + 23 ] I-I

I

1+1

-

I-I

4[

+ 2XiI -

Xi1+1

=

X iI_1

X I +I

=4[

Xl -Xl I 1+1

I

Xf+1

1+1

Xi

=4[- ~ + I + =4[

I

X iI+1

1+1

-21]

+!2 ]

(-Xf-l - 2xf

I

-Xi-I

2

I

I

Xi -Xi+1

+ Xf+l) I]

+ 23 ]

ch01

Xf+J

15:12

.1 x,+1 =4[1-1(-%+I(-3X,-1 +x, -4x,+1 +4) 01] Ell I) I] III .x,+l =4~-I(-Xf ,+2xf -2xf+,) 1] III =4[1-1 (%-1 mIJ Xf+' =ti[1-1(-2xf_l +xf -Xf+l +~)I] 1m xf+' =ti[~ -I (2xf-l - xf +2.x,+1 - 2) 1] +1)1] 11m x'+l =ti[l-I(-x' +x'-x' +~)I] 1m =4[~-I(-2xf 1+1 Xi

February 14, 2011

Table 2.

Formula for Local Rule

t+l I + I Xi =Q; X i- 1 Xi

+ XiI+1-23 ]

xt

Xf+1 =~[I-I(%-I(-2X:_l-Xf -4xf+l +5)1)1]

x.It+l

11 =Q; X.1-

Xf+l

=ti[- ~ + I

x.1+1

1l =Q; X.1-

=4[- ~ + I(2xf_1 - xl +2xl+

27

Xf+l

1

=ti[~-I (2xf -2xf -Xf+l) I] 21

Xf+l =~[

1-1 (~ -I(2xf_1 +4xf +xf+1 - 5) I) I]

1

I

1

.1[

.1[

I =ti[~-I (-2x1-1 +X~ + 2x ) I] 2

xl+1

Xf+l

=ti[~-I(-xf 2 -

I 1 .1 [XI X i + = Q; I

Xf+l

=ti[1-1

t+l

I

1 +2xf

+ 2xf+I- 2

(Xf-l - xf - Xf+l

+ ~)

I]

)1]

I

1+

(2xLl

Xt+l I

l

+ X.1+ 2X·I 1 - -23 ]

.1[

+ 2xf + Xf+l - 2) I]

+ 2X.I + X.I I - -3 ] I

t+

2

+ Xlt+ 1 -~] 2

1] Xit+l =Q; -Xi1- 1 + Xi1 + X i1+1 + 2

1-

2) I]

ch01

=4[- ~ + I(Xf-l +X: +x:+ -1) I]

=4[1-1 (-2xl_l +xl +x:+ + ~)I]

xl+1

• •1m

Formula for Local Rule .1[

.1[2XiI- 1+ XiI + X iI+1-25 ]

t+l Xi =Q;

ft'!I ~

N

15:12

N

(Continued )

February 14, 2011

Table 2.

N

(Continued )

Formula for Local Rule

x t+l

=0,

1+1 Xi

=

0,

-1[2 X i- 1 -

Xi

Xt+l

=

4[X

I

I

28

I

I

1-1

-1[2X

I I X i + --

-1

0, [

I

I

1-

I

I

~I

I 2x1-1

= 0,-1[ X iI- 1 -

Xf+l

=

1+1

I

-Xl

t+l Xi

Xi

I

1 -X-X' 1 l+ 1

2

I

I

+2 3]

=

l

1-1

2xf+l)

I]

-1[X iI

0,

1-

XiI

3] 2

-Xl I

2

+ Xl -~] 2

t+l Xi

=

4[

t+l Xi

=

4 [X iI- 1 + XiI + X iI+1 _ _21]

I

I

X i- 1

1

+ XiI -

I

X i+1

x:+ =o[~] =1 => 1

HI

+ X iI+1 + -1 ]

4[X

I-I

X~ +1) I]

]

I 1 X + = I

+ 21 ]

Xi - X i+1

+ XiI + X iI+1 (2x

Xi1+1

+ 2xf -

I

X i- 1

1-

I

(-Xf-l

4[2

x:+1= 0,[X:-l + x:+ ~

+XI -xl+I 1 -21- ] I

=

I

+~] 2

X i+1

1+1 Xi

II!tft' aM X~+1 = 4[-~+ 2 I

+ -1 ]

+ X iI+1 -21 ]

4[- ~ + I

= 0,-1[ X iI_1 -

Right Shift

+ 21 ]

All neurons Firing

ch01

=4[X

Formula for Local Rule

15:12

-~]= X~1-1 ~ 2

Xt+l I

N

February 14, 2011

Table 2.

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

that 0 ≤ φ < 1 for finite I, and when I → ∞, we have a time-1 map over the unit interval ρ: [0, 1) → [0, 1)

(8)

It is important to remember that each time-1 map is uniquely associated with one space-time pattern, or “orbit ”, from one initial bit-string configuration.

1.5. We only need to study 88 rules! Although there are 256 local rules, only 88 rules are globally independent [Chua et al., 2004] from each other. All other rules are equivalent to one of the 88 rules listed in Table 4 of [Chua et al., 2007a]. These 88 rules are listed4 in Table 3 along with an integer code M ∈ {1, 2, 3, 4, 5, 6}, where M denotes one of the following six distinct qualitative dynamics exhibited by a particular local rule N [Chua et al., 2007a], corresponding to random initial configurations:

D

Group 1 Rules Almost all space-time patterns converge to a period-1 orbit. The time-1 map corresponding to each period-1 orbit would consist of a single point attractor, or an Isle of Eden,5 on the main diagonal line, after deleting points belonging to the transient regime. Group 2 Rules Almost all space-time patterns converge to a period-2 orbit. The time-1 map of each period-2 attractor, or Isle of Eden, consists of two points, symmetrical with respect to the main diagonal line. Group 3 Rules Almost all space-time patterns converge to a period-3 orbit. The time-1 map of the period-3 attractor, or Isle of Eden, consists of three points. Group 4 Rules Almost all space-time patterns converge to a Bernoulli στ -shift attractor, or Isle of Eden, where |σ| ∈ {1, 2, 3} and |τ | ∈ {1, 2, 3, 4, 5}. We stress that the above qualitative behaviors do not depend on the length L of the bit strings, and do not depend on the initial configurations,

D

4

D

29

even though there may exist several Bernoulli attractors with different σ and τ , each with its basin of attraction. Group 5 and Group 6 Rules The space-time patterns typically have very long transients and converge to a period-T attractor with a very large period T . Moreover, the asymptotic behavior depends not only on the initial configuration, but also on the length L of the bit string. One difference between a group 5 rule and a group 6 rule is that the former is bilateral (and hence has only one globally-equivalent rule), whereas the latter is non-bilateral (and hence has three other globallyequivalent rules). The classification of each of the 256 local rules is given in Tables 7–9, 11, and 12 in [Chua et al., 2007a]. Given any rule not among those listed in the 88 globally-equivalence classes in Table 3, one can easily look up Table 4 from [Chua et al., 2007a], or Table 3 from [Chua et al., 2007b], to identify its equivalent rule, and then look up its complexity index κ (red, blue or green), and group M (1, 2, . . . or 6) from Table 3. For future reference, the complexity index κ and class M of all 256 rules are listed in Table 4. Counting the number of globally equivalent rules from each class from Tables 3 and 4, respectively, we summarize their distributions in Figs. 5 and 6, respectively.

1.6. The “Magic” rule spaces In [Cattaneo & Quaranta Vogliotti, 1997], a subset of 104, among 256, local rules have been derived and shown to exhibit “neural-like” behaviors. The authors’ approach is based on an exhaustive mathematical analysis on a bi-infinite sequence space, consuming more than 20 printed pages. The authors were so perplexed by their discovery that they dubbed these rules “magic ”. A cursory inspection of the 256 Boolean cubes listed in Table 1 would extract, in a few minutes, 104 local rules with a complexity index κ = 1, namely, those Boolean cubes whose red vertices can be separated from the blue vertices by no more than

This list is not unique in the sense that one can pick many other groups containing 88 independent rules. Our choice is obtained by scanning the 256 rules from N = 0 to N = 255 , and deleting any rule that is equivalent to a previously listed rule. 5 Robust Isles of Eden can be observed only for those rules endowed with dense Isles of Eden orbits [Chua et al., 2007a, 2007b].

February 14, 2011

30

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

Table 3. List of 88 globally independent rules. Color surrounding rule number N corresponds to complexity index κ = 1 (red), 2 (blue) or 3 (green). The integer on the lower right corner identifies the characteristic property of the rule, as specified in the color legend.

0

1 1

2 2

8

10 4

18

2

37 2

4

46

51 2

62

2

72 3

73 1

94

104 1

150 5

1

134

152

170

6

136

154

156

4

172 4

6

178

184

1

2

142

162

200

5

1

1

4

126

140

160

5

5

4

2

90

122

138

6

1

6

1

60

78

110

6

4

1

2

4

1

4

1

1

45

58

77

108

1

5

168

4

106

132

4

76

5

4

146

74

105

130 1

5

36

44

57

4

4

4

56

5

1

128

54

4

27

35

43

4

6

4

42

15

26

34

4

4

4

2

6

4

14

25

33

41

7

1

4

1

1

50 4

32

40

13 1

2

6 2

24

6

38

1

4

23

30

5

12

5

29 2

11

22 2

28

4

4

19 5

4

4

9 1

3

4

164 4

204 1

1

232 1

1

Color Legend N

1

Period-1

N

2

Period-2

N

3

Period-3

N

4

N

5

Bernoulli Complex

N

6

Hyper

31

4

1

4

1

4

1

4

1

4

1

4

1

241

225

209

193

177

161

145

129

113

97

81

65

49

33

17

1

4

6

4

6

4

5

3

5

4

6

4

4

4

2

4

2

242

226

210

194

178

162

146

130

114

98

82

66

50

34

18

2

4

4

6

4

2

4

5

4

4

4

6

4

2

4

5

4

243

227

211

195

179

163

147

131

115

99

83

67

51

35

19

3

4

4

4

6

2

4

5

3

4

4

4

4

2

4

2

4

244

228

212

196

180

164

148

132

116

100

84

68

52

36

20

4

4

1

4

1

6

1

4

1

4

1

4

1

4

1

4

1

245

229

213

197

181

165

149

133

117

101

85

69

53

37

21

5

4

4

4

1

6

5

6

1

4

6

4

1

4

2

4

2

246

230

214

198

182

166

150

134

118

102

86

70

54

38

22

6

4

4

4

2

5

6

5

4

3

6

6

2

5

4

5

4

247

231

215

199

183

167

151

135

119

103

87

71

55

39

23

7

4

4

4

2

5

6

5

6

4

4

4

2

2

4

2

4

248

232

216

200

184

168

152

136

120

104

88

72

56

40

24

8

1

1

1

1

4

1

4

1

6

1

4

1

4

1

4

1

249

233

217

201

185

169

153

137

121

105

89

73

57

41

25

9

1

1

1

2

4

6

6

6

6

5

6

5

4

6

4

4

250

234

218

202

186

170

154

138

122

106

90

74

58

42

26

10

1

1

1

1

4

4

6

4

5

6

5

4

4

4

6

4

κ = 1 (Red) 104 rules, κ = 2 (Blue) 126 rules, κ = 3 (Green) 26 rules

240

224

208

192

176

160

144

128

112

96

80

64

4

1

4

1

251

235

219

203

187

171

155

139

123

107

91

75

59

43

27

11

1

1

1

1

4

4

4

4

5

6

2

6

4

4

4

4

252

236

220

204

188

172

156

140

124

108

92

76

60

44

28

12

1

1

1

1

4

1

2

1

4

2

1

1

6

1

2

1

253

237

221

205

189

173

157

141

125

109

93

77

61

45

29

13

1

1

1

1

4

4

2

1

4

5

1

1

4

6

2

1

254

238

222

206

190

174

158

142

126

110

94

78

62

46

30

14

1

1

1

1

4

4

4

4

5

6

1

1

3

4

6

4

255

239

223

207

191

175

159

143

127

111

95

79

63

47

31

15

1

1

1

1

4

4

4

4

2

4

2

1

4

4

4

4

15:12

48

32

16

0

Table 4. List of 256 local rules. Color surrounding each rule number N corresponds to the complexity index κ = 1 (red), 2 (blue) or 3 (green) of N . The integer in the lower right corner identifies the characteristic property of the rule, as specified in Table 3.

February 14, 2011 ch01

D

February 14, 2011

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88 Equivalence Classes of Local Rules 32

I

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

1.7. Symmetries among Boolean cubes

..........

Many of the 256 Boolean cubes in Table 1 share interesting symmetrical features which give rise to important predictable dynamics and applications. We will briefly recall some of these symmetries and present new interpretations.

30

Bernoulli (J"t-shift Rules

D

1.7.1. Local complementation

D

D

D

D

Tc

D

We define the local complementation Tc

N −→ N c

D

256 Local Rules

Fig. 5. Partitioning of the 88 globally-independent rules into 6 classes.

67 Period-l

25 Period-2

..........

Rules

ff~ '$'

~~~

,,-"v r;:,~ ':o.'l.J"

,) :l.~~.::i Rules ~'l.J t======~~~-_ 18 Complex

108 Bernoulli

D

(9)

of a Boolean cube N to be the Boolean cube N c obtained by complementing the color of each vertex, i.e. 0 → 1 and 1 → 0 . Table 6 shows ten Boolean cubes and their local complements. This transformation is called local to emphasize that the D of NDand N c , with the same space-time patterns initial configuration, are not the complement of each D because the complementation is valid only for Dother one iteration. This observation is demonstrated in Fig. 5 of [Chua et al., 2004], where

ernou/Ii-sbift ules

Tc

110 −→ 145

(10)

Observe that the space-time patterns of 110 and 145 (for the same initial configuration) are not the complement of each other, except for the first iteration.8

a.-shift Rules

1.7.2. Three equivalence transformations T † , T , and T ∗

Fig. 6.

Partitioning of the 256 local rules into 6 classes.

one plane.6 These 104 rules are listed in Table 5 along with their classification number M , extracted from Table 4. A comparison of the 104 κ = 1 rules in Table 5 with those derived in [Cattaneo & Quaranta Vogliotti, 1997] shows that they are identical.7 From our Boolean cube perspective, the “magic” connotation is perhaps a bit of an anti-climax.

D D

D

6

There are exactly three global transformations that hold for all iterations, and for all initial configurations. They are the Left-Right Transformation T , and the LeftT † , the Global Complementation D Right Complementation T ∗ [Chua et al., 2004]. These three transformations, along with the identity transformation, have been shown in [Chua et al., 2004] to form an Abelian group known as Klein’s Vierergruppe. Given any local rule N , the transformed rules N †
T † (N ), N
T (N ), and N ∗
T ∗ (N ) can be derived by inspection via the simple geometrical operations illustrated in Fig. 7.

D

Since no plane is needed for rules 0 and 255 , these two rules may be reclassified with a complexity index κ = 0. Except for rule “36” listed in Tables 4 and 8 of [Cattaneo & Quaranta Vogliotti, 1997], which we believe is a typo that should be rectified to rule 136, as correctly reported in Fig. 12 of the same paper. 8 There are some rules, however, where the local complementation T c (N ) coincides with the global complementation T (N ) to be defined below. In such cases, the space-time patterns of N and N c are also complements of each other for all t. 7

February 14, 2011

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ch01

D

Chapter 1: Quasi-Ergodicity

33

Table 5. A gallery of 104 linearly-separable local rules. The red color engulfing each rule number N implies a complexity index κ = 1 for all 104 rules.

: List of 104 Linearly-Separable Boolean Function Rules. 0

2

1 1

15

17

16 4

48

80

84

81 4

128

138

136 1

176 4

213

4

241

243

4

Color Legend

4

221 1

244 4

245 4

Period-1 N Rule

224

223

4

4

1

To derive the left-right transformation N †
(N ) of any local rule N , simply obtain the 0 mirror0image of the 0 Boolean0cube N about the main diagonal plane (shown shaded in Fig. 7(a)) pass. Note that this operaing through the vertex 0m tion can be implemented by identifying each pair of symmetrically located vertices with opposite colors, and Hence 124 =
then complementing
the colors.  T † 110 , and 110 = T † 124 0. To derive the global complementation 0 N
T (N ) of any local rule N , simply identify each pair of diagonally opposite vertices that have the same color (either both red or both blue), and then change the color. For the example illustrated in Fig. 7(b), we have N = 110 . Among the four pairs of diagonally opposite vertices of 110 , we find 0m , only three  that   pairs of vertices m m m m m 7 , 2 , 5 , and 1, 6 have the same colors. Changing only the color of these three T†

232

248

247

200 1

234 1

250 1

1

251 1

Period-2 N Rule

1

2

1

207

239

253

252

4

1

1

1

175

206

238

236

2

4

1

1

127

174

205

204

1

4

4

4

79

119

171

170

4

1

4

4

1

1

1

1

168

196

117

115 4

4

1

4

1

162

192

191

113 4

1

4

4

220 4

4

242

4

160

143

187

112

47

77 1

1

4

4

76

69 1

43 4

4

14 1

42

35

68

13 1

4

4

1

2

1

4

186 2

2

142 1

95

93

34

64

12

11 4

1

4

4

4

140

179

212

4

4

1

178

208

4

4

87

32

63

59

10 1

4

2

2

85

31

23

55 2

8 4

2

4

51 2

4

21 2

7

5 1

19

50

49

4 4

4

4

4

3 4

2

1

240 1

254 1

4

255 1

1

Bernoulli N Rule

4

pairs, respectively,
we obtain its global complement 137 = T 110 , and conversely, 110 =
 T 137 . To derive the left-right complementation N ∗
∗ T (N ) of any local rule N , simply take the global 0 complementation0 first, followed 0by the left-right transformation, or vice-versa, i.e.

0

∗ N0
T ∗ (N ) = 0 T † (T (N )) = T (T †(N ))

(11)

0

For example, consider taking first the global
com- plementation of 110 to obtain 137 = T 110 0 by applyin Fig.07(b). If we 0 follow this-operation ing the left-right transformation to 137 , we would
 obtain 193 = T † 137 by reflecting the Boolean cube 137 in Fig. 7(b) about the main diagonal, as shown in Fig. 7(c), to obtain


 193 = T † 137 = T † T 110

(12)

February 14, 2011

34

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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 6.

N 2

N 3

7

6

1

0 4

5

1 5

4

7 1 5

5

4

3 7 1

0 5

2 7

6

1

0 4

90 3

5

7 1

0 5

105

2

1

4

5

150

1

2

4

5

85

1

184

1

0 4

7 5

3 7

6

3

0

5

101

5

6

1

0 4

7

2

3 7

6

170 3

0

2

3

0 4

7

6

5

6

165

2

1

2

5

105

154 3

1

0

3

0

3 7

6 4

7

6

195

2

5

2

150

1

5

145

1

2

1

0

3

0

3 7

6 4

7

6 4

7

4

5

2

3

0

60

4

5

6

1

c

2

110

1

2

3

0

3

0

N

7

6 4

7

6

3

0

6

2

210

2

4

5

2

45

6

1

0 4

7

4

3 7

6

3

0

6

N

225

2

4

c

2

30

6

Some Boolean cubes and their local complements.

2

3 7

6

1

0 4

5

71

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ch01

copy the color of vertex of the left cube

Chapter 1: Quasi-Ergodicity

0

L_~

copy the color of vertex of the left cube

e

~

) (

Tt (a) Left-Right transformation Tt

T

)

(

T

III

(b) Global complementation T

T* ) (

T* (c) Left-Right complementation T*

Fig. 7.

Geometrical constructions for deriving (a) N †
T † (N ), (b) N
T (N ), and (c) N ∗
T ∗ (N ).

35

February 14, 2011

36

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7. 2

3 7

6

1

0 4

A gallery of 16 centrally-symmetric local rules.

5

2

3 7

6

5 1• • • • +--..

4

2 7

1

0 5

2

4

1

2

4

195

7 1

5

5

4

219

We will henceforth call such perfectly symmetrical transformations (relative to the complementation transformation) perfect complementations. The set of all perfect complementary rules are listed in

3 7

6

1

0 4

5

189

7 1 5

231

(13)

5

2

3

0

1

0

126

1

2

3 7

6

3

0

6

5

4

7

6

1

2

165

It follows from the geometrical construction of the global complementary transformation T (N ) → N in Fig. 7(b) that 
pairs of diagonally
 oppo

if all four m m m m m m 0 , 7 , 2 , 5 , 3 , 4 , 1m , sitevertices 6m of N have identical colors, respectively, then

9

5

4

1.7.3. Perfect complementary rules

N
T (N ) = T C (N )
N C

1

2

3

0

3

0

3

7 0

4

7

6

3

6

102

5

6

2

60

2

153

1

5

4

7

4

7 5

5

0

3

0

1

2

129

6

4

7

6

1

0

90 3

7

6

3

0

••• II1...--t~ 66

3

36

2

•••• 1

0

4

5

2

24

~t---t~

6

1

0 4

2

4

7

6

0

6

3

2

3 7

6

1

0 4

5

255

Table 7. Since these Boolean cubes exhibit perfect symmetry in color with respect to the origin located at the center of the cube, we will henceforth call them centrally-symmetric local rules. Clearly, the local complementary space-time patterns of all centrally-symmetric local rules hold for all times.

1.7.4. Permutive rules9 There are 28 local rules whose Boolean cubes exhibit an anti-symmetry with respect to some vertical plane through the center of the cube, as illustrated in Fig. 8.

Permutive rules are originally defined by [Hedlund, 1969] on a formal topological setting. We have opted for an equivalent but geometrical definition via the Boolean cubes for pedagogical reasons. These two representations are equivalent, as shown in Appendix C.

February 14, 2011

15:12

•••

ch01

•••

•••

•••

Chapter 1: Quasi-Ergodicity

III:

III:

Left-Permutive

lEa:

Right-Permutive

37

Bi-Permutive

D D

D (a)

(b)

D

(c)

Fig. 8. Geometrical illustrations of permutive rules. (a) Rule 30 is Left-Permutive because the colors of the vertices ˘ ¯ ˘ ¯ 0m , 1m , 2m , 3m in the back face are the complement of the colors of the vertices 4m , 5m , 6m , 7m in the front ˘ ¯ m m m m in the right face are the comface. (b) Rule 154 is Right-Permutive ˘ because the colors ¯ of the vertices 1 , 3 , 5 , 7 plement of the colors of the vertices 0m , 2m , 4m , 6m on the left face. (c) Rule 150 is Bi-Permutive because it is both Left and Right Permutive.

D

A local rule N is said to be Left-Permutive, iff the vertical symmetry plane is parallel to the paper, as depicted by the “green” plane in Fig. 8(a). It is said to be Right-Permutive, iff the vertical symmetry plane is perpendicular to the paper, as depicted by the “pink ” plane in Fig. 8(b). It is said to be Bi-Permutive, iff it is both Left- and RightPermutive as depicted by the “green” and “pink ” vertical symmetry planes, respectively. A local rule N is said to be Permutive iff N is Left and/or Right-Permutive. An examination of the 256 Boolean cubes in Table 1 shows that there are only 16 Left-Permutive rules, 16 Right-Permutive rules, and 4 Bi-Permutive rules, as displayed in Tables 8–10, respectively. The union of all these local rules gives only 28 distinct Permutive rules, as exhibited in Table 11. We will show in the following sections that Permutive rules possess some remarkable properties.

D

depicted in Table 13. It is easy to prove that each rule N in Table 13 has a unique decomposition via the eight Boolean cubes basis functions in Table 12.

D

1.7.5. Superposition of local rules The eight Boolean cubes exhibited in Table 12 are independent in the sense that it is impossible to decompose any of them into the “union” of two or more simpler Boolean cubes by taking the logic “OR” operation between the colors of corresponding vertices, where “red ” is coded “1” and “blue” is coded “0”, respectively. Since each of these eight Boolean cubes contains one, and only one, red vertex, together they constitute a basis function where the “union” of two or more such rules can generate any of the remaining 256 − 8 = 248 local rules, as

1.7.6. Rules with explicit period-1 and/or period-2 orbits Recall from Fig. 6 that among the 256 local rules, 69 are endowed with robust period-1 (attractor or Isleof-Eden) orbits, and another 25 rules are endowed with period-2 orbits. It is generally impossible to predict the bit-string pattern of such period-1 or period-2 orbits without actually evolving the rule from some initial state. The purpose of this subsection is to prove a surprising and quite remarkable result asserting that the period-1 and period-2 bit strings of a large number of local rules can be predicted without carrying out any simulations. Such period-k bit string (k = 1 or k = 2) patterns are endowed upon those Boolean  m m m cubes  whose main-diagonal vertices 0 , 2 , 5 , m 7 exhibit certain color combinations. Explicit period-(1, 2) pattern theorem There are ten distinct color combinations among   the four vertices 0m , 2m , 5m , 7m on the maindiagonal plane of the Boolean cubes, labeled Type A, B, . . . , J in Tables 14(A), (B), . . . ,(J) for which the corresponding local rules have an explicit period-1 and/or period-2 bit-string pattern, regardless of the colors of the  remaining nondiagonal verm m m m tices 1 , 3 , 4 , 6 .

February 14, 2011

15:12

ch01

•••

•••

•••

•••

•••A Nonlinear Dynamics•••Perspective ••• of Wolfram’s New Kind of Science ••• • •• ••• Table 8. Sixteen left-permutive rules. ••• ••• •••

38

III •••

Ell

•••

•••

•••

•••

•••

•••

•••

•••

•••

•

&I •••

m

III

•••

•••

IBJ

11m

•••

•••

•••

•••

•••

•••

••• •••

1m

1m3

mtJ

1m

lIm

•••

•••

•••

••• ••• •••

••• ••• •••

•••

om

•

••• •••

•••

•••

1m

Proof.

It follows directly from the evolution of the Boolean cube inset on top of each table. Here, the “white” vertices denote irrelevant vertices. 

3. The Boolean cube inset in Tables 14(G)–14(J) has four color vertices on the main diagonal.

Remarks

1.7.7. Most rules harbor at least one Isle of Eden

1. The Boolean cube inset in Tables 14(A) and 14(B) have only one color vertex along the main diagonal. 2. The Boolean cube inset in Tables 14(C)–14(F) has two color vertices (with opposite colors) on the main diagonal.

Among the 256 local rules, 228 harbored at least one Isle of Eden. In particular, there are 79 (out of 88) topologically distinct rules with at least one Isle of Eden, and Table 15 displays the initial and final bit-string configurations of a typical Isle of Eden

February 14, 2011

15:12

ch01

••• ••• •••

m •••

•••

••• ••• •••

•••

•••

••• ••• •••

•••

•••

•••

Table 9.

Sixteen right-permutive rules.

•••

•••

••• ••• •••

•••

• ••• •••

•••

•••

•••

•••

•••

•••

m1

11m

1m

•••

•••

•••

•••

•••

•••

39

•••

DIm •••

•••

•••

•

••• ••• •••

IItI

IIID •••

••• Chapter 1: Quasi-Ergodicity

III

1m •••

•••

•••

•••

•••

•••

1m

11m

11m

••• ••• •••

•••

1m •••

•••

••• ••• •••

•••

11m

D D

orbit for each of them, so that readers can verify that it is indeed periodic. We end this recap section by exhibiting in Fig. 9 a period-3240 Isle of Eden of rule 30 with L = 27, a real gem that should trigger a rush to uncover Isles of Eden with even longer periods. The φn — versus — φn−1 return map of this period-3240 Isle of Eden is shown in Fig. 10(a). Note the very messy plot gives very little information. However, if we plot this Isle of Eden orbit as a

τ = 480 return map in Fig. 10(b), we see the φn — versus — φn−τ time-τ map is virtually identical to the Bernoulli mapD φn = 2φn−τ

mod 1

(14)

associated with the time-1 return map of rule 170 . Since it is well known that the Bernoulli map Eq. (14) is ergodic, Fig. 10(b) strongly suggests empirically that rule 30 is quasi-ergodic in the sense of Definition 2.1 of Sec. 2.1.

Fig. 9. Orbit of a period-3240 (with σ = 1, τ = 480) Isle of Eden of 30 with L = 27. The number inscribed inside each “capsule” is the decimal representation of a 27-bit string.

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February 14, 2011

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February 14, 2011

Table 31.

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Table 33.

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::::::::::::::::::::::::::::::::::::::::::::::1:::::: • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • •

135

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Data for generating and verifying the period of period-T orbits of 150 .

IAttractor

(x o,x T-l) ~ (initiaIJjnal) bit string on Period-T orbit

I

Xo I • • •

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5 6

T* = 3

7

T* = 7

---i~---I-=-=-=:lZ-Z-------------------------------------------------------------------------------------------------------------------------------------------------------------------------

8

T* = 4

···i~··+=·=·=IIZ·Z·Z··································

T* = 1

---i~----I-=-=-=:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

T* =2

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I

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Xo I • • • • • •

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10

T* = 6

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11

T* = 31

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12

T* = 2

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136

13

T= T*/3 = 21

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14

T* = 14

--i;;---I-=-;-=IIZ-Z-Z-=-;-;-=:Z----------------------------------------------------------------------------------------------------------------------------------------------

15

T* = 15

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16

T* =8

---i~---I-=-;-=IIZ-Z-Z-=-;-;-=:Z-Z-Z--------------------------------------------------------------------------------------------------------------------------------------

17

T* = 15

··i;~···I·=·=·=:lZ·Z·Z·=·;·=·=:Z·Z·Z·=················ T* = 14

18

_

.

--i;;-+=-;-;IIZ-Z-Z-=-;-;-;lZ-Z-Z-Z-;------------------------------------------------------------------------------------------------------------------------------

19

T* = 511

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20

T* = 12

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.

22

T* = 62

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23

T* = 2047

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T* = 63

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9

15:12

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February 14, 2011

Table 34.

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26

T= T*/3 = 42

27

T* = 4

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T* = 28

29

T* = 2 14 _1

30

--i;~;+IZ-I-IZ-I-IZ-I-IZ-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;------------------------------------------------------------------------------------~~-;+IZ-I-IZ-I-=Z-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-I-------------------------------------------------------------------------------T* = 30

31

T* = 31

32

T* = 16

33

T* = 31

137

34

T* = 30

35

T* = 4095

36

T* = 28

37 T=T*/9=29127 38

T* = 1022

39

T* = 4095

40

T* = 24

41

T* = 1023

42 I 43

T* = 127

44

T* = 124

45 46

··i;~~+IZ·I·IZ·I·=Z·I·=Z·;·IZ·;·I·=;·I·=;·I·=Z·;·=Z·;·IZ···················································· . --i;~~+IZ-I-IZ-I-=Z-I-IZ-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;--------------------------------------------------------------------------i~~~+IZ-I-IZ-I-IZ-I-IZ-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;-I--------------------------------------------------------------------··i;~~+IZ·I·IZ·I·=Z·I·=Z·;·IZ·;·I·=;·I·=;·I·=Z·;·=Z·;·IZ·;·ll··············································· . --i;~~+IZ-I-IZ-I-IZ-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;-IIZ--------------------------------------------------------------x~~;~+IZ-I-IZ-I-IZ-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-1Z-;-IIZ-I---------------------------------------------------------··i;~;+IZ·I·IZ·I·=Z·I·=Z·;·IZ·;·I·=;·I·=;·I·=Z·;·=Z·;·IZ·;·IIZ·I·=·········································· . -x;~-;-tlZ-llZ-I-=Z-I-IZ-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;-IIZ-I-=Z--------------------------------------------------l~~;;+IZ-I-IZ-I-=Z-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;-IIZ-I-=Z-I-----------------------------------------------

-x~~;JIZ·IIZ·I·IZ·I·IZ·;·IZ·;·I·=;·I·=;·I·=Z·;·=Z·;·IZ·;·IIZ·I·=Z·;·=········································ ... --i;~~+IZ-IIZ-I-IZ-I-=Z-;-IZ-;-I-Z-;-I-I;-I-=Z-;-=Z-;-IZ-;-IIZ-I-=Z-;-=Z---------------------------------------l~~;JIZ-I-IZ-I-IZ-I-IZ-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;-IIZ-I-IZ-I-=Z-I------------------------------------

I T* ~ /26 ·t;~+IZ·I·IZ·I·=Z·I·=Z·;·IZ·;·I·=;·I·=;·I·=Z·;·=Z·;·IZ·;·IIZ·I·IZ·I·=Z·I·I································ -t;3-IZ-I-IZ-I-IZ-I-=Z-;-IZ-;-I-Z-;-I-=;-I-=Z-;-=Z-;-1Z-;-IIZ-I-=Z-;-=Z-;-IZ-----------------------------t;;+IZ-I-IZ-I-IZ-I-IZ-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z-;-IZ-;-IIZ-I-IZ-I-=Z-;-IZ-;------------------------T* = 4095

T* = 4094

-t;;+IZ-I-IZ-I-=Z-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;-=Z----------------------------------------------------------------------------------------

-x~~;~+IZ·I·IZ·I·=Z·I·=Z·;·IZ·;·I·=;·I·=;·I·=Z·;·=Z·;·1 Z·;·IIZ·I·IZ·I·=Z·I·IZ·I·I····················· -x~~~dlZ-llZ-I-IZ-I-IZ-;-IZ-;-I-=;-I-I;-I-=Z-;-=Z-;-IZ-;-IIZ-I-=Z-;-=Z-;-IZ-;-I-Z------------------

ch01

28

--i;-+
z-
-
z-z-
z-z-=z-;-
z-;-;-=;-;-=;-
-=z---------------------------------------------------------------------------------------------------

15:12

25

I

(Continued )

February 14, 2011

Table 34.

47

T* T*=2

~ 51

T*

=

21

=

8

---i;---t-I;-;-IZ-;-IZ-;-;;;-;-I;-;-IZ-I-IZ-;-IZ-I-;Z-;-I-I-;-I-I;-;-IZ-;-IZ-;-IZ-;-;-I;-;----------i~~-JI;-=-IZ-;-IZ-;-;Z-;-;-I-;-;-IZ-I-IZ-;-IZ-=-;Z-;-II-; -I-I;-;-IZ-;-IZ-;-IZ-;-;-I;-;-I------

_1

2046

-X;14S+IZ-;-IZ-;-IZ-;-IZ-I-;-IZ-;-IZ-;-IZ-;-II-I-;I-;-;I-Z-I-IZ-;-;Z-;-;;-;-;Z-Z-;-IZ-;-IZ--o t=················································· = . X

T*

=

255

x~

ch01

T = T*/3 = 84 :

53

x

T* = 2 26 _1

54

o

XT_I

T* = 1022

t••• ===

.

••••••••••••••••••••••••••••••••••••••••••••••••••

X·················································· ·~::~+ii·i·ii·.·ii·.·i ••·i·i.·ii·ii·.·ii·.·iii·.·ii·.·ii·i·i·.·ii·i·ii·i·ii·.·i.·.·i·ii·.·ii···

138

55

T* = 2 20 _1

X·················································· -~~~-+ii-i-ii-.-ii-.-i ••-i-i.-ii-ii-.-ii-.-iii-.-i.-.-.-i-i-.-ii-.-ii-.-ii-.-i.-.-i-ii-.-ii---

56

T* = 56

X·················································· --~:~+ii-=-ii-=-ii-ii-ii-.-i-i-.-ii-ii-.-iii-.-ii.-.-ii-.••-i-.-ii-i-ii-i-ii-i-i.-i-i-ii-.-ii---

57 58

T* = 511

x·················································· -~-L-t-ii-i-ii-i-ii-.-i ••-i-i.-ii-ii-.-ii-.-iii-.-i.-.-i-i-i-.-ii-.-ii-.-ii-.-i.-.-i-ii-i-ii--x·················································· --~-~~-:-t-i!-i-i!-i-i!-.-i ••-i-i.-ii-ii-.-.i-.-iii-.-••-.-i-i-i-.-.i-.-.i-.-.i-.-••-.-i-••-.-••---

T* = 2 15 _2

59 I T* = 2 29 _1 60 61

15:12

49

52

-i~~-;-t;=-I;;-I-I;-;-I;;-I-III-IZ-I-IZ-;-II-I-II-Z-I-I-z-
-
z-;;I-;-
;;-
z-I;-
;--------------

T*=2 23 _1

48

(Continued )

February 14, 2011

Table 34.

T* = 60 T

=

T*/63

x·················································· --~:~+ii-i-ii-=-ii-=-i ••-i-i.-ii-ii-.-ii-.-iii-.-i.-.-i-i-i-.-ii-.-.i-.-ii-.-i.-.-i-ii-i-ii--X·················································· ·~~~·:-t-ii·i·ii·i·ii·i·ii·.·i·i.·ii·ii·.·.i·.·iii·.·i.·.· i·i·i·.·ii·.·.i·i·.i·.·i.·i·i·ii·.·ii···

February 14, 2011

15:12

ch01

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Table 34.

(Continued )

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Q

~

~

'"'

Q

~

~

;;;

Q

~

~

'"

Q

' ~

~

:;;

Q

~

~

~

~

~

"'on

Q

~

~

~

Q

~

~

~ co

Q

~

~

~

Q

~

~

on on

Q

~

~

Q

on

~

~

~

Q

~

~

-... \Q

~

II -x-

II -x-

C")

h

C"l

II -x-

II -x-



ci

ci

o

J

8

32 =1 32

Gallery 8-2

(Continued )

14:49

ρ1 =

L=5

Table 6-2.

February 14, 2011 ch02

171

8

L=6

ρ1 =

64 =1 64

(Continued )

Gallery 8-3

Table 6-2.

February 14, 2011 14:49 ch02

172

8

L=7

ρ1 =

128 =1 128

(Continued )

Gallery 8-4

Table 6-2.

February 14, 2011 14:49 ch02

173

8

L=8

ρ1 =

256 =1 256

(Continued )

Gallery 8-5

Table 6-2.

February 14, 2011 14:49 ch02

174

=

£-1

-

-

I

=

l

:xlHI

~

ti[-X t +Xt _~] 2 1-1

I

XI!1I

j

.... ••••

'3)1 • • • •

16

Xt+l

.... ••••

eI1ex xt

~ -- • - •• wi •

4

•••• •••• ~w

0.9

q>n

O.B 0.7

0.6

175

2

PI = -=025 8 .

2

PI = 3 =0.75

8

0.4

() 1/-

0.1

I)

P, =4

r/' 't' 11- 1

I

0.5

I

41 ~

t t 3o:J«JC() t:J t t t 16 t =O. 7S

___

I

/ "

/

~

~

/

fl.'l

0.5

IL

1L 31

"

ch02

24

=

1

X t +1

14:49

22

•

Basin Tree Diagrams of Rule 12 . xt

February 14, 2011

Table 6-3.

Gallery 12-1

8).~

I

PI

~ 2 1~ ~O.125

113

2 P3=-=0.125 16

12

2 = 0.3125 32

2 = 0.0625 32

(Continued )

cD. Gallery 12-2

ρ3 = 5

4 = 0.625 32

14:49

ρ2 = 5

L=5

ρ1 =

Table 6-3.

February 14, 2011 ch02

176

12

177

4 = 0.1875 64

5 = 0.46875 64

ρ2 = 3

ρ4 = 6

3 = 0.1875 64

Gallery 12-3

(Continued )

ρ3 =

1 = 0.03125 64

2 = 0.03125 64

ρ1 = 2

14:49

ρ5 = 6

L=6

Table 6-3.

February 14, 2011 ch02

178

2 = 0.109375 128 4 = 0.21875 128

ρ3 = 7

2 = 0.015625 128

ρ1 = ρ2 = 7

6 = 0.328125 128

6 = 0.328125 128

ρ4 = 7

Gallery 12-4

(Continued )

14:49

ρ5 = 7

L=7

12

Table 6-3.

February 14, 2011 ch02

3 = 0.09375 256

ρ7 = 8

ρ6 = 8

5 = 0.15625 256

179

ρ4 = 2

8 = 0.25 256

7 = 0.21875 256

ρ5 = 8

4 = 0.125 256

1 = 0.0078125 256

Gallery 12-5

(Continued )

9 = 0.140625 256

2 = 0.0078125 256

ρ8 = 4

ρ2 =

14:49

ρ3 = 8

12 L=8

ρ1 = 8

Table 6-3.

February 14, 2011 ch02

23 = 8

I I JOI • • • •

vertex I Xl 1 1-

I

Xi

I

X i+!

1

HI

February 14, 2011

XIDJ

Xi

,/

••••• ••••

q)n

3) • • • •

~

t+ 1

Xi

=

2

5

32

••••

rGJ • • • •

+-] 2

ttl i_1 +Xi -Xi+ 1

ci,[-2x

t

=

r6

/ OJ

•

0.2

<W)

-=F. 1

0

(I

0,1

w

o

°

01

0,2 03 0.4 05 0.6 0,7 0,8 09

180

I

•

PI = -=0 8 .25

~ 2

P2 =3 -=0 8 .75

~

I

I I)

5

q)n-l

o

.-~O I

~ ....... .............. 16 .--~·0"/ P2 -2- .2.=0875 1

o it e -~=0.125

Pl- 16

IGaller~ 13-1 I

/•

~

~ ~ ~

l

1

L4

lt

2

I I]

IL 31 ,

ch02

2 4 = 16

••••

•

(1.5

\..4) • • • •

(5

14:49

--

22 = 4

•

Basin Tree Diagrams of Rule 13 .

Table 6-4.

-., '\ I»

0-_

/ 0

13

ρ2 = 5

6 = 0.9375 32

2 = 0.0625 32

.~.

Gallery 13-2

(Continued )

14:49

ρ1 =

L=5

Table 6-4.

February 14, 2011 ch02

181

13

ρ3 = 2

22 = 0.6875 64

6 = 0.28125 64

2 = 0.03125 64

Gallery 13-3

(Continued )

14:49

ρ2 = 3

L=6

ρ1 =

Table 6-4.

February 14, 2011 ch02

182

13

L=7

ρ2 = 7

18 = 0.984375 128

ρ1 =

e-=e

(Continued )

Gallery 13-4

2 = 0.015625 128

Table 6-4.

February 14, 2011 14:49 ch02

183

15 = 0.46875 256

2 = 0.0078125 256

ρ1 =

Gallery 13-5

(Continued )

ρ3 = 2

67 = 0.5234375 256

14:49

ρ2 = 8

L=8

13

Table 6-4.

February 14, 2011 ch02

184

vertex I

x! 1I Xl 1-

Ix! 11x!+1

1

l+

1

X[ll]

l

\WI • • • • -

,

1) whose basins of attraction is large enough to include some initial bit string samples belonging to our sample set of at least ten random bit strings (with length L > 500). Each Gallery N − m of rule N is composed of attractors and their basins of attraction, coded in magenta (or transparent pink when they overlap), and Isles of Eden, coded in blue (a mnemonic for Isles surrounded by the blue sea). The basin tree diagrams for each rule N in Table 4 are exhibited in Tables 6-1 to 6-24 for L = 3, 4, 5, 6, 7, 8.

2.1. Each bit string has a decimal and a fractional code To enhance clarity, each binary bit string associated with a basin tree attractor, or with an Isle of Eden, is coded by an integer number equal to the decimal equivalent of an L-bit binary bit string as follow: [x0

x1

x2

···

xI ] → n


I


βi • 2(I−i) xi

i=0

(1) where I
L − 1 and βi ∈ {0, 1}. Consequently, for each string length L, there are 2L distinct binary strings labeled from n = 0 to n = 2L−1 . For ease of interpreting the evolution from an initial bit string on the basin tree diagrams, these 2L binary strings are listed in Table 7-1 for L = 3, Table 7-2 for L = 4, Table 7-3 for L = 5, Table 7-4 for L = 6, Table 7-5 for L = 7, and Ta4

285

ble 7-6 for L = 8, respectively, along with the decimal number code “n” shown in the left column, and enclosed by a circle; i.e. nk, n = 0, 1, 2, . . . , 2L − 1. For computational purposes, such as in deriving the time-1 return maps, it is necessary to convert a binary bit string from Table 7 to its associated real number φ ∈ [0, 1) via the formula [Chua et al., 2005a]: φ=

I


2−(i+1) xi

(2)

i=0

where I
L − 1. This fractional real number is displayed in the right column for each bit string listed in Table 7.

2.2. Attractor, basin of attraction, and basin tree For each rule N and length L, the basin tree diagrams exhibited in Table 6 give a complete catalog of the evolution from all possible 2L initial binary bit strings of length L. In general, they are organized into several isolated directed graphs. Each of these connected graphs is called a basin tree in [Chua et al., 2006] for conciseness, even though some of these graphs are not trees in the graph-theoretic sense. Example 2.2.1. 172 , L = 6. Gallery 172-3 catalogs the basin tree diagrams of rule 172 for L = 6. Here there are 26 = 64 distinct bit strings, coded by integers 0m , 1m , 2m , . . . , 63m , and listed in Table 7.4. This catalog is organized into 13 isolated connected component graphs. If a connected component graph contains at least a Garden of Eden,4 we will call it a basin tree. Otherwise, it is called an has four Isle of Eden. For L  = 6, rule 172    periodm m m m 1 Isles of Eden 0 , 9 , 18 , 36 , and a   m period-3 Isle of Eden 27m , 54 , 45m. Each of the remaining eight isolated connected components is a basin tree. Observe that among them, seven basin trees are trees in the graph-theoretic sense (i.e. there are no closed loops). However, the basin tree formed by the union of the six Gardens of Eden   m m m and the closed loop 46 , 23m , 43m , 53 , 58m , and 29   m m 62 , 31m , 47m , 55 , 59m , 61m is not a tree because it contains a closed loop. We will henceforth call each closed loop of a basin tree a period-T attractor,

A bit string is said to be a Garden of Eden [Moore, 1962] if it does not have a pre-image (predecessor).

February 14, 2011

286

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-1.

Decimal and fractional code for eight binary bit strings with length L = 3.

Bit-String Code Number n

3-Bit String

Decimal Fraction Representation

x0 x1 x2

φ=

1 1 1 x0 + x1 + x2 2 4 8

x0

x1

x2

0

0

0

0

0

1

0

0

1

0.125

2

0

1

0

0.25

3

0

1

1

0.375

4

1

0

0

0.5

5

1

0

1

0.625

6

1

1

0

0.75

7

1

1

1

0.875

Table 7-2.

Bit-String Code Number n

Decimal and fractional code for 16 binary bit strings with length L = 4.

4-Bit String x0 x1 x2 x3

Decimal Fraction Representation

φ=

1 1 1 1 x0 + x1 + x2 + x3 2 4 8 16

x0

x1

x2

x3

0

0

0

0

0

0

1

0

0

0

1

0.0625

2

0

0

1

0

0.125

3

0

0

1

1

0.1875

4

0

1

0

0

0.25

5

0

1

0

1

0.3125

6

0

1

1

0

0.375

7

0

1

1

1

0.4375

8

1

0

0

0

0.5

9

1

0

0

1

0.5625

10

1

0

1

0

0.625

11

1

0

1

1

0.6875

12

1

1

0

0

0.75

13

1

1

0

1

0.8125

14

1

1

1

0

0.875

15

1

1

1

1

0.9375

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-3.

Bit-String Code Number n

Decimal and fractional code for 32 binary bit strings with length L = 5.

5-Bit String

x0 x1 x2 x3 x4

Decimal Fraction Representation 1 1 1 1 1 φ = x0 + x1 + x2 + x3 + x4 2 4 8 16 32

x0

x1

x2

x3

x4

0

0

0

0

0

0

0

1

0

0

0

0

1

0.03125

2

0

0

0

1

0

0.0625

3

0

0

0

1

1

0.09375

4

0

0

1

0

0

0.125

5

0

0

1

0

1

0.15625

6

0

0

1

1

0

0.1875

7

0

0

1

1

1

0.21875

8

0

1

0

0

0

0.25

9

0

1

0

0

1

0.28125

10

0

1

0

1

0

0.3125

11

0

1

0

1

1

0.34375

12

0

1

1

0

0

0.375

13

0

1

1

0

1

0.40625

14

0

1

1

1

0

0.4375

15

0

1

1

1

1

0.46875

16

1

0

0

0

0

0.5

17

1

0

0

0

1

0.53125

18

1

0

0

1

0

0.5625

19

1

0

0

1

1

0.59375

20

1

0

1

0

0

0.625

21

1

0

1

0

1

0.65625

22

1

0

1

1

0

0.6875

23

1

0

1

1

1

0.71875

24

1

1

0

0

0

0.75

25

1

1

0

0

1

0.78125

26

1

1

0

1

0

0.8125

27

1

1

0

1

1

0.84375

28

1

1

1

0

0

0.875

29

1

1

1

0

1

0.90625

30

1

1

1

1

0

0.9375

31

1

1

1

1

1

0.96875

287

February 14, 2011

288

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-4.

Bit-String Code Number n

Decimal and fractional code for 64 binary bit strings with length L = 6.

6-Bit String

Decimal Fraction Representation

x0 x1 x2 x3 x4 x5

5

φ = Σ 2 −(i +1) xi

x0

x1

x2

x3

x4

x5

0

0

0

0

0

0

0

0

1

0

0

0

0

0

1

0.015625

2

0

0

0

0

1

0

0.03125

3

0

0

0

0

1

1

0.046875

4

0

0

0

1

0

0

0.0625

5

0

0

0

1

0

1

0.078125

6

0

0

0

1

1

0

0.09375

7

0

0

0

1

1

1

0.109375

8

0

0

1

0

0

0

0.125

9

0

0

1

0

0

1

0.140625

10

0

0

1

0

1

0

0.15625

11

0

0

1

0

1

1

0.171875

12

0

0

1

1

0

0

0.1875

13

0

0

1

1

0

1

0.203125

14

0

0

1

1

1

0

0.21875

15

0

0

1

1

1

1

0.234375

16

0

1

0

0

0

0

0.25

17

0

1

0

0

0

1

0.265625

18

0

1

0

0

1

0

0.28125

19

0

1

0

0

1

1

0.296875

20

0

1

0

1

0

0

0.3125

21

0

1

0

1

0

1

0.328125

22

0

1

0

1

1

0

0.34375

23

0

1

0

1

1

1

0.359375

24

0

1

1

0

0

0

0.375

25

0

1

1

0

0

1

0.390625

26

0

1

1

0

1

0

0.40625

27

0

1

1

0

1

1

0.421875

28

0

1

1

1

0

0

0.4375

29

0

1

1

1

0

1

0.453125

30

0

1

1

1

1

0

0.46875

31

0

1

1

1

1

1

0.484375

i=0

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-4.

Bit-String Code Number n

(Continued )

6-Bit String x0 x1 x2 x3 x4 x5

x0

x1

x2

x3

x4

Decimal Fraction Representation 5

x5

φ = Σ 2− (i +1) xi i =0

32

1

0

0

0

0

0

0.5

33

1

0

0

0

0

1

0.515625

34

1

0

0

0

1

0

0.53125

35

1

0

0

0

1

1

0.546875

36

1

0

0

1

0

0

0.5625

37

1

0

0

1

0

1

0.578125

38

1

0

0

1

1

0

0.59375

39

1

0

0

1

1

1

0.609375

40

1

0

1

0

0

0

0.625

41

1

0

1

0

0

1

0.640625

42

1

0

1

0

1

0

0.65625

43

1

0

1

0

1

1

0.671875

44

1

0

1

1

0

0

0.6875

45

1

0

1

1

0

1

0.703125

46

1

0

1

1

1

0

0.71875

47

1

0

1

1

1

1

0.734375

48

1

1

0

0

0

0

0.75

49

1

1

0

0

0

1

0.765625

50

1

1

0

0

1

0

0.78125

51

1

1

0

0

1

1

0.796875

52

1

1

0

1

0

0

0.8125

53

1

1

0

1

0

1

0.828125

54

1

1

0

1

1

0

0.84375

55

1

1

0

1

1

1

0.859375

56

1

1

1

0

0

0

0.875

57

1

1

1

0

0

1

0.890625

58

1

1

1

0

1

0

0.90625

59

1

1

1

0

1

1

0.921875

60

1

1

1

1

0

0

0.9375

61

1

1

1

1

0

1

0.953125

62

1

1

1

1

1

0

0.96875

63

1

1

1

1

1

1

0.984375

289

February 14, 2011

290

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-5.

Bit-String Code Number n

Decimal and fractional code for 128 binary bit strings with length L = 7.

7-Bit String x0 x1 x2 x3 x4 x5 x6

Decimal Fraction Representation 6

φ = Σ 2 −(i +1) xi

x0

x1

x2

x3

x4

x5

x6

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

1

0.00781

2

0

0

0

0

0

1

0

0.01563

3

0

0

0

0

0

1

1

0.02344

4

0

0

0

0

1

0

0

0.03125

5

0

0

0

0

1

0

1

0.03906

6

0

0

0

0

1

1

0

0.04688

7

0

0

0

0

1

1

1

0.05469

8

0

0

0

1

0

0

0

0.0625

9

0

0

0

1

0

0

1

0.07031

10

0

0

0

1

0

1

0

0.07813

11

0

0

0

1

0

1

1

0.08594

12

0

0

0

1

1

0

0

0.09375

13

0

0

0

1

1

0

1

0.10156

14

0

0

0

1

1

1

0

0.10938

15

0

0

0

1

1

1

1

0.11719

16

0

0

1

0

0

0

0

0.125

17

0

0

1

0

0

0

1

0.13281

18

0

0

1

0

0

1

0

0.14063

19

0

0

1

0

0

1

1

0.14844

20

0

0

1

0

1

0

0

0.15625

21

0

0

1

0

1

0

1

0.16406

22

0

0

1

0

1

1

0

0.17188

23

0

0

1

0

1

1

1

0.17969

24

0

0

1

1

0

0

0

0.1875

25

0

0

1

1

0

0

1

0.19531

26

0

0

1

1

0

1

0

0.20313

27

0

0

1

1

0

1

1

0.21094

28

0

0

1

1

1

0

0

0.21875

29

0

0

1

1

1

0

1

0.22656

30

0

0

1

1

1

1

0

0.23438

31

0

0

1

1

1

1

1

0.24219

i =0

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-5.

Bit-String Code Number n

(Continued )

7-Bit String x0 x1 x2 x3 x4 x5 x6

x0

x1

x2

x3

x4

x5

Decimal Fraction Representation 6

x6

φ = Σ 2 −(i +1) xi i=0

32

0

1

0

0

0

0

0

0.25

33

0

1

0

0

0

0

1

0.25781

34

0

1

0

0

0

1

0

0.26563

35

0

1

0

0

0

1

1

0.27344

36

0

1

0

0

1

0

0

0.28125

37

0

1

0

0

1

0

1

0.28906

38

0

1

0

0

1

1

0

0.29688

39

0

1

0

0

1

1

1

0.30469

40

0

1

0

1

0

0

0

0.3125

41

0

1

0

1

0

0

1

0.32031

42

0

1

0

1

0

1

0

0.32813

43

0

1

0

1

0

1

1

0.33594

44

0

1

0

1

1

0

0

0.34375

45

0

1

0

1

1

0

1

0.35156

46

0

1

0

1

1

1

0

0.35938

47

0

1

0

1

1

1

1

0.36719

48

0

1

1

0

0

0

0

0.375

49

0

1

1

0

0

0

1

0.38281

50

0

1

1

0

0

1

0

0.39063

51

0

1

1

0

0

1

1

0.39844

52

0

1

1

0

1

0

0

0.40625

53

0

1

1

0

1

0

1

0.41406

54

0

1

1

0

1

1

0

0.42188

55

0

1

1

0

1

1

1

0.42969

56

0

1

1

1

0

0

0

0.4375

57

0

1

1

1

0

0

1

0.44531

58

0

1

1

1

0

1

0

0.45313

59

0

1

1

1

0

1

1

0.46094

60

0

1

1

1

1

0

0

0.46875

61

0

1

1

1

1

0

1

0.47656

62

0

1

1

1

1

1

0

0.48438

63

0

1

1

1

1

1

1

0.49219

291

February 14, 2011

292

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-5.

Bit-String Code Number n

(Continued )

7-Bit String x0 x1 x2 x3 x4 x5 x6

x0

x1

x2

x3

x4

x5

Decimal Fraction Representation 6

x6

φ = Σ 2 −(i +1) xi i =0

64

1

0

0

0

0

0

0

0.5

65

1

0

0

0

0

0

1

0.50781

66

1

0

0

0

0

1

0

0.51563

67

1

0

0

0

0

1

1

0.52344

68

1

0

0

0

1

0

0

0.53125

69

1

0

0

0

1

0

1

0.53906

70

1

0

0

0

1

1

0

0.54688

71

1

0

0

0

1

1

1

0.55469

72

1

0

0

1

0

0

0

0.5625

73

1

0

0

1

0

0

1

0.57031

74

1

0

0

1

0

1

0

0.57813

75

1

0

0

1

0

1

1

0.58594

76

1

0

0

1

1

0

0

0.59375

77

1

0

0

1

1

0

1

0.60156

78

1

0

0

1

1

1

0

0.60938

79

1

0

0

1

1

1

1

0.61719

80

1

0

1

0

0

0

0

0.625

81

1

0

1

0

0

0

1

0.63281

82

1

0

1

0

0

1

0

0.64063

83

1

0

1

0

0

1

1

0.64844

84

1

0

1

0

1

0

0

0.65625

85

1

0

1

0

1

0

1

0.66406

86

1

0

1

0

1

1

0

0.67188

87

1

0

1

0

1

1

1

0.67969

88

1

0

1

1

0

0

0

0.6875

89

1

0

1

1

0

0

1

0.69531

90

1

0

1

1

0

1

0

0.70313

91

1

0

1

1

0

1

1

0.71094

92

1

0

1

1

1

0

0

0.71875

93

1

0

1

1

1

0

1

0.72656

94

1

0

1

1

1

1

0

0.73438

95

1

0

1

1

1

1

1

0.74219

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-5.

Bit-String Code Number n

(Continued )

7-Bit String x0 x1 x2 x3 x4 x5 x6

x0

x1

x2

x3

x4

x5

Decimal Fraction Representation 6

x6

φ = Σ 2 −(i +1) xi i=0

96

1

1

0

0

0

0

0

0.75

97

1

1

0

0

0

0

1

0.75781

98

1

1

0

0

0

1

0

0.76563

99

1

1

0

0

0

1

1

0.77344

100

1

1

0

0

1

0

0

0.78125

101

1

1

0

0

1

0

1

0.78906

102

1

1

0

0

1

1

0

0.79688

103

1

1

0

0

1

1

1

0.80469

104

1

1

0

1

0

0

0

0.8125

105

1

1

0

1

0

0

1

0.82031

106

1

1

0

1

0

1

0

0.82813

107

1

1

0

1

0

1

1

0.83594

108

1

1

0

1

1

0

0

0.84375

109

1

1

0

1

1

0

1

0.85156

110

1

1

0

1

1

1

0

0.85938

111

1

1

0

1

1

1

1

0.86719

112

1

1

1

0

0

0

0

0.875

113

1

1

1

0

0

0

1

0.88281

114

1

1

1

0

0

1

0

0.89063

115

1

1

1

0

0

1

1

0.89844

116

1

1

1

0

1

0

0

0.90625

117

1

1

1

0

1

0

1

0.91406

118

1

1

1

0

1

1

0

0.92188

119

1

1

1

0

1

1

1

0.92969

120

1

1

1

1

0

0

0

0.9375

121

1

1

1

1

0

0

1

0.94531

122

1

1

1

1

0

1

0

0.95313

123

1

1

1

1

0

1

1

0.96094

124

1

1

1

1

1

0

0

0.96875

125

1

1

1

1

1

0

1

0.97656

126

1

1

1

1

1

1

0

0.98438

127

1

1

1

1

1

1

1

0.99219

293

February 14, 2011

294

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-6.

Bit-String Code Number n

Decimal and fractional code for 256 binary bit strings with length L = 8.

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

Decimal Fraction Representation 7

φ = Σ 2 −(i +1) xi

x0

x1

x2

x3

x4

x5

x6

x7

0

0

0

0

0

0

0

0

0

0

1

0

0

0

0

0

0

0

1

0.00390625

2

0

0

0

0

0

0

1

0

0.0078125

3

0

0

0

0

0

0

1

1

0.01171875

4

0

0

0

0

0

1

0

0

0.015625

5

0

0

0

0

0

1

0

1

0.01953125

6

0

0

0

0

0

1

1

0

0.0234375

7

0

0

0

0

0

1

1

1

0.02734375

8

0

0

0

0

1

0

0

0

0.03125

9

0

0

0

0

1

0

0

1

0.03515625

10

0

0

0

0

1

0

1

0

0.0390625

11

0

0

0

0

1

0

1

1

0.04296875

12

0

0

0

0

1

1

0

0

0.046875

13

0

0

0

0

1

1

0

1

0.05078125

14

0

0

0

0

1

1

1

0

0.0546875

15

0

0

0

0

1

1

1

1

0.05859375

16

0

0

0

1

0

0

0

0

0.0625

17

0

0

0

1

0

0

0

1

0.06640625

18

0

0

0

1

0

0

1

0

0.0703125

19

0

0

0

1

0

0

1

1

0.07421875

20

0

0

0

1

0

1

0

0

0.078125

21

0

0

0

1

0

1

0

1

0.08203125

22

0

0

0

1

0

1

1

0

0.0859375

23

0

0

0

1

0

1

1

1

0.08984375

24

0

0

0

1

1

0

0

0

0.09375

25

0

0

0

1

1

0

0

1

0.09765625

26

0

0

0

1

1

0

1

0

0.1015625

27

0

0

0

1

1

0

1

1

0.10546875

28

0

0

0

1

1

1

0

0

0.109375

29

0

0

0

1

1

1

0

1

0.11328125

30

0

0

0

1

1

1

1

0

0.1171875

31

0

0

0

1

1

1

1

1

0.12109375

i =0

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i +1) xi i=0

32

0

0

1

0

0

0

0

0

0.125

33

0

0

1

0

0

0

0

1

0.12890625

34

0

0

1

0

0

0

1

0

0.1328125

35

0

0

1

0

0

0

1

1

0.13671875

36

0

0

1

0

0

1

0

0

0.140625

37

0

0

1

0

0

1

0

1

0.14453125

38

0

0

1

0

0

1

1

0

0.1484375

39

0

0

1

0

0

1

1

1

0.15234375

40

0

0

1

0

1

0

0

0

0.15625

41

0

0

1

0

1

0

0

1

0.16015625

42

0

0

1

0

1

0

1

0

0.1640625

43

0

0

1

0

1

0

1

1

0.16796875

44

0

0

1

0

1

1

0

0

0.171875

45

0

0

1

0

1

1

0

1

0.17578125

46

0

0

1

0

1

1

1

0

0.1796875

47

0

0

1

0

1

1

1

1

0.18359375

48

0

0

1

1

0

0

0

0

0.1875

49

0

0

1

1

0

0

0

1

0.19140625

50

0

0

1

1

0

0

1

0

0.1953125

51

0

0

1

1

0

0

1

1

0.19921875

52

0

0

1

1

0

1

0

0

0.203125

53

0

0

1

1

0

1

0

1

0.20703125

54

0

0

1

1

0

1

1

0

0.2109375

55

0

0

1

1

0

1

1

1

0.21484375

56

0

0

1

1

1

0

0

0

0.21875

57

0

0

1

1

1

0

0

1

0.22265625

58

0

0

1

1

1

0

1

0

0.2265625

59

0

0

1

1

1

0

1

1

0.23046875

60

0

0

1

1

1

1

0

0

0.234375

61

0

0

1

1

1

1

0

1

0.23828125

62

0

0

1

1

1

1

1

0

0.2421875

63

0

0

1

1

1

1

1

1

0.24609375

295

February 14, 2011

296

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i+1) xi i =0

64

0

1

0

0

0

0

0

0

0.25

65

0

1

0

0

0

0

0

1

0.25390625

66

0

1

0

0

0

0

1

0

0.2578125

67

0

1

0

0

0

0

1

1

0.26171875

68

0

1

0

0

0

1

0

0

0.265625

69

0

1

0

0

0

1

0

1

0.26953125

70

0

1

0

0

0

1

1

0

0.2734375

71

0

1

0

0

0

1

1

1

0.27734375

72

0

1

0

0

1

0

0

0

0.28125

73

0

1

0

0

1

0

0

1

0.28515625

74

0

1

0

0

1

0

1

0

0.2890625

75

0

1

0

0

1

0

1

1

0.29296875

76

0

1

0

0

1

1

0

0

0.296875

77

0

1

0

0

1

1

0

1

0.30078125

78

0

1

0

0

1

1

1

0

0.3046875

79

0

1

0

0

1

1

1

1

0.30859375

80

0

1

0

1

0

0

0

0

0.3125

81

0

1

0

1

0

0

0

1

0.31640625

82

0

1

0

1

0

0

1

0

0.3203125

83

0

1

0

1

0

0

1

1

0.32421875

84

0

1

0

1

0

1

0

0

0.328125

85

0

1

0

1

0

1

0

1

0.33203125

86

0

1

0

1

0

1

1

0

0.3359375

87

0

1

0

1

0

1

1

1

0.33984375

88

0

1

0

1

1

0

0

0

0.34375

89

0

1

0

1

1

0

0

1

0.34765625

90

0

1

0

1

1

0

1

0

0.3515625

91

0

1

0

1

1

0

1

1

0.35546875

92

0

1

0

1

1

1

0

0

0.359375

93

0

1

0

1

1

1

0

1

0.36328125

94

0

1

0

1

1

1

1

0

0.3671875

95

0

1

0

1

1

1

1

1

0.37109375

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i +1) xi i=0

96

0

1

1

0

0

0

0

0

0.375

97

0

1

1

0

0

0

0

1

0.37890625

98

0

1

1

0

0

0

1

0

0.3828125

99

0

1

1

0

0

0

1

1

0.38671875

100

0

1

1

0

0

1

0

0

0.390625

101

0

1

1

0

0

1

0

1

0.39453125

102

0

1

1

0

0

1

1

0

0.3984375

103

0

1

1

0

0

1

1

1

0.40234375

104

0

1

1

0

1

0

0

0

0.40625

105

0

1

1

0

1

0

0

1

0.41015625

106

0

1

1

0

1

0

1

0

0.4140625

107

0

1

1

0

1

0

1

1

0.41796875

108

0

1

1

0

1

1

0

0

0.421875

109

0

1

1

0

1

1

0

1

0.42578125

110

0

1

1

0

1

1

1

0

0.4296875

111

0

1

1

0

1

1

1

1

0.43359375

112

0

1

1

1

0

0

0

0

0.4375

113

0

1

1

1

0

0

0

1

0.44140625

114

0

1

1

1

0

0

1

0

0.4453125

115

0

1

1

1

0

0

1

1

0.44921875

116

0

1

1

1

0

1

0

0

0.453125

117

0

1

1

1

0

1

0

1

0.45703125

118

0

1

1

1

0

1

1

0

0.4609375

119

0

1

1

1

0

1

1

1

0.46484375

120

0

1

1

1

1

0

0

0

0.46875

121

0

1

1

1

1

0

0

1

0.47265625

122

0

1

1

1

1

0

1

0

0.4765625

123

0

1

1

1

1

0

1

1

0.48046875

124

0

1

1

1

1

1

0

0

0.484375

125

0

1

1

1

1

1

0

1

0.48828125

126

0

1

1

1

1

1

1

0

0.4921875

127

0

1

1

1

1

1

1

1

0.49609375

297

February 14, 2011

298

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i +1) xi i =0

128

1

0

0

0

0

0

0

0

0.5

129

1

0

0

0

0

0

0

1

0.50390625

130

1

0

0

0

0

0

1

0

0.5078125

131

1

0

0

0

0

0

1

1

0.51171875

132

1

0

0

0

0

1

0

0

0.515625

133

1

0

0

0

0

1

0

1

0.51953125

134

1

0

0

0

0

1

1

0

0.5234375

135

1

0

0

0

0

1

1

1

0.52734375

136

1

0

0

0

1

0

0

0

0.53125

137

1

0

0

0

1

0

0

1

0.53515625

138

1

0

0

0

1

0

1

0

0.5390625

139

1

0

0

0

1

0

1

1

0.54296875

140

1

0

0

0

1

1

0

0

0.546875

141

1

0

0

0

1

1

0

1

0.55078125

142

1

0

0

0

1

1

1

0

0.5546875

143

1

0

0

0

1

1

1

1

0.55859375

144

1

0

0

1

0

0

0

0

0.5625

145

1

0

0

1

0

0

0

1

0.56640625

146

1

0

0

1

0

0

1

0

0.5703125

147

1

0

0

1

0

0

1

1

0.57421875

148

1

0

0

1

0

1

0

0

0.578125

149

1

0

0

1

0

1

0

1

0.58203125

150

1

0

0

1

0

1

1

0

0.5859375

151

1

0

0

1

0

1

1

1

0.58984375

152

1

0

0

1

1

0

0

0

0.59375

153

1

0

0

1

1

0

0

1

0.59765625

154

1

0

0

1

1

0

1

0

0.6015625

155

1

0

0

1

1

0

1

1

0.60546875

156

1

0

0

1

1

1

0

0

0.609375

157

1

0

0

1

1

1

0

1

0.61328125

158

1

0

0

1

1

1

1

0

0.6171875

159

1

0

0

1

1

1

1

1

0.62109375

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i +1) xi i=0

160

1

0

1

0

0

0

0

0

0.625

161

1

0

1

0

0

0

0

1

0.62890625

162

1

0

1

0

0

0

1

0

0.6328125

163

1

0

1

0

0

0

1

1

0.63671875

164

1

0

1

0

0

1

0

0

0.640625

165

1

0

1

0

0

1

0

1

0.64453125

166

1

0

1

0

0

1

1

0

0.6484375

167

1

0

1

0

0

1

1

1

0.65234375

168

1

0

1

0

1

0

0

0

0.65625

169

1

0

1

0

1

0

0

1

0.66015625

170

1

0

1

0

1

0

1

0

0.6640625

171

1

0

1

0

1

0

1

1

0.66796875

172

1

0

1

0

1

1

0

0

0.671875

173

1

0

1

0

1

1

0

1

0.67578125

174

1

0

1

0

1

1

1

0

0.6796875

175

1

0

1

0

1

1

1

1

0.68359375

176

1

0

1

1

0

0

0

0

0.6875

177

1

0

1

1

0

0

0

1

0.69140625

178

1

0

1

1

0

0

1

0

0.6953125

179

1

0

1

1

0

0

1

1

0.69921875

180

1

0

1

1

0

1

0

0

0.703125

181

1

0

1

1

0

1

0

1

0.70703125

182

1

0

1

1

0

1

1

0

0.7109375

183

1

0

1

1

0

1

1

1

0.71484375

184

1

0

1

1

1

0

0

0

0.71875

185

1

0

1

1

1

0

0

1

0.72265625

186

1

0

1

1

1

0

1

0

0.7265625

187

1

0

1

1

1

0

1

1

0.73046875

188

1

0

1

1

1

1

0

0

0.734375

189

1

0

1

1

1

1

0

1

0.73828125

190

1

0

1

1

1

1

1

0

0.7421875

191

1

0

1

1

1

1

1

1

0.74609375

299

February 14, 2011

300

14:49

ch02

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i +1) xi i =0

192

1

1

0

0

0

0

0

0

0.75

193

1

1

0

0

0

0

0

1

0.75390625

194

1

1

0

0

0

0

1

0

0.7578125

195

1

1

0

0

0

0

1

1

0.76171875

196

1

1

0

0

0

1

0

0

0.765625

197

1

1

0

0

0

1

0

1

0.76953125

198

1

1

0

0

0

1

1

0

0.7734375

199

1

1

0

0

0

1

1

1

0.77734375

200

1

1

0

0

1

0

0

0

0.78125

201

1

1

0

0

1

0

0

1

0.78515625

202

1

1

0

0

1

0

1

0

0.7890625

203

1

1

0

0

1

0

1

1

0.79296875

204

1

1

0

0

1

1

0

0

0.796875

205

1

1

0

0

1

1

0

1

0.80078125

206

1

1

0

0

1

1

1

0

0.8046875

207

1

1

0

0

1

1

1

1

0.80859375

208

1

1

0

1

0

0

0

0

0.8125

209

1

1

0

1

0

0

0

1

0.81640625

210

1

1

0

1

0

0

1

0

0.8203125

211

1

1

0

1

0

0

1

1

0.82421875

212

1

1

0

1

0

1

0

0

0.828125

213

1

1

0

1

0

1

0

1

0.83203125

214

1

1

0

1

0

1

1

0

0.8359375

215

1

1

0

1

0

1

1

1

0.83984375

216

1

1

0

1

1

0

0

0

0.84375

217

1

1

0

1

1

0

0

1

0.84765625

218

1

1

0

1

1

0

1

0

0.8515625

219

1

1

0

1

1

0

1

1

0.85546875

220

1

1

0

1

1

1

0

0

0.859375

221

1

1

0

1

1

1

0

1

0.86328125

222

1

1

0

1

1

1

1

0

0.8671875

223

1

1

0

1

1

1

1

1

0.87109375

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules Table 7-6.

Bit-String Code Number n

(Continued )

8-Bit String x0 x1 x2 x3 x4 x5 x6 x7

x0

x1

x2

x3

x4

x5

x6

Decimal Fraction Representation 7

x7

φ = Σ 2 −(i +1) xi i=0

224

1

1

1

0

0

0

0

0

0.875

225

1

1

1

0

0

0

0

1

0.87890625

226

1

1

1

0

0

0

1

0

0.8828125

227

1

1

1

0

0

0

1

1

0.88671875

228

1

1

1

0

0

1

0

0

0.890625

229

1

1

1

0

0

1

0

1

0.89453125

230

1

1

1

0

0

1

1

0

0.8984375

231

1

1

1

0

0

1

1

1

0.90234375

232

1

1

1

0

1

0

0

0

0.90625

233

1

1

1

0

1

0

0

1

0.91015625

234

1

1

1

0

1

0

1

0

0.9140625

235

1

1

1

0

1

0

1

1

0.91796875

236

1

1

1

0

1

1

0

0

0.921875

237

1

1

1

0

1

1

0

1

0.92578125

238

1

1

1

0

1

1

1

0

0.9296875

239

1

1

1

0

1

1

1

1

0.93359375

240

1

1

1

1

0

0

0

0

0.9375

241

1

1

1

1

0

0

0

1

0.94140625

242

1

1

1

1

0

0

1

0

0.9453125

243

1

1

1

1

0

0

1

1

0.94921875

244

1

1

1

1

0

1

0

0

0.953125

245

1

1

1

1

0

1

0

1

0.95703125

246

1

1

1

1

0

1

1

0

0.9609375

247

1

1

1

1

0

1

1

1

0.96484375

248

1

1

1

1

1

0

0

0

0.96875

249

1

1

1

1

1

0

0

1

0.97265625

250

1

1

1

1

1

0

1

0

0.9765625

251

1

1

1

1

1

0

1

1

0.98046875

252

1

1

1

1

1

1

0

0

0.984375

253

1

1

1

1

1

1

0

1

0.98828125

254

1

1

1

1

1

1

1

0

0.9921875

255

1

1

1

1

1

1

1

1

0.99609375

301

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where T is the number of branches in the loop. The set union of all branches belonging to a basin tree, and all branches belonging to the associated period-T attractor, will henceforth be called the basin of attraction of the attractor.5 Based on the above definitions, a basin tree is just the union of an attractor and its basin of attraction. Observe that an Isle of Eden is any connected component graph in the basin tree diagrams that has an empty set as its basin of attraction. It is often useful to think of an attractor as analogous to the steady state regime of a dissipative system, and to think of any path from a basin of attraction which converges to an attractor as analogous to the associated transient regime. Hence, the union of all possible transient regimes of an attractor is its basin of attraction. Likewise, it is useful to think of an Isle of Eden as analogous to a transientless steady state regime 6 of a conservative (i.e. lossless) system.  For  rule  m 172 , the four period-1 Isles of Eden 0 , 9m,     18m, 36m, and the single period-3 Isle of Eden   m 27m , 54 , 45m, are both isolated Isles of Eden. In

bit in the circular architecture in Fig. 1(a) as our reference bit i = 0. Since any one of the remaining L − 1 bits could also have been chosen as the origin, it follows that any generic property of a particular bit string [x0 x1 x2 · · · xI ] that does not depend on the choice of the origin should be exhibited also by the corresponding bit string obtained by shifting it one bit to the left, namely, [x1 x2 · · · xI x0 ]. Likewise, bit strings [x2 x3 · · · xI x0 x1 ], [x3 x4 · · · xI x0 x1 x2 ], . . . , and [xI x0 x1 · · · xI−1 ] would all be isomorphic to each other. Observe, however, that if a bit string exhibits certain symmetry in its patterns, then not all the “L” left-shifted bit strings are distinct. For example, in the case of the alternating pattern [0 1 0 1 · · · 0 1], there are clearly only two distinct patterns. Now, since the left-shift operation on a binary bit string with the decimal code

contrast, all Isles of Eden of rule 204 in Gallery 204-1 to Gallery 204-5 are dense.

is equivalent to multiplying Eq. (3) by 2, the decimal code of the left-shifted bit string is simply given by

2.3. Explicit formula for generating isomorphic basin trees A cursory inspection of the basin tree diagrams in Table 6 reveals that for each string length L, there are many basin trees that are topologically identical from a graph-theoretic perspective. We will henceforth refer to them as isomorphic basin trees and consider them as members of the same generic basin tree family. For example, let us examine the basin tree of 172 for L = 6 (recall Example 2.2.1). we find  the three period-1 Isles    Here, m, and 36 m to be isomorphic of Eden 9m, 18 and count them as only one distinct Isle of Eden. Similarly, we the  find that the basin tree containing  m in Gallery bit strings 4m , 5m , 6m , 7m , 37m , 38m , 39 172-3 has six isomorphic copies. Consequently, we would summarize that Gallery 172-3 has only five distinct isomorphic basin trees. The existence of isomorphic copies of basin trees comes from our arbitrary choice of a particular 5

n=

I


βi • 2(I−i) xi

(3)

i=0

n
=

I


βi • 2(I+1−i) xi = 2n

mod 2L − 1

(4)

i=0

We will now summarize this general result as follows: Theorem 2.3.1. Explicit Basin Tree Generation Formula. For each bit string nk of length L in a basin tree, there exists an isomorphic basin tree
obnk maps to bit string nm where bit string tained by multiplying the decimal number n by 2 mod 2L − 1; namely,

n
= 2n

mod 2L − 1

(5)

Let us illustrate this remarkably simple formula Example 2.3.1. 172 , L = 6. Substituting L = 6 into Eq. (5), we obtain n
= 2n

mod 63

(6)

Let us choose nk = 9mas our bit string “seed” for generating all other Isles of Eden isomorphic

To avoid introducing yet another name, we will sometimes abuse our terminology by including only the branches in the transient regime as the basin of attraction of an attractor. From the context, it will be clear which definition is applicable. 6 Unlike continuous systems, however, an Isle of Eden can either be isolated, or dense [Garay & Chua, 2008].

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Chapter 2: Period-1 Rules



to 9m :

tor 1. 9 → 2 • 9 2. 18 →  2 • 18 3. 36 → 2 • 36

mod 63 = 18 mod 63 = 36 mod 63 = 9

(7a) (7b) (7c)

, 18m , and It follows that the three bit strings 9m m 36 in Gallery 172-3 are isomorphic period-1 Isles of Eden.  , 5m , 6m , 7m , Consider next the basin tree 4m  m in Gallery 172-3. Multiplying each of 37m , 38m , 39 these seven bit strings   by 2 mod 63, we obtain the m m m m m m m 8 , 10 , 12 , 14 , 11 , 13 , 15 , isomorphic string which can be clearly identified in Gallery 172-3. Iterating this left-shifting operation, we easily obtain the remaining four isomorphic period-1 bit strings. Let us apply next our explicit formula (5) to the basin tree  the basin of at formed by the union of m m m m m m traction 46 , 23 , 43 , 53 , 58 , 29 , and the attrac-

ρm


 m m 61m , 62 , 31m , 47m , 55 , and 59m :   m m 46m , 23m , 43 , 53m , 58m , 29   m m , 46 , 23m , 43m , 53 , 58m → 29m   m m 61m , 62m , 31 , 47m , 55m , 59   m m → 59m , 61 , 62m , 31m , 47 , 55m

2.5. Genotype and phenotype Isomorphic basin trees share the same “seed”, and consequently they have the same digraph representation, except for the bit-string numbers. However, also non-isomorphic basin trees can have the same digraph representation, as it happens for all period1 Isles of Eden. This phenomenon can be effectively described by using the concepts of “genotype” and “phenotype”, usually employed in biology, to indicate respectively the intrinsic and the observable characteristics of a basin tree. Therefore, we can say that isomorphic basin trees have the same genotype, and hence the same phenotype, and nonisomorphic basin trees have different genotypes, but they can have the same phenotype.

(8)

(9)

Note that in this case, the bit string seed has generated itself: it follows that this basin tree is unique in the sense that it is not isomorphic to any other bit string. Moreover, Eq. (9) proves that this period-6 attractor executes a Bernoulli shift with σ = 1 and τ = 1.

2.4. Robustness coefficient ρ To estimate the robustness of each distinct attractor and each distinct Isle of Eden, we simply calculate the ratio of the total number of bit strings, for each bit length L, over the total number 2L of bit strings:

No. of bit strings in attractor “m” or Isle of Eden “m” 2L

We will henceforth call the number ρm the robustness coefficient of the attractor “m”, or the Isle of Eden “m” of the local rule N . The reader should verify that the robustness coefficients of the period-1 attractors, or period-1 Isles of Eden, of all 25 period-1 rules are significantly larger than the robustness coefficients of the coexisting period-T attractors and/or period-T Isles of Eden, where T > 1.

303

(10)

For instance, for rule 4 and L = 4 there are six different period-1 Isles of Eden, belonging to two different groups: the first one contains the bit string 1m , and its isomorphic bit strings 2m , 4mand 8m ; the second group contains the bit string 5m , and its iso. Therefore, these six bit strings morphic “twin” 10m present two different genotypes, but they have exactly the same phenotype. In Table 6, we display all basin tree rules grouped by genotype, and hence the robustness shown is a sort of genotypic robustness, which describes the robustness of each genotype. In Table 8, for the sake of brevity, we often grouped basin trees according to their phenotype, and hence we show the phenotypic robustness, which describes the robustness of each phenotype. For this reason, the values of seeds and robustness do not always correspond in these two tables. An emblematic example is given by rule 204 : in Table 6-23, we find that for any bit string length L there are several nonisomorphic basin trees, each with its own genotypic robustness; however, in Table 8-23 we preferred to show the phenotypic robustness, in order to emphasize that all orbits have exactly the same digraph representation, because they are all are period-1 Isles of Eden.

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3. Omega-Limit Orbits Given any local rule N , and any initial bit string [x0

x1

x2

···

xL−1 ]0 → φ0 ∈ [0, 1)

(11)

a fundamental problem is determining the timeasymptotic behavior [x0

x1

x2

···

xL−1 ]n → φn ∈ [0, 1)

(12)

as n → ∞. For finite L, the evolution must converge to a period-T orbit, where T < 2L . As is now well known [Chua et al., 2007a], and illustrated extensively in the basin tree diagrams in Tables 6-1 to 6-24, the periodic orbit can either be an attractor or an Isle of Eden. Both correspond to the familiar steady state regimes in continuous systems. In this paper, we will often develop analytical methods that apply to both attractors and Isles of Eden of a rule N and we will henceforth refer to both types of asymptotic orbits as Omega-limit orbits, abbreviated ω-limit orbits.7 We have opted for this classical terminology in anticipation of our forthcoming papers involving bi-infinite bit strings L → ∞ where the time asymptotic steady state orbits (as n → ∞) may not be periodic. For example, the ω-limit orbit of the right-copycat rule 170 [Chua et al., 2005a] for the bi-infinite case is not periodic if we pick a bi-infinite bit string made of a concatenation of any pair of irrational numbers inside the unit interval as the initial bit string. In fact, since all orbits of 170 are Isles of Eden, there are no transient regimes and any initial bit string belongs to its ω-limit orbit.

3.1. General observations on ω -limit orbits from Table 6 The basin tree diagrams exhibited in the 24 Galleries in Tables 6-1 to 6-24 provide numerous concrete examples illustrating the following common observations which apply to all one-dimensional cellular automata with finite string length L: 1. The evolution from any one of the 2L possible initial bit strings must converge to either an 7

attractor, or an Isle of Eden of period8 T < 2L . Each basin tree diagram includes all initial bit strings that converge to an attractor or to an Isle of Eden. In the former case, the basin of attraction would include all bit strings belonging to the transient regime and the attractor. Since we are concerned mostly with time-asymptotic behaviors, the transient points have been deleted from the “time-1 return maps” displayed in the upper right-hand corner of the first page (Gallery N − 1) of each Gallery in Table 6. 2. In the case of an Isle of Eden, there is no transient regime and every initial bit string converges to an ω-limit orbit, which is in fact, an Isle of Eden. 3. Each basin tree, attractor, or Isle of Eden, has in general L isomorphic copies9 in the sense that except for the node labeling integers, they can all be mapped onto the same digraph, as demonstrated in Example 2.3.1, via the explicit formula (5). 4. The robustness of each attractor, or Isle of Eden, in the basin tree Galleries can be measured by the robustness coefficient ρ defined in Eq. (10). This number measures the “size” of the basin of attraction for each attractor, or Isle of Eden, listed in Table 6, regardless of whether it has other isomorphic copies, which we will henceforth identify as a single attractor, or a single Isle of Eden, respectively. Consequently, we will henceforth multiply the robustness coefficient ρ calculated from Eq. (10) by the number of isomorphic copies when referring to the robustness of an isomorphic attractor, or Isle of Eden.

3.2. Portraits of ω-limit orbits from the basin tree diagrams exhibited in Table 6 For ease of future reference, all relevant qualitative properties of the ω-limit orbits of each of the 24 rules (listed in Table 4) belonging to Group 1 (period-1 rules) have been extracted, organized and exhibited in 24 portraits in Tables 8.1, 8.2, . . . , 8.24.

Our terminology is inspired by the deep concepts of α-limited set and ω-limited set, from classical nonlinear dynamical systems [Birkhoff, 1927], where they denote the time asymptotic evolutions of symbolic maps as n → −∞, or n → ∞, respectively. Birkhoff had chosen the symbols “α” and “ω” because they represent the first and the last letters in the Greek alphabet, respectively. We will be concerned only with the latter (n → ∞) because cellular automata are generally time irreversible [Chua et al., 2006]. 8 When the ω-limit orbit is an Isle of Eden, the evolution converges in one iteration because it does not have a transient regime. In the bi-infinite case (n → ∞), an attractor, or an Isle of Eden, may be aperiodic, i.e. (T → ∞). 9 For those attractors and Isles of Eden which exhibit some forms of symmetry in their binary string patterns, some of the L isomorphic copies may coincide with each other.

Table 8-1.

Portraits of ω-Limit Orbits of Rule 4 .

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Table 8-1.

(Continued )

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Table 8-1.

(Continued )

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Table 8-2.

Portraits of ω-Limit Orbits of Rule 8 .

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308

Table 8-3.

Portraits of ω-Limit Orbits of Rule 12 .

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309

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Table 8-3.

(Continued )

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Table 8-3.

(Continued )

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Table 8-4.

Portraits of ω-Limit Orbits of Rule 13 .

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312

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(Continued )

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Table 8-4.

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Table 8-5.

Portraits of ω-Limit Orbits of Rule 32 .

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314

Table 8-6.

Portraits of ω-Limit Orbits of Rule 36 .

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315

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(Continued )

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Table 8-6.

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316

Table 8-7.

Portraits of ω-Limit Orbits of Rule 40 .

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317

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(Continued )

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Table 8-7.

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318

Table 8-8.

Portraits of ω-Limit Orbits of Rule 44 .

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319

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Table 8-8.

(Continued )

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320

Table 8-9.

Portraits of ω-Limit Orbits of Rule 72 .

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321

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(Continued )

14:49

Table 8-9.

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Table 8-10.

Portraits of ω-Limit Orbits of Rule 76 .

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323

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Table 8-10.

(Continued )

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Table 8-10.

(Continued )

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Table 8-10.

(Continued )

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(Continued )

14:49

Table 8-10.

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Table 8-11.

Portraits of ω-Limit Orbits of Rule 77 .

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328

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Table 8-11.

(Continued )

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Table 8-11.

(Continued )

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(Continued )

14:49

Table 8-11.

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Table 8-12.

Portraits of ω-Limit Orbits of Rule 78 .

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332

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(Continued )

14:49

Table 8-12.

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Table 8-13.

Portraits of ω-Limit Orbits of Rule 104 .

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334

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Table 8-13.

(Continued )

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Table 8-14.

Portraits of ω-Limit Orbits of Rule 128 .

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336

Table 8-15.

Portraits of ω-Limit Orbits of Rule 132 .

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337

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Table 8-15.

(Continued )

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Table 8-15.

(Continued )

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(Continued )

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Table 8-15.

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Table 8-16.

Portraits of ω-Limit Orbits of Rule 136 .

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341

Table 8-17.

Portraits of ω-Limit Orbits of Rule 140 .

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342

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Table 8-17.

(Continued )

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Table 8-17.

(Continued )

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(Continued )

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Table 8-17.

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Table 8-18.

Portraits of ω-Limit Orbits of Rule 160 .

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346

Table 8-19.

Portraits of ω-Limit Orbits of Rule 164 .

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347

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Table 8-19.

(Continued )

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(Continued )

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Table 8-19.

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Table 8-20.

Portraits of ω-Limit Orbits of Rule 168 .

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350

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Table 8-20.

(Continued )

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(Continued )

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Table 8-20.

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Table 8-21.

Portraits of ω-Limit Orbits of Rule 172 .

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353

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Table 8-21.

(Continued )

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Table 8-21.

(Continued )

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(Continued )

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Table 8-21.

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Table 8-22.

Portraits of ω-Limit Orbits of Rule 200 .

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357

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Table 8-22.

(Continued )

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(Continued )

14:49

Table 8-22.

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Table 8-22.

(Continued )

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(Continued )

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Table 8-22.

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Table 8-23.

Portraits of ω-Limit Orbits of Rule 204 .

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362

Table 8-24.

Portraits of ω-Limit Orbits of Rule 232 .

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Table 8-24.

(Continued )

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Table 8-24.

(Continued )

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(Continued )

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Table 8-24.

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Chapter 2: Period-1 Rules

For each bit string length L, identified in column 1, only distinct isomorphic attractors and Isles of Eden are displayed in Table 8: they are identified in column 2 by a red integer i. For example, for L = 5, rule 4 has three distinct ω-limit orbits, identified as i = 1, 2, 3, respectively, in column 2 (last three rows of the first page of Table 8.1). Among these three ωlimit orbits, there is only one isomorphic (topologically distinct) Isle of Eden, even though the basin tree diagram in Gallery 4-2 shows five period-1 Isles of Eden; namely, { 5 , 9 , 10 , 18 , 20 }. Observe that the four Isles of Eden 9 , 10 , 18 , and 20 can be generated from Isle of Eden 5 via formula (5). Consequently, only the bit string 5 is displayed in Table 8.1 and identified as x0 ( 5 ), where the subscript ‘0’ denotes the initial bit string to generate the Isle of Eden, which in this case corresponds also with the final bit string, because T = 1. Since Eq. (5) is equivalent to a left shift of 1 bit to the left, it follows that we can generate the binary strings for the Isles of Eden 10 , 20 , 9 and 18 by shifting the bit string 5 by one, two, three, and four bits, respectively. The four isomorphic Isles of Eden 9 , 10 , 18 , and 20 are displayed in column 6, connected by arrows generated from 5 . Following the same format, we found from Gallery 4-2 that for L = 5, there are two isomorphically distinct attractors, identified as i = 2, and i = 3 under L = 5 in column 2 of Table 8.1. For attractor i = 2, Gallery 4-2 shows that there are five isomorphic copies, namely, { 1 , 2 , 4 , 8 16 }. Observe that all of them can be generated from attractor 1 via formula (5). Consequently, only the binary bit string 1 is displayed in column 6 of Table 8.1 (identified by x0 ( 1 ). The isomorphic copies 2 , 4 , 8 and 16 are connected by arrows originating from 1 . The remaining attractor 0 , identified as i = 3 in column 2, is found to be a degenerate case possessing the maximum

10

367

symmetry so that all five isomorphic copies are identical to each other, namely, the bit string 0 . It is displayed (identified as x0 ( 0 )) in the last row of the first page of Table 8.1. The information extracted from Gallery 4-2 as described above is collected in columns 1 through 6 in Table 8.1. In addition, column 7 (Bernoulli Parameters) shows the three relevant Bernoulli parameters (β, σ, τ ) [Chua et al., 2005a] for each rule N . For period-1 rules,10 we have σ = 0, τ = 1, and β > 0 (indicated by a + sign). Column 8 of Table 8.1 specifies the integer δmax defined as the distance from the farthest Garden of Eden to the attractor of each basin tree diagram. Clearly, δmax = 0 for all Isles of Eden. For rule 4 , δmax = 1 for all L = 3, 4, . . . , 8 since every bit string in the transient regime of 4 is a Garden of Eden and hence it takes only one iteration to converge from any of these “transient-regime” initial states to an attractor. For many rules, however, δmax > 1. The last column 9 is devoted to the robustness coefficient for each isomorphic attractor, or Isle of Eden. The robustness coefficient ρi for each orbit is  in the form ρi =
calculated and displayed k × number of bit strings/2L so that the number k of isomorphic copies of attractors, or Isles of Eden, can be identified directly. Note that ρi is equal to the integer listed in the magenta column 3 if the ω-limit orbit is an attractor, or in the blue column 4 if the ω-limit orbit is an Isle of Eden. Following the same format delineated above, the portraits of the ω-limit orbits of all 24 nontrivial period-1 (Group 1) rules from Table 4 are exhibited.

3.3. Concatenated ω -limit orbit generation algorithm A careful analysis of the ω-limit orbits of rule 4 for L = 8 displayed in column 6 of the third page of Table 8.1 shows that each of them can be decomposed into two ω-limit orbits of shorter lengths L
and L

such that11 L = L
+ L

.

A period-1 rule is a degenerate Bernoulli rule with σ = 0, indicating no shift at all. Alternately, we can also assign σ = L for each period-1 rule. These are trivial cases and are included for consistency. There are some rules in Table 8, however, with nontrivial (β, σ, τ ) parameters. 11 Recall our color code in Table 8: blue for Isles of Eden and magenta for attractors.

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Example 3.3.1. Isle of Eden 17 . This 8-bit string can be decomposed into a concatenation of two 4-bit strings as follows: 1

Γ′(4)

1

= 17

The symbol denotes concatenation, forming a chain of two shorter 4-bit strings. If we introduce , Γ

(4) = 1m , and Γ(8) = 17m the symbols Γ
(4) = 1m to mean a bit string with 4, 4, and 8 bits respectively, then we can code the above concatenation as follow:

≡

Γ′′(4) = Γ(8)

′

Γ (4) = (L=4)

′′

Γ(8) = 17 (L=8)

Γ (4) = (L=4)

Example 3.3.2. Isle of Eden 21 . Using the same notations as above, we have

Γ′(4)

Γ′′(4) = Γ(8)

≡

≡ ′

Γ (4) = (L=4)

′′

Γ(8) = 21 (L=8)

Γ (4) = (L=4)

m where Γ
(4) = 1m , Γ

(4) = 5m , and Γ(8) = 21 .

Example 3.3.3. Isle of Eden 37

Γ′(4)

Γ′′(4) = Γ(8)

≡

≡ Γ′(4) = (L=4)

Γ′′ (4) = (L=4)

Γ(8) = 37 (L=8)

where Γ
(4) = 2m , Γ

(4) = 5m , and Γ(8) = 37m . Example 3.3.4. Isle of Eden 85

Γ′(4)

≡

Γ′′(4) = Γ(8)

′

Γ (4) = (L=4)

′′

Γ(8) = 85 (L=8)

Γ (4) = (L=4)

where Γ
(4) = 5m , Γ

(4) = 5m , and Γ(8) = 85m . Example 3.3.5. Attractor 1

Γ′(4)

Γ′′(4) = Γ(8)

≡

≡ ′

Γ (4) = (L=4)

where Γ
(4) = 0m , Γ

(4) = 1m , and Γ(8) = 1m .

′′

Γ (4) = (L=4)

Γ(8) = 1 (L=8)

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Chapter 2: Period-1 Rules

369

Note that this decomposition is not unique, since we can also concatenate L
(5) and L

(3)

′

Γ (5)

′′

Γ (3) = Γ(8)

≡

≡ Γ′(5) = (L=5)

Γ′′ (3) = (L=3)

Γ(8) = 1 (L=8)

where L
(5) = 0m , L

(3) = 1m , and L(8) = 1m . Example 3.3.6. Attractor 5

Γ′(4)

Γ′′(4) = Γ(8)

≡

≡ ′

Γ (4) = (L=4)

′′

Γ(8) = 5 (L=8)

Γ (4) = (L=4)

where Γ
(4) = 0m , Γ

(4) = 5m , and Γ(8) = 5m . Example 3.3.7. Attractor 9

Γ′(5)

Γ′′(3) = Γ(8)

≡

≡ ′

Γ (5) = (L=5)

′′

Γ(8) = 9 (L=8)

Γ (3) = (L=3)

where Γ
(5) = 1m , Γ

(3) = 1m , and Γ(8) = 9m . Example 3.3.8. Attractor 0

Γ′(4)

Γ′′(4) = Γ(8)

≡

≡ ′

Γ (4) = (L=4)

where Γ
(4) = 0m , Γ

(4) = 0m , and Γ(8) = 0m . The above examples illustrate that certain period-T ω-limit orbits of length L
when concatenated with another period-T ω-limit orbit of length L

result in a new period-T ω-limit orbit of length L = L
+ L

. The following theorem is a rigorous formulation of this result. Theorem 3.3.1. ω-Limit Orbit Generation Algorithm. Given a period-T ω-limit orbit Γ
(L
) of rule N with L
bits, and a period-T ωlimit orbit Γ

(L

) of rule N with L

bits, where L
and L

need not be equal, we can construct another period-T ω-limit orbit Γ(L) of rule N with length L = L
+ L

by the concatenation depicted in Fig. 4, subject to the three Seamless Concatenation Conditions specified in Fig. 4(d).

′′

Γ (4) = (L=4)

Γ(8) = 0 (L=8) 8

Corollary 3.3.1. Boundary Color Matching Condition. Any two period-T ω-limit orbits (of lengths L
and L

, respectively) with identical left boundary colors, and identical right boundary colors, can be concatenated to obtain a new periodT ω-limit orbit of length L = L
+ L

. Since the colors of corresponding left (resp. right) bits are identical, all three conditions of Theorem 3.3.1 are automatically satisfied.


Proof.

For instance, note that Examples 3.3.1, 3.3.2, 3.3.4, 3.3.7 and 3.3.8 satisfy the hypotheses of Corollary 3.3.1 and can be concatenated without resorting to the more time-consuming verifications required by Theorem 3.3.1.

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Fig. 4. Illustration of Seamless Concatenation Conditions. (a) Period-T ω-limit orbit Γ
(L
) of length = L
bits. (b) Period-T ω-limit orbit Γ

(L

) of length = L

bits. (c) Concatenation between Γ
(L
) (on the left) and Γ

(L

) (on the right) gives a new ω-limit orbit Γ(L) of the same period T, with length equal to L = L
+ L

bits. (d) Three seamless concatenation conditions: (1) Seamless Interface; (2) Seamless left-boundary; and (3) Seamless right-boundary.

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For some local rules, Theorem 3.3.1 holds under less restrictive conditions, because the three seamless concatenation conditions specified in Fig. 4(d) are always satisfied. This result is formulated in the following corollary. Corollary 3.3.2. For rules 0 , 51 , 204 , and 255 , any two period-T ω-limit orbits of length L
and L

, respectively, where L
and L

need not be equal, can be concatenated to obtain a new period-T ω-limit orbit of length L = L
+ L

. For N = {0, 51, 204, 255}, T N (000) = T N (001) = T N (100) = T N (101), and T N (010) = T N (011) = T N (110) = T N (111); hence, all three conditions of Theorem 3.3.1 are automatically satisfied.


Proof.

This result is not surprising, because rules 0 and 255 have a global attractor, whereas rule 51 and rule 204 have exclusively Isles of Eden. Moreover, there are local rules for which one of the three seamless concatenation conditions is always satisfied, as proven in the following “twin” corollaries. Corollary 3.3.3. For rules 0 , 17 , 34 , 51 , 68 , 85 , 102 , 119 , 136 , 153 , 170 , 187 , 204 , 221 , 238 , and 255 , any two period-T ω-limit orbits of length L
and L

, respectively, where L
and L

need not be equal, with identical left boundary colors can be concatenated to obtain a new period-T ω-limit orbit of length L = L
+ L

. Proof. For N = {0, 17, 34, 51, 68, 85, 102, 119, 136, 153, 170, 187, 204, 221, 238, 255}, the concatenation conditions 1(a) and 3 are always satisfied, because: T N (000) = T N (100), T N (001) = T N (101), T N (010) = T N (110), T N (011) = T N (111). The restriction on the left boundary of the ω-limit orbits guarantees that also the rest of conditions of Theorem 3.3.1 are satisfied.


Corollary 3.3.4. For rules 0 , 3 , 12 , 15 , 48 , 51 , 60 , 63 , 192 , 195 , 204 , 207 , 240 , 243 , 252 , 255 , any two period-T ω-limit orbits of lengths L
and L

, respectively, where L
and L

need not be equal, with identical right boundary colors can be concatenated to obtain a new period-T ω-limit orbit of length L = L
+ L

. For N = {0, 3, 12, 15, 48, 51, 60, 63, 192, 195, 204, 207, 240, 243, 252, 255}, the concatenation Proof.

371

conditions 1(b) and 2 are always satisfied, because: T N (000) = T N (001), T N (010) = T N (011), T N (100) = T N (101), T N (110) = T N (111). The restriction on the right boundary of the ω-limit orbits guarantees that also the rest of conditions of Theorem 3.3.1 are satisfied.
Only 10 of the 28 local rules listed in the last three corollaries are globally independent: 0 , 3 , 12 , 15 , 34 , 51 , 60 , 136 , 170 , and 204 . Four of them — 0 , 12 , 136 , and 204 — belong to Group 1 (period-1 rules); rule 51 is a period-2 rule; four rules — 3 , 15 , 34 , 170 — belong to Group 4 (Bernoulli rules); finally, rule 60 is a hyper Bernoulli-shift rule. We close this section by noting that by applying Theorem 3.3.1 and its Corollaries, we can generate an infinite variety of period-1 orbits for all 25 period-1 rules from Table 4, for both finite L and infinite L.

4. Rules of Group 1 have Robust Period-1 ω -Limit Orbits The classification of CA local rules into six groups proposed in [Chua et al., 2007b] was empirical: on the one hand, we analyzed the basin tree diagrams for finite bit strings with lengths up to 8, and on the other hand, we studied the behavior of random bit strings for large L (up to 400 bits) through extensive computer simulations. In this section, we give a formal definition of robust period-1 ω-limit orbits, and prove analytically that all rules belonging to group 1, which are the subject of this article, have robust period-1 ω-limit orbits. It is useful to recall that, in this context, the expression “basin of attraction of a period-1 ωlimit orbit” includes the periodic orbit itself and all bit strings not belonging to the periodic orbit, if any, that converge to the periodic orbit. Note that throughout the paper we sometimes abused our terminology by including only the branches in the transient regime as the basin of attraction of an attractor ; however, the precise definition has always been clear from the context. For finite L, local rules belonging to Group 1 harbor mainly, or exclusively, period-1 ω-limit orbits (attractors or Isles of Eden), as experimentally observed in [Chua et al., 2007b]. We found that these periodic orbits are robust, in the sense that they appear for any L and their basins of attraction include most of the 2L bit strings. Since, by

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definition, the union of all basins of attraction coincides with the whole space of 2L orbits, all bit strings, except for a few, are in the basin of attraction of a period-1 ω-limit orbit. These concepts can be easily formalized for biinfinite bit strings as follows: Definition 4.1. Robust period-1 ω-limit orbits: A local rule has robust period-1 ω-limit orbits if all bit strings, except for a finite subset, belong to the basin of attraction of a period-1 ω-limit orbit when L → ∞.

The “finite subset” we mention in the definition includes all period-T ω-limit orbits (attractors or Isles of Eden) with T = 1, which are nonrobust for period-1 rules in the sense that they appear only for some values of L and their basin of attraction includes only bit strings containing prescribed patterns, as will be shown in the following. Restricting Definition 4.1 to finite L, we find that a local rule having robust period-1 ω-limit orbits is characterized by the fact that most of the 2L bit strings belong to the basins of attraction of period-1 ω-limit orbits, which is exactly the behavior described previously. The expression “most of”

Table 9.

Input

4

may seem vague, but in the following it will be evident that period-1 ω-limit orbits are dominant even for low values of L (e.g. L < 9). Moreover, for a period-1 rule, the probability that a random bit string is in the basin of attraction of a period-1 ω-limit orbit tends to 1, when L → ∞. These two elements confirm that the empirical approach we took in [Chua et al., 2007b] was effective in uncovering the intrinsic nature of local rules belonging to Group 1. So far our classification has been based on exhaustive computer simulations; here, we prove rigorously the following theorem: Theorem 4.1. All and only local rules belonging to

Group 1 have robust period-1 ω-limit orbits.

For the reader’s convenience, we divided this theorem into 24 lemmas, one for each rule of Group 1 (except for 0 , which is trivial) providing a detailed and rigorous proof for each of them. Such proofs are based exclusively on the properties of the truth tables, summarized in Tables 9 and 10, and they also include the definition of the maximum transient length δmax . There are two ways to demonstrate that a local rule has robust period-1 ω-limit orbits: either

Period-1 rules harboring exclusively period-1 ω-limit orbits (attractors and Isles of Eden).

8

12

36

72

76

78

128 132 136 140 200 204

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Chapter 2: Period-1 Rules Table 10.

Period-1 rules harboring non-robust period-T ω-limit orbits, T
= 1 (attractors and Isles of Eden).

Input 13

32

40

44

77

we prove that period-T ω-limit orbits, T = 1, can arise only for subsets of bit strings (for example, bit strings containing exclusively particular patterns), or we prove that all bit strings containing a prescribed pattern are in the basin of attraction of a period-1 ω-limit orbit (when L → ∞, all bit strings contain all patterns). The structure of the proof is the same for all 24 lemmas: first, we identify the firing patterns of the rule; second, we find the general form for all period-T ω-limit orbits, which can be directly deduced from the firing pattern; third, we prove that only period-1 ω-limit orbits are robust by showing that only small subsets of bit strings belong to the basin of attraction of period-T attractors with T > 1; and last, we summarize the results. After each lemma, there is an example focusing on the particular properties of the rule just analyzed. Note that the inverse implication of Theorem 4.1 — only Group 1 rules have robust period-1 12

373

104 160 164 168 172 232

ω-limit orbits — will be proved in our forthcoming papers, which will be devoted to rules of Groups12 2–4. Rules belonging to Group 1 can be divided into two classes: those having exclusively period-1 ωlimit orbits, and those harboring also non-robust ω-limit orbits with T > 1. The first class is composed of 13 rules — namely 4 , 8 , 12 , 36 , 72 , 76 , 78 , 128 , 132 , 136 , 140 , 200 , and 204 — and their behavior is analyzed in Lemmas 4.1–4.13; the second class is composed of 11 rules — namely, 13 , 32 , 40 , 44 , 77 , 104 , 160 , 164 , 168 , 172 , and 232 — and their behavior is analyzed in Lemmas 4.14–4.24. In the following, we use the expression “isolated
” (respectively, “isolated
”) to indicate a
(resp.,
) bit whose both neighbors are
(resp.,
), and the expression “runs of k
” (respectively, “runs of k
”) to indicate k consecutive
(resp.,
) bits, which can be also denoted by

· · ·
(resp.,

Rules of Groups 5 and 6 do not have robust period-1 ω-limit orbits even for low values of L, as proved in [Chua et al., 2007a] and [Chua et al., 2007b].

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· · ·
). The symbol “
” means that the value of the bit cannot be determinate a priori, but we need the whole information about its neighbors. Lemma 4.1.

Rule 4 . Rule 4 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 1. The only firing pattern of rule 4 is


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, it is isolated. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor; moreover, all bit strings not containing runs of k
, k > 1, are period1 ω-limit orbits (attractors or Isles of Eden). Runs of k
, k > 1, are transformed into runs of k
in one iteration; hence, all bit strings containing a run of k
, k > 1, are in the basin of attraction of a period-1 attractor, and the maximum transient length is δmax = 1. In conclusion, all bit strings not being period-1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Proof.

Example 4.2. Rule 8 . For rule 8 and L = 6,

bit strings containing at most runs of k
, k = 1, converge to

· · ·
in one iteration — e.g.





→





→





— whereas bit strings containing runs of k
, k > 1, converge to the period-1 attractor

· · ·
in two iterations — e.g.





→





→





→





. Compare this result with Example 4.1. Lemma 4.3. Rule 12 . Rule 12 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 1.

the bit string





is a period-1 ω-limit orbit (attractor),





→





, because it does not contain runs of k
, k > 1. The bit string





is not a period-1 ω-limit orbit because it has a run of three
, and it belongs to the basin of attraction of





:





→





→





.

Proof. The firing patterns of rule 12 are


and


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, its left neighbor is
. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor; moreover, all bit strings not containing runs of k
, k > 1, are period1 ω-limit orbits. Runs of k
, k > 1, are transformed into a
followed by a run of k−1
in one iteration, because


→


; hence, all bit strings containing a run of k
, k > 1, are in the basin of attraction of a period-1 attractor, and the maximum transient length is δmax = 1. In conclusion, all bit strings not being period-1 ω-limit orbits are in the basin of attraction of period-1 attractors.


Lemma 4.2. Rule 8 . The bit string

· · ·
is a global attractor for rule 8 ; the maximum transient length for this rule is δmax = 2.

Example 4.3. Rule 12 . For rule 12 and L =

Example 4.1. Rule 4 . For rule 4 and L = 6,

The only firing pattern of rule 8 is


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, its left and right neighbors are
and
, respectively. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor. All bit strings not containing runs of k
, k > 1, converge to the period-1 attractor

· · ·
in one iteration, because


→
; all bit strings containing runs of k
, k > 1, converge to the period-1 attractor

· · ·
in two iterations, because


→


→


,


→


, and


→


. In conclusion, all bit strings converge to the global attractor

· · ·
in at most two iterations.


Proof.

6, the bit string





is a period-1 ω-limit orbit (attractor),





→





→





, because it does not contain runs of k
, k > 1. The bit string





is not a period-1 ω-limit orbit because it has runs of k
, k > 1, and it belongs to the basin of attraction of





:





→





→





. Compare this result with Example 4.1.

Lemma 4.4. Rule 36 . Rule 36 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 2.

The firing patterns of rule 36 are


and


(see Table 9): consequently,
changes if, and Proof.

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Chapter 2: Period-1 Rules

only if, it is isolated, and
remains unchanged if, and only if, it is isolated. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor; moreover, all bit strings containing exclusively runs of k
, k > 1, and isolated
are period-1 ω-limit orbits. Runs of k
, k > 2, are transformed into runs of k
in one iteration, because


→
,


→
, and


→
. Runs of k

, k arbitrary, are transformed into runs of 2k
in two iterations (the left boundary can be either

or
, the right boundary can be either
or

); here, the four possible cases are listed:



· · ·


→



· · ·


→



· · ·


;



· · ·



→



· · ·



→



· · ·



;


· · ·


→


· · ·


→


· · ·


;


· · ·



→


· · ·



→


· · ·



. Hence, all bit strings containing runs of k
, k > 1, and/or isolated
are in the basin of attraction of a period-1 attractor, and the maximum transient length is δmax = 2. In conclusion, all bit strings not being period-1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Example 4.4. Rule 36 . For rule 36 and L = 6,

the bit string





is a period-1 ω-limit orbit (attractor),





→





, because it contains exclusively runs of at least k
, k > 1, and isolated
. The bit string





is not a period1 ω-limit orbit because it contains isolated
, k > 1, and runs of
, k > 1, but it is in the basin of attraction of





:





→





→





. Bit strings containing the pattern



and/or



have δmax = 2, like





→





→





→





. Lemma 4.5. Rule 72 . Rule 72 has exclusively

period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 2.

Proof. The firing patterns of rule 72 are


and


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, only one of its neighbors is
. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor; moreover, all bit strings containing exclusively runs of k
, k arbitrary, and runs of k
, k = 2, are period-1 ω-limit orbits.

375

Isolated
are transformed into
in one iteration, because


→


; runs of k
, k > 2, are transformed into runs of k
in two iterations, because


· · ·


→


· · ·


→


· · ·


. Hence, all bit strings containing runs of k
, k = 2, are in the basin of attraction of a period-1 attractor, and the maximum transient length is δmax = 2. In conclusion, all bit strings not being period-1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Example 4.5. Rule 72 . For rule 72 and L = 6, the bit string





is a period-1 ω-limit orbit (attractor),





→





→





, because it contains exclusively runs of k
, k arbitrary, and runs of k
, k = 2. The bit string





is not a period-1 ωlimit orbit and it belongs to the basin of attraction of





,





→





→





, because it contains runs of k
, k = 2. Lemma 4.6. Rule 76 . Rule 76 has exclusively

period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 1.

The firing patterns of rule 76 are


,


, and


(see Table 9): consequently,
never changes, and
changes if, and only if, both its neighbors are
. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor; moreover, all bit strings containing exclusively runs of k
, k arbitrary, and runs of k
, k < 3, are period-1 ω-limit orbits. Runs of k
, k ≥ 3, are transformed — in one iteration — into a sequence beginning and ending with
, and containing a run of k − 2
, because


· · ·


→


· · ·


. Hence, all bit strings containing runs of k
, k ≥ 3, are in the basin of attraction of a period-1 attractor, and the maximum transient length is δmax = 1. In conclusion, all bit strings not being period-1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Proof.

Example 4.6. Rule 76 . For rule 76 and L =

6, the bit string





is a period-1 ω-limit orbit (attractor),





→





, because it contains exclusively runs of k
, k arbitrary, and runs of k
, k < 3. The bit string

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belongs to the basin of attraction of the period-1 ω-limit orbit





,





→





→





, because it contains a run of three
. Compare this result with Example 4.5.

Lemma 4.8.

Rule 128 . Rule 128 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L/2. The only firing pattern of rule 128 is


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, both its neighbors are
. Since


→
and


→
, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits: the former is an Isle of Eden, the latter is an attractor. Runs of k
, k arbitrary, converge to runs of k
in (k + 1)/2 iterations, because


· · ·


→


· · ·


. Hence, all bit strings containing runs of k
, except for

· · ·
, converge to

· · ·
; since kmax = L − 1, where kmax is the longest run of
, the maximum transient length is δmax = L/2. In conclusion, all bit strings, except for

· · ·
, are in the basin of attraction

· · ·
.
Proof.

Lemma 4.7. Rule 78 . Rule 78 has exclusively

period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 2 • L/2 − 1.

The firing patterns of rule 78 are


,


,


, and


(see Table 9): consequently,
changes if, and only if, its left and right neighbors are
and
, respectively, whereas
changes if, and only if, both its neighbors are
. Since


→
and


→
, the bit string

· · ·
is a period-1 attractor; moreover, all bit strings containing exclusively isolated
and runs of k
, k < 3, are period-1 ω-limit orbits. Runs of k
, k ≥ 3, are transformed — in one iteration — into a sequence starting and ending with
, and containing a run of k − 2
, because


· · ·


→


· · ·


. Runs of k
bits, k > 1, are transformed into a run of k/2

, when k is even, and into a sequence beginning with
and followed by a run of (k − 1)/2

, when k is odd. Hence, all bit strings nonisolated
and runs of k
, k ≥ 3, are in the basin of attraction of a period-1 attractor, and they converge in 2 • (k + 1)/2 − 1 iterations; since kmax = L − 1, where kmax is the longest run of
, the maximum transient length is δmax = 2 • L/2 − 1. In conclusion, all bit strings not being period-1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Proof.

Example 4.8. Rule 128 . For rule 128 and L = 6, the maximum transient length is δmax = 3, and it occurs for any bit string containing five consecutive
, like





:





→





→





→





→





. For L = 5, the maximum transient length is δmax = 2, and it occurs for any bit string containing four consecutive
, like




:




→




→




→




. Lemma 4.9.

Rule 132 . Rule 132 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = (L − 1)/2. The firing patterns of rule 132 are


and


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, both its neighbors are either
or
. Since


→
and


→
, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits: the former is an Isle of Eden, the latter is an attractor. Moreover, all bit strings containing exclusively runs of k
, k arbitrary, and isolated
are period-1 ω-limit orbits. Runs of k
, k even, are transformed into runs of k
in k/2 iterations, whereas runs of k
, k odd and k > 1, are transformed into a sequence Proof.

Example 4.7. Rule 78 . For rule 78 and L = 6, the maximum transient length is δmax = 5, and it is obtained for any bit string containing a run of five
, like





:





→





→





→





→





→





→





. For L = 7, the maximum transient length is also δmax = 5, and it is obtained for any bit string containing a run of six
, like






:






→






→






→






→






→






→






.

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Chapter 2: Period-1 Rules

beginning and ending with (k − 1)/2
, and containing an isolated
, in (k −1)/2 iterations. Hence, all bit strings containing nonisolated
are in the basin of attraction of a period-1 attractor, and they converge in k/2 iterations; since kmax = L − 1, where kmax is the longest run of
, the maximum transient length is δmax = (L − 1)/2. In conclusion, all bit strings not being period1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Example 4.9. Rule 132 . For rule 132 and L =

6, the maximum transient length is δmax = 2, and it occurs for any bit string containing runs of five
, like





:





→





→





→





. For L = 5, the maximum transient length is δmax = 2, and it occurs for any bit string containing runs of four
, like




:




→




→




→




. Compare this result with Example 4.8.

Lemma 4.10.

Rule 136 . Rule 136 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 1. The firing patterns of rule 136 are


and


(see Table 9): consequently,
never changes, and
remains unchanged if, and only if, its right neighbor is
. Since


→
and


→
, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits: the former is an Isle of Eden, the latter is an attractor. Runs of k
, k arbitrary, converge to runs of k
in k iterations, because

· · ·


→

· · ·


. Hence, all bit strings containing runs of k
, except for

· · ·
, converge to

· · ·
; since kmax = L − 1, where kmax is the longest run of
, the maximum transient length is δmax = L − 1. In conclusion, all bit strings not being period1 ω-limit orbits are in the basin of attraction of period-1 attractors.


Proof.

Example 4.10. Rule 136 . For rule 136 and L = 6, the maximum transient length is δmax = 5, and it occurs for any bit string containing a run of five
, like





:





→





→





→





→





→





→





. For L = 5, the maximum transient length is δmax = 4, and it occurs

377

for any bit string containing a run of four
, like




:




→




→




→




→




→




. Compare this result with Example 4.8. Lemma 4.11.

Rule 140 . Rule 140 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 2. The firing patterns of rule 140 are


,


, and


(see Table 9): consequently,
never changes, and
changes if, and only if, its left and right neighbors are
and
, respectively. Since


→
and


→
, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits (Isles of Eden). Moreover, all bit strings containing exclusively runs of k
, k arbitrary, and isolated
are period-1 ω-limit orbits. Runs of k
, k > 1, are transformed into a sequence beginning with
followed by a run of k − 1
in k − 1 iterations, because

· · ·


→

· · ·


. Hence, all bit strings containing runs of k
, k > 1, are in the basin of attraction of a period-1 attractor; since kmax = L−1, where kmax is the longest run of
, the maximum transient length is δmax = L − 2. In conclusion, all bit strings not being period1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Proof.

Example 4.11. Rule 140 . For rule 140 and L = 6, the maximum transient length is δmax = 4, and it occurs for any bit string containing runs of five
, like





:





→





→





→





→





→





. For L = 5, the maximum transient length is δmax = 3, and it occurs for any bit string containing runs of four
, like




:




→




→




→




→




. Compare this result with Examples 4.9 and 4.10. Lemma 4.12.

Rule 200 . Rule 200 has exclusively period-1 ω-limit orbits; the maximum transient length for this rule is δmax = 1. The firing patterns of rule 200 are


,


, and


(see Table 9): consequently,
never changes, and
changes if, and only if, it is isolated. Proof.

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Since


→
and


→
, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits: the former is an Isle of Eden, the latter is an attractor. Moreover, all bit strings containing exclusively runs of k
, k arbitrary, and runs of k
, k > 1, are period-1 ω-limit orbits. Isolated
are transformed into
in one iteration, because


→


; hence, bit strings containing isolated
are in the basin of attraction of a period-1 attractor. In conclusion, all bit strings not being period1 ω-limit orbits are in the basin of attraction of period-1 attractors.
Example 4.12. Rule 200 . For rule 200 and L = 6, the maximum transient length is δmax = 1, and it occurs for any bit string containing the pattern


, like





→





→





. Lemma 4.13. Rule 204 . Rule 204 has exclusively

period-1 Isles of Eden; hence, the maximum transient length for this rule is δmax = 0. The firing patterns of rule 204 are all, and only, those containing a
in the central position (see Table 9): consequently, both
and
never change, and no evolution is possible for any string.
Proof.

Example 4.13.

Rule 204 . The bit string







contains all eight 3-bit patterns13 ; since







→







, we can state that all bit strings of arbitrary length L are period-1 ω-limit orbits (Isles of Eden) and δmax = 0 (see also [Chua et al., 2005]). Lemma 4.14. Rule 13 . Rule 13 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 2.

The firing patterns of rule 13 are


,


, and


(see Table 10): consequently,
changes if, and only if, both its neighbors are
, and
remains unchanged if, and only if, its left neighbor is
. Since


→
and


→
, the bit strings

· · ·
and

· · ·
constitute a period-2 ωlimit orbit (Isle of Eden),

· · ·
→

· · ·
→

Proof.

13



· · ·
. Moreover, all bit strings containing exclusively runs of k
, k < 3, and isolated
are period-1 ω-limit orbits. Runs of k
, k ≥ 2, are transformed into a sequence beginning and ending with
, and containing a run of k − 2
, because

· · ·

→

· · ·

; runs of k
, k > 1, are transformed into a sequence beginning with
and followed by k − 1
in one iteration, because


· · ·

→


· · ·

; eventually, after at most kmax iterations, where kmax is the longest run of
followed by one or two
, all bit strings not containing exclusive runs of k
, k < 3, and isolated
converge to a period-1 attractor containing exclusively runs of k
, k < 3, and isolated
; since kmax = L − 2, the maximum transient length is δmax = L − 2. In conclusion, all bit strings, except for

· · ·
and

· · ·
, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Example 4.14. Rule 13 . For rule 13 and L = 6, the maximum transient length is δmax = 4, and it occurs for any bit string containing a run of four or five
, like





—





→





→





→





→





→





— and





—





→





→





→





→





→





. For L = 5, the maximum transient length is δmax = 3, and it occurs for any bit string containing a run of three or four consecutive
, like




—




→




→




→




→




— and




—




→




→




→




→




. Lemma 4.15.

Rule 32 . Rule 32 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax =  L2 . The only firing pattern of rule 32 is


(see Table 10): consequently,
changes if, and only if, it is isolated, and
always changes. Since


→
and


→
, the bit strings



· · ·

and



· · ·

, which can occur only for even values of L, constitute a period2 ω-limit orbit (Isle of Eden),



· · ·

→



· · ·

→



· · ·

. Moreover, the bit string

· · ·
is a period-1 ω-limit orbit (attractor) because


→
and


→
; there are

Proof.

This is the only bit string with L = 8 having such a property, along with its “twin”







.

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no other period-1 ω-limit orbits besides it. Runs of k
, k arbitrary, are transformed into runs of k
in one iteration. Runs of k

, k arbitrary, are transformed into runs of k − 1

in one or two iterations, depending on the boundaries (the left boundary can be either

or
, the right boundary can be either
or

); here, the four possible cases are listed:



· · ·




→



· · ·




;



· · ·



→



· · ·



;


· · ·




→


· · ·




;


· · ·



→


· · ·



. Eventually, after at most kmax + 1 iterations, where kmax is the largest run of k

, all bit strings containing runs of k
, k arbitrary, and/or runs of k

, converge to

· · ·
; since kmax = (L − 1)/2, when L is odd, and kmax = (L−2)/2, when L is even, the maximum transient length is δmax = kmax + 1 =  L2 . In conclusion, all bit strings, except for



· · ·

and



· · ·

when L is even, are in the basin of attraction of

· · ·
.
Example 4.15. Rule 32 . For rule 32 and L = 6,

the maximum transient length is δmax = 3, and it occurs for the bit string





(and those obtained by rotating it):





→





→





→





→





. For L = 7, the maximum transient length is δmax = 4, and it occurs for the bit string






(and those obtained by rotating it):






→






→






→






→






→






. Lemma 4.16.

Rule 40 . Rule 40 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 1. Proof. The firing patterns of rule 40 are


and


(see Table 10): consequently,
changes if, and only if, it is isolated, and
remains unchanged if, and only if, its left and right neighbors are
and
, respectively. Since


→
and


→
, the bit strings



· · ·

and



· · ·

, which can occur only for even values of L, constitute a period2 ω-limit orbit (Isle of Eden),



· · ·

→



· · ·

→



· · ·

. All bit strings containing exclusively isolated
and runs of k
, k < 3, constitute a Bernoulli period-T ωlimit orbit (Isle of Eden) with σ = 1 and τ = 1

379

(left shift), where T = L/n, n ∈ N, because


→
,


→
,


→
, and


→
. Finally, the bit string

· · ·
is a period-1 attractor, because


→
and


→
; there are no other period-1 ω-limit orbits besides it. Runs of k
, k ≥ 3, are transformed into a run of at least k−1
in one iteration; runs of k
, k > 1, are transformed into runs of at least k +1
. Hence, all bit strings containing runs of
, k > 1, and/or runs of k
, k ≥ 3, converge to the period-1 attractor

· · ·
; the maximum transient length is δmax = L − 1. In conclusion, all bit strings not belonging to a nonrobust period-T ω-limit orbit, T > 1, belong to the basin of attraction of the period-1 attractor

· · ·
.
Example 4.16. Rule 40 . For rule 40 and L = 4,

the maximum transient length is δmax = 3, and it occurs for the bit string



(and those obtained by shifting it):



→



→



→



→



. For L = 5, we find the following Bernoulli period-5 Isle of Eden:




→




→




→




→




→




, which, as proved previously, contains exclusively isolated
and runs of k
, k < 3. In contrast, the bit string




does not satisfy this condition, and hence it converges to

· · ·
:




→




→




→




→




. Lemma 4.17.

Rule 44 . Rule 44 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 2. The firing patterns of rule 44 are


,


, and


(see Table 10): consequently,
changes if, and only if, it is isolated and
remains unchanged if, and only if, its left neighbor is
. The bit string

· · ·
is a period-1 attractor, because


→
and


→
; moreover, all bit strings containing exclusively runs of k
, k > 1, and isolated
are period-1 ω-limit orbits. All bit strings containing exclusively isolated
and runs of k
, k = 2, are Bernoulli period-3 ω-limit orbits (Isles of Eden) with σ = 1 and τ = 1 (left shift), because


→
,


→
, and


→
. Runs of k
, k > 2, are transformed into a sequence composed of a
followed by a run of k − 1
in one iteration, because


· · ·

→


· · ·

; all bit strings containing runs of k
, k > 1, converge to a period-1 attractor in Proof.

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at most L − 1 iterations, because

· · ·

→

· · ·

. Finally, bit strings containing exclusively isolated
and
converge to

· · ·
in one iteration, because


→
and


→
. Hence, all bit strings containing isolated
and runs of k
, k = 2, or runs of k
, k > 1, and nonisolated
belong to the basin of attraction of a period-1 attractor. In conclusion, all bit strings not belonging to a nonrobust period-T ω-limit orbit, T > 1, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Example 4.17. Rule 44 . For rule 44 and L =

5, the bit string




has maximum transient length, δmax = 3:




→




→




→




→




. For L = 6, the bit string





is a Bernoulli period-3 Isle of Eden,





→





→





→





.

Lemma 4.18.

Rule 77 . Rule 77 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L/2 − 1. The firing patterns of rule 77 are


,


,


and


(see Table 10): consequently,
changes if, and only if, both its neighbors are
, and
changes if, and only if, both its neighbors are
. Since


→
and


→
, the bit strings

· · ·
and

· · ·
constitute a period-2 ωlimit orbit (Isle of Eden),

· · ·
→

· · ·
→

· · ·
. Moreover, all bit strings containing exclusively runs of k

, k
< 3, and k


, k

< 3, are period-1 ω-limit orbits. Runs of k
, k > 2, are transformed — in one iteration — into a sequence beginning and ending with
and containing a run of k − 2
; runs of k
, k > 2, are transformed — in one iteration — into a sequence beginning and ending with
and containing a run of k − 2
. Hence, runs of k identical bits, k ≥ 3, are disrupted in (K + 1)/2 − 1 iterations, and bit strings containing them converge to a period-1 attractor containing exclusively runs of k

, k
< 3, and k


, k

< 3; since kmax = L−1, where kmax is the longest run of identical bits, the maximum transient length is δmax = L/2 − 1. In conclusion, all bit strings, except for

· · ·
and

· · ·
, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Proof.

Example 4.18. Rule 77 . For rule 77 and L = 6,

the maximum transient length is δmax = 2 and it occurs for any bit string containing a run of five
or
, like





→





→





→





and





→





→





→





. For L = 5, the maximum transient length is δmax = 1 and it occurs for any bit string containing a run of four
or
, like




→




→




and




→




→




. Lemma 4.19. Rule 104 . Rule 104 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 2.

The firing patterns of rule 104 are


,


and


(see Table 10): consequently,
changes if, and only if, it is isolated, and
remains unchanged if, and only if, only one of its neighbors is
. Since


→
and


→
, the bit strings



· · ·

and



· · ·

, which can occur only for even values of L, constitute a period2 ω-limit orbit (Isle of Eden),



· · ·

→



· · ·

→



· · ·

. Bit strings containing exclusively isolated
and runs of k
, k = 3, are Bernoulli period-2 ω-limit orbits (Isles of Eden) with σ = 2 and τ = 1, as it can be easily verified. Moreover, the bit string

· · ·
is a period-1 ω-limit orbit (attractor) because


→
and


→
. Finally, all bit strings containing exclusively runs of k
, k > 1, and runs of k
, k = 2, are period-1 ω-limit orbits. Runs of k
, k > 1, are stable because

→

and they tend to expand until they do not find a run of k
, k > 1 (list of the possible cases for the right boundary, the cases for the left boundary are symmetrical):
· · ·


→
· · ·

, and
· · ·


→
· · ·


. Runs of k
, k = 2, never change when their boundaries are runs of k
, k > 1, i.e.





→





. Runs of k
, k > 2, tend to disappear when their boundaries are runs of k
, k > 1, because





→





→





, and


· · ·




→


· · ·




→


· · ·




. Hence, all bit strings containing runs of k
, k > 1, are in the basin of attraction of a period-1 attractor. Therefore, there remains to analyze only bit strings containing isolated
. All bit strings containing runs of k
, k > 3, and isolated
are in Proof.

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Chapter 2: Period-1 Rules

the basin of attraction of a period-1 attractor, because


· · ·


→


· · ·


, and hence, in one iteration they exist for runs of k
, k > 1. Finally, all bit strings containing at least a run of k
, k < 3, and isolated
, except for



· · ·

and



· · ·

, converge to the attractor

· · ·
, because





→





→





; the maximum transient length is δmax = L − 2. In conclusion, all bit strings not belonging to a nonrobust period-T ω-limit orbit, T > 1, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Example 4.19. Rule 104 . For rule 104 and L = 6,





is a period-1 ω-limit orbit (Isle of Eden) because it contains exclusively runs of k
, k > 1, and runs of k
, k = 2. For L = 8, the bit string







and







(and the “twin” pair obtained by shifting them) form Bernoulli period-2 ω-limit orbits (Isles of Eden) with σ = 2 and τ = 1,







→







. Lemma 4.20.

Rule 160 . Rule 160 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 1.

The firing patterns of rule 160 are


and


(see Table 10): consequently,
changes if, and only if, both its neighbors are
, and
remains unchanged if, and only if, both its neighbors are
. Since


→
and


→
, the bit strings



· · ·

and



· · ·

, which can occur only for even values of L, constitute a period-2 ωlimit orbit (attractor). The bit string

· · ·
is a period-1 ω-limit orbit (attractor) because


→
, and the bit string

· · ·
is a period-1 ω-limit orbit (Isle of Eden) because


→
; there are no other period-1 ω-limit orbits besides these two. All bit strings containing exclusively isolated
and runs of k
, k odd, are in the basin of attraction of the period-2 attractor



· · ·

→



· · ·

→



· · ·

; note that this behavior is clearly nonrobust, due to the very particular conditions for which it happens, and it can occur only for L even. Runs of k
, k even, are transformed into a sequence starting and ending with
and containing a run of k − 2
; hence, after k/2 iterations they form runs of at least two
. But runs of k
, k > 1, tend to form runs of k + 2
, because

· · ·

→

· · ·

; therefore, all bit strings containing a run of k
, k > 1, converge to

Proof.

381

the attractor

· · ·
at most in L − k iterations. Hence, all bit strings not having exclusively runs of k
, k odd, and isolated
, except for

· · ·
, belong to the basin of attraction of the period-1 attractor

· · ·
; the maximum transient length is δmax = L − 1. In conclusion, all bit strings not belonging to a nonrobust period-T ω-limit orbit, T > 1, belong to the basin of attraction of the period-1 attractor

· · ·
.
Example 4.20. Rule 160 . For rule 160 and L = 5, the maximum transient length δmax = 4 occurs for the string




(and those obtained by shifting it):




→




→




→




→




→




. For L = 6, the bit string





belongs to the basin of attraction of the period-2 attractor, because it contains exclusively isolated
and runs of k
, k odd:





→





→





→





→





. Lemma 4.21.

Rule 164 . Rule 164 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 1.

The firing patterns of rule 164 are


,


, and


(see Table 10): consequently,
changes if, and only if, it is isolated, and
changes if, and only if, only one of its neighbors is
. The bit strings





and





form a Bernoulli period-2 ω-limit orbit with σ = 3 and τ = 1,





→





→





; in general, all L multiples of 6 have such orbit. The bit strings

· · ·
and

· · ·
are period-1 ωlimit orbits (attractors) because


→
and


→
. Moreover, all bit strings containing exclusively runs of k
, k > 1, and isolated
are period-1 ω-limit orbits. Runs of k
, k > 1, are stable because

→

and they tend to expand until they do not find an isolated
(list of the possible cases for the right boundary and the cases for the left boundary are symmetrical):
· · ·


· · ·

→
· · ·


· · ·

and
· · ·



→
· · ·


→
· · ·


, and
· · ·



→
· · ·



. Hence, all bit strings containing runs of k
, k > 1, are in the basin of attraction of a period-1 ω-limit orbit. Runs of k
, k even, tend to converge to a run of at least two
after k/2 steps, because


· · ·


→


· · ·


; therefore, also bit strings containing runs of k
, k even, are in the Proof.

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basin of attraction of a period-1 ω-limit orbit. Finally, bit strings containing exclusively isolated
and runs of k
, k odd, are in the basin of attraction of period-T attractors, T > 1; note that this behavior is clearly nonrobust, due to the very particular conditions for which it happens; the maximum transient length is δmax = L − 1. In conclusion, all bit strings not belonging to a nonrobust period-T ω-limit orbit, T > 1, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Example 4.21. Rule 164 . For rule 164 and L = 6,

the bit string





is a period-1 ω-limit orbit (attractor) because it contains exclusively runs of k
, k > 1, and isolated
,





→





; the bit string





does not satisfy this condition because it also contains isolated
, and it belongs to the basin of attraction of





:





→





→





→





. The bit strings





and





form a Bernoulli period-2 ω-limit orbit (attractor),





→





→





. One of the bit strings belonging to its basin of attraction is





,





→





, because it contains exclusively isolated
and runs of k
, k odd. Lemma 4.22. Rule 168 . Rule 168 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 2.

The firing patterns of rule 168 are


,


, and


(see Table 10): consequently,
changes if, and only if, it is isolated, and
remains unchanged if, and only if, its right neighbor is
. Since


→
and


→
, the bit strings



· · ·

and



· · ·

, which can occur only for even values of L, constitute a period-2 ω-limit orbit (Isle of Eden). All bit strings containing exclusively isolated
and runs of k
, k arbitrary, form a Bernoulli period-T ω-limit orbit (Isle of Eden) with σ = 1 and τ = 1 (left shift), where T = L/n, n ∈ N, because


→
,


→
,


→
,


→
, and


→
. Moreover, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits: the former is an attractor, the latter an Isle of Eden. Besides these two, there are no other period-1 ω-limit orbits. Runs of k
, k > 1, tend to form runs of at least k + 1
, because

· · ·
→

· · ·
. Hence, Proof.

all bit strings containing a run of k
, k > 1, converge to the period-1 attractor

· · ·
in at most L − k iterations; the maximum transient length is δmax = (L − kmin ) = L − 2, where kmin is the length of the shortest run of
. In conclusion, all bit strings — except for



· · ·

and



· · ·

when L is even, and

· · ·
— are in the basin of attraction of

· · ·
.
Example 4.22. Rule 168 . For rule 168 and L = 5, the bit string




belongs to a Bernoulli period-5 Isle of Eden, because it does not contain runs of k
, k > 1:




→




→




→




→




→




. Also the bit string




satisfies this condition, and hence it is part of a Bernoulli period-5 Isle of Eden too:




→




→




→




→




→




. The maximum transient length δmax = L − 2 = 3 occurs for bit strings containing runs of two
, like




:




→




→




→




→




. Compare this result with Example 4.16. Lemma 4.23. Rule 172 . Rule 172 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L − 2.

The firing patterns of rule 172 are


,


,


, and


(see Table 10):
changes if, and only if, it is isolated, and
changes if, and only if, its left and right neighbors are
and
, respectively. All bit strings containing exclusively isolated
and runs of k
, k > 2, form Bernoulli periodT ω-limit orbits with σ = 1 and τ = 1 (left shift), where T = L/n, n ∈ N, because


→
,


→
,


→
, and


→
. The bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits, because


→
and


→
: the former is an Isle of Eden, the latter is an attractor for L even and an Isle of Eden for L odd. All bit strings containing exclusively runs of k
, k > 1, and isolated
are period-1 ω-limit orbits. All bit strings containing exclusively isolated
and
, which can occur only for L even, converge to the period-1 attractor

· · ·
in one iteration, because


→
and


→
. Proof.

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Chapter 2: Period-1 Rules

Bit strings containing exclusively isolated
and runs of k
, k arbitrary, belong to the basin of attraction of the Bernoulli period-T attractor; note that this behavior is clearly nonrobust, due to the very particular conditions for which it happens. All bit strings containing at least a run of k
, k > 1, are in the basin of attraction of a period-1 attractor having exclusively runs of k
, k > 1, and isolated
: this is because runs of k
, k > 1, are preserved (
· · ·
→
· · ·
) and runs of k
, k > 1, tend to be disrupted at most in two iterations, because


· · ·
→


· · ·
; the maximum transient length is δmax = L − 2. In conclusion, all bit strings not belonging to a nonrobust period-T ω-limit orbit, T > 1, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Example 4.23. Rule 172 . For rule 172 and L =

5, we find the following Bernoulli period-5 ωlimit orbit (attractor):




→




→




→




→




→




. The bit string




is not part of an ω-limit orbit, because it does not contain exclusively runs of k
, k > 2, and isolated
, but it is in the basin of attraction of the previous Bernoulli period5 attractor:




→




→




→




→




→




→




. The maximum transient length δmax = L − 2 = 5 occurs for the bit string




(and those obtained by shifting it):




→




→




→




→




→




. Compare this result with Example 4.24. Lemma 4.24.

Rule 232 . Rule 232 has robust period-1 ω-limit orbits; the maximum transient length for this rule is δmax = L/2 − 1.

Proof. The firing patterns of rule 232 are


,


,


, and


(see Table 10): consequently, both
and
change if, and only if, they are isolated. Since


→
and


→
, the bit strings



· · ·

and



· · ·

, which can occur only for even values of L, constitute a period-2 ωlimit orbit (Isle of Eden). Moreover, the bit strings

· · ·
and

· · ·
are period-1 ω-limit orbits (attractors), because


→
and


→
. Finally, all bit strings containing no isolated
and
are period-1 ω-limit orbits.

383

Bit strings containing isolated
and
must also have at least two adjacent
or
: if there are adjacent
but no adjacent
, the bit string converges to the period-1 attractor

· · ·
, because


→


and


→


; if there are adjacent
but no adjacent
, the bit string converges to the period-1 attractor

· · ·
, because


→


and


→


; if there are both adjacent
and
, the bit string converges to a period-1 attractor containing no isolated
and
. In all cases, the bit string converges at most in kmax iterations, where kmax is the longest run of alternate bits present in the bit string; since kmax = (L − 2)/2 for L even, and kmax = (L − 1)/2 for L odd, the maximum transient length is δmax = L/2 − 1. In conclusion, all bit strings, except for



· · ·

and



· · ·

, either are period-1 ω-limit orbits or belong to the basin of attraction of a period-1 attractor.
Example 4.24. Rule 232 . For rule 232 and L = 6,

the maximum transient length is δmax = 2, and it occurs for all bit strings containing a run of two alternate bits, like





:





→





→





. For L = 7, the maximum transient length is δmax = 3, and it occurs for all bit strings containing a run of two alternate bits, like






:






→






→






→






→






. The results of these 24 lemmas are summarized in Table 11. As already mentioned, period-1 rules can be classified into two groups: those harboring exclusively period-1 ω-limit orbits and those harboring also nonrobust period-T ω-limit orbits, with T = 1. Note that rule 0 and rule 8 are the only ones having a global attractor, which means that all bit strings are in the basin of attraction of

· · ·
for any L. Rules 128 and 136 have a similar property, because all bit strings are in the basin of attraction of

· · ·
, except for

· · ·
which is an Isle of Eden. Therefore, rules 0 , 8 , 128 , and 136 have only one period-1 attractor, whose basin of attraction contains all (for rules 0 and 8 ), and all but one (for rules 128 and 136 ) bit strings, respectively. As for rules harboring also nonrobust period-T ω-limit orbits, with T = 1, we find that only four of them — namely, 32 , 40 , 160 , and 168 — have only one period-1 attractor (

· · ·
for rules 32 and 40 , and

· · ·
for rules 160 , and 168 ). In addition to it, these rules have also

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A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science

nonrobust period-T Isles of Eden, and rule 160 has also a nonrobust period-2 attractor, for L even. The eight rules we just listed, which appear in the first two rows of Table 11, are unique in having only one robust period-1 attractor. This means that when L → ∞, the probability for a random bit string to be in the basin of attraction of

· · ·
— for rules 0 , 8 , 32 , 40 , 128 , and 136 — or

· · ·
— for rules 160 , and 168 — tends14 to 1. These rules coincide with Class 1 in Wolfram’s classification [Wolfram, 2002]; moreover, as already noticed by [Li & Packard, 1990], they form a cube in which the distance between adjacent vertices is one bit. Rules 36 and 78 , which are in the third row of Table 11, are the only ones having multiple robust period-1 attractors, but no Isles of Eden. All other rules, listed in the fourth row of Table 11, have multiple robust period-1 attractors and Table 11.

14

at least one period-T Isle of Eden. The only exception is rule 204 , which has exclusively period-1 Isles of Eden. It is important to emphasize that almost one third of the period-1 rules have nonrobust period-T orbits with T > 1, but this feature often passed unperceived in the past. For instance, classical works on the classification of CA local rules, like [Li & Packard, 1990] and [Wolfram, 2002], included rules 32 , 40 , 160 and 168 in the same class as 0 , 8 , 128 and 136 whereas, from our point of view, the first four rules have a richer dynamical behavior, because they harbor nonrobust periodic orbits with T > 1. We can focus on rules 40 and 168 , which have Bernoulli period-T Isles of Eden with σ = 1 and τ = 1, where T divides the bit string length L, as shown in the respective basin tree diagrams in Table 6. In other words, these rules behave like rule

Period-1 rules classified according to the properties of their period-1 ω-limit orbits.

To be precise, for rules 0 and 8 this probability is 1 for any L.

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Chapter 2: Period-1 Rules

170 for certain particular subsets of strings, i.e. strings containing exclusively prescribed patterns. In the following, we will see how easily this phenomenon can be explained. In Table 12, the truth tables of rules 170 , 40 and 168 are compared. It is straightforward to notice that rules 40 and 170 have in common six rows, namely


,


,


,


,


and


. Therefore, if we restrict our analysis to bit strings containing any pattern except for


and


, we cannot observe any difference between 40 and 170 . It is also easy to find that such subsets are formed by all bit strings containing at most two consecutive
bits and one consecutive
bits. Rule 168 has a very similar behavior, because all rows of its truth table coincide with those of rule 170 , except for


. Similarly as for 40 , we can restrict our analysis to all bit strings not containing the pattern


, and in this case rule 168 will be indistinguishable from 170 . This

Table 12.

Input

Truth tables of rules 170, 40 and 168.

170

40

168

385

subset is formed by all bit strings having at most one consecutive
(no condition for
bits). As proved in [Chua et al., 2005], rule 170 is chaotic in the Devaney sense, and hence, for the aforementioned subsets of bit strings, rules 40 and 168 also have such a property. This is exactly the same result arrived at by other authors [Ohi, 2007] and [Ohi, 2008] who used the classical theory of cellular automata. Finally, we want to draw attention on the fact that this feature is not peculiar to rules 40 and 168 , but it can be verified for other rules on the last column of Table 11 (e.g. rule 172 ).

5. Concluding Remarks This article has been entirely devoted to period1 rules, presenting the basin tree diagrams for all of them and formalizing important concepts, like ω-limit orbits and their robustness. Moreover, we introduced a methodology to prove rigorously that all CA local rules belonging to Group 1 have robust period-1 ω-limit orbits for any finite, and infinite, bit-string length. This novel approach confirms the results obtained empirically in the previous chapters of our quest. By using our technique, we have been able to give more accurate results than those presented in similar works, like [Li & Packard, 1990] and [Wolfram, 2002]. For example, both these works “put in the same box” rules 0 , 8 , 32 , 40 , 128 , 136 , 160 and 168 because they have only one robust period-1 attractor. However, we made a finer classification, summarized in Table 11, proving that 0 and 8 are the only rules having a global attractor, whereas 128 and 136 have a global attractor, except for one bit string that forms a period-1 Isle of Eden. The most important aspect, though, is that we showed that for rules 32 , 40 , 160 and 168 there exist nonrobust periodic orbits, including period-T Bernoulli attractors and Isles of Eden with σ = 1 and τ = 1. This is extremely important, because it means that for certain particular subsets of strings, i.e. strings containing only prescribed patterns, these four rules behave like rule 170 , which is proved to be chaotic in the Devaney sense [Chua et al., 2005]. Rule 204 is unique in having exclusively period-1 Isles of Eden. In our forthcoming paper, we will also observe that 204 has a counterpart in Group 2, since rule 51 has exclusively period-2 Isles of Eden.

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Finally, all other rules have multiple robust period-1 ω-limit orbits. Nevertheless, for this last class of CA rules, we were able to make a further distinction between those harboring nonrobust period-T , T > 1, ω-limit orbits and those having exclusively period-1 ω-limit orbits. In this last category, we find rules 36 and 78 , whose peculiarity is not having any Isles of Eden, as proved in [Chua et al., 2008]. Note that this article is the first of a series of three: the other two will be dedicated to period-T

rules, T > 2 and Bernoulli rules, respectively. These three works have a common purpose, which is to formalize the results given in the previous parts of our saga on Cellular Automata. At the end of the series, we will provide a classification showing the robust and nonrobust behavior of all rules belonging to the first four groups, which can summarize at a glance the whole universe of period-T and Bernoulli CA local rules.

October 23, 2010

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References

♦

REFERENCES ♦

Ë kind of science. Part V: Fractals everywhere,” Int. J. Bifurcation and Chaos 15, 3701–3849. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2006] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part VI: From time-reversible attractors to the arrow of time,” Int. J. Bifurcation and Chaos 16, 1097–1373. Chua, L. O., Guan, J., Sbitnev, V. I. & Jinwook, S. [2007a] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part VII: Isles of Eden,” Int. J. Bifurcation and Chaos 17, 2839– 3012. Chua, L. O., Karacs, K., Sbitnev, V. I., Guan, J. & Jinwook, S. [2007b] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part VIII: More Isles of Eden,” Int. J. Bifurcation and Chaos 17, 3741–3894. Chua, L. O., Pazienza, G. E., Orzo, L., Sbitnev, V. & Shin, J. [2008] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part IX. Quasi-erogodicity,” Int. J. Bifurcation and Chaos 18, 2487–2642. Courbage, M., Mercier, D. & Yasmineh, S. [1999] “Travelling waves and chaotic properties in cellular automata,” Chaos 9, 893–901. Courbage, M. & Yasmineh, S. [2001] “Wavelengths distribution of chaotic travelling waves in some cellular automata,” Physica D 150, 63–83.

Birkhoff, G. D. [1927], Dynamical Systems (Amer. Math. Soc., Providence, RI). Cattaneo, G. & Quaranta Vogliotti, C. [1997] “The ‘magic’ rule spaces of neural-like elementary cellular automata,” Theoret. Comput. Sci. 178, 77–102. Chua, L. O. [1996] CNN: A Paradigm for Complexity (World Scientific, Singapore). Chua, L. O., Yoon, S. & Dogaru, R. [2002] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part I: Threshold of complexity,” Int. J. Bifurcation and Chaos 13, 2377–2491. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2003] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part II: Universal neuron,” Int. J. Bifurcation and Chaos 13, 2377–2491. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2004] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part III: Predicting the unpredictable,” Int. J. Bifurcation and Chaos 14, 3689–3820. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005a] “A nonlinear dynamics perspective of Wolfram’s new kind of science. Part IV: From Bernoulli shift to 1/f spectrum,” Int. J. Bifurcation and Chaos 15, 1045–1183. Chua, L. O., Sbitnev, V. I. & Yoon, S. [2005b] “A nonlinear dynamics perspective of Wolfram’s new 387

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Garay, B. M. & Chua, L. O. [2008] “Isles of Eden and the ZUK theorem in Rd ,” Int. J. Bifurcation and Chaos 18, 2951–2963. Hedlund, G. A. [1969] “Endomorphisms and automorphisms of the shift dynamical system,” Th. Comput. Syst. 3, 320–375. Ireland, K. & Rosen, M. [1990] A Classical Introduction to Modern Number Theory (SpringerVerlag). Li, W. & Packard, N. [1990] “The structure of the elementary cellular automata rule space,” Compl. Syst. 4, 281–297. Martin, O., Odlyzko, A. M. & Wolfram, S. [1984] “Algebraic properties of cellular automata,” Commun. Math. Phys. 93, 219–258. Moore, E. F. [1962] “Machine models of selfreproduction,” in Mathematical Problems in the Biological Science, Proc. Symp. Applied Mathematics 14 (Providence RI), pp. 17–33. Myhill, J. [1963] “The converse of Moore’s Gardenof-Eden theorem,” Proc. Amer. Math. Soc. 14, 685–686.

Ohi, F. [2007] “Chaotic properties of the elementary cellular automaton rule 40 in Wolfram’s class I,” Compl. Syst. 17, 295–308. Ohi, F. [2008] “Chaotic properties of elementary CA Rule 168,” in Proc. Automata 2008. Pivato, M. & Yassami, R. [2008] “The spatial structure of odometers in certain cellular automata,” Journ’ees Automates Cellulaires 1, 119–126. Shereshevsky, M. A. [1992] “Ergodic properties of certain surjective cellular automata,” Mh. Math. 114, 305–316. Shirvani, M. & Rogers, T. D. [1991] “On ergodic one-dimensional cellular automata,” Commun. Math. Phys. 136, 599–605. Wolfram, S. [2002] A New Kind of Science (Wolfram Media, Inc., Champaign IL, USA). Wuensche, A. & Lesser, M. [1992] The Global Dynamics of Cellular Automata: An Atlas of Basin of Attraction Fields of One-Dimensional Cellular Automata (Addison-Wesley, Reading, MA).

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index

♦

INDEX ♦

Ë additive rules, 1, 3, 83, 111–114, 116, 152 all neurons quenched, 13 alternating symmetry duality, 90, 115, 116 analytical theory, 157 anti-additive rules, 111–114 anti-symmetry, 36 attractors, 1, 115, 117, 118, 142–145, 151, 285, 304, 367, 372, 378, 382

30 , 40 45 , 98, 99 60 , 117, 118, 142, 146, 150 90 , 117, 143, 147, 150 105 , 117, 144, 148, 150 110 , 108, 109 137 , 33 150 , 117, 145, 149, 150 154 , 98, 99 Group 1 Rules , 29 Group 2 Rules , 29 Group 3 Rules , 29 Group 4 Rules , 29 8-bit code, 96 8-dimensional parameter space, 11 25 globally-independent period-1 rules, 162 25 robust period-1 rules, 164 67 globally-equivalent robust period-1 local rules, 161 67 robust period-1 rules, 160 88 equivalence classes, 32 88 globally independent rules, 30 104 linearly-separable local rules, 33 256 local rules, 31, 32, 159 ω-Limit Orbit Generation, 369 ω-limit orbits, 304, 367, 370, 371, 372, 380, 385

basin of attraction, 29, 106, 285, 302, 371, 375, 380, 381, 384 basin tree, 285, 304 basin tree diagrams, 1, 157, 163, 304 Basin Tree Diagrams of Rule 4 , 165 Basin Tree Diagrams of Rule 8 , 170 Basin Tree Diagrams of Rule 12 , 175 Basin Tree Diagrams of Rule 13 , 180 Basin Tree Diagrams of Rule 32 , 185 Basin Tree Diagrams of Rule 36 , 190 Basin Tree Diagrams of Rule 40 , 195 Basin Tree Diagrams of Rule 44 , 200 Basin Tree Diagrams of Rule 72 , 205 Basin Tree Diagrams of Rule 76 , 210 Basin Tree Diagrams of Rule 77 , 215 Basin Tree Diagrams of Rule 78 , 220 Basin Tree Diagrams of Rule 104 , 225 Basin Tree Diagrams of Rule 128 , 230 Basin Tree Diagrams of Rule 132 , 235 Basin Tree Diagrams of Rule 136 , 240

Abelian group, 32 additive cellular automata, 111 additive rule N , 111 389

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Index

Basin Tree Diagrams of Rule 140 , 245 Basin Tree Diagrams of Rule 160 , 250 Basin Tree Diagrams of Rule 164 , 255 Basin Tree Diagrams of Rule 168 , 260 Basin Tree Diagrams of Rule 172 , 265 Basin Tree Diagrams of Rule 200 , 270 Basin Tree Diagrams of Rule 204 , 275 Basin Tree Diagrams of Rule 232 , 280 basin-tree diagrams, 106, 108–110 basin-tree generation formula, 1, 106, 109 basins of attraction, 106 basis characteristic functions, 45 Bernoulli στ -orbits, 107, 152 Bernoulli στ -shift, 109 Bernoulli στ -shift attractor, 29, 107, 108 Bernoulli στ -shift orbit, 106, 108, 110 Bernoulli στ -shift rules, 32, 152, 159, 163 Bernoulli basin-tree diagrams, 1 Bernoulli Isle of Eden, 105, 106 Bernoulli map, 39 Bernoulli orbits, 106 Bernoulli parameters, 116, 152, 367 Bernoulli period-T attractor, 383 Bernoulli period-5 Isle of Eden, 382 Bernoulli period-14 attractor, 108 Bernoulli shift, 303 Bernoulli velocity, 1 bi-infinite sequence space, 29 bi-permutive, 37, 155 bi-permutive rules, 42, 155 Boolean cube 137 , 33 Boolean cube basis functions, 44 Boolean cube of rule N , 285 Boolean cube representation, 3 Boolean cubes, 5, 9, 29, 32, 34, 36, 38, 111, 112, 154, 155, 163 Boolean operations, 83 cellular automata, 1, 79, 97, 157, 386 cellular neural networks (CNN), 3 centrally-symmetric, 99 centrally-symmetric local rules, 36 centrally-symmetric strictly-dissipative local rules, 100 chaotic in the Devaney sense, 385 characteristic function, 285 complemented right shift, 13 complex and hyper Bernoulli rules, 12 complex and hyper Bernoulli-shift, 1 complex and hyper Bernoulli-shift rules, 59, 60, 62, 67, 106, 152 complex Bernoulli shifts, 1 complex Bernoulli-shift rules, 2, 32, 159, 163

complex or hyper Bernoulli-shift rules, 116 complexity index, 30–32, 113, 159 complexity index κ, 29 concatenated ω-limit orbit generation algorithm, 367 concatenation, 370 conservative, 99 conservative local rules, 98 conservative rules, 1 conservative tout court, 99 decimal code, 286 degenerate Bernoulli rule, 367 dense Isles of Eden orbits, 29 difference equation, 11 dissipative rules, 1 equivalence transformations, 32 ergodicity, 59 Euler totient function, 113, 115 explicit formula, 302 explicit formula for generating the truth table, 285 firing patterns, 375, 377, 379, 381–383 firing/quenching patterns, 155 formula for local rule, 13 fractal patterns, 80–83, 85–89, 91–93, 95, 97 fractality, 80, 91 fractals, 1, 79, 83 fractional code, 286 Gardens of Eden, 60, 76, 78, 79, 101, 103, 104, 285 genotype, 303 genotypic robustness, 303 geometrical constructions, 35 global attractor, 105, 106, 374 global complementation, 32–33, 35, 161, 164 global equivalence, 113 global equivalence theorem, 163 globally independent rules, 29, 97 globally-equivalence classes, 29 group 1, 105, 106, 163, 304, 371, 372 group 1 rules, 367 group 4, 152 group 5, 152 group 6, 152 group 4 rules, 60 group 5 rules, 29 group 6 rules, 29 group four, 108 groups 5 and 6, 373 Hedlund, 36, 154 hyper Bernoulli shifts, 1, 32, 113, 159, 163 index of complexity, 3, 9, 12, 163 inequality (31), 152

February 12, 2011

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index

Index

Isles of Eden, 1, 29, 38, 39, 57, 97–99, 101, 103–106, 113, 116–118, 152, 285, 302–304, 367, 368, 372, 378, 381, 382, 386 Isles of Eden digraph, 97 isolated Bernoulli Isles of Eden, 109 isolated Isles of Eden, 60 isomorphic attractors, 367 isomorphic basin trees, 302 isomorphic copies, 304 Klein’s Vierergruppe, 32 Lamerey (cobweb) diagrams, 60–62, 76 left — right complementation, 161, 164 left — right transformation, 161, 164 left-permutive, 37, 98 left-permutive rules, 38, 154 left-right complementation, 32, 33, 35 left-right transformation, 32, 33, 35 left-shift Bernoulli-map, 59 local complementation, 32 local complements, 34 local rule, 158 local rule number, 95 magic, 32 magic rule spaces, 29 maximum period, 116, 152 maximum period T , 142–145 maximum period-T , 117 maximum period of attractors, 113 maximum period of orbits, 115, 116 maximum transient length, 374–381, 383 multiplicative order, 150, 153 multiplicative order function, 113, 115 multiplicative suborder, 150, 153 multiplicative suborder function, 113, 115 neural-like behaviors, 29 no Isle of Eden rules, 102 non centrally-symmetric rules, 99, 101 non-bilateral, 29 nonlinear difference equation, 11 nonlinear dynamics, 1 nonrobust period-T Isles of Eden, 384 nonrobust period-T ω-limit orbit, 373, 379, 383 non-spatially periodic finite strings, 152 non-strongly quasi-ergodic rules, 78 nontrivial additive rules, 113, 116, 152 notations, 4, 158 number of fractal patterns, 96

omega-limit orbits, 304 orbit, 29 perfect complementary rules, 36 perfect complementations, 36 perfect symmetry, 36 perfectly-complementary local rules, 99 period-1 attractor, 163, 375, 380 period-1 orbit, 29 period-1 rules, 32, 157, 159, 163, 304 period-1 spatially-alternating orbits, 52, 56 period-1 spatially-homogeneous orbits, 49–51, 55, 56 period-1 ω-limit orbits, 372, 374–378, 382, 384, 386 period-2 attractor, 29, 381 period-2 rules, 32, 159, 163 period-2 spatially alternating orbit, 54 period-2 spatially-homogeneous orbits, 53, 55 period-3 attractor, 29 period-3 rule, 32 period-3240 Isle of Eden, 39, 40, 59 period-n Isle of Eden, 98 period-T attractors, 382 permuting map, 154 permutive, 98, 303 permutive rules, 1, 36, 37, 43, 154 phenotypic robustness, 303 Portraits of ω-Limit Orbits of Rule 4 , 305 Portraits of ω-Limit Orbits of Rule 8 , 308 Portraits of ω-Limit Orbits of Rule 12 , 309 Portraits of ω-Limit Orbits of Rule 13 , 312 Portraits of ω-Limit Orbits of Rule 32 , 314 Portraits of ω-Limit Orbits of Rule 40 , 317 Portraits of ω-Limit Orbits of Rule 44 , 319 Portraits of ω-Limit Orbits of Rule 72 , 321 Portraits of ω-Limit Orbits of Rule 76 , 323 Portraits of ω-Limit Orbits of Rule 77 , 328 Portraits of ω-Limit Orbits of Rule 104 , 334 Portraits of ω-Limit Orbits of Rule 128 , 336 Portraits of ω-Limit Orbits of Rule 132 , 337 Portraits of ω-Limit Orbits of Rule 136 , 341 Portraits of ω-Limit Orbits of Rule 140 , 342 Portraits of ω-Limit Orbits of Rule 164 , 347 Portraits of ω-Limit Orbits of Rule 168 , 350 Portraits of ω-Limit Orbits of Rule 200 , 357 Portraits of ω-Limit Orbits of Rule 204 , 362 preimage, 98 qualitative behaviors, 29 quasi-ergodic, 62, 76 quasi-ergodic phenomenon, 152 quasi-ergodic rule, 60 quasi-ergodic space-time patterns, 12

391

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Index

quasi-ergodic vignettes, 67 quasi-ergodicity, 1, 3, 59, 61, 76, 152 right-copycat rule, 106 right-permutive, 37, 98, 154 right-permutive rules, 39, 155 robust Bernoulli attractors, 109 robust period-1 global attractor, 105 robust period-1 ω-limit orbits, 372 robustness, 11 robustness coefficient, 303, 367 rule 30 , 39, 154 rule 60 , 84, 116, 146 rule 90 , 86, 116, 147, 151 rule 105 , 88, 116, 148 rule 110 , 61, 82 rule 150 , 90, 116, , 149 rule 170 , 106, 385 rule number, 93 rule number N , 285 scale-free decomposition, 150, 153 scale-free order, 116, 150, 151, 153 scale-free order ξL , 150 scale-free phenomena, 1, 3 scale-free property, 116, 146–149, 152 seamless concatenation conditions, 370 seamless interface, 370 seamless left-boundary, 370 seamless right-boundary, 370 semi-dissipative local rules, 98 shift maps, 1 single point attractor, 29 space-time patterns, 12, 29, 67 spatially periodic bi-infinite strings, 152 stratification of characteristic functions, 95 strictly-dissipative, 99 strictly-dissipative local rules, 98 strictly-dissipative rules, 99

strongly quasi-ergodic, 76 strongly quasi-ergodic rules, 79 strong quasi-ergodicity, 76 superposition, 45 superposition of local rules, 37 surjectivity, 79 symbols, 4, 158 symmetries, 32, 302 T , 35 T ∗ , 35 T † , 35 time-1 characteristic functions, 67, 77, 79–82, 84–95 time-1 maps, 29, 285 time-1 return maps, 1, 59, 67 time-τ return map, 12, 59 transient components, 285 transient points, 77 transient-regime, 367 trivial additive rules, 113 truth table, 13 truth table of rule N , 285 type A Boolean cube, 49 type B Boolean cube, 50 type C Boolean cube, 51 type D Boolean cube, 52 type E Boolean cube, 53 type F Boolean cube, 54 type G Boolean cube, 55 type H Boolean cube, 55 type I Boolean cube, 56 type J Boolean cube, 56 universal difference equation, 12 universal formula, 4, 10, 12, 158 Vierergruppe, 115 weak quasi-ergodicity, 60, 77

A NONLINEAR DYNAMICS PERSPECTIVE OF WOLFRAM'S NEW KIND OF SCIENCE Volume IV

Volume IV continues the author’s odyssey on l-D cellular automata as chronicled in Volumes I, II and III, by uncovering a novel quasi-ergodicity phenomenon involving orbits meandering among omega-limit orbits of complex (group 5) and hyper (group 6) Bernoulli rules. This discovery is embellished with analytical formulas characterizing the fractal properties of characteristic functions, as well as explicit formulas for generating colorful and pedagogically revealing isomorphic basin tree diagrams. Many new results were derived and proved by uncovering subtle symmetries endowed by various subsets of the 256 Boolean cubes. For the first time, rigorous analyses were used to identify 67, out off 256, local rules whose asymptotic behaviors consist of robust period-l orbits. The highlight of this continuing odyssey is the discovery of an isolated period-3240 Isle of Eden hidden among the dense omega-limit orbits of Wolfram’s remarkable “random number generating” rule 30. This is the largest gem known to-date and readers are challenged to uncover even larger ones. The cover cartoon is drawn by Valery Sbitnev.

about the author Leon Chua is a foreign member of the Academia Europea and a recipient of eight USA patents and 12 Docteur Honoris Causa. He has received numerous international awards, including the first IEEE Kirchhoff Award, the Neural Networks Pioneer Award, and the “Top 15 Cited Authors” Award based on the ISI Citation Index in Engineering from 1991 to 2001. When not immersed in science, he relaxes by searching for Wagner’s leitmotifs, musing over Kandinsky’s chaos, and contemplating Wittgenstein’s inner thoughts.

ISBN-13 978-981-4317-30-6 ISBN-10 981-4317-30-6

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