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

3/3/11 8:59 AM

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

7835 tp.indd 2

•

LONDON

•

SINGAPORE

•

BEIJING

•

SHANGHAI

•

HONG KONG

•

TA I P E I

•

CHENNAI

3/3/11 8:59 AM

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.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

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

Typeset by Stallion Press Email: [email protected]

Printed in Singapore.

February 26, 2011

10:47

contents

To

CYNTHIA

who thrives on games, from ergodic to quasi-ergodic

This page is intentionally left blank

February 26, 2011

10:47

contents

♦

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

February 26, 2011

viii

10:47

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

February 26, 2011

10:47

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

This page is intentionally left blank

February 26, 2011

10:47

contents

♦

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

February 14, 2011

15:12

ch01

♦

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

February 14, 2011

2

15:12

ch01

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

15:12

ch01

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 :<+1 =4[I-1(-~+1(3:<-1-:< +2:<+1)1)1]

:<+1 =ti[l-1 (~ -I (-4:<-1 +:< - 2xf+1 +1) DI]

•

Xf+l = ti[-3xf-l +xf - X,+1 + ~ ]

ch01

17

III x,+1 =ti[~-1(2x,_1-2xf+xf+l)l] III Xf+1 =ti[-l+I(-x,-l- xf+Xf+1 +~)I] &I x,+1 =ti[~-1(2x,_1-4xf-xf+l+1)1]

Formula for Local Rule

15:12

N

(Continued )

February 14, 2011

Table 2.

N

N

Formula for Local Rule X,-1 -

xi+

1

4['

1-

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

+ 21 ]

= 4[~ -I (-2xi_1 -xi -4xi+1 +3) I]

I

•

x:+! = .i[%-I(-4x:_ -x, -2x:+! +4)1]

IIIIIIftII 11IM

Xt+1 =

1%1I

1

I

4 [1-1 (x'

I-I

- x' I

+ 2x' -.!.) ] 2 1

x:+ = .i[~ -I (x:_ + x:+

HI

1

1

1

-1)

I]

1If-----

18

xJ+I = 4[1-1 (~-I 2 (-3xJ- I +4xJ - 2x1+1 + 1) I) I] Xit+1

= g,-l[' X i _1 + Xi' - 2' X i+I -

XJ+I =

xi+

1

Xit+1

4

[

-XJ+I

1]

2

I Complemented +21] = ---r X i+1::::::> Left shift

= 4[1-1 (-xi_I-xi -2xi+1 + ~)I]

7]

= g,-l[' -Xi _1 - Xi'3 - X it+1 + 2

•

1 xi+ =

4[~ -I (-2xi_1 + xi -

XJ+l

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

=

Xit+1 =

4[

+ 1) 1]

'+ Xit - 3' 5 ] X i+1 + 2

-Xi_I

xi+ =

4[~ -I (-2xi_1 - xi -

Xit+1 =

g,

1

2xi+1

-l[-Xit

1-

X it+1

+ 23 ]

2xi+1

+ 2) 1]

ch01

Xit+1 =

x:+ ~]

Formula for Local Rule

15:12

x,+! = .i[

I

(Continued )

February 14, 2011

Table 2.

Formula for Local Rule

t![~ -I (-2x:_1+x! +x:+1+1) I]

x!+!

=

t![~ -I (-x:_1+x: + x:+1) 1]

X!+I

=

t![~ -I (-x!-1 -2x! -4x!+1 +4)1]

I!ID DID DID

X~+I = 4[1-I(x~

Xf+1 = 4[1-1 (-xf_1 +2xf +xf+1 -

x!+l

I

=

~) I]

t![~ -I (-x:_ 1-4x: -2x:+1+4)1] I-I

-XlI -2xHI +~)I] 2 l

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

DID 1m

Formula for Local Rule XII +1= 41~-1 (-x~I-I -XlI -XlHI +2) 2 Xf+l

=

I]

ti[~ -I (1-1 (-xf_1 + xf - xf+1 + 1) DI]

am

x:+1=

IrII

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

• •am

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

aD

x:+1=

t![J-I (x:_1+ x! +2X!+1 - ~) I]

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

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

ch01

19

• •III •ImIJ

xt 1=

N

15:12

N

(Continued )

February 14, 2011

Table 2.

N

(Continued )

Formula for Local Rule

[

I

_

I _

1]

I

IfttI' IIaII x:+ =4[1-1 (2x:- +X: +x:+l-~)I] 1

__

Xi+1 2

I-I

I

x:+ = 4[1-1 (%-1 C-4x:_ +2x: -x:+ +3)1)1] 1

1

I

I

1

20

1[

X:+1=4[1-1 (%-1(-4X:-1 - x: +2xf+, +3) DI]

4[Xi_ -Xi - 3Xi+ + 2"5 ] I

1

I

I

1

X:+ 1 = 4[~ -I (-x:_ 1 - 2x: - 2x:+1 + 2) I] XI +I = 4[-XI -Xl1+1 I

2

Xf+l

=4[- ~ +1 (-xf-, +xf +xf+, -1)1]

Xf+'

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

X:+ 1

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

1If-----

Xi1+1 = a XiI_1 - 3XiI -XiI+1 + 2"5 ]

Xi1+1 =

lID lID III

1

I

+2] 2

1

1

111m Xf+l =4[~ -I (2xf_, +2xf +xf+, - 3) 1]

1m 1m

x,+1

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

X:+ 1

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

Xi1+1

=a

1

1[-Xi- -Xi -Xi+ +2"5 ] I

I

1

I

1

ch01

Xl+l = 4[X I -Xl -Xl1+1 +~] 2 I

Formula for Local Rule

15:12

1+1

Xi = 4 2xi_1 Xi

N

February 14, 2011

Table 2.

am

N

Formula for Local Rule t _~] 4[Xt +xHI 2

21

Xit+1 =

g,

-l[ Xit-1 + Xit + Xit+1--5 ]

Xt+l

Xf+l =

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

xi+1=

4[~ -I (-Xi-1 -2r, +4x:+1-1) I]

Xf+l =

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

Xit+l

g,

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

X:+1= J,[l-I(-%+1 (-Xf-I- 2X: +3X:+1- 2)1)1]

III lID III lID r,+1 lilt

Xf+1 =

2

I

Xf+1 = 4[~

1]

-I (2xf_1 - xf - 2xf+1 + 1) 1]

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

Xf+l =

=

I

I]

I

=

Xit+l =

-l[-Xit-1 + Xit + 2Xit+1 -2"3 ]

-l[-Xit_1+ 2Xit + Xit+1 -2"3 ]

g,

X:+1 = J,[1-1(%-I(-2x:_1 -4x: +3x:+ 1 +2)1)1]

4[~ -I (r,-1- xi - Xi+1 + 1)1]

Xit+l

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

Xf+l =

=

-l[-Xit-1 + Xit + Xit+1-2"1 ]

g,

4[-3xf_l +Xf +Xf+l + ~]

ch01

1m

Formula for Local Rule

15:12

N

(Continued )

February 14, 2011

Table 2.

N

(Continued )

Formula for Local Rule

N

Formula for Local Rule

=

xi+'

22

1

X!+l

-

=

-I(x!_,-x!

=

1

=

=

1

1

2

=

(X!_l +4x! - 2X!+1 - 2)

X!+l

=

(-x!

x!+'

=

=

2

1

=

(-X!_l- x! 2xi+l

=

1

+

=

x!+'

1

-xi +2xi+l-

1

2

2

-1

1

=

=

1

1

ch01

1m x:+1 4[~-I(-x: 1-2x: +4x:+1) I] III +r,+l -1)1] 1m ~[~ x:+1 4[1-I(x:_ -2x: +x:+1-!)I] III

15:12

1] III ~[~ -I lID ~[~ -I +r,+l) 1] ~)I] lID ~[H(r,-l x:+1 4[~-I(-x:_l +2x: -2x:+ +1)1] III x:+1 4[1-1 (x:_1+2x: - x:+1- ~) I] x:+ 4[~ -I (-x:_ -x: +x:+ +1)I] am III ~[H + + ~) I] x:+1 4[~-I(x: +2x: -2x:+ -l)l] 1m lID 11m X:+ =4[%-I (~-I (--4X:-l - 2X: 4X:+l -~) D[ lID x:+ 4[- ~+ I(x:_1-x: -x:+1)I] x:+1 4[- ~+I (-2x:_ +X: +x:+ +1)I] x:+ 4[- ~+I(X:-l +X: +x:+1-2)1] lID lID x:+1= 4[!-I(-2x: 2 - 1 +X: -x:+1+2)1]

February 14, 2011

Table 2.

Formula for Local Rule t

t+l _

Xi

t+l

=

23

Xf+l =

__ 3

2

]

~[t X i- 1 -

t+ 2Xit+ -23 ]

Xi

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

1m xf+! =~[~-I(-xf-l+xf+!)I] Xf+' = ~[H(xf-l -xf -2xf+, + ~)I] =

2

4[1-lex:_l +X: -2x:+1 + ~)I]

1 X:+ =

1

1)

1m xf+]

IJ;

4

x:+ =

1

l1li xf+l =~[-%+ I(-2xf-1 +xf - 2xf+1 + I] •

~[t 5] X i- 1 + Xit+ 2X it+1 - -

Xi(+1 =

4[~-le2x:_l +X: -4x:+1 +1)1] 2 IJ;

Formula for Local Rule

~[~ -I (-2xf_, +xf +2xf+,- 1)I]

t+l

Xi

Xf+l

X:+

1

x:+1

-21] =

t

Xi +1

=> 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

15:12

ch01

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

15:12

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

15:12

ch01

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

February 14, 2011

15:12

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.

D Mwr'l.vr~

-- -- --

.l'MI_I'V~ KreIN!""I!! }--<j.l','t(J~l!v

[I'IMI,OI/,-

>--i rH,YMJf!

}o-{,WIV'f>.

~

1

---

~

{r,ctflrhl }+(f!nv~'Jt

-- --

,rffl(M06,

(S''''''U,! J-+{!lIN,I/(,f),)

t'JIMr;fI/

r;:(1,-'),-6\

Ilf,ilSi:V

/)/)WY/()i:!

f,O;'-Vf~',

~'It'OI}-UI

_,-r,-y,fll

_cYlllhdi

U61.'MM

LLlrM

~''''\,IK!

\"~,CBI

I~"tln'\

MlI,Ylfi_

_:IIX9c fJ!

nW!l11/J1

1I,\'~YVIr>

!!

YlMiI~(h

!~\~_N"

\'t)SI.VYi:{

Ifi>//'If6

'h.fH,-rl{

U,'fVIOU()!

u{v/r/.V'I

_~'~'Vt,~'f

f'I.I'I.,'

>11.1/111},'

thM!Vh

rll/()
cINJ.\'I/(HJ!

wlrt."JI

Irt6WI~

r~'t!/OOIl ~1~,-,-'1.'1>--(YIVt!.OIIl]

~ Ilctorwo

ItI"rro_1"

6c(J0I}!r

wtl},n,

,\'!HWfWJI

I!\n,_hc

r,,(,"~,,1

Uf~'_,-,O/'

1-~V{.WI.:.

6,~'Hfi

fl.lfi>lll

t.lfO/61

,~,6~i'j,(J_

rVfMIi'Y1

o<'JJlILlh
1'11'-1-1)/

(Jfl/,r>I~

-1~"AlJ1r"

(:::::~:;:J~

h'-f()~'1!

~;"'I,I!"'.fU~U;-If~Y.r::_/'WflfV~

~

,V\l~'MI,1

BW'f)\ 'IIMII.~1

Irhlilyn

~

.I'IIM

~~tllfh.CV(C.::-:~:~~:-~1 rM,-0,6~

6()(1!T~O}'

t!1f6}-ru

1,-_01,',-,11/

1I,,-,f6,1';;

'61.1I.~I!I

6tOl't_1'

6,',"10If,1

60_IIfU,

6,-'111_1,1'1

1',n,lojJl

II'MLwtll

6r.-n,'e'Y

(,_MWll!

,S_,I~!l

lr'Jou"JII

~,c\lo~1

/'\'r,IJ'!iY

IWHOSII

i:1I>!l1~'1

,Yll'i:IWI

O,reMM

II~_/(',I'I{

Wi:IiI!ro

i%~tno

tl!/OW,YI{

6ts,-v~.r

r}'If';I~U

fl'I/ylf6wt

liM,60f';

t61/1fOI/,

,',vr;:rll

,';001/_111

,V61/1J01fr

I~W\'Ot,-,

ItN,In

,Wa6,Y

6NIW/H

I~VSt6I/,

~I"-'VWI

Ifl';6tOOI

WO.llf9!/

r'IJUI.~6

,-'i6a'gWI

o{o;;Y9lrYfHn';;s6(ri:I).~1'ltlllr')l

Oi:i';;f!'<-~

_fi:r.tS!i

","1',,-6(,-

(l/WI9i:ti:l

Ilft"t,.

rt'.YUII

M!lI>~..

Wi'tll%I')

l'/fi:lfl'lll

')i:IlIrJ,

IM.!'f"

IJr,,-!','lt

UO!iti://U

Iilrt/"'lM

,llsILt,-,

,10r,-'Il'

'1.,,11,-6

!lM/(J.(I~~ III"""M!

~ '-.'W/lO~!

~:;~:::: ' ) ,-Wf,'IH)+-{_,y6WMV)+-{,,-W;'-\'M)

>(.\'V60~t!"f )+-{16i<-Y._' )+-{r6~,-\'I/_,)

~:"(:::I;::.Ir-.J

{WI'Y.\'I)!<\')-(tlll!IS0I6)--(S6"IOH'W)--{WAW>l6)-(I!!ThW6

~

('~,-i:I).l'1) }-(otMlr.!' }-{llf\'Y!I\'f,1

.(._Ii'M!'.}--{MWi'/,lr}--{i:.Wfl/l't)

16t>o,on ' )

M,_fO,Y, -(W!·'~I0~'-)+-('Hllo;,\').--{rll9,-II',\'~)~

S"-,l/!WI

i:~c,~,lc

1),'9'1·,,'-1

Iri.\'cOIl'l'-

f,f(lOWn'

,'!
o,YtWtv9

Jr'd,he

Mm,'1l'6.

_11,Y.OII)

fl)!'rMI.!'

.ff"(,I)'

i'fr'!."IIM

,WIj'IIKJ

fWlj',r,"1I1

lIII.i'.I""

,YI'MrIXJI)

f~.'(,111I1

j'tll),WI

_,'(,lfI6,1

~i:.-tW~Gf

r.-rf~'nJ

.-t9ff9'"

,head

,'/lfflfll"

!,I!lU/().

,"M)!'rUIi

O'l.c9t'lt

,'M11IfL

Leller,

Inll(lf~"

S''-fl
IS'yOI'Io,'\'

WMd'WlnJ

rO_f!·!.t\J

f(1619fOI'

f6ei:~'9r,.

W.'1LiMt

'1f.-rtftll

,'MOltfL

c,'Mdt

IZII,9()f1

O,f'l,L(!1'

M,'c9\,I',

~6.6111Ilm}.{,"II"IIj'r,

,ti:9friJ,

t;I"91-111

1'91'1''-'11',

.-/f69M,

'1n".-It'l,)

trllll,f,YI

--

o{l)tI,lllff)--(MIfi:ly/l >-~tl~Jllf}-(Mf\l>!'I!'}-(r\'fr,NXJt 'l9S,OLr

,·MIMi:.

,-~",\'Mtn

,6S,tofi:

~"tc~'HM

9/11'.i:tIII

IIlftllS\"! ~

tn'l,-t

:.t.-fi:L'I:--:tWtWM.-O ~

"\OON,'I19i:r

II(,/IMlll)

f.IIW!\lWlI 6cM!fI9ffl

h,,'W,'Wf

HIL,,'d'

Srm.69i:

(6f\l\'j't,!i.}-("'!1I9;'1,Yf) rntr,t~'

i:U:Iii/lOI

M,'ff>i:.'1

t>.~',-Mfl

M9INfM

~

o{fII.fIl.~r}-(k'~'!'h~'t\fr)--(rf~,,"i:11I>--<,'t~/lIt,;:. )--(M,j'lJt"'~'tl).

'.JMi:

,-,',"",1'.

f,IMI'"

\1=.1'(,"%

i:I@~IW

IlJfll'/illll/

1I,,'Vi~1 rJII9~'\i
_w,~"lnl ~;tfr,
,'I'M",',.

f!J60_11 )+(lIrHlfOi:. )+-(tli:IMti:1 )+(thtrtW )+-(,'Itftril'l )+-(1I.6_t_SI)+-( 1/(IIM6,'~ )+-(liOfi:riO;OI )+(tl,'fl,'S!' )+-(9ff'lI,Ii:)+-(rWI,"'116 )+-(t,J-HMOI )+(9tllf>trillY )+-{LttH'9)+--( 11'1',-16.-0 )+(lIe'l'-!'i:Sf)+-( .W'ri'l'-~'; )+-(t!_.I~i:S\'~' , ) 11!\lII),t, ~'--ti:rM(I/

t'-/~fl'!~'9

J,r.\lfi'1J

_,-.f.tf!

~_/fL'I ~',..,r.'fi'

IIiN/llhl

crt,'-Oll,

ri,1I6tttl'!

.,·MIWt!

1),'l/il'l!\IlI

i'lJr_~I\I.

fWf.f-,"~1

ttNit.rf

'1t9'1/'!'" M~,~'Hi:I

'fn!·.-.-n

109JMi:

e'lMtW"

tW6IM,'/ri

~'ri:t()M"1'!

IS'ge',''111

tri.tof'l ~

,Y.~-,""I"

W,,'rfI<M
.11t!o-'Ml.

M,1)."'lJr

~1'lIfl).tt

INii'~'6,"1I

lMwll)lI1

IJrl,-,'I'!'fJ

,,,'~r,I'i:I),1

/IIIIJIVVII/JI

~ lV'fJ,-.'_" /~OIWO~I:~ ,r,Llld

-r,I'i:IIM.

-- -- -- -- -- -- -- -1J"I',~fl)

flJfr~,'U

tlN)!',"tfJ

1)~,!iI\'W~

V#!\"Uf,

f'l/il'IJfJlfJ

----

.I)/L.,tM/9()flt~")-(9!"f60111M

fJ;'VlI/,,'f }--( ~lIt""!Ifl) )-( IIt,-,-i/ rr

KJ~t,i:I,"

-- --

rl'~IJ,I!~>

-%lllJ,'1;:

(,MO,!'!il}--(k"HttO,V)

-- ----

,",Ot,..nl ...~·(J',"'I-'"

in~'HIIXJ}-(("I'Hl/lflt}--(;;'\'\III"tflJ

("MllfW,-t}-(I,~I(-fH)

.r~,·f,r;~

}--(f"I>Ii;",-t)

S~::_::,CI::l ' )

,tM'ri9;',) 1f()/('~1W

;"'\"\"'(,~'9)+-(t'M,.,trl'

,/(I!iIlWml)+(\,rfclfI9MIJt'ifJfMWe (,-VIJ'l/Jt"I, }-(O/V,,!liWI }-(f,-!il.''''t~)

"J\'.-lItf}--(';;'OU'\I')

-- --

-nMlWif

(fMHff!i}-(I\IIIi.W''''f)

f"Lt'i,f\','1

)

.;;.w~M,Yf

Of//l(il'\:~ ~II()J,'/V\I!

~ ,,,,,-,()/Ot

-- -- -- -- -- -1)!'
/<,IO;;""f(l

fd\l,~;,

,tll'lll1l,Y

'f.'dlXJ

~ roWf,-.t

,¥'l/J,I'tnr

l

--. ) Hn'tl,rt

"YrHi/ftf}--(MfM,,'\I,)

-- --

-- --

[.rNl;'9/r}-(II'~Mfrt

\'\III"cllJt}-(1II9,1/<,I')

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science fVIt'lttt!")

40

--

9'lt'l"HfMMt6,'r9v

~ '9,~/LII'~

~ Y,~tnrL~

Mlltllir

}--(rrMV'!<-r

}-(1J~~frr;'t

r'lJc,-11.

-- -- --

(/lII9frtll,Y >--<1,~frW~,-1 }--o{r.Y,Ylf,·

~ ~

ch01

(rW_~rrtr

tot,\",MY,

-'1%1'1.1111;(

15:12

rl.Ll.MI~,-

,vfhtIl!1MMIN,vtt'MI,I/IJIi<J~1

).

February 14, 2011

~

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

~G"o;m:;;;II!();;;:"e:>'--C;'CiI~--GB':@->(ij'!!2D..cz~~-8~;D--c§~·C:~CB--G~D--G:~B--G~B--G:~ry@~~-8~9--G~~>CZ'~"i)-<-G!'!E">--. 0-(

~.fl(..nH)

Fig. 9.

(Continued )

41

February 14, 2011

15:12

ch01

•• •

•• •

•• •

•• •

••

•• •

10

•• • •• •

•• •

•• •

Table 10.

Four bi-permutive rules.

•• • •• •

••

••

•• •

••

III

•• •

42

10

February 14, 2011

15:12

ch01

. .......

.......

......-•••11111-+---1....

.... .--

•••lIltt-+---H

;~~~....

•••

III :L-Permutive

•••

Table 11.

•

: L-Permutive

..:.... •••

~-+~I....

...

III :L-Permutive ....

m: •••

~~.....

...

...

•••permutive rules. Twenty-eight

...

•••

..-~........

••

··11111-+---1....

ILl: L-Permutive

•••

.~t--

•••

...

... ...

•••

.... .--

...

Chapter 1: Quasi-Ergodicity

• •• ~t--~.

•

: L-Permutive

•••

•••

...

•••

R-Permutive

. .......

• .oIt--......

..1r.1~~

...

• ••I..--"'--e.

. : R-Permutive

...

•••

• ••

...

•••

ED: R-Permutive ..IIIIt-+-.,.::.H

... ...

• •••I..--"'--e.

1.%11: B-Permutive 1mI: R-Permutive 1m: R-Permutive 1m: B-Permutive

...

.~~~......

•••

. . . ---t.....

•••

•••

...

•••

...

...

•••

111m: R-Permutive

. : L-Permutive

1m: L-Permutive

...

...

...

..:....

.~t-~

~~.....

...

•••

...

•••

...

...

···IIII~~

•••

~I.-~~.

IIP): R-Permutive

•••

...

•••

... •••

...w~~~

...

• •••I..--"'--e.

IIIJ: B-Permutive 1m: R-Permutive 1m: R-Permutive 1m: B-Permutive

..

.~~,-

...

~....

...

••• ~---t.....

...

•

•••

: R-Permutive . . .- .

...

+--+-4.....

•••

...

•••

...

. .• •rlt-+---H

...

• •••I..---e•

1m: R-Permutive 1iZJJ: R-Permutive I£DJ: L-Permutive •••

.~t--

•••

...

...

•••

...

...

1m: L-Permutive Bm: L-Permutive

... ...

...

... . : L-Permutive

• •• • •• II"t-+---1~.

...DDJ:

...

~.--~.

L-Permutive

43

44

4

6

4

6

5

16

0

7 1 4

6 5

7

5

7

32

0

2

3

0

2

4

6

2

5

1

2

1

0

7

3

1

3

1

3

4

6

4

6

5

7

5

7

64

0

2

4

0

2

Eight Boolean cube basis functions.

1

3

1

3

4

6

4

6

5

7

5

7

128

0

2

8

0

2

1

3

1

3

15:12

2

Table 12.

February 14, 2011 ch01

45

21

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

4

4

4

4

4

4

4

2

2

4

4

4

2

2

2

4

18 18 18 18 18 18 18 18

8

2

31

16

16

2

43

1

2

42

8

8

8

41

16

1

8

40

4

16

2

1

39

16

4

4

37

1

4

36

2

35

1

2

34

32

32

32

32

32

32

32

32

32

32

32

53

16

65

64

63

62

61

60

59

58

57

56

55

54

52

51

16

33

8

8

32

4

4

16

50

8

49

16

4

48

16

16

47

16

8

46

16

4

8

8

45

16

64 128 44

32

16

16

32 1

2 1

1

30

29

28

2

27

1

2

26

8

1

25

8

8

4

4

4

24

2

23

1

2

2

22

1

2

64 128

38

32

16

16

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

into superposition of eight basis characteristic functions.

4

4

4

4

4

4

4

4

4

4

4

4

4

8

8

8

8

8

8

8

8

8

8

8

8

8

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

64

64

64 128

15:12

3

2

1

0

2

N

Decomposition of χ1

D

1

Table 13.

February 14, 2011 ch01

46

87

86

85

84

83

82

81

80

79

78

77

76

75

74

73

72

71

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

64

8 8

4 4

64 64 64 64

16 16 16 16

4 4 4 64

64

16

16

64

16

4

64

16

64

64

8

64

8

4

64

8

64

64

8

8

4

64

8

64

4

4

109

2

2

2

4 1

1

1

1

4

108

107

106

105

104

103

102

4

4

101 2

2

2

2

4 1

1

1

1

4

100

99

98

97

96

95

94

4

93 1

4

92

2

64

4 4

2

1

91

64

4

64 64

16 16

32 32

8

32

32

32

32

32

32

32

32

32

32

32

64

16

32

64

16

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

16

16

64

16

64 128 64

32

16

16

8

8

8

8

8

8

8

8

8

8

8

2

8

90

4

64

2

4

64 1

1

8

64 128

89

32

64

16 8

8 88

4

(Continued )

131

130

129

128

127

126

125

124

123

122

121

120

119

118

117

116

115

114

113

112

111

110

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

4

4

4

4

4

4

4

4

4

4

4

8

8

8

8

8

8

8

8

8

8

8

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

16

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

128

128

128

128

64 128

15:12

70

69

68

67

66

1

Table 13.

February 14, 2011 ch01

128 128

4

128

8

137

128

8

2

139

47

128

16

128

16

4

8

153

1

8

152

2

1

151

16

128

128

128

16

4

2

150

16

128

16

4

149

1

128

16

4

148

2

1

147

128

16

2

146

128

16

1

145

128 128

8

4

128

16

8

144

2

143

1

2

142

4

128

8

4

141

1

128

8

4

140

1

128

8

2

138

1

128

8

175

174

173

172

171

170

169

168

167

166

165

164

163

162

161

160

159

158

157

156

155

154

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

2

2

4

4

4

4

4

4

4

4

4

4

4

4

4

8

8

8

8

8

8

8

8

8

8

8

8

8

8

8

16

16

16

16

16

16

16

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

(Continued )

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

64 128

197

196

195

194

193

192

191

190

189

188

187

186

185

184

183

182

181

180

179

178

177

176

1

1

1

1

1

1

1

1

1

1

1

1

32 32

16 16

2

2

2

2

2

4

4

8

8

4 4

16

8 4

16

16

16

16

8 8

16

16 8

8

16

16

4

4

2

2

4

2

4

8

32

16

2

32

32

32

32

32

32

32

32

32

32

32

16

2

16

32

16

4

32

16

8

32

4

16

2

64

64

64

64

64

64

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

64 128

15:12

136

2

135

1

2

134

1 4

64 128

128

32

4

16

133

8 128

4 4

2

132

1

Table 13.

February 14, 2011 ch01

48

219

218

217

216

215

214

213

212

211

210

209

208

207

206

205

204

203

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

8

4

64 64 64 64 64 64 64 64 64 64

16 16 16 16 16 16 16 16 16 16

4 4

8

8

8

64

16

4

64

64 16

8

8

4

4

64

8

4 64

64

8

64

64

8 8

64

8

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

64

64

64 128

8

32

128

16

64

8

4

4

4

2 2

4

2

241

240

239

238

237

236

235

234

233

232

231

230

229

228

227

226

225

224

223

222

221

220

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

2

2

4

4

4

4

4

4

4

4

4

4

4

4

4

8 16 16

32

32

32

32

32

8 8

32

32

32

32

32

32

32

32

32

32

32

32

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

64

16 16

64

16

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

128

64 128 64

32

32

16

16

8

8

8

8

8

8

8

8

8

8

(Continued )

255

254

253

252

251

250

249

248

247

246

245

244

243

242

1

1

1

1

1

1

1

1

2

2

2

2

2

2

2

2

2

4

4

4

4

4

8

8

8

8

8

8

8

16

16

16

16

16

16

16

16

16

16

4

16

16

16

16

16

8

8

4

4

4

32

32

32

32

32

32

32

32

32

32

32

32

32

32

32

64

64

64

64

64

64

64

64

64

64

64

64

64

64

128

128

128

128

128

128

128

128

128

128

128

128

128

128

64 128

15:12

202

201

200

199

198

1

Table 13.

February 14, 2011 ch01

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

49

Table 14(A). There are 128 local rules endowed with a period-1 spatially-homogeneous blue orbit  · · ·  for all L = 3, 4, 5, . . . , n.

Type A 2 6

0

4

3 7

1

5

2

3

2

7

6

4

7

22

4

4

66 3 7 1

0 5

7 1 5

2 0 5

2 6

2 0

6

1

1

5

6

1

0 4

5

1

0 4

5

7

6

1

0 4

5

242

1

1

4

5

244

1

4

1 5

246

5

2

2 7

6

1

0 4

5

248

1 5

2

1

4

5

250

4

1

1 5

1

0 4

5

254

1

0 5

2

5

236

1

5

2

238

1

0 4

5

218 1

5

3 7

6

3

0 4

1

0

196

7

6

3 7

216 1

0 4

2

4

5

2

5

6

3

0

3 7

6

1

7

4

5

1

0

174

5

6

3 7

4

7

2

214

7

6

1

0

1

194 3

1

5

2 6

3

0

0

152

5

6

3 7

4

3

0

4

7

4

3

234 3

7

6

2

4

232 1

5

2

212

7

2

1

0 4

1 5

6

5

2

192 3

2 6

1

172 3

0

5

130

0

7

1

0

4

7

4

7

6 4

7

6

3

0

1

5

2

5

2

190

5

6

0

6

3 7

6

3

2

1

0

2

150

170 3

7

4

7

4

7

252

2

3

2

3

5

1

7

1

5

4

3

4

168

5

0

6

5

6

5

2

1

0

4

210 1

0

0

4

5

6

7

6

3

2

1

0

5

108

2

148 3

0

6

1

0

128 1

3 7

6

3 7

3

4

2

4

2

4

7

6

5

7

6

5

2

1

1 5

126

146

3

2

230 3

0

6

0

2

188 1

0 4

7

6

2

7

5

106 1

0

1

0

86

7

6

3 7

6 4

0 4

7

4

2

3

6

3

6

3

4

5

4

7

6

228 3

1

5

2

6

166

208 3

0 4

0 4

3

0 4

7

6

226 3

0

1

0

1 5

206 3

1

7

6

1

3 7

6

5

2

5

2

5

2

124

5

2

186 3

0

0

0

1

0

144

3 7

4

7

6 4

7

6 4

7

6

2

204

5

2

1 5

2

1

5

6

7

4

5

104 3

6

3 7

4

5

2

184 3

0

0

6

164 3

7

4

7

6

224 3

0

2

3

0

1

5

4

7

6 4

7

6

1 5

2

0

0 4

1

2

5

64 1

2

1

0 4

7 0

3 7

6

84 1

4

2

3

4

7 0

1 5

6

3

6

102

5

2

3 7

6

5

2

1

5

2

162 6

0

5

122 3

1

0

3

0 4

5

2

7

2

82

7

6

1

4

5

62

7

1

0

42

0 4

3 7

6

3

6

3

0

3

4

7

6

142

1

0 4

1

7

4

3 7

6

182 3

0

2

3 7

4

202

5

2

6

1

7

4

1

5

2

5

140

160

5

6

222 3

0

2

3

0 4

220 2

1

7

6

0 4

180

5

2

1

0

6

5

2

120 1

1

5

6

80 2

100

5

2

5

3

0

3

0

1

0

1

2

2

4

2

60

7

6 4

7

6 4

7

4

3 7

4

3 7

6

3 7

4

200 3

7

6

1

6

6

5

2

1

118 1

5

6

5

2

98

5

2

1

2

1

7

4

5

40

0

3

78 3

0

5

2

5

3

6

1

0

1

0

20

7

6

3 7

4

3

4

2

58 3

0

1

4

7

6 4

7

6

3

0

138

5

2

3

0 4

0

0

2

4

7

4

3 7

4

158

7

6

6

3 7

4

5

2

198 2

1

6

178 3

7

6

0

4

5

2

3 7

1

136

5

2

176 2

0

0

5

2

1 5

6

5

2

76

7

2

116 3

7

4

156 3

0

6

3 7

4

154 7

5

2

5

2

1

1

0 4

4

96 3

1

0

7 0

5

2

38 3

6

56 3

5

36 2

1

0

5

2 6

18

1

4

1

0 4

7 0

3 7

3

6

5

3

5

2 6

16

1

4

1

0

2

7 0

3 7

4

3

6

7

6 4

7

6

3

0 4

7

6

114

134 3

7

0

4

1

2

5

2

5

6

94

5

3 7

4

7

4

1

0

3

0

5

7

6

1

2

3

5

2

74

2

92

112 1

5

6

4

1

2

54 3

0

7

4

7

6

72 1

2

1

0 4

5

3

0

1

2

3

0

52 3

0

5

1

2

34

2 6

5

2 6

14

5

32 1

0

4

1

0

3

0

3 7

4

7

6

5

3

5

2 6

12

1

0

1

0

2

7

6

3 7

4

3

4

7

6 4

7

6 4

7

6 4

7

3 7

4

2

3

6

132

6

5

2

110

4

1

4

2

70

7 0

3

0

6

5

90

2

4

4

5

2

50 1

0

1

0

3

5

2

30 3

7

6 4

7

6

3

6

88

6

5

2

5

2

68

2

4

1

0

5

6

7

6

1

1

1

0 4

2

48 3

5

5

2 6

10 3 7

6

1

4

2

28 3

0

1

3 7

0

5

8 3

0

2 6

1

0 4

7

6 4

7

6 4

2

7 0

4

1 5

3

5

2

46

2

6

7

4

44

5

2

26

0

5

1

0

3

6

1

3 7

6 4

2

7 0

4

5

3

4

3 7

6

6

2

24

2 6

1

0 4

5

6

7

6 1

0

3

1

0

5

2

3 7

4

2

3

2 6

1

4

2

2

3 7

0

5

0

4

2 6

1

0

5

6

3 7

6

1

0 4

There are 128 Local Rules with a type A boolean cube defined by a blue color at vertex 0 ( ).

2

3 7

6

1

0 4

5

240

February 14, 2011

15:12

ch01

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

50

Table 14(B). There are 128 local rules endowed with a period-1 spatially-homogeneous red orbit  · · ·  for all L = 3, 4, 5, . . . , n.

Type B 2 6

0

4

1

3 7 1

0 4

7

5

2

2

1 5

2

7

6

1

0 4

5

2

7

6

1

0 4

5

1

0 4

5

2

4

1

2

183 7

6

1

0 4

5

7 1

0 4

5

7

6

1

0 4

5

2

7

6

1

0 4

5

2

1

0 4

5

249

5

2

2

1

0 4

5

250

5

2

4

5

251

1

2

1

0 4

5

252

5

2

2 7

6

1

0 4

5

253

1 5

2

2 7

6

1

0 4

5

254

1 5

2

1

0 4

5

255

1

0 5

2

1

0 4

5

246

2

1

2

5

237 1

247

1

0 4

7 5

3 7

6

3

0 4

5

226

5

6

1

0 4

3

2

3 7

6

236 3

7

6

1

0 4

235

7

6

1 5

5

215

7

6

1

0 4

5

2

3 7

6

225 3

0 4

2

3

0 4

7

6

3

245 3

7

6

2

4

244 3

5

2

234 3

0 4

1

0

5

224

7

6

1

0 4

3

4

7

6

243 3

5

2

233 3

0 4

1

0

1 5

1

7

6

5

204

5

2

1

0 4

214 3

3 7

6

3

0 4

7

6

223 3

5

2

1

7

6

213 3

0

1

0

193

5

2

5

2

203 3

1

0

3

0

3 7

4

7

6 4

7

6 4

7

6 4

7

6 4

7

6

242 3

7

6

1

0 4

1 5

2

232 3

5

2

222 3

0 4

7

6

241 1

5

5

2

212 1

0

1

0

3 7

6 4

7

6

231

5

2

1

0

1 5

2

7

4

5

2

202 3

6

211

221 3

7

6

3

0

3

0

2

4

7

6 4

7

6

1 5

2

240 3

7

6

1

0 4

220

230 3

4

1

2

1

0

5

2 6

192 3

7

4

201

5

2

1 5

1

0

191

5

182

7

4

1

0

3

6

3 7

6 4

2

1

2

2

181

5

6

5

3

0

3

0

4

7

6 4

7

6

3

0

3

0

5

3

0

1

0

2

4

7

6 4

7

6

5

2

210

7

6

5

2

5

200 1

0 4

3

4

7

6 4

7

6

239 3

7

6

1

0 4

1 5

2

229 3

5

2

219 3

0

1

0 4

7

6 4

7

6

238 2

1 5

2

228 3

5

1 5

1

0

3 7

6

7

4

199

209 3

1

1

2

190 3

5

1

0

5

171 3

7

6

180 3

0 4

2

5

2

5

6

4

7

6

189 3

0

3

0

4

7

6 4

7

6 4

7

6

218 3

0 4

1

0 4

7

6

227 2

1 5

1 5

2

2

198

208 3

1 5

2

5

1

0

2

170 1

0

4

2

1

0

5

3

3 7

6

1

4

7

6

179 3

7

6

188 3

0

3

0

4

7

6 4

7

6 4

7

6

217 3

5

2

1 5

2

207 3

0 4

216 2

4

7

6

1

0

5

2

7

6

206 3

3

5

5

178 1

1

2

2

7

169 3

0

5

5

160 3

0

1

0

159 6

3 7

6 4

2

1

0

149

5

3

5

2

1

0

1

0 4

7

6

3 7

6

3

4

7

6 4

7

6 4

2

2

168

5

3

0

5

2

1

4

7

6

187

197

2

1

0 4

205 6

196

1

2

5

2

4

7 0

1

0

5

2

158 3

5

2

148 1

0 4

7

6

3

6

177 3

0

3

0 4

4

7

6 4

7

6

5

7

5

2

186 1

0

1

2

7

6

3

6

4

5

2

1

0

5

2

167 3

7

6

176 3

0

1

0

5

1

0

3

5

138

7

4

7

6

157 1

4

1

0

1

1

0

3

6

3 7

4

2

5

2

2 6

137

147 3

7

6 4

7 0

5

2

2

3

6

1

166 3

5

2

4

5

3

0

5

156 3

0

1

0

1

0

1

4

7

6

146

7

4

7

6 4

7

6 4

7

6

3

4

2

5

2

1

2

195 1

0 4

5

2

185

5

3

1

0

5

2

175

7

6

3

0 4

4

3

4

7

6

7

6

5

2

194 2

1

2

184 3

5

1

0

5

2

165 3

7

6

174 3

0

1

0

4

164

7

6

5

2

1

0

1 5

2

3 7

0

5

136

7

4

2 6

1

4

3

6

3 7

0

5

2

3

6

155 3

5

2

1

0

145 3

0

1

0

2 6

135

7

6

3 7

4

3

4

7

6 4

7

6

1

4

3

4

7

6 4

5

2

5

2

173 3

0

1

0

5

2

7 0

163 3

7

6 4

7

6

4

3

6

1

5

2

154

2

7

1

0 4

1 5

2

144 3

7

6

153

0

5

5

3

6

1

0 4

2

1

2

172

5

2

5

2 6

134 3

0 4

1

4

7

6

143 3

7

6

152

162 3

1

0

2

1 5

2

3 7

0

5

133 3

0 4

2 6

1

4

7

6

142

7

4

1 5

2

3 7

0

5

132 3

0 4

3

6

7 0

4

7

6

2

3

6

161 2

1 5

2

1 5

2 6

1

4

7

6

141

151 3

7 0

3

0 4

2

3 7

0

5

131 3

4

7

6

150 2

1 5

2 6

1

4

2 6

3 7

0

5

130

140 3

2 6

1

0

3

0 4

3 7

4

7

6

139 2

5

2 6

129 3

0

1

0 4

7

4

3 7

6

128 6

There are 128 Local Rules with a type B boolean cube defined by a red color at vertex 7 ( ).

5

2 6

3

2

3 7

6

1

0 4

5

248

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

51

Table 14(C). There are 64 local rules endowed with two period-1 spatially-homogeneous orbits  · · ·  and  · · ·  for all L = 3, 4, 5, . . . , n.

Type C 2 7

6

1

0 4

5

2

3 7

6

2

5

4

2

5

4

144 2

3

5

2

4

3 1 5

2

4

2

3

4

2

5

4

7

6 4

5

2

2

1

0 5

7

6

2

5

240

5

242

1 5

2

244

2

246

2

5

248

5

250

2

5

238

7

2

3 7

6 1

252

1

0 4

3

5

3 7

6 1

0 4

5

222

5

6 1

0 4

4

3

2

1

0

236 3

3 7

6

7 0

5

2

1 5

4

7

6 1

0 4

1 5

1

206 3

2

234 3

4

7

6

3 7

0

220 3

0

1

0 4

7

4

7

6 1

5

5

232 3

0 4

2

1

0

5

6

2 6

5

6 1

5

190

204

7

1

0

3

2

3 7

4

7

4

218 3

2 6

1

0

3

0 4

7

4

7

6 1

5

2

1 5

2

5

6

5

6 1

3

2

5

174

5

6 1

2

216

230 3

0 4

4

3

1

0

188

202 3

0

3

0

4

7

6

7

4

7

6 1

0 4

2

5

2

1 5

6

228 3

3

4

7 0

200

214 3

2 6

1

0

5

3 7

4

7 0

186 3

1

2

1

0

2 6

5

6

5

158

172 3

1

0

3

0

3 7

4

7

4

7

4

7

4

7

4

7 0

2

1

0

1 5

6

5

6 1

5

4

7

6 1

0 4

1 5

3

2

2 6

184

198 3

2

226 3

4

7

6

4

7 0

212 3

0

1

0 4

7

4

224 2

5

6

2 6

5

6 1

210 3

3

2

5

3

0

182

196

7

4

7

4

4

1

5

170

7

1

2

1

0

2 6

5

6

5

142

156 3

1

0

3

0

3 7

4

7

4

7

4

168 2

1

2

1 5

6

2 6

5

6

5

140

154 3

2 6

1

0

3

0

3 7

4

7

4

7 0

3

0

1

2

4

7

4

7 0

3

0

208 6

5

6 1

0

2

1

2

1 5

6

194 3

2

180 3

5

6

2 6

5

6 1

0

3

0

3 7

5

138

152

166

7

4

7 0

192 2

1 5

6 1

0

2

4

2 6

1

0

3

0

3 7

4

7

6

5

4

164

178

7

1 5

6

2

1

2

5

136

150 6

2 6

1

0

3

0

3 7

4

7

6

3

0

3

0

2

4

7

4

7

6

176 6

5

162

7 0

2

1

0

1 5

6

5

134

148 3

2 6

1

0

3

0

3 7

4

7

4

7

4

160 6

1

2

1

0 4

2 6

5

6

5

132

146

7

6

2 6

1

0

3

0

3 7

4

7

6 1

0

5

130 3

7

6

2 6

1

0 4

128 2

3 7

6 1

0 4

There are 64 Local Rules whose Boolean Cube has a type C color combination at vertex ( ) and vertex ( )

3

1

0 4

5

254

February 14, 2011

52

15:12

ch01

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

Table 14(D). There are 64 local rules endowed with two period-1 spatially-alternating orbits  · · ·  and  · · ·  for all even L = 4, 6, 8, . . . , 2n.

Type D 2 7

6

1

0 4

5 2

3 7

6

2

5

2

5

20 2

3

4

2

1

0 5

4

2

3

4

2

1

0 5

4

7

6 4

5

4

5

148 2 7

6 4

2

5

4

7

6

5

212

5

4

5

213

5

214

5

2

5

215

4

2

5

220

1 5

221

2

5

5

207

7

2

3 7

6 1

222

1

0 4

3

5

3 7

6 1

0 4

1 5

159

7

6

7

4

3

2

3

0

206 3

0 4

2 6

5

4

7

6 1

0 4

5

5

143 1

0

205 3

7

6 1

0 4

5

4

7

6 1

0

1

0

3

2

3 7

6

158 3

5

2

1 5

4

7

6 1

204 3

5

2

3

0

157 3

0 4

7

6

4

7

6 1

2

1

0 4

2

199 3

5

4

7

6

1

95

2

1

3 7

0

142 3

0

2

1

0 4

7

6

156 3

0 4

7

6 1

0 4

5

2

4

7

6

198 3

7

6

2

1

0

5

2

1

0

5

5

6

5

6

141 3

3

2

1

1

0

94

7

4

7

6 1

151 3

5

2

4

3 7

4

7 0

3

0

140

7

4

7

6 1

2

1

0

2

197 3

5

150 3

0

196 2

4

7

6 1

0

5

4

5

6 1

0

1

2

2

79

2

93 3

5

6

5

6 1

0

3

0

3

78

7

4

7

6

3

0

5

2

1 5

6 1

1

92

135

7

2

1

0 4

7

4

3 7

31

2

1 5

6

2

1 5

6

5

6

30

77 3

0

3

2

4

7

4

7

4

3

0

149 3

5

6

1

0

134 1

2 6

5

6 1

2

3

2

5

3

0

3

1

0 4

7

4

7 0

76

87

7

4

7

4

4

2

1

0

1

3 7

15

2 6

5

6

5

14

29 3

2 6

1

0

3

0

3 7

4

7

4

7

4

7 0

3

0

3

0

5

6

5

6 1

0

1

2

1

2

2

86 3

1 5

6

1

2

71 3

0

133 3

4

7

4

7

4

7

6

1

0

132 2

2 6

5

6 1

0

3

2

5

2 6

5

6

5

13

28 3

2 6

1

0

3

0

3 7

4

7

4

7 0

70

85 3

1

0

2

1

2

1 5

6

5

6

2 6

12

23 3

3 7

0

3

0 4

7

4

7 0

84 2

5

6

1

2

1

2 6 4

7

6

5

6

5

2

22

69

7

3

0

3

1

0 4

7

4

7 0

68 6

1

3 7

7

2 6

5

6

5

6

21

7

6

2 6

1

0

3

0 4

3 7

4

7

6 1

0 4

5

5 3

7

6

2 6

1

0 4

4 2

3 7

6 1

0 4

4

There are 64 Local Rules whose Boolean Cube has a type D color combination at vertex ( ) and vertex ( )

3

1

0 4

5

223

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

53

Table 14(E). There are 64 local rules endowed with a period-2 spatially-homogeneous orbit  · · ·  ↔  · · ·  for all L = 3, 4, 5, . . . , n.

Type E 2

There are 64 Local Rules whose Boolean Cube has a type E color combination at vertex ( ) and vertex ( )

3 7

6

1

0 4

5 2

3 7

6

1

0 4

2

5

5

5

17 2

3

4

2

5

4

33 2

3

4

2

5

4

7

6 4

5

4

5

4

5

4

97 2 7

6

2

5

113

4

2

5

115

4

5

117

2

5

119

4

2

5

121

5

123

2

3 7

6 1

125

5

111

7

5

1

0 4

3

0 4

3 7

6

5

6 1

0 4

2

1

2

5

95

109 3

1

0 4

7

4

7

6 1

0 4

5

3 7

6

3

0

107 3

7

6 1

0 4

5

2

1 5

6 1

0

5

79 3

2

1

0

93 3

3 7

4

7

4

7

6 1

105 3

5

2

2

1

0

5

6

5

6 1

91 3

0 4

7

6 1

0 4

5

2

4

7

6

103 3

7

6 1

0 4

5

2

1

0

5

89 3

7

1

63 3

2

3 7

0

77 3

0

2

4

7

4

5

6 1

0

1

47 3

2

1 5

6 1

4

5

6

3 7

0

61 3

2

1

0

75 3

0 4

7

6 1

101 3

7

6 1

0 4

5

5

2

4

7

6

87 3

0

99 3

4

7

6 1

0

5

5

2 6

7

4

7 0

73 1

1

2

3

2

5

31

5

6

5

6 1

2

3

1

0

45

59 3

0

3

0

4

7

4

7

6 1

2

5

2

71

7

5

6

4

7 0

57 1

2

2 6

1

0

5

3 7

4

7 0

43 3

1

2

1

0

2 6

5

6

5

15

29 3

1

0

3

0

3 7

4

7

4

7

4

7

4

7

4

85 3

2

3

0

3

0 4

7

6 1

0

5

2

5

6 1

83 3

1

2

5

6

1 5

6 1

2

69

7

4

7

6

4

3

1

2

41

55 3

0

3

0

81 2

5

6 1

0

1

2

4

7

6

67 3

5

2

4

7 0

53 3

0

2 6

1

0

5

2 6

5

6

5

13

27 3

2 6

1

0

3

0

3 7

4

7

4

7 0

39 3

1

2

1

0

2 6

5

6

5

11

25 3

2 6

1

0

3

0 4

7

4

7

4

7

4

7

6

5

6

65 2

1

2

1

0

2

51 3

2

1 5

6

1

3 7

4

7

6

5

6

37 3

0

49 2

4

7

6 1

0

5

2

23 3

5

9

0 4

7 0

35

7

6

2 6

1

0

5

2 6

1

0

3 7

6

3 7

4

7

21 3

5

2

1

0 4

7

6 1

0

5

2 6

1

0

3 7

6

3 7

4

5

19

7

6

5

2

1

0 4

2 6

1

0

3 7

6 1

0 4

2

3 7

4

3 3

7

6

2 6

1

0 4

1 2

3 7

6

1

0 4

5

127

February 14, 2011

54

15:12

ch01

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

Table 14(F). There are 64 local rules endowed with a period-2 spatially alternating orbit  · · ·  ↔  · · ·  for all even L = 4, 6, 8, . . . , 2n.

Type F 2 7

6

1

0 4

5 2

3 7

6 4

2

5

4

2

4

3

2

5

4

7

4

5

4

5

4

112 2 7

6 4

2

5

4

160 2

5

176 2 7

6

5

224 2

3

2

5

240

2

1 5

241

3 1

0 4

5

242

2

1

2

2

2

5

248

5

249

1

2

1

0 4

2

2

3 7

6 1

250

5

235 3

5

1

0 4

7 0

4

3 7

6

5

6

5

187

234 3

1

0

3

0 4

7

6 1

0 4

5

3 7

4

7

6

233 3

1 5

2

1

0 4

7

6 1

243

5

2 6

186 3

5

171 3

0

1

0 4

7

4

7

6

232 3

5

2

1

0

1 5

3 7

6

5

6

185 3

2

1

2

5

123

170 3

0 4

7

4

7 0

4

1

4

7

4

7

6

5

6

5

6

2

184 3

5

169 3

0 4

227

7

6

1

0 4

226 3

0 4

1 5

4

7

6

7

6

5

1

0

3

0

3 7

6

5

6 1

0

2

1

2

5

107

122 3

1

0 4

7

4

7

6

168

5

2

2

1

2

5

3 7

6

3

0

121 3

0

179 3

0 4

7

6 1

0

5

6

3

0 4

7

6

225

7

6

1

4

7

4

7

6

5

2

1

0 4

2

178 3

5

2

163 3

0 4

7

6 1

0 4

5

2

4

7

6

177 3

5

1

0

5

2

1 5

6 1

0

5

59 3

2

1

0

106 3

3 7

4

7

4

7

6 1

1

0

105

120 3

5

2

2 6

5

6 1

5

43 3

2

1

0

58

7

4

7

4

7

6 1

2

1

0 4

2

162 3

5

4

3 7

4

7 0

3

0

3

0

115 3

0 4

7

6 1

0 4

5

2

4

7

6

161 3

7

6

2

1

0

5

5

6 1

0

5

6 1

2

2

1

2

1 5

6

57 3

2 6

42 3

0

104 3

4

7

4

7

4

7

6 1

1

0

99

114 3

5

2

2 6

5

6 1

5

3 7

0

41 3

2

2 6

1

0

56

7

4

7

4

7

6 1

0

5

4

3 7

4

7 0

3

0

3

0

113 3

5

6 1

0

5

6 1

2

2

1

2

1 5

6

51 3

2 6

40 3

0

98 3

4

7

4

7

4

7

6 1

0

5

2

1

0

97 3

2 6

5

6 1

5

3 7

0

35 3

2

2 6

1

0

50

7

4

7

6

4

3 7

4

7 0

3

0

96 2

5

6 1

0

2

1

2

1 5

6

49 3

2 6

34 3

0

48 6

4

7

6 1

2

5

3 7

0

33

7 0

2 6

1

0

32 6

3 7

6 1

0

4

There are 64 Local Rules whose Boolean Cube has a type F color combination at vertex ( ) and ) vertex (

3

1

0 4

5

251

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

55

Table 14(G). There are 16 local rules endowed with two period-1 spatially-homogeneous orbits  · · ·  and  · · ·  for all L = 3, 4, 5, . . . , n, as well as two period-1 spatially-alternating orbits  · · ·  and  · · ·  for all even L = 4, 6, 8, . . . , 2n.

Type G 2

There are 16 Local Rules whose Boolean Cube has a type G color combination at vertex ( ), vertex ( ), vertex ( ), and vertex ( )

3 7

6

1

0 4

5

2

3

2

7

6

6 1

0 4

4

132

5

3

2

7

5

4

196

5

2

5

4

198

5

2

5

4

204

5

1 5

2

4

206

5

2

5

4

212

5

2

5

4

214

5

158 3

2

5

3 7

6 1

0

1

0 4

7

6 1

3 7

6

156 3

0

2

1

0 4

7

6 1

3 7

6

150 3

0

2

1

0 4

7

6

3 7

6

148 3

0

2

1

0 4

7

6 1

3 7

6

142 3

0

2

1

0 4

7

6 1

3 7

6

140 3

0

2

1

0 4

7

6 1

0

3 7

6

134

2

4

2

1

0

5

6

3 7

1

0 4

220

5

222

Table 14(H). There are 16 local rules endowed with two period-2 spatially-homogeneous orbits  · · ·  and  · · ·  for all L = 3, 4, 5, . . . , n, as well as two period-2 spatially-alternating orbits  · · ·  and  · · ·  for all even L = 4, 6, 8, . . . , 2n.

Type H 2 7

6

1

0 4

5

2

3 7

6

2

5

2

5

97

99

5

2

5

105

5

2

5

107

5

2

5

113

5

2

5

115

5

2

5

121

5

59 2

3 7

6 1

0 4

1

0

3 7

6

3 7

4

57 1

0 4

2 6

1

0

3 7

6

3 7

4

51 1

0 4

2 6

1

0

3 7

6

3 7

4

49 1

0 4

2 6

1

0

3 7

6

3 7

4

43 1

0 4

2 6

1

0

3 7

6

3 7

4

41 1

5

2 6

1

0

3

0 4

3 7

4

7

6 1

0

5

35 3

7

6

2 6

1

0 4

33 2

3 7

6 1

0 4

4

There are 16 Local Rules whose Boolean Cube has a type H color combination at vertex ( ), vertex ( ), vertex ( ), and vertex ( )

3

1

0 4

5

123

February 14, 2011

56

15:12

ch01

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

Table 14(I). There are 16 local rules endowed with two period-1 spatially-homogeneous orbits  · · ·  and  · · ·  for all L = 3, 4, 5, . . . , n, as well as a period-2 spatially-alternating orbit  · · ·  ↔  · · ·  for all even L = 4, 6, 8, . . . , 2n.

Type I 2

There are 16 Local Rules whose Boolean Cube has a type I color combination at vertex ( ), vertex ( ), vertex ( ), and vertex ( )

3 7

6

1

0 4

5

2

3

2

7

6 4

5

4

5

2

3 7 1

0 5

2

4

2

3 1

0 4

226

170

5

2

4

232

2

4

234

2

5

4

240

5

2

5

4

242

5

186 3

2

5

3 7

6 1

0

1

0 4

7

6 1

3 7

6

184 3

0

2

1

0 4

7

6 1

4

5

5

3 7

6

178

7 0

2

1

0

3

6 1

0

5

3 7

6

176 3

7

6

2

1

0 4

5

3 7

6 1

0 4

7

6 1

5

224

5

2

3 7

6

168 3

0

2

1

0 4

7

6

3 7

6

162

2

4

1

0

160 6

7

6 1

0

3

1

0 4

248

5

250

Table 14(J). There are 16 local rules endowed with two period-1 spatially-alternating orbits  · · ·  and  · · ·  for all even L = 4, 6, 8, . . . , 2n, as well as a period-2 spatially-homogeneous orbit  · · ·  ↔  · · ·  for all L = 3, 4, 5, . . . , n.

Type J 2 7

6

1

0 4

5

2

3 7

6

2

5

5

69

4

5

71

5

2

4

5

77

5

2

4

5

79

5

2

4

5

85

5

2

4

5

87

5

2

4

5

93

5

31 2

3 7

6 1

0

1

0

3 7

6

3 7

4

29 1

0

2 6

1

0

3 7

6

3 7

4

23 1

0

2 6

1

0

3 7

6

3 7

4

21 1

0

2 6

1

0

3 7

6

3 7

4

15 1

0

2 6

1

0

3 7

6

3 7

4

13 1

0

2 6

1

0

3 7

6

3 7

4

7 1

0

5

2

3 7

6

2 6

1

0 4

5 2

3 7

6 1

0 4

4

There are 16 Local Rules whose Boolean Cube has a type J color combination at vertex ( ), vertex ( ), vertex ( ), and vertex ( )

3

1

0 4

5

95

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity Table 15. Gallery exhibiting the initial (t = 0) and final bit string (t = T − 1) for an Isle of Eden of the 79 topologically-independent rules having at least one of them.

N

L

T

1

9

2

2

9

3

3

9

18

4

9

1

5

9

2

6

8

8

7

8

1

9

9

3

10

8

4

11

9

2

12

8

1

13

9

2

14

9

18

15

9

18

18

8

2

19

8

2

22

8

2

23

9

2

25

9

2

26

8

16

Initial bit string x0 and final bit string xT-1

x0 x1 x0 x2 x0 x17 x0 x0 x0 x1 x0 x7 x0 x0 x0 x2 x0 x3 x0 x1 x0 x0 x0 x1 x0 x17 x0 x17 x0 x1 x0 x1 x0 x1 x0 x1 x0 x1 x0 x15

N

L

27

9 18

28

6

2

29

9

2

30

8

1

32

8

2

33

8

2

34

8

2

35

9 18

37

7

38

9 18

40

9

9

41

9

9

42

9

9

43

9 18

44

9

45

9 504

50

8

2

51

9

2

54

8

8

56

9

9

Initial bit string x0 and final bit string xT-1

T

7

3

x0 x17 x0 x1 x0 x1 x0 x0 x0 x1 x0 x1 x0 x1 x0 x17 x0 x6 x0 x17 x0 x8 x0 x8 x0 x8 x0 x17 x0 x2 x0 x503 x0 x1 x0 x1 x0 x7 x0 x8

57

February 14, 2011

58

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 15.

N

L

T

57

9

3

58

9 18

62

9

3

72

9

1

73

9

2

74

9

1

76

9

1

77

9

2

94

8

2

104 9

1

105 8

4

106 9

9

108 9

2

110 8

8

122 4

2

128 9

1

130 9

3

132 9

1

134 9

1

136 9

1

Initial bit string x0 and final bit string xT-1

x0 x2 x0 x17 x0 x2 x0 x0 x0 x1 x0 x0 x0 x0 x0 x1 x0 x1 x0 x0 x0 x3 x0 x8 x0 x1 x0 x7 x0 x1 x0 x0 x0 x2 x0 x0 x0 x0 x0 x0

(Continued )

N

L

T

138 9

9

140 9

1

142 9 18 146 9

2

150 8

4

152 9

1

154 9 72 156 9

2

160 9

1

162 9

1

164 9

1

168 9

9

170 9

9

172 9

3

178 9

1

184 9

9

200 9

1

204 9

1

232 9

1

Initial bit string x0 and final bit string xT-1

x0 x8 x0 x0 x0 x17 x0 x1 x0 x3 x0 x0 x0 x71 x0 x1 x0 x0 x0 x0 x0 x0 x0 x8 x0 x8 x0 x2 x0 x0 x0 x8 x0 x0 x0 x0 x0 x0

February 14, 2011

15:12

ch01

m

¢n-480

~

¢n

I -,----..----------,.

epn Chapter 1: Quasi-Ergodicity

59

0.5

0.5

O~""""'---'-"""""+--'--'-"""T"""~

o

0.5

ep n-l

(a)

o~..........---.--.--.f--,---.---.-.,........j o

(b)

Fig. 10. (a) Time-1 return map of period-3240 Isle of Eden. (b) Time-τ return map of the same orbit with τ = 480 resembles the left-shift Bernoulli-map.

2. Quasi-Ergodicity In the previous parts of our saga about Cellular Automata we introduced two kinds of graphical representations for the local rules: the time-1 return map and the time-1 characteristic function [Chua et al., 2005a]. The first one illustrates the evolution from a specific initial bit string as a Lamerey (cobweb) diagram for all times; the second one shows the output bit string corresponding to any possible input for one iteration. In general, a local rule has a unique time-1 characteristic function but many time-1 return maps, one for each initial condition. Therefore, a question arises: Can the time-1 return map and the time-1 characteristic function coincide? In other words, is it possible to extract information about the temporal behavior from the time-1 return map of a Cellular Automaton by looking at the spatial representation from its time-1 characteristic function? The correspondence between the temporal and the spatial behaviors of an important class of dynamical systems is a well-known concept dubbed “ergodicity”, and it is fundamental for a number of branches of mathematics and physics. Since a thorough treatment of ergodicity would go beyond the purpose of this introductory section, we will opt for a pedagogically more transparent approach by introducing the empirical concept “quasi-ergodicity”10 to stress its difference with “ergodicity”, which would require an in-depth measure-theoretic analysis. 10

Remarkably, “quasi-ergodicity” captures a peculiarity unique among the complex and hyper Bernoulli rules, thereby adding a further raison d’ˆetre for the classification summarized in Sec. 1.5, besides those based on the period, or the Bernoulli στ -shift, of the attractors defined in [Chua et al., 2007a], which differ from the complex (group 5) and hyper (group 6) Bernoulli-shift rules in the fact that they are independent from the initial configuration and the length L of the bit strings.

2.1. Only complex and hyper Bernoullishift rules are quasi-ergodic By visual inspection, we noticed that for certain rules the time-1 return maps corresponding to different initial conditions are indistinguishable, and they tend to be very similar to the time-1 characteristic function too. As we mentioned before, this behavior is somehow related to “ergodicity” but, in order to emphasize that this phenomenon has been observed empirically, we preferred to introduce the property of “quasi-ergodicity” via the following definition. Definition 2.1. Quasi-ergodicity. A rule is said to

be quasi-ergodic iff given an arbitrary point P belonging to an attractor ΛP , it is possible to find a point Q arbitrarily close to it which belongs to another attractor ΛQ which is visually indistinguishable from ΛP .

D

A graphical illustration of the quasi-ergodic property is given in Fig. 11, for a generic rule N .

“Quasi-ergodicity” is an expression widely used also in statistical physics; however, in this context we employ it in a different, albeit related, way.

February 14, 2011

60

15:12

ch01

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

1

Φn

I

0.75

lIE lIE I point P belonging to attractor ΛP through P

Q

0.5

-ε

ε

point Q arbitrarily arbitrarly close close (|P-Q ε) to P belongs to (|P-Q|< another attractor ΛQ

0.25 0

P

0

0.25

0.5

0.75 Φ n-1 1

Fig. 11. For a quasi-ergodic rule, given an arbitrary point P (in red) belonging to the attractor ΛP in the time-1 return map, there is a nearby point Q (in blue) belonging to another attractor ΛQ which is empirically indistinguishable from ΛP .

D

Figure 12 shows one of the consequences of quasiergodicity of rule 110 : Since the time-1 return maps corresponding to three different initial conditions cling arbitrarily close to each other, the Lamerey (cobweb) diagrams (Figs. 12(a)–12(c)) obtained from their superposition in Fig. 12(d) are visually indistinguishable. We cannot overemphasize that Definition 2.1 is only an empirical definition for the new phenomenon to be described in this section. In particular, we have found that only the complex and hyper Bernoulli-shift rules, corresponding to the fifth and sixth group in Sec. 1, are quasiergodic. This observation is remarkable because it introduces a particular feature possessed by only rules belonging to these two groups, within the more general class of Bernoulli-shift rules. So far, our distinction between the Bernoulli στ -shift rules from group 4 and the complex and hyper Bernoulli-shift rules from groups 5 and 6 is that the two Bernoulli parameters σ and τ from group 4 do not depend on the initial configuration, or the length L. Table 16 shows the Lamerey (cobweb) diagrams for the 18 complex and hyper Bernoulli-shift rules. Observe that the map (starting from a random initial condition) tends to cover the whole horizontal axis — except for some gaps formed by

sub-intervals madeD of Gardens of Eden, and isolated Isles of Eden which are not visible due to printing resolution. D Furthermore, the vignettes in Table 17 show that for these 18 rules the time-1 return map for an arbitrary initial condition tends to be very simiDlar (except for rule 73 ) to the time-1 characteristic function, as required by quasi-ergodicity. An accurate analysis reveals that rule 73 is an exception to this general behavior, because different initial conditions generate different time-1 return maps. Nevertheless, we noticed that rule 73 has only two kinds of possible time-1 return maps corresponding to two sets of mutually exclusive initial conditions (except for, as usual, a D Isles finite number of isolated periodic points or of Eden) exhibited in Table 17. Moreover, all initial conditions belonging to the same set give indistinguishable time-1 return maps. Hence, instead of having quasi-ergodicity all over the axis φn−1 , there are two regions, each exhibiting quasiergodicity. Therefore, we say that rule 73 is weakly quasi-ergodic, according to the following D definition.

D

Definition 2.2. Weak quasi-ergodicity. A rule is

said to be weakly quasi-ergodic iff it is quasi-ergodic D in a finite number of disjoint regions, whose union gives the whole axis φn−1 . Among the 256 local rules, 73 and its global topologically-equivalent rule 109 are the only D ones to exhibit weak quasi-ergodicity. The property of weak quasi-ergodicity of rule 73 is evident from Fig. 13, in which the two different time-1 return maps are depicted in Figs. 13(a) and 13(b) and then their union in Fig. 13(c) is compared with the time-1 characteristic function χ173 in Fig. 13(d). Finally, we can identify a further type D of rules within the quasi-ergodic ones thanks to the following observation. While it is true that the time-1 return map of most quasi-ergodic rules tends to cover almost everywhere the unit interval φn−1 ∈ [0, 1], sometimes relatively large compact regions are excluded (notice the gapD for D rule D 18 D in TaDbigD ble 16 when φ > 0.8); in general, this behavior is observable by increasing the length L of the bit strings. Nonetheless, for some rules we cannot see any gap in a generic time-1 return map independently from the length L of the bit strings. These rules — namely 30 , 45 , 60 , 90 , 105 , 106 ,

February 14, 2011

15:12

ch01

l ,

L = 601 Chapter 1: Quasi-Ergodicity

61

0.7;,

0.5

0.2:>

0.25

l ,

L = 601

l , L

4>"nlil (a)

0.5

0.75

= 601

(b)

0.7;,

0.7;,

0.5

0.5

0.2;,

0.2;,

0.75

4>,,-1

0.25

0.75

D

(c)

(d)

Fig. 12. Lamerey (cobweb) diagrams corresponding to three different initial conditions for rule 110 : Because of quasiergodicity, when the diagrams are superimposed they are practically indistinguishable.

February 14, 2011

62

15:12

ch01

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

Table 16. Lamerey (cobweb) diagrams for 18 complex and hyper Bernoulli-shift rules: Since these rules are quasi-ergodic, the qualitative features of their Lamerey (cobweb) diagrams do not depend on the initial condition.

1 <1>/1

0.75

18

22

L=203

L=101

0.5

0.25

0.25

0.5

0.75 ffi

'+' /1-1

1

0.25

1

_

0.75

30

1 _ - - - -..........- - - - _

26

L=201

<1>/1

0.5

0.75 ffi

'+'1'1-1

1

L=101

0.5

0.25

0.5

0.75 ffi

'+'1'1-1

1

0.25

0.5

0.75 ffi

'+' /1-1

1

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity Table 16.

l~~~ 41 L=601 0.75

(Continued )

1== 45

0.5

0.5

0.25

0.25

0.25

0.5

0.75 th

'+' /1-1

L=101

0.75

1

0.25

lr---_

0.5

0.75 th

'+' /1-1

1

lr------r----:

54

L=201

0.75

60

L=101

0.5

0.25

0.5

0.75 th

'+' /1-1

1

0.25

0.5

0.75 th

'+'/1-1

1

63

February 14, 2011

64

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 16.

(Continued )

-

1

1 ./

./

,///

<1>/1

0.75

73

<1>/1

L=80 /

//

73

0.75

./

/

,"

/

L=94 /

/' ./

/

/ "

1/

//'

0.5

/

/

0.5

1///

/11

,,/

/

0.25

/

./

./

./

/

0.25

./

"

./

./

00

0.25

0.5

0.75

/1-1

0.25

1

lr--<1>/1

0.75

0.5

0.75 It.

'V /1-1

1

1

90

105

L=101

L=201

0.5

0.5

0.25

0.5

0.75 It.

'V /1-1

1

0.25

0.5

0.75 It.

'V /1-1

1

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity Table 16.

(Continued )

1__----.--
0.75

106

110

L=101

0.5

0.5

0'25~

0.25

0.25

0.5

0.75 th

'V 1'1-1

1

0.25

0.75

0.5

0.75 th

'V 11-1

1

l r -......._

1 r----.=:::;

L=601

122

L=201

0.75

126

L=201

0.5

0.5W1E

0.25

0.25

0.25

0.5

0.75 th

'V 11-1

1

0.25

0.5

0.75 th

'V 11-1

1

65

February 14, 2011

66

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 16.

(Continued )

Jr------"""'T"'""--------,.

146 0.5

L=301

1

<j)n 0.75

150

L=201

E=

0.25

lr----

154

L=601

0.5

0.75 th

't' n-1

1

February 14, 2011

15:12

ch01

("")

a

--

on

r-

6

""II

-e-c

t-

on

Table 17. Quasi-ergodic vignettes of 18 complex and hyper Bernoulli-shift rules. For each rule it is depicted: (a) the time-1 characteristic function, where χ(φ) is in red (resp. blue) if the last bit of the string is 0 (resp. 1), and purple when both points coincide due to printer resolution; (b) the time-1 return map corresponding to the initial condition indicated in (c); (c) the space-time pattern corresponding to a fixed initial condition.

N

II

6

0

.r;

f0

N

6

II

....J

(".~

:c-

IIr

-e-

6

0

0

-.:t -.:t 0\ -.:t

0

("") V")

-e-

r("") N 0\ 00 V") ("")

r-

'0>

~

r("")

0

-e-

'<:;

("")

"J

II

f::

0

0

on

V)

0

~

rr-

o on

e: r--;

~

~

0\

0\

'-

s:: ":- ..e;, ~

0

~

0 II

~

0\ 00 00

~

-

~

~

':0:::.

'('oJ

II

-e
~

§]~'

<:;;

~

~ "" - I~

1.0 N 0 V)

r-

II

t-

'"'I

II

.

f-

..

g

' ~

""

~

'.

--C)

\...l

'"

-~c::,;

':>

..

'

.

"

.'

0

,,~'.J

.

-e-~ V)

6

~

"

",

.'

co

.

0\ N V")

co 0

.

' V)

N

I~

..,...... ~

-.:t -.:t

00

0

("") ("") ("")

0

0

....'.l

"'~ ~4

..j

...~

-N II

-e
-

')

..-- .•",

.

l

~.. '4

..

,

~

~ .....<1

......01

..."1

~.. 'J

..-I/. ,

.

I ..

~

"'~

..

.... .....

r,

67

(Continued )

(tJ ~¢J = 2- 11 y,.(q,) printed red (bl/le) ~ (b) L=27' 4 = 108 T=3239

b) L=IOO, T=2 17 =lJI,072 I

1ft

X~

'i'

0.7 )

l

,~

.,.-:

f

0..,

0I 0

t.. ' ~,.

,

I'~

1,,1. l ( ....

0.25

I 0.5

!

0.75

. •

~

J

0.5

:'

•

. [7 '"

If'," .. t..J#~

'

"' 0.25

J....

0.75 ~ 1

'1/ -

o

7

X~,. ~'....... •

'.

.

• / __I

....

...

"

0.5

0.5

r

..

7r 7-.

I Iii:

I

,

t

~'

0.2

,,\

0.25

; ; ' "..

"

"

.--,

o

/

.'

.

,

.'

0.5


0.25

':r 0 0.75

0

0.25

0.5

0.75

¢

-+--7,

'"

I

0,"::

o

X

~

""l'

1

I_

t...; : 025

•

I::, ?II

,'I 0.5

¢

075

n-I

n-I


't'T

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

I ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

,+"

't'T

••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••_1

ch01

'I!' f~ , l ....

1/ oj

.... t..

'"

0.2 )

I"...

15:12

] ,.j!laJ'~·~D~Cf.,.,~~I'~ kl

,

February 14, 2011

Table 17.

-

-/I

(a) t.<j> - 2

X(¢!) printed red (blue) iflast bit of ¢ = 0 (I)

I

~ r~ -t

XIm 0.7~<

./

f,

t

~

'i

~

0.2

{

o

0.5

~t

..',!

Il' ,/' 'i

0.25

\

..

+&

~

~

. 'tt.

,,-

t

.t

..

0.75

0.5

0.25

t '' i .. .t.. 0.75

,
I

o

- ..

...

/ .-

o

/ \

1/ .. 0.5

/ '

.

...

G>

. ,, .

(b) L=lOO, T=2'7=131,072

rlI

0.5

L.

V

&

0.25 -1-"" Y

0.2

A

'''I

•

....,

..- I ~

00

0.75

0.25

0.5

0.75 G> I

oV o

"

I

0.25

0.5

G>

I~i~~"

0.75

n-I

n-I

(c)

iflastbitof¢!=O(l)

0.)

. •

0.25

I


(c)


<1>0

<1>0

69

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

T 1••--••••••••__•••__••••••••__•••__••••••••__•••__••••••••__•••_ ••••••••••__•••••


T

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

ch01

O. ,.

'" t ..

c'

n

ILl ~hr~,~J ,":U ;;JS]4~l3Z1 (a) tHjJ = 2-1/ x(q») printed red (blue)

T=2'7 =1 31,072

15:12

I~

~

1

IIIE.m (b) L=l 00,

(Continued )

February 14, 2011

Table 17.

(a) Li<jJ = 2~tt 1

a

X(4l) printed red (bIUe)m iflastbitof4l=O(l) ~ (b) L=IOO, T=z!7=13J,072

IT]

f,~? r"."4." ~

,.

I <1> 0

0.75

I-t;

... ..

.,

. .", . .. . ~

'

0.5

...

0.25

......

~

.."

"

•....no

......

~~

••

Or 0

0.25

0.25

0.5

0.75

•


1

0

.

.

1.

o

0.25

0.7

J/

1/ 1/ ~

[7 x~Cf\~ ,..

'"

......

' f

" ..

~"'"

"

~~ lI-"'"

• .:!

.

0.2

~

"'f

1 "

o

0.5 <1> 0.75 0-1

,.",., IA"

0./5

~

~

0.25

0.5

0.5

r

0.25

I

~J.~

'1-"

,.""jo

0.75


o

1

70

iEE! EI

EI

iUl I

I(

'1"',


-

,.

X

..

I

,

I 0.25

"

,''\'

iIEEiiil·.'EI I,EEEII IIIII IBiEiBil!llllil

..

..

iI Ii -

iii

ii

-

IE

, I

0.395283707836447

Ii

,

I

0.5 0.75 n-l

,

0

I...

..,...

!ill. iiii i Ei -

1l1li

OJ

fti

IEIIIIIIIE.IIIIIEIEIC_II"I':IIIEIIIII_III IIIII IEIIIEI!I!llliill, .liiliil-EliiI-IIII!I!I!II· 1!IEliil!lc IIIIIII!II

:111

or

1;&11

o

00

iEI ii EEl . iE !iii

, 7

~

,. I'jo

'-\

m OJ

,

1'...

,.~

#'"

1 \'..

(c)
,.~

,

I $~ID.~"\J/"0

x

O. -l

~ (0) L=IOO 1'=2 17=131072

x(
iflastbitofifl=O(l)~'

ch01

"'o,

r..

0.5

.. $o .. t

(a) M = 2~11

15:12

'I

X~~ lEI

(Continued )

February 14, 2011

Table 17.

ti

III : iii iEi .

-

,.

ti

il

III III - II . i!i

. . ._. ._. . . . . . _._ . . . . . . _._._ . ...............

.

(a)

L1~ 1

= 2- 11

(b) L= 100, T=i 7 = 131,072

bit of <jl- 0 (1)

I

1

-

I

I

..

I

I

I ¥~

•

0.75

I

...

~

.

•

..

"X

I

.

...

0.75

.1

/...

0.5;

'oJ

~

\"

01 0

..

!

,-..

...

I

eM

0.2$

'II

Va-

0.$

I

0.7$

'\ I


0.25 ...

o-v:

., 1

0

A

~

.•

I

i

0.25

OJ

.1

~.' J~
.,..

0.2j;

.

0.75

u

\l' .~

\. ~_\a\ 00

..

0.25

\1'

"lJ~

\., '\

O.bl x

~,

v~

'r \.

I

XIZIl .

'II,

...;

0.5

~'"~

-~

0.5

\.'

0.75

0.25

\" \'

-~


o

1

(c) 71




/

1/ /r"l\ ~.

.' •

.....

o

0.25

0.5

...
0.75

n-I

n-I

(C)

1/

..

\'

\

\l'

\.

0.75

\.,

'

. • \ -.,\...

~

ch01

0.5

I

1r>'

1


I

~

XIZll 0.751 11'

red (blue) . 17 (a) L1<jl = 2- 11 ifx<4»lastprinted bit of If = 0 (1 ) (b) L = 100, T = 2 = I3 I ,072

(c)


15:12

K

I

X<'I» printed red (blue)

if last

(Continued )

February 14, 2011

Table 17.

(a) M = 2- 11 ~(ij))printed red (blue) ~ 1

iflast bit ofij)= 0 (1)


'[IJ '~.v' I.,

0~5

•

t

r

0.25

I


;;

&.. ,

I

co

Y.

'i

,i;

0.25

A

Y

1"

I

I

.... I ,

• I ....

,'i,

I

I

1

ch01

0.5

J2J

\'

15:12

(c)

17

T=2 =131,072

I

I.

o r, o

~o

(Continued )

,

- ' ( b ) L=100,

~ X~~ ~

1

February 14, 2011

Table 17.

I

l

o

0.5 . 0.75 n-l

0.459111017087255

0.25

0.5


0.75

n-I

(c) ~o


0.210557453751221

72

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

•••••_ ••••••••••••••••••••••••••••••••••••••••••••••••-_._._-_._••••_•••


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

T

I••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

a ~ =2- 11 x(¢)prinledred (blue)

(V

I

I

XlmI

.rIT"(b) L= 100, I

I

-.

•

0.75

i

i

(a) M = 2- 11 xJ~)prif1ted

T=2 17 = 131,072

&MIiM n

I"

0.751

(Continued )

..

>.

i

7

7l

XrriijIIL~j;~·J3: '" '" I

1

I

;

'1

J

I

W

II

..

I

red (blue)

A

~ a~

I

A

.--

P\

r

r

.1.'

0.5

I

0.25 -f----:J'

0.25

0.5

,. r

Y

I,'

~

.. I

05

0.75

0.5

I~

.-A

0.2:J;

0 025

'I' n

.' •

~5

'I.

0.5

,.1;'

f

....... , t •

0.25

0

0.25

0.5

0.75


I

o

------:."'1, ,

7

o

JI.

t/ '/

<1>0


0.454963695923757;

~.. .

'l,

it'"

•"

(c)

0.25


¢T

••••••••••••_

0.5

0.75

0.993892907368291

.................... • • • • • • • • • •_ . _ • • _

•••••••••••••••_

•••••_

II

';.

73

<1>T

,,-

11-1

¢o

\

#1.

.. ....

~

,

n-I

(c)

,'i. .'

• • • • • • • • • •_

••••••••_

•••••_

• • • • •1

ch01

•

'.

L=23'4 = 92 T=1057

I"

A

0.51

r

I'WII' (b)

15:12

I

ifZast bit of ~ = 0 (I) i . Ji i 11"


February 14, 2011

Table 17.

(Continued )

(a) t. = 2- 11 x(t!Jlprinted red (hltre) I'I:ft (b) L = 100 T = 2 18 = 262/44

I.

I

"1 ~I

~ ~, ,.~

ft.' If'

,.

~' .if,

l 0

"

"". If ,.\

~'

_. 0.2:'

. 0.'

"\

~

I

,....

0.25 ~

0.75 I

.~;/ o

1/ '/

0.25

0.7)

,,

"

,.

. '\

o. •

II:IP.I

I

0.75

"

" 0.5

0.25

,.

:f'

. -0.2:,

-0.5

0.75 I

o

262,/44

. "JV / . I

/

.-

;/

f

I· I I

a

0.25

0.5

0.75

n-l

n-l

(c)

.

=

.

,"

\

o

I

n 0.75

"

'y- ,

l .

1

~.

~ 4 j

-' j

l'

,"",-1

18

l.a!.I(b) L = 100, T = 2

r - - ....~

...

0.2 \

/

0.5

_

~

fi

I

a

if last bit old;! = 0 (I)

I

["""-;f" I f/ XIillJO-:....~"U ,-

Il

It

.

, ,"

0.5

I'

,. ..

n

~,.-

Ii'

,.j.

0.75 ~

,,\

1

(aJ t.<jl = 2- 11 X(d;!) printed red (hllle)

,

ch01

0

I

fIo-

If""

0.5I~"

,

fl.'

"I~

,~

.\ .1

0.2

,..

,

<1>0 = 0.587057386127713; T= 0.952197075820593

(c)

74


<1>0 = 0.700012154501655;


15:12

XIilll 0,7 ,

~

iI/astbitold;!=O(I) .I> I .... Mil.'",

February 14, 2011

Table 17.

0.809338561752269

a' L'l. = 2- 11 X(<j» printed red (hlue)

0')


iflaslbitof<j>=O(l)

0.7.5

•

W

I

...

' "1· ~

:

n_

.

0.7)

It

~.

J••

I"

J

0.5

• l.

0.2..

0

0.25

0

0.25

0.5

T = 2 18 = 262,144

0.75 <1> 1

o

l/ o

/ ~.

~

0.25

7

7

ch01

0.5f

(Continued )

15:12

'1 0

xlillJ ---d:. , 1I

I...E.fiI (b) L = 100,

February 14, 2011

Table 17.

,

i>~

0.2.51

I

. to

.

~-~

/.

.

11

I

'I

T

...

I..

«I

o ar

I

05 ~

i

075

<1>0=

0.4285713689631575;

<1>T=

Ii 0.25

I

;a , i ~i

« II '

I

05 ~

0.75 n-t

n-t

(c)

,

0.172138288850949

(c)

<1>0 = 0.25169916877883; <1>T=

0.210557453751221

¢o

¢o 75

¢ T , ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••

1

th

'l'T

I

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

.

r

February 14, 2011

15:12

ch01

(aJ 76

,:1,.<1> =

J

2-1/ x(¢!) printed red (blue) if fa:>t bit oj = 0 (l)

~ (b)

I.Di.I I

L = 100 T = 2 17 = 131 072

'

,

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

Xl[ill]

I,'"

O.7:>-rI,

III,

I

'..

Table 17.

~n

,

• (

)'

0.5 I'

o <1> J

(c)

<1>0 =

0.250062064340869;

1/

--p I I

0.25

o

<1>1'=

~

0.25

J'

1/

/

.

(Continued ) 075

.

;.

i ,: 0.5 ~ 0.75 n-1

0.154807393439345


T

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

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

DD

150 , 154 — will henceforth be called strongly quasi-ergodic.11 Definition 2.3. Strong quasi-ergodicity. A quasiergodic rule is said to be strongly quasi-ergodic iff its time-1 return maps tend to cover the whole unit interval φ ∈ [0, 1], except for isolated points, independently from the bit string length L. The gaps are created by three different mechanisms: First, the transients, because we do not display them in the Lamerey (cobweb) diagrams; 11

second, the Isles of Eden; and third, the Gardens of Eden.

2.2. Quasi-ergodicity and Gardens of Eden From Definition D D2.3, it is clear that a necessary condition for strong quasi-ergodicity is the absence of non-isolated compact sets of Gardens of Eden since, as we mentioned previously, they generate gaps on the horizontal axis. As an illustration, Table 18

It is also possible to prove that strongly quasi-ergodic rules, along with 15 and 170 , are actually ergodic, according to a formal mathematical definition [Shirvani et al., 1991; Shereshevsky, 1992].

February 14, 2011

15:12

ch01

III 1

.."

0.75

. 0.5

0.25

o

III

L=100

V

""

4"....

V VI

•. '" • ......

V

L=100 Chapter 1: Quasi-Ergodicity

77


-I-....:.<..---+---+---------:,f-------I

0.5

-1:----+----,1£----+-------1

~

~

..

0.25

-1_-'---_+-_ _+--_ _+--

----1

~

~

~

..

\

I

o

0.25

0.5


o

0.75

0.25


0.5

0-1

III .."


r@]

(a)

.."

0.75

III

L=100 1

....

•,..

""

..

0.5

--\.,

0.25 ~

~ ..

0

...,

\"

. ....

~

"',

0.25

~

-~

,

~

\:~

\1,

0.25

0.5


\,'\

-,

~

~~\ ~, \'\

"\.':\.,

\., ~\.:\ o

.

~ ~,

~':\.\

~

0 0.75

.,

~

\.\

• "-

'\.

0.5

,

(b)

\t, ~

....

0.25

..."

0.75

*...

0

1',

~

0.5

0.75

0-1

-,

~

0.75

D

(c)

(d)

Fig. 13. Rule 73 exhibits a weaker form of quasi-ergodicity with 2 distinct time-1 maps (a) and (b), whose union in (c) resembles the time-1 characteristic function in (d). Points in (d) not present in (c) are transient points.

February 14, 2011

15:12

ch01

lEI Strings starting with (111 ... ) are GoE => 0.875 S 9< 1 78

Xl A Nonlinear Dynamics Perspective of Wolfram’s New I Kind ofI Science I I I I I

HI

I

III

0

Table 18.

O.125 0.25 0.375

0.5

0.625 0.75

I

0.875

lei>

1

Non-strongly quasi-ergodic rules have compact sets of Gardens of Eden.

Strings starting with (010101001 ... ) are GoE =>0.33 < ¢ < 0.332 Xl

I

'III 0

I

I

0.125 0.25 0.375

0.5

0.625 0.75

0.875

lei>

0.625 0.75

0.875

1

III Strings starting with (1111 ... ) are GoE => 0.93 < ¢ < 1 Xl

m

I

'm 0

I

I

I

0.125 0.25 0.375

0.5

lei>

Strings starting with (10111 ... ) are GoE=>0.719s¢<0.75 Xl

I

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

lei>

III Strings starting with (10101 ... ) are GoE::::} 0.656 ~ 9< 0.687 Xl

m

•

I

I

I

0

I

I

I

0.125 0.25 0.375

I

0.5

I

I

0.625 0.75

0.875

14>

1

Strings starting with (100111 ... ) are GoE=> 0.609 s Xl

'ID

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

.4>

11m Strings starting with (01010... ) are GoE=> 0.3125 ~ X}

¢ < 0.625

¢< 0.344

I

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

lei>

1m Strings starting with (00100... ) are GoE => 0.125 S ¢ < 0.156

11m Strings• starting with (010... ) are GoE => 0.25 < ¢ < 0.375 Xl

I

I

0

Xl

1m

I

I

I

0.125 0.25 0,375

I

0.5

I

I

0.625 0.75

I

0.875

I

I

0

I

I

I

0.125 0.25 0.375

I

0.5

I

I

0.625 0.75

I

0.875

lei>

1

14>

1

I!rd Strings starting with (111001 ... ) are GoE => 0.89 < ¢ < 0.906 Xl

1m

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

lei>

February 14, 2011

15:12

Ell III

ch01

X. Xl

Table 19.

ImI]

0

X• Xim

lID

XII X. Xl

0.125 0.25 0.375

0.5

0.625 0.75

0.875

l1li

I

0.125 0.25 0.375 I

I

I

0.5 I

1

Chapter 1: Quasi-Ergodicity

Strongly rules sets Iof Gardens I quasi-ergodic I do not Ihave compact I I of Eden.I I

• •ImJ t. 111m

0

0.625 0.75 I

I

I

0.875

1

I

I

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

1

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

1

I

0 I

I

I

I

0.125 0.25 0.375 I

I

I

0.5

I

I

I

I

0.875

1

I

I

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

1

0

0.125 0.25 0.375

0.5

0.625 0.75

0.875

1

I

0

I

I

I

I

0.125 0.25 0.375

gives examples of non-isolated compact set of Gardens of Eden for those quasi-ergodic rules that are not strongly quasi-ergodic. In contrast, Table 19 shows that strongly quasi-ergodic rules can have only isolated Gardens of Eden. Finally, we note that all strongly quasi-ergodic rules are also permutive (see Tables 8 and 9). This follows from [Hedlund, 1969], who proved that in one-dimensional CA only permutive rules are surjective, in the sense that they have no Gardens of Eden when L → ∞.12 Hence, only permutive rules can satisfy our necessary condition for strong quasi-ergodicity.

I

I

0.625 0.75

I

0.5

I

I

I

0.625 0.75

79

I

I

0.875

I

functions of one-dimensional Cellular Automata, and some partial explanations for this phenomenon were given. In this section we prove a general theorem that gives more thorough results and provides an analytical formula for finding the fractal patterns.

3.1. All time-1 characteristic functions are fractals

3. Fractals in 1D Cellular Automata

In order to specify explicitly the number of bits contained in a generic string x, we introduce the notation . . (15) xI+1 = (x0 . . . xk−1 .. ..xI )

In [Chua et al., 2005b] it was shown that fractals arise naturally in the time-1 characteristic

to indicate the set of bit strings composed of I + 1 elements in which the first k elements and the last

12

When L is finite, injectivity implies surjectivity and vice versa, as proved in [Moore, 1962] and [Myhill, 1963].

February 14, 2011

80

15:12

ch01

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

. . one are known. For example, x6 = (01.. ..1) is equivalent to the set {(010001), (010011), (010101), (010111), (011001), (011011), (011101), (011111)}. Since any bit string x can be uniquely associated with a real number φ(x) ∈ [0, 1), the set of bit strings defined by Eq. (15) corresponds to a subset of points of the unit interval [0, 1], with the formula

D (xI+1 ) = φI+1 (x0 . . . xk−1... ...xI ) φI+1 =

D

I


D i=0

xi 2−(i+1)

(16)

Similarly, the time-1 characteristic function for the local rule N corresponding D to the set of bit strings φI+1 (xI+1 ) is defined by χ1N

,I+1

D

(xI+1 ) = χ1N

D =

I
i=0

,I+1

(a)

. . (x0 . . . xk−1 .. ..xI )

T N (xi−1 xi xi+1 )2−(i+1)

D

(17)

where T N (xi−1 xi xi+1 ) denotes the application of the local rule DN to the triplet pattern (xi−1 xi xi+1 ). To avoid clutter, we will usually delete the commas (xI+1 ) by separating bits, and abbreviate χ1

D

D

N ,I+1

χI+1 (xI+1 ).

Theorem 3.1. Fractality of the time-1 characteris-

tic function χ1 .

(b)

N

The time-1 characteristic function

χ1 N

of any

rule N exhibits a fractal behavior in any subinterval of φ ∈ [0, 1). Given an arbitrary natural number I > 1, .. .. we define the two bit strings xα I+1 = (0. .0) and .. .. xβ I+1 = (1. .0). According to the notation introduced previously, xα I+1 is the set of all bit strings composed by I + 1 elements with x0 = 0 and xI = 0, whereas xβ I+1 is the set of all bit strings composed by I + 1 elements with x0 = 1 and xI = 0. Their equivalent decimal representations . . . . are φα = φI+1 (0.. ..0) and φβ = φI+1 (1.. ..0), and it is easy to prove that 0 ≤ φα < 1/2 and 1/2 ≤ φβ < 1. Since φα ∪ φβ = [0, 1) as I → ∞, we can divide the time-1 characteristic function χ1 Proof.

D

D

N

for a generic rule N into two parts, one for each . . interval: χα = χI+1 (0.. ..0) in correspondence to φα

Fig. 14. The four fractal patterns χα , χβ , χγ and χδ and their position in the time-1 characteristic function.

. . and χβ = χI+1 (1.. ..0) in correspondence to φβ [see Fig. 14(a)]. An analogous procedure can be followed when xI = 1, leading to the definition of the two . . . . strings xγI+1 = (0.. ..1) and xδI+1 = (1.. ..1), and their equivalent decimal representations φγ = . . . . φI+1 (0.. ..1) and φδ = φI+1 (1.. ..1), where 0 < φγ < 1/2 and 1/2 ≤ φδ < 1. Also in this case the time-1 characteristic function can be divided into . . two parts: χγ = χI+1 (0.. ..1) in correspondence to . . φγ and χδ = χI+1 (1.. ..1) in correspondence to φδ [see Fig. 14(b)]. Let us consider a generic set of bit strings xk+I+1 , each composed of k +I +1 elements, having

February 14, 2011

15:12

ch01

D

D in common the first k + 1 and the last element k bits







xk+I+1 = (x0 x1 . . . xk−1

I+1 bits

  .. .. xk . .xk+I )

i=0

(18)

According to the formula (16), we can associate to the set xk+I+1 a unique interval of values in [0, 1) called φk+I+1 , in which a certain χk+I+1 can be defined. Our purpose is to explore the behavior of the function χk+I+1 in subintervals of φk+I+1 , thereby showing the emergence of fractal patterns. In general, splitting φk+I+1 into n parts corresponds to introducing log 2 n bits after xk in the set of bit strings xk+I+1 . For example, we can subdivide φk+I+1 into four regions just by adding two extra bits — called xleft and xright — after xk in (18), and creating the new set of bit strings x
k+I+3  x
k+I+3

k+2 bits





= (x0 x1 . . . xk xleft  =

k+I+3 bits



  .. .. xright . .xk+I )

i=0

i=k+2

T N (x
i−1 x
i x
i+1 )2−(i+1)

T N (x
i−1 x
i x
i+1 )2−(i+1)

(21)

Let us analyze D separately the two terms of this last equation. By using the formula (20), we find that the first Dto term is equivalent k+1
i=0

T N (x
i−1D x
i x
i+1 )2−(i+1) 1 = T N (xDk+I−1 x0 x1 ) 2 k−1


D

T N (xi−1 xi xi+1 )2−(i+1)

1

o

T (xk−1 xk xleft ) 2k+1 N D 1 + k+2 T N (xk xleft xright ) 2

D

+



χk+I+3 (x
k+I+3 )

k+I+2


i=1

(19)

The first of the four subregions corresponds to the case (xleft xright ) = (00), the second subregion to (xleft xright ) = (01), the third subregion to (xleft xright ) = (10), and finally the fourth subregion to (xleft xright ) = (11). The expression for the D function χk+I+3 (x
time-1 characteristic k+I+3 ) can be found by using the formula (17)

81

T N (x
i−1 x
i x
i+1 )2−(i+1)

+

The correspondence between xk+I+1 and x
k+I+3 can be obtained by comparing (18) and (19) 
x0 = x0    ..    .   
 x k = xk     x
k+1 = xleft (20)
 x = x right  k+2     x
k+3 = xk+1     ..    .  
xk+I+2 = xk+I

=

+

I+1 bits

(x
0 x
1 . . . x
k+I+2 )

k+I+2


k+1


=

Chapter 1: Quasi-Ergodicity

(22)

Since for any local rule N the value D T N (xi−1 xi xi+1 ) is either 1 or −1, from Eq. (22)

k+1


−(i+1) is a it follows that i=0 T N (xi−1 xi xi+1 )2 rational number, with 0≤

k+1
i=0

T N (x
i−1 x
i x
i+1 )2−(i+1) ≤

D

k+2


2−j < 1

j=1

(23)

The second term of the formula (21) can be written as D k+I+2
i=k+2

T N (x
i−1 x
i x
i+1 )2−(i+1)

=

D

I
j=0

=

=

T N (x
j+k+1 x
j+k+2 x
j+k+3 )2−(j+k+3)

1 2k+2 1 2k+2

D

I
j=0



T N (x
j+k+1 x
j+k+2 x
j+k+3 )2−(j+1)

1 T (xleft xright xk+1 ) 2 N

March 12, 2011

10:18

ch01

-0 82

o

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

1 +-T0 (xright xk+1xk+2 ) 4 N I−1


+

1 +o 2

I+1




1 2k+2

D

T N (xj+k−1 xj+k xj+k+1 )2−(j+1)

j=2

TN

 (xk+I−1 xk+I x0 )

. . χI+1 (xright .. ..xleft )

(24)

because the actual outcome of the term (1/2I+1 )T N (xk+I−1 xk+I x0 ) is negligible for I  1, and (xk+1 . . . xk+I ) are arbitrary bits, as stated in (18). Therefore, using (22) and (24), we can rewrite the expression (21) as follows:

o

χk+I+3 =

k+1
i=0

T N (x
i−1 x
i x
i+1 )2−(i+1)



+





vertical shift

1

. χI+1 (xright ..

2k+2

  

.. .xleft )

(25)

scale factor

This means that in each subregion there is a . . scaled and shifted copy of χI+1 (xright .. ..xleft ).

χ 1

In particular, in the first subregion χI+1 . . . . (xright .. ..xleft ) = χI+1 (0.. ..0) = χα , in the second .. .. . . subregion χI+1 (xright . .xleft ) = χI+1 (1.. ..0) = . . χβ , in the third subregion χI+1 (xright .. ..xleft ) = . . χI+1 (0.. ..1) = χγ , and in the fourth subregion . . . . χI+1 (xright .. ..xleft ) = χI+1 (1.. ..1) = χδ . Since k and I are arbitrary, it follows that D the time1 characteristic function of a generic rule N is composed of the four fractal patterns χα , χβ , χγ and χδ (in this order) in any subinterval of φ ∈ [0, 1).


χ

xI = 0

110

As an example, let us consider rule 110 : Its fractal patterns for the case xI = 0 are shown in . . Fig. 15(a), where χα = χI+1 (0.. ..0) is plotted in . . red and χβ = χI+1 (1.. ..0) is in purple; the fractal patterns for the case xI = 1 are in Fig. 15(b), . . where χγ = χI+1 (0.. ..1) is plotted in blue and . . χδ = χI+1 (1.. ..1) is in cyan. According to Theorem 3.1, these four patterns appear naturally in any subinterval of φ ∈ [0, 1). For instance, let us consider φ ∈ [0.25, 0.5), which corresponds to the set of bit strings xI+2 = . . (01.. ..0); the restriction of χ to φ ∈ [0.25, 0.5) is depicted in Fig. 15(a). Let us divide this interval into four parts by adding two additional bits: the first subregion corresponds to the case . . xI+4 = (0100.. ..0) and 0.25 ≤ φI+4 (xk+I+4 ) <

1

0.75

0.75

0.5

0.5

0.25

0.25

xI = 1

110

D 0 0

0.25

0.5 (a)

0.75

1

φ

0 0

0.25

0.5

0.75

1

φ

(b)

Fig. 15. Time-1 characteristic function of rule 110 : (a) case xI = 0, (b) case xI = 1. The fractal pattern χα , χβ , χγ and χδ are red, purple, blue, and cyan, respectively.

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

χ 1

χ

x I = 0 and 1

110

0.875

0.9375

0.84375

0.875

0.8125

0.8125

0.78125

83

x I = 0 and 1

110

o 0.75 0.25

0.3125

0.375

0.4375

0.5

φ

0.75 0.375

0.40625

(a) Fig. 16.

. χI+4 (0101..

. χI+4 (0110..

0.46875

0.5 φ

(b)

Fractality of χ1

110

: The four fractal patterns appear in two arbitrary subintervals of φ ∈ [0, 1).

. . 0.3125, the second subregion to xI+4 = (0101.. ..0) and 0.3125 ≤ φI+4 (xI+4 ) < 0.375, the third . . subregion to xI+4 = (0110.. ..0) and 0.375 ≤ φI+4 (xI+4 ) < 0.4375, and the fourth subregion to . . xI+4 = (0111.. ..0) and 0.4375 ≤ φI+4 (xI+4 ) < 0.5. The analytical expressions for χI+4 (xI+4 ) in the four subregions can be derived from the formulas (22) and (25), and the results are illustrated in Fig. 16(a). In particular, . χI+4 (0100..

0.4375

1 1 .. .0) = T 110 (001) + T 110 (010) 2 4 -0 -0 1 1 . . + T 110 (100) + χI+1 (0.. ..0) 8 8 -0 3 1 1 1 1 = + + χα = + χα 4 8 2 4 8 1 1 .. .0) = T 110 (001) + T 110 (010) 2 4 -0 -0 1 1 . . + T 110 (101) + χI+1 (1.. ..0) 8 8 -0 1 1 1 1 7 1 = + + + χβ = + χβ 2 4 8 8 8 8 1 1 .. .0)  T 110 (001) + T 110 (011) 2 4 1 1 . . + T 110 (110) + χI+1 (0.. ..1) 8 8

1 1 1 1 γ + + + χ 2 4 8 8 7 1 = + χγ 8 8 1 1 .. .0)  T 110 (001) + T 110 (011) 2 4 1 1 . . + T 110 (111) + χI+1 (1.. ..1) 8 8 1 1 1 = + + χδ 2 4 8 3 1 = + χδ 4 8 =

. χI+4 (0111..

As expected, the time-1 characteristic function χI+4 in the four subregions is a scaled and shifted version of χα , χβ , χγ and χδ , respectively. In Fig. 16(b) a further subdivision of the horizontal axis 0.4375 ≤ φI+6 (xk+I+6 ) < 0.5 is depicted, and also in this case the four fractal patterns emerge naturally.

DOD

D

3.2. Fractals in CA additive rules Further analytical results can be given for nontrivial additive rules — namely 60 , 90 , 105 and 150 — which will be extensively analyzed in Sec. 6. These rules can be either described in terms of Boolean operations (see Sec. 6), or by using absolute values and standard arithmetic operations. The relationship between these two kinds of representations can

February 14, 2011

84

15:12

ch01

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

D

be derived straightforwardly; for example

Dxi xi+1 ) = |xi−1 − xi | 60 : xn+1 = xni−1 ⊕ xni → T 60 (xi−1 i

(26)

= xni−1 ⊕ xni+1 → T 90 (xi−1 xi xi+1 ) = |xi−1 − xi+1 | 90 : xn+1 i

(27)

105 :

= xni−1 ⊕ xni ⊕ xni+1 → T 105 (xi−1 xi xi+1 ) = 1 − ||xi−1 + xi + xi+1 − 1| − 1| xn+1 i

(28)

150 :

= xni−1 ⊕ xni ⊕ xni+1 → T 150 (xi−1 xi xi+1 ) = ||xi−1 +Dxi + xi+1 − 1| − 1| xn+1 i

(29)

D D these four In the following we consider individually rules, finding for each of them an analytical formula for the lines delimiting their fractal patterns. D

D

3.2.1. Rule 60

The time-1 characteristic function χ 60 can be obtained from Eq. (26): χ 60 =

D

I


patterns are presented in red, purple, blue and cyan, respectively. The function χ 60 can be bounded by using the following D formula, whose proof is given in Appendix A

I

I







−(i+1) −(i+1) 1− xi−1 2 + xi 2 − 1

i=0

T 60 (xi−1 xi xi+1 )2

−(i+1)

=

D

|xi−1 − xi |2−(i+1)

(30)

i=0

i=0

χ 60 is plotted in Fig. 17(a) for xI = 0, and in Fig. 18(a) for xI = 1. The four fractal

χ 1

i=0

D

i=0

⇒ sup 60 ≥ χ 60 ≥ inf where sup 60 (inf of χ 60 .

χ

xI = 0

60

D

≥ χ 60 D

I D

I





≥ xi−1 2−(i+1) − xi 2−(i+1)

i=0 I


D

1

0.75

0.75

0.5

0.5

0.25

0.25

60

(31)

60

) is the supremum (infimum)

xI = 0

60

D 0

0

0.25

0.5 (a)

0.75

1

φ

0

0

0.25

0.5

0.75

1

φ

(b)

Fig. 17. Time-1 characteristic function of rule 60 , case xI = 0: The fractal pattern χα is in red and the fractal pattern χβ is in purple.

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

χ 1

χ

xI = 1

60

1

0.75

0.75

0.5

0.5

0.25

0.25

85

xI = 1

60

D 0

0

0.25

0.5

0.75

1

φ

0

0

0.25

(a)

0.5

0.75

1

φ

(b)

Fig. 18. Time-1 characteristic function of rule 60 , case xI = 1: The fractal pattern χγ is in blue and the fractal pattern χδ is in cyan.

Moreover, the term I
i=0

I

−(i+1) i=0 xi−1 2

D

xi−1 2−(i+1) =

D

I−1


j=−1

appearing in (31) can be simplified as follows

D

I−1

1 1
1 1 xj 2−(j+2) = xI + xj 2−(j+1)  xI + φ 2 2 2 2

(32)

j=0

D

In the last step we have discarded a term proportional to 2−(I+1) , which is extremely small for I  1. The analytical expression for sup 60 and inf 60 can be found by combining (31) and (32)

D

60

I

 I





 1

 −(i+1) −(i+1)  : 1 − x 2 + x 2 − 1 sup

= 1 − |3φ + xI − 2|, i−1 i  60 

2  i=0 i=0

 I

1


DI 


−(i+1) −(i+1)   : x 2 − x 2 inf

= |φ − xI |.

i−1 i  60

2

i=0

i=0 D -

Therefore, the equations for the upper and lower bounds are



3

D 

 (a): sup 60 = 1 − 2 φ − 1

, φ ∈ (0, 1) for xI = 0,   (b): infD = 1 φ, φ ∈ (0, 1) 60 2



3 1



 (c): sup 60 = 1 − 2 φ − 2 , φ ∈ (0, 1) DxI = 1, for  1 1  (d): inf φ ∈ (0, 1) 60 = 2 − 2 φ, and the letters correspond to the lines in Figs. 17(b) and 18(b), respectively. Furthermore, in Fig. 19 we show the emergence of the fractal behavior since the four subpatterns appear when the function χ 60 is restricted to arbitrary subintervals of the axis φ.

February 14, 2011

86

15:12

ch01

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

χ 1

χ

xI = 0

60

0.9375

0.984375

0.875

0.96875

0.8125

0.953125

D 0.75 0.5

D0.5625

0.625

φ 0.6875

(a) Fractality of χ1

Fig. 19.

60

0.75

xI = 0

60

1

φ

0.9375 0.625

0.640625 0.65625 0.671875 0.6875 (b)

D

: The four fractal patterns appear in two arbitrary subintervals of φ ∈ [0, 1).

D

D

3.2.2. Rule 90 The time-1 characteristic function χ 90 can be obtained from Eq. (27): χ 90 =

I


T 90 (xi−1 xi xi+1 )2−(i+1) =

i=0

I


|xi−1 − xi+1 |2−(i+1)

(33)

i=0

It is plotted in Fig. 20(a) for xI = 0, and in Fig. 21(a) for xI = 1. As usual, the four fractal patterns are depicted in different colors.

χ 1

χ

xI = 0

90

1

0.75

0.75

0.5

0.5

0.25

0.25

xI = 0

90

D 0 0

0.25

0.5 (a)

0.75

1

φ

0 0

0.25

0.5

0.75

1

φ

(b)

Fig. 20. Time-1 characteristic function of rule 90 , case xI = 0: The fractal pattern χα is in red and the fractal pattern χβ is in purple.

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

χ 1

χ

xI = 1

90

1

0.75

0.75

0.5

0.5

0.25

0.25

87

xI = 1

90

D 0

0

0.25

D

D

0.5

0.75

1

φ

0

0

0.25

0.5

(a)

0.75

1

φ

(b)

Fig. 21. Time-1 characteristic function of rule 90 , case xI = 1: The fractal pattern χγ is in blue and the fractal pattern χδ is in cyan.

Similarly as for rule 60 , the time-1 characterisD for rule 90 can be bounded as follows: tic function

I I







−(i+1) −(i+1) 1− xi−1 2 + x 2 − 1

D D Di+1

i=0

≥ χ 90

i=0

I I







≥ xi−1 2−(i+1) − xi+1 2−(i+1)

D

i=0

⇒D sup 90 ≥ χ 90 ≥ inf

D

90

i=0

90

(34)

The term Ii=0 xi+1 2−(i+1) appearing in (34) can be simplified as follows: I
i=0

xi+1 2−(i+1) =

I+1


xj 2−j + x0 − x D0  2φ − xD0

j=1

(35) Here, we have discarded a term proportional to 2−I , which is extremely small for I  1. The analytical expression for sup 90 and inf 90 can be found by combining (34), (32) and (35)



 I I





1 

−(i+1) −(i+1)   : 1 − x 2 + x 2 − 1 = 1 − |5φ − 2x0 + xI − 2|, sup

i−1 i+1  90 

2  i=0 i=0

D I I 




1
 

−(i+1) −(i+1)  inf 90 : xi−1 2 − xi+1 2

= |3φ − 2x0 − xI |.  

 i=0 i=0 D - 2

Therefore, the equations for the upper and lower bounds when xI = 0 are



5

 D

  (a): sup 90 = 1 − 2 φ − 1 , for (x0 , xI ) = (0, 0)  D -3   (b): inf 90 = φ, 2



5



,  (c): sup = 1 − φ − 2  90

 2 for (x0 , xI ) = (1, 0)



3

 

(d): inf 90 = φ − 1

, 2



1 φ ∈ 0, 2  1 φ ∈ 0, 2  1 φ∈ ,1 2  1 φ∈ ,1 2

February 14, 2011

15:12

ch01

D 88

--

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

D

-

-

where the letters correspond to the lines in Fig. 20(b), whereas when xI and lower bounds are

 D

5 -1



  (e): sup 90 = 1 − 2 φ − 2 , for (x0 , xI ) = (0, 1)

D 

31-

 

(f): inf 90 = φ − , 2 2



5 3



D   (g): sup 90 = 1 − 2 φ − 2 , for (x0 , xI ) = (1, 1)  3 3   (h): inf 90 = − φ + , 2 2

= 1 the equations for the upper

D



1 φ ∈ 0, 2  1 φ ∈ 0, 2  1 φ∈ ,1 2  1 φ∈ ,1 2

lines in Fig. 21(b). where the letters correspond to the D The fractality of χ 90 is evident from Fig. 22, in which we show that the four subpatterns appear in two arbitrary subintervals of the axis φ.

D

D

3.2.3. Rule 105 The time-1 characteristic function χ 105 can be obtained from Eq. (28): χ 105 =

I


D T 105 (xi−1 xi xi+1D )2−(i+1) =

i=0

I


(1 −D||xi−1 + xi + xi+1 − 1| − 1|)2−(i+1)

(36)

i=0

It is shown in Fig. 23(a) for xI = 0, Dand in Fig. 24(a) for xI = 1. Also in this case, the four fractalDpatterns are plotted in different colors. As usual, the time-1 characteristic function can be bounded; namely, sup 105 ≥ χ 105 ≥ inf

(37)

105

The analytical calculation of the upper and lower bounds for the four patterns is tedious due to the complicated expression for χ 105 . Therefore, here we give the explicit formulas for sup 105 and

χ 0.5

xI = 1

90

χ 0.125

0.375

0.09375

0.25

0.0625

0.125

0.03125

xI = 1

90

D 0 0

0.125

0.25

0.375

(a) Fig. 22.

0.5 φ

0 0.25

0.28125

0.3125

0.34375

0.375φ

(b)

Fractality of χ1

90

: The four fractal patterns appear in two arbitrary subintervals of φ ∈ [0, 1).

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

χ 1

χ

xI = 0

105

1

0.75

0.75

0.5

0.5

0.25

0.25

89

xI = 0

105

D 0 0

0.25

0.5

0.75

1

φ

0 0

0.25

(a)

0.5

0.75

1

φ

(b)

Fig. 23. Time-1 characteristic function of rule 105 , case xI = 0: The fractal pattern χα is in red and the fractal pattern χβ is in purple.

χ 1

χ

xI = 1

105

1

0.75

0.75

0.5

0.5

0.25

0.25

xI = 1

105

D 0 0

0.25

0.5 (a)

0.75

1

φ

0 0

0.25

0.5

0.75

1

φ

(b)

Fig. 24. Time-1 characteristic function of rule 105 , case xI = 1: The fractal pattern χγ is in blue and the fractal pattern χδ is in cyan.

February 14, 2011

15:12

ch01

D D 90

inf

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science 105

: When xI = 0 they are

for (x0 , xI ) = (0, 0)

for (x0 , xI ) = (1, 0)

D  D - 7    = − φ + 1, (a): sup   105  6     D -7 (b): inf 105 = − φ + 1,  2      7 1  D   (c): inf 105 = 6 φ − 3 ,    (d):           (e):     (f):           (g):

1 φ ∈ 0, 2  2 φ ∈ 0, 7  2 1 φ∈ , 7 2  D 7 1 6 φ∈ , sup 105 = φ, 6 2 7  D - 7 6 sup 105 = − φ + 4, φ ∈ ,1 2 7  7 1 4 , inf 105 = −- φ + 2, φ ∈ 2 2 7 D  7 2 4 inf 105 = φ − , φ∈ ,1 6 3 7 D

where the letters correspond to the lines in Fig. 23(b); when xI = 1 the   D 105 = 7 φ + 1 ,  (h): sup   6 2         D105 = − 7 φ + 5 ,  (i): sup   2 2 for (x0 , xI ) = (0, 1)  D  7 1   (j): inf  105 = − 2 φ + 2 ,        D 7 1   (k): inf 105 = φ − , 6 6  7 1   (l): supD105 = φ + ,    6 6      7 7 = − φ +D , for (x0 , xI ) = (1, 1) (m): sup 105 D  2 2  D D     0  7 7   (n): inf 105 = − φ + , 6 6 where the letters correspond to the lines in Fig. 24(b). As we did for 60 and 90 , we show that also the time-1 characteristic function for 105 is a fractal, showing in Fig. 25 that the D four subpatterns appear in two arbitrary subintervals of the axis φ.

o

3.2.4. Rule 150 The time-1 characteristic function χ 150 can be obtained from Eq. (29):

χ 150 =

I
i=0 I

o


=

i=0



formulas are  3 φ ∈ 0, 7  3 1 φ∈ , 7 2  1 φ ∈ 0, 7  1 1 φ∈ , 7 2  1 5 φ∈ , 2 7  5 φ∈ ,1 7  1 φ∈ ,1 2

T 150 (xi−1 xi xi+1 ) 2−(i+1)

0

||xi−1 + xi + xi+1 − 1| − 1| 2−(i+1)

D(38)

It is plotted in Fig. 26(a) for xI = 0, and in Fig. 27(a) for xI = 1. We can easily draw analogies between 150 and 105 because of the alternating symmetry duality between the two rules. Following the analogy with Eq. (37), we found that χ 150

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

χ 1

χ

xI = 1

105

0.625

0.875

0.59375

0.75

0.5625

0.625

0.53125

91

xI = 1

105

o 0.5 0

0.125

0.25

0.375

0.5

φ

0.5 0.25

0.28125

(a)

χ 1

0.34375

0.375φ

(b)

Fractality of χ1

Fig. 25.

0.3125

105

: The four fractal patterns appear in two arbitrary subintervals of φ ∈ [0, 1).

χ

xI = 0

150

1

0.75

0.75

0.5

0.5

0.25

0.25

xI = 0

150

D 0 0

0.25

0.5 (a)

0.75

1

φ

0 0

0.25

0.5

0.75

1

φ

(b)

Fig. 26. Time-1 characteristic function of rule 150 , case xI = 0: The fractal pattern χα is in red and the fractal pattern χβ is in purple.

February 14, 2011

92

15:12

ch01

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

χ 1

χ

xI = 1

150

1

0.75

0.75

0.5

0.5

0.25

0.25

xI = 1

150

D 0 0

0.25

0.5

0.75

0

φ

1

0

0.25

0.5

(a)

0.75

1

φ

(b)

Fig. 27. Time-1 characteristic function of rule 150 , case xI = 1: The fractal pattern χγ is in blue and the fractal pattern χδ is in cyan.

D

is bounded; namely,

D

D

D

D sup 150 ≥ χ 150 D ≥ inf- 150

The explicit formulas for sup 150 and inf

for (x0 , xI ) = (0, 0)

for (x0 , xI ) = (1, 0)

150

(39)

for theD case xI = - 0 are-

    (a):         (b):           (c):    (d):             (e):     (f):           (g):

D sup

150

supD 150



-7 = φ,

φ∈

2

7 4 = -− φ + ,

inf D 150 =

6

3

7φ, 6

-

2 0, 7





 2 1 φ∈ , 7 2   1 φ ∈ 0, 2

 1 4 φ∈ , 150 2 2 7   D - 7 5 4 ,1 sup 150 = − φ + , φ ∈ 6 3 7   7 1 inf 150 = − φ + 1, φ∈ ,1 6 2   7 6 φ∈ ,1 inf 150 = φ − 3, 2 7

D sup

-7 = φ − 1,



February 14, 2011

15:12

ch01

D

-

D

Chapter 1: Quasi-Ergodicity

-

where the letters correspond to the lines in Fig. 26(b); when xI = 1 the  D -7 1   (h): sup = φ+ ,  150  2 2      D - 7  7   = − φ+ , (i): sup  150  6 6 for (x0 , xI ) = (0, 1)  7 1   = -− φ + , (j): infD   150  6 2      7- 3   (k): infD 150 = 2 φ − 2 ,   D -7  1   = φ− , (l): sup  150  6 6      7 5 (m): inf 150 = − φ + , for (x0 , xI ) = (1, 1)  6 6      7 5    (n): inf 150 = φ − ,   2 2

formulas are   1 φ ∈ 0, 7   1 1 φ∈ , 7 2   3 φ ∈ 0, 7   3 1 φ∈ , 7 2

o

o and the letters correspond to the lines in Fig. 27(b). o Finally, also for 150 the four subpatterns appear D in two arbitrary subintervals of the axis φ, as shown in Fig. 28. Note that the time-1 characteristic function for 150 (and consequently, the four fractal patterns and their bounds) can be obtained from that of 105 by means of a reflection through the axis χ 105 = 1/2.

χ 0.5

D

 φ∈ 

1 ,1 2



 1 5 φ∈ , 2 7   5 φ∈ ,1 7

D

3.3. From the time-1 characteristic function to the rule number The analytical characterization of the fractality of χ1 for a generic rule N makes it possible to N

obtain the rule number directly from the time1 characteristic function diagram. Here we consider strings composed by I + 1 bits, and their decimal representation φI+1 defined as in (16). Let

χ

xI = 0

150

0.125

0.375

0.09375

0.25

0.0625

0.125

0.03125

xI = 0

150

o 0

0

0.125

0.25

0.375

(a) Fig. 28.

0.5

φ

0

0

0.03125

0.0625

0.09375

0.125

(b)

Fractality of χ1

150

93

: The four fractal patterns appear in two arbitrary subintervals of φ ∈ [0, 1).

φ

February 14, 2011

94

15:12

ch01

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

us differentiate between xI = 0 and xI = 1: For the first case, we divide the axis φ into four intervals: φ0,I = φI+1 (00 . . . 0) corresponding to 0 ≤ φ0,I < 1/4, φ0,II = φI+1 (01 . . . 0) corresponding to 1/4 ≤ φ0,II < 1/2, φ0,III = φI+1 (10 . . . 0) corresponding to 1/2 ≤ φ0,III < 3/4, and φ0,IV = φI+1 (11 . . . 0) corresponding to 3/4 < φ0,IV-<0 1. In each of these intervals, the time-1 characteristic function for a generic rule N and its representation through fractal patterns can be defined by means of the formula (25). For example, in the first subregion

D

. . and φ1,IV = φI+1 (11.. ..1) corresponding to 3/4 < φ1,IV ≤ 1. - 0 for the time-1 characteristic The expression function in the first interval can be found through Eq. (25) -0 . χ1,I = χI (00..

1 .  (T N (100) + χI−1 (0.. 2 1 1 = T N (100) + χα 2 2

χ0,I

o

therefore χ0,I >

o

1 ⇔ T N (000) = 1 2

o

This means that the value of T N (000) is completely determined by the position of the characteristic function in the first quarter of the diagram for xI = 0: If-the 0 function χ0,I is below 1/2 then T N (000) = 0; if it is above 1/2 then T N (000) = 1. Following an analogous procedure we find that

o

. χ0,II = χI (01..

.. .0)

o

therefore

o

. χ1,II = χI (01..

⇔ T N (101) = 1

o

.. .0)

. χ1,III = χI (10..

1 1 1 - 0 T N (010) + χγ → χ0,III > 2 2 2

o

o

χ

. = χI (11..

.. .0)

o

⇔ T N (110) = 1

o

o

1 1 1  T N (011) + χδ → χ0,IV > 2 2 2 ⇔ T N (011) = 1

and then the values for T N (001), T N (010) and T N (011) depend on the position of the time-1 characteristic function in the second, third and fourth regions, respectively. As for the case xI = 1, we can also divide the . . axis φ into four intervals: φ1,I = φI+1 (00.. ..1) cor. . responding to 0 < φ1,I < 1/4, φ1,II = φI+1 (01.. ..1) corresponding to 1/4 < φ1,II < 1/2, φ1,III = . . φI+1 (10.. ..1) corresponding to 1/2 < φ1,III < 3/4,

.. .1)

1 1 1 =- 0 T N (110) + χγ → χ1,III > 2 2 2

⇔ T N (010) = 1 0,IV

.. .1)

1 1 1 - 0 T N (101) + χγ → χ1,II > 2 2 2

o

χ

o

1 ⇔ T N (100) = 1 2

o

⇔ T N (001) = 1 . = χI (10..

χ1,I >

In this case the value of T N (100) is completely determined by the position of the characteristic function in the first quarter of the diagram for xI = 1: If the points of χ1,I are below 1/2 then T N (100) = 0; if the points of χ1,I are above 1/2 then T N (100) -=01. We can extend the same result to the other three subregions obtaining

1 1 1 =- 0 T (001) + χβ → χ0,II > 2 N 2 2

0,III

.. .0))

o

o

1 1 = χI+1 (00 . . . 0) = T N (000) + χα 2 2

.. .1)

o

. χ1,IV = χI (11.. =

.. .1)

o

o

1 1 1 T (111) + χδ → χ1,IV > 2 N 2 2

⇔ T N (111) = 1 and then the values for T N (101), T N (110) and T N (111) depend on the position of the time-1 characteristic function in the second, third and fourth regions, respectively. In conclusion, the number of the local rule can be found straightforwardly by visual inspection of the time-1 characteristic function diagrams, as summarized in Fig. 29. Observe that this procedure

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

1

0

χ

xI = 0

0

1

000

001

010

011

1

1

0

0

φ

0

0

100

(a) Fig. 29.

110

111

D

3.4. Number of fractal patterns Often a rule does not exhibit four distinct fractal 0 patterns because some of them can coincide. For example, in rule 110 , whose time-1 characteristic function is drawn in Fig. 15, χα and χγ are exactly 0

χγ :

101

φ

Correspondence between time-1 characteristic function and local rule number.

D

χβ :

1

(b)

generalizes the concept of Stratification of characteristic functions illustrated in [Chua et al., 2005b].

χα :

xI = 1

1

{

χ

95

 ..  α,I   χ = χI+1 (00.    χα,II = χI+1 (01...  ..  β,I   χ = χI+1 (10.    χβ,II = χI+1 (11...  ..  γ,I   χ = χI+1 (00.    χγ,II = χI+1 (01...

the same; in fact, this rule has only three distinct patterns. In order to identify the number of different fractal patterns for each local rule, we divide χα , χβ , χγ and χδ for a generic rule N into two subpatterns, indicating the first one with the superscript I and the second -one with the superscript II: For 0 by χα,I and χα,II , χβ by example, χα is composed χβ,I and χβ,II etc. The analytical -expressions for the subdivisions can be found by using 0 Eq. (25):

- 1 1 .. . .0) = T0 N (000) + χI (0.. 2 2

- 1 .. α .0) = (T0 N (000) + χ ) 2

(40) - 1 1 .. . .0) = T0 N (001) + χI (1.. 2 2

1 1 .. . T (010) + χI (0.. .0) =- 0 2 N 2

- 1 .. β .0) = (T0 N (001) + χ ) 2 - 1 .. γ .1) = (T0 N (010) + χ ) 2

(41) - 1 1 .. . .0) = T0 N (011) + χI (1.. 2 2

1 1 .. . .1) = T N (100) + χI (0.. 2 2

- 1 .. δ .1) = (T0 N (011) + χ ) 2

1 .. α .0) = (T N (100) + χ ) 2 (42)

1 1 .. . .1) = T N (101) + χI (1.. 2 2

1 .. β .0) = (T N (101) + χ ) 2

February 14, 2011

96

15:12

ch01

-0

- 0

-0

- 0

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

o

o

χδ :

o.

o

o

.   χδ,I = χI+1 (10..   δ,II  χ

o

Therefore, if T N (000) = T N (100) and T N (001) = T N (101) it follows that χα = χγ . This kind of representation allows us to find easily all the cases in which two (or more patterns) coincide. For instance, from the formulas (40) and (42) we can see that χα = χγ when T N (000) = T N (100) and T N (001) = T N (101). Moreover, one of the two patterns can be a shifted copy of another one: This happens when T N (000) = T N (100) = 0 and T N (001) = T N (101) = 1, for which χγ = χα − (1/2), or when

o

o

o

o

o

o

..1) = 1 T (110) + 1 χI (0... ...1) = 1 (T (110) + χγ ) 2 N 2 2 N 1 1 1 . . . . = χI+1 (11.. ..1) = T N (111) + χI (1.. ..1) = (T N (111) + χδ ) 2 2 2

o

o

o

o (43)

D

T N (000) = T N (100) = 1 and T N (001) = T N (101) = 0, for which χα = χγ − (1/2). Thanks to this procedure, it is possible to classify all the rules according to the number of fractal patterns they exhibit. Representing each rule via an 8-bit code N ↔ (β7 β6 β5 β4 β3 β2 β1 β0 ), βj ∈ {0, 1}, D namely,

D

o

o

N = β7 · 27 + β6 · 26 + β5 · 25 + β4 · 24 + β3 · 23 + β2 · 22 + β1 · 21 + β0 · 20 , where β0 ≡ T N (000) . . . β7 ≡ T N (111), it is possible to prove that rule N contains:

 (β7 β6 β3 β2 ) = (0011)      (β7 β6 β3 β2 ) = (1100)    (β5 β4 β1 β0 ) = (0011) 4 patterns ⇔ (β5 β4 β1 β0 ) = (1100)      (β3 β2 ) = (β7 β6 )    (β β ) = (β β ) 1 0 5 4  (β5 β4 β1 β0 ) = (0011)    (β5 β4 β1 β0 ) = (1100) 3 patterns ⇔ (β3 β2 ) = (β7 β6 )    (β β ) = (β β ) 1 0 5 4

or

 (β7 β6 β3 β2 ) = (0011)    (β7 β6 β3 β2 ) = (1100) (β3 β2 ) = (β7 β6 )    (β β ) = (β β ) 1 0 5 4

 (β7 β6 β5 β4 ) = (0011)       (β7 β6 β5 β4 ) = (1100) 2 patterns ⇔ (β5 β4 ) = (β7 β6 )    (β3 β2 ) = (β7 β6 )    (β1 β0 ) = (β5 β4 )

or

or

or

 (β7 β6 ) = (00)    β4 = β5 (β3 β2 ) = (00)    (β β ) = |1 − (β β )| 1 0 5 4

 (β7 β6 β3 β2 ) = (0011)    (β5 β4 β1 β0 ) = (0011) (β5 β4 β1 β0 ) = (1100)    (β β ) = (β β ) 1 0 5 4

or

 (β7 β6 β3 β2 ) = (0011)    (β7 β6 β3 β2 ) = (1100) (β5 β4 β1 β0 ) = (0011)    (β β ) = (β β ) 3 2 7 6

 β6 = β7       (β5 β4 ) = (β7 β6 ) β3 = |1 − β7 |    β2 = β3    (β1 β0 ) = (β5 β4 )

or

or

or

 (β7 β6 β3 β2 ) = (1100)    (β5 β4 β1 β0 ) = (0011) (β5 β4 β1 β0 ) = (1100)    (β β ) = (β β ) 1 0 5 4

or

 (β7 β6 β3 β2 ) = (0011)    (β7 β6 β3 β2 ) = (1100) (β5 β4 β1 β0 ) = (1100)    (β β ) = (β β ) 3 2 7 6

 (β5 β4 ) = (β7 β6 )       β4 = β5 (β3 β2 ) = (β7 β6 )    β1 = |1 − β5 |    β1 = β0

 β7 = β6    (β5 β4 ) = |1 − (β7 β6 )| (β3 β2 ) = (β5 β4 )    (β β ) = (β β ) 1 0 7 6

or

 β7 = β6    (β5 β4 ) = (β7 β6 ) (β3 β2 ) = |1 − (β7 β6 )|    (β β ) = (β β ) 1 0 7 6

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

  (β5 β4 ) = (β7 β6 ) 1 pattern ⇔ (β3 β2 ) = (β7 β6 )  (β β ) = (β β ) 1 0 3 2

or

 β6 = β7    (β5 β4 ) = (β7 β6 ) (β3 β2 ) = |1 − (β7 β6 )|    (β β ) = (β β ) 1 0 3 2

The 39 globally independent rules (equivalent to 100 local rules) with four distinct fractal patterns are listed in Table 20; the 36 globally independent rules (equivalent to 120 local rules) with three distinct fractal patterns are listed in Table 21; the eight globally independent rules (listed equivalent to 28 local rules) with two distinct fractal patterns are listed in Table 22; the five globally independent rules (listed equivalent to eight local rules) with only one fractal pattern are listed in Table 23. Note that this last class coincides with the trivial additive rules.

or

97

 β6 = β7     (β5 β4 ) = |1 − (β7 β6 )| (β3 β2 ) = (β7 β6 )     (β β ) = |1 − (β β )| 3 2 1 0

Table 22. List of eight globally independent local rules with two distinct one fractal patterns.

3 46

12 60

29 34 136 184

Table 23. List of five globally independent local rules with only one fractal pattern.

0

15

51

170 204

Table 20. List of 39 globally independent local rules with four distinct fractal patterns.

5 6 9 10 22 23 24 26 27 36 37 40 41 43 54 57 58 73 74 77 78 90 94 104 105 108 122 126 130 134 142 146 150 156 160 164 172 178 232 Table 21. List of 36 globally independent local rules with three distinct fractal patterns.

1 2 13 14 30 32 44 45 DOD 76 106 140 152 13

4 7 8 11 18 19 25 28 33 35 38 42 50 56 62 72 110 128 132 138 154 162 168 200

4. New Results about Isles of Eden The notion of “Isle of Eden” was first introduced in [Chua et al., 2007a], and it was clear D fromDthe beginning that it is of crucial importance in the analysis of one-dimensional Cellular Automata. In this section we give some new definitions that allow us to classify the local rules according to the presence of Isles of Eden. Moreover, we also prove that 45 and 154 have Isles of Eden if, and only if, L is odd.13 This result is not totally new, since it was presented in [Chua et al., 2007b]. However, the proof was based on a graph-theoretical tool called Isles of Eden digraph, which is powerful but complex, because it requires the knowledge of further concepts like the De Bruijn graphs. Here we give an alternate proof for this theorem, based exclusively on induction, and simpler than the one known so far. Finally, we collect all the rules with no Isle of Eden for any length L, giving a rigorous proof of this property for all of 0 them. This result is remarkable, because for the first time we find all the rules that have no everywhere invertible orbit in the state transition graph.

o

DOD

Obviously, this result holds also for the local rules that are global equivalent to 45 and 154 , which are 75 , 89 , 101 and 210 , 180 , 166 , respectively.

February 14, 2011

98

15:12

ch01

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

4.1. Definitions and basic lemmas

D definitions allows us to make a disThe following tinction among rules based on the presence of Isles of Eden. These names are chosen via an analogy with the harmonic oscillator.

D

Definition 4.1. Strictly-Dissipative local rules. A

local rule N is said to be strictly-Dissipative iff it has no Isle of Eden for any L.

D

Definition 4.2. Conservative local rules. A local rule N is said to be Conservative or nonDissipative iff every orbit is an Isle of Eden. Definition 4.3. Semi-Dissipative Local Rules. A local rule N is said to be Semi-Dissipative iff it has at least one Isle of Eden for some length L.

D

D

Let us recall next two basic lemmas, whose formal proofs can be found in [Chua et al., 2007b].

D

Lemma 4.1. A bit string D x = (x0 x1 . . . xL−1 ) is a

period-n Isle of Eden of local rule N if, and only if, x has a unique preimage under T n . N

D

D

Lemma 4.2. A local rule N is conservative if, and only if, every bit string x = (x0 x1 . . . xL−1 ) has a unique preimage under T N , for any L ∈ N.

D

D

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

D

Proof.

can be used to determine, step by step, all remaining unknown bits xti .


D

DNow, we are ready to enunciate the following theorem, equivalent to Theorem 4.1 in [Chua et al., 2007b], giving an alternate proof based exclusively D on mathematical induction.

D

Theorem 4.1. Every orbit of local rules 45 and

154 is an Isle of Eden, if, and only if, L is an odd integer. Necessity: If L is even then all orbits of 45 or 154 are not Isles of Eden. From Lemma 4.2, it follows that we can prove this statement by showing that there are at least two bit strings having D different D D the same output. We can find a counterexample intuitively, since any string containing k times the pattern (01) has the same output as a string containing k times the pattern (00): In fact, in both cases the output is a string containing k times the pattern (11), because T 45 (010) = T 45 (101) = T 45 (000) = 1. However, the same result can be used through an inductive procedure, introducing the notation (x)L to indicate that a string x is composed by L elements, (x)L = (x0 x1 . . . xL−1 ). Let us consider the two following strings with length L = 2k: Proof.

(xα)tL = (xα0 xα1 . . . xαL−1 ), with  0, if i = 2n − 1 α xi = 1, if i = 2n

From Table 11, we can see that both 45 and 154 belong to the class of permutive rules. We recall that for left-permutive rules

D

T N ( xti−1 xti xti+1 ) = xt+1 i

D

⇒ T N ( x ti−1 xti xti+1 ) = x t+1 i and for right-permutive rules

and (xβ )tL = (xβ0 xβ1 . . . xβL ), with xβi = 0, ∀ i ∈ {0, L − 1}.

D

T N ( xti−1 xti xti+1 ) = xt+1 i ⇒ T N ( xti−1 xtix ti+1 ) = x t+1 i

D

It is easy to notice that for a permutive rule, given and the local rule N , we can unixti , xti+1 , xt+1 i vocally identify xti−1 . This result was generalized in [Wuensche et al., 1992] through the following lemma. Lemma 4.3. If N is permutive, then knowing xt+1

and any two adjacent bits xti and xti+1 of xt , the bit string xt is unambiguously determined.

If the local rule N is left-permutive, then

t At+1 n−1 defines xn−2 ; if N is right-permutive then t+1 t An defines x0 . In both cases, the same procedure

For k = 1, L = 2 we have (xα)t2 = (01) and = (xβ )t+1 = (11). (xβ )t2 = (00), and (xα)t+1 2 2 Let us assume that for L = 2n we have β t+1 α t (xα)t+1 2n = (x )2n with (x )2n = (0101 . . . 01) and β t (x )2n = (0000 . . . 00) (induction hypothesis), and let us analyze the case L = 2(n + 1) = 2n + 2. The strings (xα)t2n+2 and (xβ )t2n+2 are (xα)t2n+2 = (0101 . . . 01) = ((xα)t2n 0 1) and (xβ )t2n+2 = (0000 . . . 00) = ((xβ )t2n 0 0)

February 14, 2011

15:12

ch01

D

D

D

0

0 D

Noticing D

T 45 (000)

that =

T 45 (010)

0

D

=

1, it follows that

T 45 (101)

=

(xα)t+1 2n+2

=

(xβ )t+1 2n+2 . The proof for rule 154 is very similar, except for the fact that T 154 (010) = T 154 (101) =

0

T 154 (111) = 1, and where the string xβ should be defined as follows

D

(xβ )L = (xβ0 xβ1 . . . xβL ), with xβi = 1, ∀ i ∈ {0, L − 1}. Sufficiency: When L is odd, all orbits of rules 45 and 154 are Isles of Eden. According to Lemma 4.2, it is sufficient to prove that in this case there is no Garden of Eden. We apply the mathematical induction as follows: Let L = 2n + 1 be the length of the string: Our assumption holds for n = 1 ⇒ L = 3; we assume it is true for a generic n = k ⇒ L = 2k + 1 (induction hypothesis), and then, we will confirm that it is true for n = k + 1 ⇒ L = 2k + 3 too. Then, for all L = 2k + 1 (xα)t2k+1 = (xα0 xα1 . . . xα2k ) and (xβ )t2k+1 = (xβ0 xβ1 . . . xβ2k ) such that β t+1 α t β t (xα)t+1 2k+1 = (x )2k+1 ⇔ (x )2k+1 = (x )2k+1 . Now, we analyze the case L = 2k +3, supposing that there is a Garden of Eden. This implies that there are two different strings with the same image: (xα)t2k+3 = (xα0 xα1 . . . xα2k+2 ) and (xβ )t2k+3 = (xβ0 xβ1 . . . xβ2k+2 ), with (xα)t2k+3 = (xβ )t2k+3 and β t+1 (xα)t+1 2k+3 = (x )2k+3 . The two strings can be decomposed into = ((xα)t2l+1 xα2l+1 . . . xα2k+2 ) and (xα)t2k+3 (xβ )t2k+3 = ((xβ )t2l+1 xβ2l+1 . . . xβ2k+2 ). If (xα)t2l+1 and (xα)t2l+1 have periodic boundaries, that is xα2l = xα2k+2 , xα0 = xα2l+1 and xβ2l = xβ2k+2 , xβ0 = xβ2l+1 , then according to the induction hypotheses, (xα)t2l+1 = (xβ )t2l+1 . But for Lemma 4.3, since we know the string (xα)t+1 2k+3 and two α t adjacent pixels of (x )2k+3 , the remaining pixels xα2l+1 . . . xα2k+2 are fixed; the same happens with (xβ )2k+3 , xβ2l+1 . . . xβ2k+2 . Since (xα)t+1 2k+3 = β t+1 α t β t (x )2k+3 and (x )2k+1 = (x )2k+1 , it follows that j = l . . . k : xα2j+1 = xβ2j+1 , xα2j+2 = xβ2j+2 . But this means that (xα)t2k+3 = (xβ )t2k+3 , contrary to the hypothesis.14
14

0 Chapter 1: Quasi-Ergodicity

0

99

Note that this method cannot be used find 0to0 0Isles of Eden, 0 but only to confirm that all orbits are actually Isles of Eden. This result about 45 and 154 is similar to that obtained0in [Chua0et al., 2007a] for 105 and 150 , which are conservative for any L0that is not0divisible by 3. There are also some rules that are conservative tout court, like 15 , 170 , 204 and 51 (and their global equivalent rules). This follows directly from the nature  of these rules, since they consist in a mere shift 15 and 170 or in transformation  depending only on the central pixel 204 and 51 .

o

4.3. There are exactly 28 0 strictly-dissipative local rules

o

While some rules — like 45 and 154 — are conservative, others – like 60 [ChuaDO et al., 2007b]0and 90 [Chua et al., 2007a] — are strictly-dissipative, which means that they have no Isle of Eden. In addition to the already mentioned 60 and 90 , we found that all the centrally-symmetric (also called perfectlycomplementary) local rules are strictly-dissipative (see Table 24) as well as rules 8 , 46 , and 78 and their equivalent rules (see Table 25). These results are proved in Theorems 4.2 and 4.3, respectively, and all the 28 strictly-dissipative rules (nine topologically independent) are listed in Table 26. Theorem 4.2. All centrally-symmetric local rules

are strictly-dissipative. By definition, any triplet of a centrallysymmetric rule produces the same output as its complement; therefore, any bit string xt has the same output as its complement x t . Then, the proof follows from Lemma 4.1.
Proof.

o

For example, we can show how this theorem applies to rule 24 , which is centrally-symmetric. Its firing patterns are


and


, and they are the complement of the other; as for the quenching patterns, they are also complement pairs:


and


,


and


,


and


. Therefore, any bit string has the same output as its complement, and hence rule 24 has no Isles of Eden.

o

DO

0

Rules 8 , 46 and 78 are the only non centrally-symmetric rules to be strictlydissipative. Theorem 4.3.

To be precise, we can always choose a (xα )t2k+1 according to the previous conditions when L > 17; for the values of L ≤ 17 we checked the theorem experimentally.

February 14, 2011

100

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 24.

T †[ N ]

N 0

2

4

4

4

4

1

0 4

5

2

5

4

219

2

3

195

7

6

4

2 7

6

2

129

5

7 1 5

2

3 7

6

1

0 4

5

153

5

2

3 7

6

1

0 4

3

0

1

0

1 5

4

3

3 7

6

5

4

5

4

1

0

1

1

2

165

7 0

3

0

5

3

7

6

1

0

7

6

2

2 6

1

0

102

189

5

6

4

3

3 7

4

7

6

2 0

5

3

1 5

6

1

1

2

4

7

3

0

126

231

7

6

90

3

0

5

2

2 6

1

0

60

66

7

6

7

4

3

3

0

5

2

2 6

1

0

36

255

3

T *[ N ]

T [N ]

7

6

24

Sixteen centrally-symmetric strictly-dissipative local rules.

5

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity Table 25.

Twelve non-centrally-symmetric strictly-dissipative local rules.

T †[ N ]

N 8

2 7

6 4

3 7

6 4

3 7

6

1

0 4

5

ODD

92

3 7

6

5

o

o

Lemma 4.4. Rule 8 has no Isle of Eden.

We recall that the only firing pattern of rule 8 is


. Then, from Fig. 30 we see that all the strings containing more than one consecutive red pixel — pattern

— are Gardens of Eden. Moreover, strings D belonging to anDIsle of Eden cannot contain an isolated red pixel — pattern


— because, for the uniqueness of the firing pattern, their predecessor would necessarily have the pattern

. Consequently, the only possible Isle of Eden would include exclusively the pattern


; but since T 8 (000) = T 8 (111) = 0, the string (00 . . . 0) has two predecessors, which are (00 . . . 0) and (11 . . . 1), and this would contradict Lemma 4.1.
Proof.

D

Lemma 4.5. Rule 46 has no Isle of Eden.

2 7

1

0

D

4

2

5

3 7 1

0 4

3

1 5

6

1 5

6

1

0

We found, through computer simulations, at least one Isle of Eden for each non centrallysymmetric rule, as shown in Table 15. The only exceptions are rules 8 , 46 , 78 , for which we provide an analytical proof of their strict dissipativity in the next three lemmas, respectively.


Proof.

141

209

7

4

7

4

3

3

0

5

2

2 6

1

0

5

253

7

6

1

2

4

139

7

4

3

0 4

3

0

5

2

2

2 6

1 5

6

1

0

78

116

239

7

4

T *[ N ]

T [N ] 3

0

5

2

2 6

1

0

46

64

3

101

197

5

2

3 7

6

1

0 4

5

D D

We recall that the firing patterns of rule 46 are


,


,


, and


. First of all, we notice that strings with only either red or blue pixels have the same successor, which is a string containing only blue pixels, because T 46 (000) = T 46 (111) = 0. From Lemma 4.1, it follows that a necessary condition for a string to belong to an Isle of Eden contains the pattern

. Nevertheless, strings including the pattern


are Gardens of Eden, as proved in Fig. 31, and this implies that any string belonging to an Isle of Eden has to include at least two consecutive red pixels. However, not even the pattern



can be part of an Isle of Eden, because it has the same successor as a different string, as shown in Fig. 32. Therefore, the last case to analyze is the case



whose predecessors either give rise to contradictory conditions or are Gardens of Eden, as detailed is Fig. 33.
Proof.

D

Lemma 4.6. Rule 78 has no Isles of Eden.

February 14, 2011

15:12

ch01

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

102

Table 26.

0

2

3 7

6 4

7

6 4

3 7

6 4

3 7

6 4

3 7

6 4

4

7

6

1

0 4

5

7

4

3

239

3 1

0 5

253

7 1

0 4

5

1 5

2

3 7 1

0 4

3

3 7

6

5

6

2

219

1

2

5

4

3

1

0

7

4

7

6

2

3 7

6

1

0

5

189

5

6

1

2

4

209

2 0

7 0

1 5

4

3

3 7

6

1

2

4

3

0

5

2

2

139

5

6

1 5

6

1

0

231

197

7

6

7

4

3

165

2 0

7 0

1 5

4

3

3 7

6

5

4

3

0

5

2

2

102

1

2

2 0

7

6

1 5

6

1

0

195

153

129

1 5

4

3

0

7

4

2

7

6

1 5

4

3

0

5

2

2

66

7

6

1 5

6

1

0

141

126

92

3

0 4

3

0

7

4

2

2 6

1 5

4

3

0

5

2

2

36

7

6

1 5

6

1

0

116

90

64

7

4

3

0 4

3

0

5

2

2

2 6

1 5

6

1

0

78

60

24

7

4

3

3

0

5

2

2 6

1

0

46

8

Twenty-eight no Isle of Eden rules.

255

5

2

3 7

6

1

0 4

5

February 14, 2011

15:12

ch01

---~ Chapter 1: Quasi-Ergodicity

0 __ 0 D

D Fig. 30. Rule 8 : Strings containing the pattern

are Gardens of Eden, because their predecessor would give rise to contradictory conditions on the pixel .

We recall that the firing patterns of 78 are


,


,


, and


.

Proof.

103

We start proving that strings containing more than three consecutive red pixels are Gardens of Eden (see Fig. 34). The following step consists in showing that strings containing exactly three adjacent red pixels — pattern


— do not belong to an Isle of Eden because there is always another string with the same successor, as evident from Fig. 35; a similar situation happens for strings containing exactly two adjacent red pixels because their predecessor is not unique, as implied by Fig. 36. Finally, we need to see what happens for strings including an isolated red pixel, which is the pattern


. In order to perform a thorough analysis, we divide this case into four subcases, according to what the neighbors for the pattern


are: the case




, analyzed in Fig. 37; the case

D

D

Fig. 31. Rule 46 : Strings containing the pattern


are Gardens of Eden, because all the possible predecessors would give rise to contradictory conditions on the pixel .

D

Fig. 32.

Rule 46 : The pattern



has the same successor as a different string, hence it cannot belong to an Isle of Eden.

D

Fig. 33. Rule 46 : Strings with more than two consecutive red pixels (pattern



) cannot belong to an Isle of Eden because their predecessor either gives rise to contradictory conditions (cases (a)–(d)), or contains an isolated red pixel (cases (e) and (f)) and then it is a Garden of Eden (see Fig. 31).

February 14, 2011

104

15:12

ch01

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

D

Fig. 34. Rule 78 : Strings containing more than three consecutive red pixels are Gardens of Eden, because all of their possible predecessors would give rise to contradictory conditions.

D

11Fig. 35. Rule 78 : Strings containing exactly three consecutive red pixels cannot belong to an Isle of Eden, because it is always possible to find another bit string with the same successor.

D

DD DO

leg]

(rl)

(r:)

(b)

(a)

~

~eg]

DD

/

Fig. 36. Rule 78 : Strings containing exactly two consecutive red pixels have at least two different predecessors, and hence they cannot belong to an Isle of Eden.

0111111

D

D

Fig. 37. Rule 78 : The pattern




cannot belong to an Isle of Eden because its predecessor either gives rise to contradictory conditions (cases (a)–(c)), or contains more than three consecutive red pixels (case (d)) and then it is a Garden of D Eden (see Fig. 34).

D




, analyzed in Fig. 38; the case




, analyzed in Fig. 39; and finally the case




, analyzed in Fig. 40. As shown in the respective figures, none of these patterns can be in a string that is part of an Isle of Eden.

The last possibility would of Eden composed by a single be either (00 . . . 0) or (11 . . . 1); sibility is excluded by the fact

D

D

be having an Isle string, which can however, this posthat T 78 (000) =

T 78 (111) = 0, as it happened for 8 and 46 .




February 14, 2011

15:12

ch01 (b)

(1'\)

~....

~..

(c)

.DD DD••~~

~ D

/

•

DD Chapter 1: Quasi-Ergodicity

105

D

D

Fig. 38. Rule 78 : The pattern




cannot belong to an Isle of Eden because its predecessor either gives rise to contradictory conditions (cases (a)–(c)), or contains more than three consecutive red pixels (case (d)) and then it is a Garden of Eden (see Fig. 34).

4.4. Isles of Eden for rules of group 1

0••••••0 0.....0

We mentioned in Sec. 1 that our classification of local rules into six groups is based mainly on the properties of attractors (number, period etc.), although it is also supported by other features, like quasi-ergodicity. According to our classification

~/

• • •

D

Fig. 39. Rule 78 : The pattern




cannot belong to an Isle of Eden because it has at least two different predecessors.

principle, rules of group 1 have a robust period-1 global attractor, which means that most D of the 2L bit strings belong to a unique basin of attraction. Here we want to show that, at least for some rules of group 1, the only possible attractor is the robust one, and all strings not belonging to it form Isles of Eden. Let us take into consideration rule 40 , whose firing patterns are


and


. Rule 40 has a robust attractor in 0; in other words, for any L most of bit strings will fall into the basin of attraction of L elements

 x = ( 00 . . . 0 ). Which strings do not belong to this attractor, and what happens with them? In [Ohi, 2007] the following theorem is proved.15

D

D

Theorem 4.4. For rule 40 , x = (00 . . . 0) is a global attractor, except for the strings with at most one consecutive 0 and at most two consecutive 1 s, which form a Bernoulli Isle of Eden with σ = 1 and τ = 1.

The following proof is far simpler than the one in [Ohi, 2007], mainly because the original paper also includes other results requiring a more complex notation and treatment. The first part of the proof consists in showing that any string containing more than one consecutive 0 or more than two consecutive 1’s is in the basin of attractor of x = (00 . . . 0). As for the first case, we consider, without loss of generality, the string (. . . 100 . . .), which is the “border” between a “1” and the run of 0’s; from Fig. 41, it is evident that the “border” tends to move towards the left, and then all the 1’s will be substituted, step by step, by 0’s. As for the second case, we consider, without loss of generality, the string (. . . 0111 . . .), which is the “border” between a “0” and the run of 1’s; from Proof.

D

Fig. 40. Rule 78 : The pattern




cannot belong to an Isle of Eden because, after a finite number of steps, it evolves into the pattern




which cannot belong to an Isle of Eden, as shown in Fig. 39. 15

This theorem is a summary of the results presented in several lemmas terminology and notation.

and theorems in [Ohi, 2007] using a different

February 14, 2011

106

15:12

ch01

o

!

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

Theorem 4.5. For rule 168 , x = (00 . . . 0) is a

D

global attractor, except for the strings with more than one consecutive 0, which form a Bernoulli Isle of Eden with σ = 1 and τ = 1.16

••!

The first part of the proof is very similar to thatDof Theorem 4.4, but to refer only D we need D to Fig. 41, discarding the results in Fig. 42. Also D D D the second part is very similar, but in this case D D D the hypothesis excludes the triplets (000), (001) D (100), and then we need to analyze only the and triplets (010), (011), (101), (110) and (111), for which T 168 (010) = 0 = T 170 (010), T 168 (011) = 1 = T 170 (011), T 168 (101) = 1 = T 170 (101), T 168 (110) = 0 = T 170 (110) and T 168 (111) = 1 = T 170 (111). The rest of the proof is identical to the one for the previous theorem.
Proof.

Fig. 41. Rule 40 : All strings containing the pattern


are in the basin of attraction of

· · ·
.

D

D

Fig. 42. Rule 40 : The pattern



evolves into the pattern


after one iteration, and then it is in the basin of attraction of

· · ·
(see Fig. 41).

Fig. 42, we can see that after one step we always obtain a string with a run of at least two 0’s which, as proved previously, will eventually evolve into the string x = (00 . . . 0). The second part of the theorem states that strings containing no more than one consecutive 0 and at most two D consecutive 1’s belong 0 to a Bernoulli attractor. By hypothesis, the D D D triplets (000), (001), (100), (111) will never be present in D D D these kinds of strings, and hence we need to ana0 (110). D only the tripletsD(010), (011), (101) and lyze It is straightforward to notice that with such kind of restriction rule 40 is equivalent to rule 170D be0 = cause T 40 (010) = 0 = T 170 (010), T 40 (011) 1 = T 170 (011), T 40 (101) = 1 = T 170 (101) and T 40 (110) = 0 = T 170 (110). Since rule 170 is a right-copycat rule (or equivalently, a left shift), for these initial conditions the dynamics of rule 40 corresponds exactly to the dynamics of rule 170 , which is a Bernoulli orbit with σ = 1 and τ = 1. Moreover, the basin of attraction of this orbit is empty, because either a string fulfills the conditions of the Theorem, and then it belongs to the orbit, or D it does not, and then it converges to 0. A very similar theorem holds for rule 168 , belonging to group 1 too. In this case the firing patterns are


,


, and


.


o

5. How to Find Analytically the Basin-Tree Diagrams for Bernoulli Attractors Our previous papers [Chua et al., 2007a] and [Chua et al., 2007b] were in large part devoted to showing the basin-tree diagrams, all of them found by brute force, of the complex and hyper Bernoulli-shift rules. In general, these kinds of rules have many Bernoulli orbits, either attractors or Isles of Eden, whose number increases with the length L of the bit strings. Therefore, it is particularly important to characterize them analytically, in order to avoid long and tedious simulations, which make it practically impossible to find the basins of attraction when L is greater than 30. Here we present a method that allows us to reconstruct the whole basin-tree diagram of a Bernoulli-shift attractor by analyzing only a fraction of all bit strings with a fixed L.

5.1. Bernoulli-στ basin-tree generation formula

o

By definition, given any string belonging to a Bernoulli στ -shift orbit, after τ iterations we find the same string shifted σ positions towards left, if σ is positive.17 For example, Fig. 43 depicts a period T = 14 Bernoulli στ -shift orbit for rule 110 , in which σ = 1 and τ = 4.

16 Note that this statement is very similar to the one for 40 , but the limitation on the number of consecutive 1’s has been removed. 17 In the following we consider only the case σ > 0, because any attractor with σ < 0 is equivalent to another one with σ
= L − |σ| > 0 and same τ .

February 14, 2011

15:12

ch01

D

0 1 ••••••• 2 ••••••• 3 •••••••

Chapter 1: Quasi-Ergodicity

iterations, nτ = T τ (n0 ), it follows that

4 •••••••

N

5 ••••••• ~

6 •••

II

7 •••

h

8 •••

.........

nτ = n0 · 2σ

(J=

+1

(mod 2L − 1)

""'pII"

•••• •• •••• ••

np·τ (mod T ) = n0 · 2p·σ

D

0≤p<

0

(mod2L − 1),

T , lcm(τ, T )

p∈N

In general, a left shift by σ positions in the binary representation x = (x0 x1 . . . xL−1 ) is equivalent to a multiplication  by 2σ in the decimal reprei sentation n, where n = L−1 i=0 xL−1−i 2 . Therefore, if n0 is a (decimal) element of a Bernoulli-στ orbit, and nτ the result after τ p = 1:

n1·4(mod 14) = n0 · 21·1

p = 2:

2·1

n2·4(mod 14) = n0 · 2

mod (2 − 1) ⇒ n8 = 31 · 4

p = 3:

n3·4(mod 14) = n0 · 23·1

mod (27 − 1) ⇒ n12 = 31 · 8 (mod127) = 121

p = 4:

n4·4(mod 14) = n0 · 24·1

mod (27 − 1) ⇒ n2 = 31 · 16

p = 5:

5·1

mod (2 − 1) ⇒ n6 = 31 · 32

6·1

mod (2 − 1) ⇒ n10 = 31D · 64 (mod127) = 79

p = 6:

n6·4(mod 14) = n0 · 2

(45)

where the parameter p indicates the current iteration. Obviously, iteration T /lcm(τ, T ) would give n0 itself. For example, this procedure can be applied to the Bernoulli στ -shift attractor (with τ = 2 and T = 14) of rule 110 when L = 7, as depicted in Fig. 44. From what was explained previously, there will be lcm(τ, T ) = 2 different subgroups, each with length T /lcm(τ, T ) = 7. Starting from an arbitrary element of a period-14 Bernoulli στ -shift orbit (cyan circles in Fig. 44), like n0 = 31, and using the formula (45), we obtain the values:

Fig. 43. Example of Bernoulli στ -shift attractor with σ = 1, τ = 4, for rule 110 and L = 7.

n5·4(mod 14) = n0 · 2

(44)

Furthermore, this formula can be modified to find not only nτ , but also all other elements of the orbit generated by n0 . A Bernoulli στ -shift orbit with period T harbors exactly lcm(τ, T ) different subgroups, where lcm indicates the least common multiple, each with length T /lcm(τ, T ), henceforth called order of the subgroup. This means that formula (44) can be iterated (T /lcm(τ, T )) − 1 times to obtain a corresponding number of strings, using at each iteration the result nτ as new value for n0 . As a consequence, Eq. (44) can be generalized as follows

9 ••• 10 • • 11 • • 12 • • 13 • • 14 • •

107

mod (27 − 1) ⇒ n4 = 31 · 2 7

(mod127) = 62 (mod127) = 124

7

(mod127) = 115 (mod127) = 103

7

Finally, for p = T /lcm(τ, T ) = 7, n7·4( mod 14) = n0 = 31, as expected. p = 7:

n7·4( mod 14) = n0 · 27·1

mod (27 − 1) ⇒ n0 = 31 · 128

(mod127) = 31

The seven elements belonging to the second subgroup, which can be identified in the period-14 Bernoulli orbit in Fig. 44, can be found by using a different starting point. Since T 110 (31) = 49, n0 = 49 is a natural candidate; then, it follows that p = 1:

n1·4( mod 14) = n0 · 21·1

mod (27 − 1) ⇒ n4 = 49 · 2

(mod127) = 98

p = 2:

n2·4( mod 14) = n0 · 22·1

mod (27 − 1) ⇒ n8 = 49 · 4

(mod127) = 69

February 14, 2011

108

15:12

ch01

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

L=7

Bernoulli (σ=1,τ=4) Period-14 Attractor

110

D

Fig. 44.

Basin-tree diagram of the Bernoulli στ -shift attractor with σ = 1, τ = 4, for rule 110 and L = 7.

p = 3:

n3·4( mod 14) = n0 · 23·1

mod (27 − 1) ⇒ n12 = 49 · 8

(mod127) = 11

p = 4:

n4·4( mod 14) = n0 · 24·1

mod (27 − 1) ⇒ n2 = 49 · 16

(mod127) = 22

p = 5:

n5·4( mod 14) = n0 · 25·1

mod (27 − 1) ⇒ n6 = 49 · 32

(mod127) = 44

p = 6:

n6·4( mod 14) = n0 · 26·1

mod (27 − 1) ⇒ n10 = 49 · 64

(mod127) = 88

p = 7:

n7·4( mod 14) = n0 · 27·1

mod (27 − 1) ⇒ n0 = 49 · 128

(mod127) = 49

D

Also in this case, when p = T /lcm(τ, T ) = 7 we find the initial value n0 = 49, as expected. Thanks to the formula (45), which generalizes the analogue presented in [Chua et al., 2006] about the στ -Bernoulli rules of group four, all 14 elements of the Bernoulli στ -shift orbit were found starting just from n0 = 31 and n0 = 49. Remarkably, this formula can be also used to find elements that do not belong to the orbit but to

D

the D basin tree. This is because in general T

N

T

D

N

xn −−−→ xn+1 ⇒ σ(xn ) −−−→ σ(xn+1 )

(46)

where σ(xn ) is the string obtained by shifting σ times the bit string xn . For example, since T 110 (13) = 31, we can expect that a shifted version of the bit string representing 31 is obtained by transforming under the rule N the shifted bit

February 14, 2011

15:12

ch01

o

o

Chapter 1: Quasi-Ergodicity

109

string representing 13. p = 1: n1·4( mod 14) = n0 · 21·1

mod (27 − 1) ⇒ n4 = 13 · 2

(mod127) = 26

and indeed T

T

110

110

13 −−−−→ 31 ⇒ σ(13) = 26 −−−−→ σ(31) = 62 The values for n0 = 13 are p = 2:

n2·4( mod 14) = n0 · 22·1

mod (27 − 1) ⇒ n8 = 13 · 4

p = 3:

n3·4( mod 14) = n0 · 23·1

mod (27 − 1) ⇒ n12 = 13 · 8 (mod127) = 104

p = 4:

n4·4( mod 14) = n0 · 24·1

mod (27 − 1) ⇒ n2 = 13 · 16

(mod127) = 81

p = 5:

n5·4( mod 14) = n0 · 25·1

mod (27 − 1) ⇒ n6 = 13 · 32

(mod127) = 35

p = 6:

n6·4( mod 14) = n0 · 26·1

mod (27 − 1) ⇒ n10 = 13 · 64 (mod127) = 70

(mod127) = 52

and, as usual, for p = T /lcm(τ, T ) = 7 we obtain the initial value n0 = 13, as expected. p = 7:

n7·4( mod 14) = n0 · 27·1

mod (27 − 1) ⇒ n0 = 13 · 128

To sum up, all the information about a Bernoulli attractor can be retrieved from only one element of each subgroup of the orbit, like 31 and 49, and their basins of attraction (in this example the basins are formed by 15 bit strings altogether, as depicted in Fig. 45). Then, using only these 17 elements we are able to draw the whole basin-tree diagram, which is composed of 119 bit strings! For obvious reasons, we christen Eq. (45) as the “Bernoulli στ -shift basin-tree generation formula”, and it can be applied to any rule having Bernoullishift attractors.18

(mod127) = 13

[Chua et al., 2007b]), and suppose that no information about the form of the basin-tree diagrams is known. We can use x0 = (0000001) = 1 as the

Detail of Bernoulli (σ=1, τ=4) Period-14 Attractor, L=7

110

5.2. A practical application of the Bernoulli στ -shift basin tree generation formula The procedure described previously allows us to find all elements of a Bernoulli basin tree starting from a limited subset of them. In fact, in some cases, this method is even more powerful, because it allows us to obtain a basin tree without knowing any of its elements! This happens when all of the possible shifts of a bit string x do not belong to the same basin tree as x, but form a different, though topologically equivalent, basin tree. Let us illustrate this case through an example, considering rule 110 with L = 6 (see Table 11 in

D

o

18

Fig. 45. Detail of the basin-tree diagram of the Bernoulli στ -shift attractor with σ = 1, τ = 4, for rule 110 and L = 7. These 17-bit strings allow us to reconstruct the whole basintree diagram, thanks to formula (45).

Rules belonging to groups 4–6 have robust Bernoulli στ -shift attractors. In addition, rules belonging to groups 1–3 may also harbor isolated Bernoulli στ -shift Isles of Eden and attractors.

February 14, 2011

15:12

ch01

D 110

D

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

D

Since x5 is a shifted version of x2 — in particular x5 = σ 4 (x2 ) — both strings belong to a Bernoulli στ -shift orbit with τ = 5−2 = 3 and σ = 4. By using the formula (45) with n0 = 7, it is possible to notice that n9 = n0 = 7, which means that the period of the attractor is the minimum T ≡ 0 (mod 9) ⇒ T = 9. Therefore, in the Bernoulli στ -shift orbit there are lcm(τ, T ) = 3 different subgroups, each with length (order) T /lcm(τ, T ) = 3. The elements of the first subgroup, found using n0 = 7, are as follow:

starting point and find its successors iteratively:

D

x1 = T 110 (x0 ) = (000011) = 3,

D

x2 = T 110 (x1 ) = (000111) = 7, x3 = T 110 (x2 ) = (001101) = 13, x4 = T 110 (x3 ) = (111111) = 31, x5 = T 110 (x4 ) = (110001) = 49.

D

p = 1:

n1·3( mod 9) = n0 · 21·4

mod (26 − 1) ⇒ n3 = 7 · 16

p = 2:

n2·3( mod 9) = n0 · 22·4

mod (26 − 1) ⇒ n6 = 7 · 256

(mod 63) = 28

p = 3:

n3·3( mod 9) = n0 · 23·4

mod (26 − 1) ⇒ n0 = 7 · 4096

(mod 63) = 7

(mod 63) = 49

As for the second subgroup, we use n0 = 13 because T 110 (7) = 13, and obtain 1·4

D

p = 1:

n1·3( mod 9) = n0 · 2

mod (26 − 1) ⇒ n3 = 13 · 16

p = 2:

n2·3( mod 9) = n0 · 22·4

mod (26 − 1) ⇒ n6 = 13 · 256

p = 3:

n3·3( mod 9) = n0 · 23·4

mod (26 − 1) ⇒ n0 = 13 · 4096

(mod 63) = 19 (mod 63) = 52 (mod 63) = 13

Finally, for the third subgroup n0 = 31 because T 110 (13) = 31, and obtain p = 1: n1·3( mod 9) = n0 · 21·4

mod (26 − 1) ⇒ n3 = 31 · 16 (mod 63) = 55

p = 2: n2·3( mod 9) = n0 · 22·4

mod (26 − 1) ⇒ n6 = 31 · 256

(mod 63) = 61

p = 2: n2·3( mod 9) = n0 · 22·4

mod (26 − 1) ⇒ n6 = 31 · 256

(mod 63) = 31

Therefore, the basin tree corresponding to strings 7, 13 and 31 is sufficient to reconstruct the whole basin-tree diagram exhibited in Table 11 of [Chua et al., 2007b]. The binary representation of 7, which we used as the “seed” for the first subgroup of bit strings in the orbit, is





, and it has six possible shifts:





↔ 14,





↔ 28,





↔ 56,





↔ 49,





↔ 35, and





↔ 7 itself. Surprisingly, only three of them form part of the orbit, namely 7, 28 and 49. What about the others? Obviously, they are included in a different attractor, which, for the formula (46), must be strictly related to the one we already know. In particular, the “missing” strings are 14, 56 and 35, and observe that 14 = σ 1 (7), 56 = σ1 (28), and 35 = σ 1 (49). Therefore, this “twin” basin tree can be found by applying the operator σ 1 (one position left shift) to every string of the first basin tree, and keeping the same topology, as evident from Table 11 of [Chua et al., 2007b].

For instance, the Bernoulli στ -shift orbit of the first basin tree is 7 → 13 → 31 → 49 → 19 → 55 → 28 → 52 → 61 → 7, or equivalently





→





→





→





→





→





→





→





→





→






Applying the operator σ1 , we are able to find the Bernoulli στ -shift orbit of the “twin” basin tree





→





→





→





→





→





→





→





→





→






or, using the decimal representation, 14 → 26 → 62 → 35 → 38 → 47 → 56 → 41 → 59 → 14.

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

x has only two possible shifts, because σ 2 (x) = x. But the string x
obtained by changing only the last pixel from 1 to 0, x
= (0101 . . . 00), will have clearly s = L shifts, where L can be very large. However, the parameter s for a string x can be obtained experimentally as s = mint≤L {σ t (x) = x}.

Through a similar procedure, we can draw the whole D “twin” basin tree starting from the first one. In general, let us consider a string x with length L which can be shifted s times, corresponding to s different bit strings, with s ≤ L. If x belongs to D a subgroup with order o under the transformation T N (x), then o must divide s and there will be s/o “twin” orbits. In the example just presented, the string x = (000111) corresponding to n = 7 could be shifted six times, s = 6, and the order of the subgroup, under the transformation T 110 (x), is o = 3; therefore, there will be s/o = 6/3 = 2 “twin” orbits, which we actually found. Unfortunately, calculating a priori the number of shifts s of a string x may not be easy, since it involves certain deep concepts from coding theory. Sometimes, modifying only one bit in a string can change dramatically the number of possible shifts s. For example, let us consider a bit string x with length L made of L/2 repetitions of the pattern 01; namely, string x = (0101 . . . 01). It is evident that (abc) = (000):

xn+1 i

(abc) = (001):

xn+1 i

(abc) = (010):

xn+1 i

(abc) = (011):

xn+1 i

(abc) = (100):

xn+1 i

=0

(mod 2) ⇒

=

xni+1

=

xni

=

xni

=

xni−1

xn+1 i

(mod 2) ⇒ (mod 2) ⇒

+ xni+1

DO 0= xni−1 + xni+1 (abc) = (101): xn+1 i

The behavior of additive cellular automata can be described through the expression xn+1 = T N (xni−1 xni xni+1 ) i

= axni−1 + bxni + cxni+1

(mod 2)

(47)

where a,D b, c ∈ {0, 1}. Since Eq. (47) contains three free parameters, there are 23 = 8 distinct additive rules, which can be represented through simple Boolean expressions as follows

o

= 0 ↔ Rule 0

xn+1 i

=

=

(mod 2) ⇒ xn+1 i

xni+1

xni

o

o

↔ Rule 170

↔ Rule 204

xn+1 i =

o

=

xni−1

xni

⊕

xni+1

D D

↔ Rule 102

o

↔ Rule 240

(mod 2) ⇒ xn+1 = xni−1 ⊕ xni+1 ↔ Rule 90 i

(abc) = (110):

= xni−1 + xni xn+1 i

(abc) = (111):

xn+1 = xni−1 + xni + xni+1 i

D

0

(mod 2) ⇒ xn+1 = xni−1 ⊕ xni ↔ Rule 60 i (mod 2) ⇒ xn+1 = xni−1 ⊕ xni ⊕ xni+1 ↔ Rule 150 i

Each additive rule N has a complementary antiadditive rule N c = 255 − N . The eight additive and the eight anti-additive rules are represented in Table 27, along with their Boolean cubes and Boolean expressions. Table 28 shows that four additive rules are globally equivalent to their corresponding anti-additive rules, whereas in the remaining four cases additive and anti-additive rules exhibit the so-called alternating symmetry duality [Chua et al., 2007a] described in Sec. 6.2. Additive rules can be displayed on the vertices of a cube19 as illustrated in Fig. 46(a). Drawing the 19

D 6. Old Theorems and New Results for Additive Cellular Automata

xn+1 i

(mod 2) ⇒

111

D

axis a passing through the vertices containing rules 0 and 150D , all the other rules O 0 are included into two planes, A and B, perpendicular to a. Thanks toDthis it is possible to separate the rules Drepresentation, 0 into four different subsets, according to the plane they belong to: The first subset contains rule 0 ; the second subset contains rules belonging to plane A, namely 170 , 204 and 240 ; the third subset contains rules belonging to plane B, namely 60 , 90 and 102 ; finally, the fourth subset contains rule 150 . Rules belonging to the same subset have the same complexity index κ (κ = 1, κ = 2, and κ = 3, respectively) and the same number of inputs

o

This cube shows the connection among additive rules, and it should not be confused with the 256 Boolean cubes in Table 1.

February 14, 2011

112

15:12

ch01

A Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science Table 27. Eight additive and eight anti-additive rules, their Boolean cubes, and their analytical formulas.

8 Additive Rules N 2

x

1

0

4 n +1 i

3 7

6

0

x 7

1

= x ⊕x

n +1 i

x

2

1

0

4

4

6

n +1 i

x

7

4

x

1

2

2 6

x

7

6

x

6

x

7

x

1

5

=x

6

51

5

n +1 i

x 3 7

0

1

7

6

15

5

=x

x

3 7

0

4 n +1 i

1

n i

2

5 n i −1

3

0

4

=x

1

n i +1

2

n i

=x

0

4 n +1 i

3 7

6

85

3

0

4 n +1 i

= x ⊕ x ⊕ xin+1 n i

2

=x

2

240

1

1

5

n i −1

5

4 n +1 i

x

n i +1

2

204

n +1 i

3

0

4 n +1 i

0

4 n i +1

3 7

6

105

=x ⊕x ⊕x

170

5

= x ⊕ xin+1

5 n i

1

n i

3

0

n i −1

0

4

n i +1

3 7

6

153

5

2

n +1 i

2

=x ⊕x

150

5

= x ⊕ xin+1

x

1

1

n i −1

3

n i

x

0

4

0

3 7

6

165 n +1 i

7

6

n +1 i

= x ⊕ xin

5 2

102

5

2

xin +1 = xin−1 ⊕ xin+1

1

n i −1

3 7

6

90

0

4 n i

3 7

6

5 n i −1

x

=1 2

195

1

5

3

0

4

0

4 n +1 i

3 7

6

255

=0

6

n +1 i

2

5

2

60

c

8 Anti-Additive Rules N

5

=x

n i −1

1

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity Table 28. rules.

Relationship between additive and anti-additive

Additive Rule

Anti-additive Relationship between the two rules rule

0

255

Global equivalence

60

195

Global equivalence

90

165

Global equivalence

102

153

Global equivalence

150

105

Alternating symmetry

170

85

Alternating symmetry

204

51

Alternating symmetry

240

15

Alternating symmetry

D

0

n (n = 0, n = 1, n = 2, and n = 3, respectively); moreover, the Boolean cube corresponding to one element of a subset can be transformed into that corresponding to another element of the same subset through a rotation. A similar procedure can DDDDDOOO be followed for anti-additiveDDDO rules, as illustrated in Fig. 46(b). Remarkably, except for rule 0 and 255 , the number of inputs of any additive or anti-additive rule coincides with its complexity index.20 There are only nine globally-independent addiDDO tive rules: 0 , 15 , 51 , 60 , 90 , 105 , 150 , 170 , 204 . Five of them — namely 0 , 15 , 510, 170 and 204 — have complexity index κ = 1, and their behavior can be easily Danalyzed; for this reason, we dubbed them “trivial additive DO rules”. In contrast, 0 the remaining four local rules — 60 , 90 , 105 and 150 — have complexity index κ = 2 rules  60 and 90 or κ = 3 rules 105 and 150 . According to the classification introduced in [Chua et al., 2007a], rule 60 belongs to group 6 (hyper Bernoulli-shift rules), and rules 90 , 105 and 150 belong to group 5 (complex Bernoulli-shift rules); therefore, they exhibit complex behaviors, and we dubbed them “nontrivial additive rules”. In the following, we will take into consideration only these four nontrivial additive rules, for

o o

D

o

D

o

D

20

D

113

which it has been possible to find several remarkable results.

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

D

Some partial results about additive rules were given in [Chua et al., 2007a] and [Chua et al., 2007b]. For example, the periods of the attractors of rule 90 found by brute force for 1 ≤ L ≤ 100 were listed in Table 25 of [Chua et al., 2007a]. Unfortunately, this table is incomplete, because the period of the attractors corresponding to certain L is so large that it exceeds the simulation time. Nevertheless, [Martin et al., 1984] contains several valuable results about additive rules, which allow us to find all of the missing values of the table under consideration. For the reader’s convenience, we summarized such results in two theorems using a plain style, our notation and our nomenclature. First of all, we need to introduce three concepts from number theory that will be extensively used in the following. Definition 6.1 (Euler totient function). Given a positive integer n, the Euler totient φ(n) is the number of positive integers less than n that are coprime to n. Definition 6.2 (Multiplicative order function). Given a positive integer n, the multiplicative order of 2 (mod n) is the minimum positive integer o(n) for which 2o(n) = 1 (mod n) Definition 6.3 (Multiplicative suborder function). Given a positive integer n, the multiplicative suborder of 2 (mod n) is the minimum positive integer s(n) for which 2s(n) = ±1 (mod n) Remark 6.1. It is possible to prove that φ(n) ≤ n−1, s(n) ≤ (n − 1)/2, and s(n)|o(n)|φ(n), where the bar “|” denotes “divides”. The values of φ(n), o(n) and s(n) for n odd and n < 100 are listed in Table 29.21 Now, we are ready to present the two aforementioned theorems, whose proofs can be found in [Martin et al., 1984].

We could assign complexity index κ = 0 to 0 and 255 , but this would mean introducing a fourth value for κ exclusive to only two rules, and not being consistent with the notation used in our previous works. 21 Note that the multiplicative order and suborder functions of 2 (mod n) are not defined for n even.

February 14, 2011

114

15:12

ch01

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

A

(a)

A

III

(b) Fig. 46.

Geometrical interpretation of additive (a) and anti-additive (b) rules.

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

115

Table 29. Values of the Euler’s totient function φ(n), multiplicative order function o(n), and multiplicative suborder function s(n), for n < 100, n odd. n

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

φ (n) : Euler's o(n) : Multiplicative s(n) : Multiplicative totient function order function suborder function

D

1 2 4 6 6 10 12 8 16 18 12 22 20 18 28 30 20 24 36 24 40 42 24 46 42

0

1 2 4 3 6 10 12 4 8 18 6 11 20 18 28 5 10 12 36 12 20 14 12 23 21

D

1 1 2 3 3 5 6 4 4 9 6 11 10 9 14 5 5 12 18 12 10 7 12 23 21

0

Theorem 6.1. (Maximum period of the orbits of

rules 90 and 150 ). Let L be the length of the bit strings and let TL be the maximum period of the orbit (attractor or Isle of Eden) of rules 90 and 150 , then

(a) If L is odd, then TL divides the quantity T ∗ = 2s(L) − 1, where s(L) is the multiplicative suborder of 2 (mod n); (b) If D L = 2n , then x = (00 . . . 0) is a global attractor ; (c) If D L is even but not of the form L = 2n , then TL = 2 · TL/2 . Theorem 6.2. (Maximum period of the attractors

of rule 60 ). Let L be the length of the bit strings and let TL be the maximum period of the attractor of rule 60 , then

(a) If L is odd, then TL divides the quantity T ∗ = 2o(L) − 1, where o(L) is the multiplicative order of 2 (mod n); (b) If L = 2n , then x = (00 . . . 0) is a global attractor; (c) If L is even but not of the form L = 2n , then

n

51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

φ (n) : Euler's o(n) : Multiplicative s(n) : Multiplicative totient function order function suborder function

32 52 40 36 58 60 36 48 66 44 70 72 40 60 78 54 82 64 56 88 72 60 72 96 60

n

8 52 20 18 58 60 6 12 66 22 35 9 20 30 39 54 82 8 28 11 12 10 36 48 30

0

8 26 20 9 29 30 6 6 33 22 35 9 20 30 39 27 41 8 28 11 12 10 36 24 15

TL = 2 · TL/2 . Although these theorems do not include any statement about rule 105 , in [Chua et al., 2007a] it was proved that rules 105 and 150 exhibit an alternating symmetry duality, which is a kind of relationship weaker than the topologically conjugated transformations from the Vierergruppe0defined in [Chua et al., 2004] but still sufficient to relate the orbits of the two rules. The alternating symmetry duality can be described as follows: Given two bit strings xn =
n
n (xn0 xn1 . . . xnL−1 ) and x
n = (x
n 0 x1 . . . xL−1 ), generated respectively by rules 105 and 150 from the same initial state, x and x
obey the following relations

o

1 n n ] + (−1)n xD x
n i = [1 − (−1)D i 2

(48)

1 xni = [1 − (−1)n ] + (−1)n x
n i 2

(49)

x
n Therefore, xn = x
n for n even and xn = for n odd; in other words, χ2105 = χ2150 , where

February 14, 2011

D

15:12

ch01

o

D

o

116

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

χ2

is the time-2 characteristic function of the local

N

o

o

o

rule N . For L odd, Theorem 6.1 states that TL is also odd for rule 150 . By applying the alternating symmetry duality, it is straightforward to see that TL has to be even for 105 , otherwise we would obtain a contradiction. In particular, when L is odd then TL for 105 has to be twice as that of TL for 150 . For L even, Theorem 6.1 states that TL is also even for rule 150 . Therefore, starting from any element of the orbit and going through the whole orbit and taking every other element, we will eventually go back to the first element after TL /2 steps. As a consequence of the alternating symmetry duality, the orbit for rule 105 contains all these TL /2 elements of the orbit for rule 150 , replacing the remaining TL /2 with new ones. In conclusion, for L even TL for rule 105 is equal to TL for rule 150 . We summarize these results in the following theorem:

o

o

o

o

o

o

o

Theorem 6.3. (Maximum period of the orbits of rule 105 ). Let L be the length of the bit strings and let TL be the maximum period of the orbit (attractor or Isle of Eden) of rule 105 , then

(a) If L is odd, then TL divides the quantity TL∗ = 2 · (2s(L) − 1), where s(L) is the multiplicative suborder of 2 (mod n); (b) If L is even, then TL divides the quantity TL∗ = 2 · (2s(L/2) − 1). It is important to notice that the parameter TL∗ does not give necessarily the maximum period TL of the orbits, because in general TL |TL∗ ; however, we found experimentally that TL = TL∗ for most L. The practical implications of Theorems 6.1–6.3, and Remark 6.1 are noteworthy. First D of all,0they state that TLD can assume very few values0because it has to divide TL∗ , and then resort to brute force, and numerical simulations is extremely minimal; second, we discover that the maximum period of the orbits for additive rules is not 2L as hypothesized in our L−1 previous papers, but 2 2 −1 for rules 90 and 150 , L+1 2L−1 for rule 60 , and 2 2 − 2 for rule 105 . The values of TL∗ for the nontrivial additive rules for L odd and L < 100 are listed in Table 30. In addition to this, we also give four tables — Table 31 refers to rule 60 , Table 32 refers to rule 90 , Table 33 refers to rule 105 , and Table 34 refers to rule

D

22

o

D

150 — containing the actual value of TL along with a bit string belonging to the orbit and its result after TL − 1 iterations, so that the reader can check the correctness of our simulations. In some cases the value of TL is so large that no simulation was possible. However, in these cases the orbits are so robust that practically every bit string has the indicated TL . Finally, a graphical representation of the relationship between the length L of the bit string and the maximum period TL of the orbit for nontrivial additive rules is depicted in Figs. 47–50.

6.2. Scale-free property for additive rules All nontrivial additive rules are either complex or hyper Bernoulli-shift rules; hence, their Bernoulli parameters σ and τ , and consequently the periods of their orbits, depend crucially on L. In principle, given L and theDO corresponding0 maximum period TL of the orbits of a nontrivial additive rule, it is not possible to extract any information about the maximum period TL
of the orbits for L
= L; in other words, TL /L is unrelated to TL
/L
. Nevertheless, in [Chua et al., 2007a] it was empirically noticed that for rules 90 , 105 and 150 there exist certain values of L and L
for which D
(log(TL
)−log(TL ))/(log(L )−log(L)) = 1, or equivalently TL /L = TL
/L
. This means that such rules exhibit a scale-free property as L → ∞, and hence there are some L
= L for which it is possible to know TL
using only the information about TL ; the presence of a scale-free property also for rule 60 is confirmed in [Chua et al., 2007b]. The scale-free property can be easily noticed looking at Figs. 51–54 and noticing that most of the points lie on diagonal lines with slope 1 (here we draw just a few of them). A question naturally arises: For what values of L and L
does the scale-free property hold? In order to characterize quantitatively this phenomenon, we give the following definition: Definition 6.4 (Scale-free order). Let L be the length of the bit strings and let TL∗ be the maximum period of the orbits (attractors or Isles of Eden) of a nontrivial additive rule, then its scale-free order ξL is defined as22 ξL =

TL∗ 2s(L) − 1 = L L

We have chosen the symbol ξL to denote the scale-free order to recall the parallel straight lines visually representing the scale-free property.

D

D

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49

D

1 1 3 7 7 31 63 15 15 511 63 2,047 1,023 511 16,383 31 31 4,095 262,143 4,095 1,023 127 4,095 8,388,607 2,097,151

1 2 6 14 14 62 126 30 30 1,022 126 4,094 2,046 1,022 32,766 62 62 8,190 524,286 8,190 2,046 254 8,190 16,777,214 4,194,302

Rule 105: Rules 90 and 150: * * s( L) TL = 2 · (2s ( L ) − 1) TL = 2 − 1 o( L)

51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99

−1 L

1 3 15 7 63 1,023 4,095 15 255 262,143 63 2,047 1,048,575 262,143 268,435,455 31 1,023 4,095 36 2 -1 4,095 1,048,575 16,383 4,095 8,388,607 2,097,151

T =2 * L

Rule 60: 510 134,217,726 2,097,150 1,022 1,073,741,822 2,147,483,646 126 126 17,179,869,182 8,388,606 35 2·(2 -1) 1,022 2,097,150 2,147,483,646 39 2·(2 -1) 268,435,454 41 2·(2 -1) 510 536,870,910 4,094 8,190 2,046 36 2·(2 -1) 33,554,430 65,534

255 67,108,863 1,048,575 511 536,870,911 1,073,741,823 63 63 8,589,934,591 4,194,303 235-1 511 1,048,575 1,073,741,823 39 2 -1 134,217,727 241-1 255 268,435,455 2,047 4,095 1,023 236-1 16,777,215 32,767

Rule 105: Rules 90 and 150: * * s( L) TL = 2 · (2s ( L ) − 1) TL = 2 − 1 255 52 2 -1 1,048,575 262,143 258-1 60 2 -1 63 4,095 266-1 4,194,303 35 2 -1 511 1,048,575 1,073,741,823 39 2 -1 54 2 -1 82 2 -1 255 268,435,455 2,047 4,095 1,023 36 2 -1 48 2 -1 1,073,741,823

T = 2o ( L ) − 1 * L

Rule 60:

Maximum period-T of attractors and/or Isles of Eden of local rules 60 , 90 , 105 and 150 , for L < 100, L odd.

D

L

Table 30.

February 14, 2011 15:12 ch01

117

L lIsle ofEden

...

Data for generating and verifying the period of period-T orbits of 60 .

(x 0 ,.x T-l ) ~ (initiaIJinal) bit strinK on Period- T orbit

Attractor T*

=

3

4 5

T*

=

1

T*

15

=

~

I

X2

I·iii·iii·iii···············································

Xo

I ••••

.

---~Q_--j-~-~-~-~-~--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------XI4

•••••

I•••••• Xs I • • • • • • Xo

T*

=

6

7 8 9

T*

=

7

T*

=

1

T*

=

63

10

T*

=

30

11

T= T*/3

---~Q_--j-~-~-~-~-~-~~-~-~-~-~----------------------------------------------------------------------------------------------------------------------------------------------------------

12

T*

12

13

T= T*/5

--i;~--+=-=-=-=-=-=I=-=-=-=-=------------------------------------------------------------------------------------------------------------------------------------------------------

14

T*

=

14

15 16 17

T*

=

15

T*

=

255

·~~·+=·=·=·=·=·=I=·=·=·=·=·=;I=·=·····················

18

T*

=

126

19

T= T*/27

-~~~-+=-=-=-=-=-=I=-=-=-=-=-=;I=-=-=-------------------------------------------------------------------------------------------------------------------------------x~;~J=-=-=-=-=-=I=-=-=-=-=-=;I=-=-=-=---------------------------------------------------------------------------------------------------------------------------

T*

=

1

20

T*

=

60

21

T*

=

63

22

T=T*/3

23 I

I T*

=

2047

···i~··+=·=·=·=·=·=I·································· Xo

__

.

I ••••••••

~o~ __j-=-=-=-=-=-=I=-=------------------------------------------------------------------------------------------------------------------------------------------------------------------

---~Q_--j-~-~-~-~-~-~~-~-~-~-------------------------------------------------------------------------------------------------------------------------------------------------------------X29

X340

••••••••••

•••••••••••

·~~~·+=·=·=·=·=·=I=·=·=·=·=·=························· ··i;~·+=·=·=·=·=·=I=·=·=·=·=·=I·······················

. .

--i;~-+=-=-=-=-=-=I=-=-=-=-=-=;I-----------------------------------------------------------------------------------------------------------------------------------------Xo

I ••••••••••••••••

··i~~·+=·=·=·=·=·=I=·=·=·=·=·=;I=·=·=·=·=·············

.

.

--i~o~-+=-=-=-=-=-=I=-=-=-=-=-=;I=-=-=-=-=-=-------------------------------------------------------------------------------------------------------------------~;~--j-=-=-=-=-=-=I=-=-=-=-=-=II=-=-=-=-=-=-=---------------------------------------------------------------------------------------------------------------x;~~-J=-=-=-=-=-=I=-=-=-=-=-=;I=-=-=-=-=-=-=I-----------------------------------------------------------------------------------------------------------

ch01

118

6

=

15:12

3

D

February 14, 2011

Table 31.

24

T*

24

=

T

26

T= T*/5

27

T

28

T*/41

=

T*/19

=

T*

T= T*/565

119

30

T*

=

30

31 32 33

T*

=

31

T*

34

T*

35

T*

36

T*

T* = = = =

=

1

1023 510 4095

252

37

T= T*/21255

38

T= T*/27

39

T*

40

T*

41

T= T*/25

42

T*

43

T= T*/3

44

T

=

T*/3

T*

=

4095

T*

=

4094

4H 46

I

= =

=

4095 120

126

i~~~;~I-IIZ-I-I;-=ll=-IIZ-ll;-I-IZ-=-ll=-ll---------------------------------------------------------------------------------------x~:;JIIZ-I-I;-=-ll=-llZ-llZ-I-IZ-=-IZ-=-IZ-I-------------------------------------------------------------------------------------i;~~~~I-IIZ-I-I;-=lll-IIZ-ll;-I-IZ-=-I=-=-IZ-I-I----------------------------------------------------------------------------------

-i;;---I-IIZ-I-I;-=ll=-IIZ-ll;-I-IZ-=-I=-=-lll-I-I-------- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -t~~+IIZ-I-I;-=-ll=-llZ-llZ-I-IZ-=-I=-=-IZ-I-I-I=---------------------------------------------------------------------------i;~---I-IIZ-I-I;-=lll-ll=-ll;-I-IZ-=-ll=-lll_=_lll------------------------------------------------------------------------i:~---I-IIZ-I-I;-=ll=-Il=-Il;-I-IZ-=-Il=-IZ-I-I-I=-I-1-------------------------------------------------------------------~

I ••••••••••••••••••••••••••••••••

x~:;JIIZ-I-I;-=-ll=-llZ-llZ-I-IZ-=-I=-=-IZ-Z-I-I=-I-IZ-1-------------------------------------------------------------x~~~+IIZ-I-I;-=-III-ll=-ll;-I-IZ-=-ll=-IZ-I-I-III-IZ-1-1---------------------------------------------------------x~:~JIIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ-------------------------------------------------------x;~~--I-IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ-I---------------------------------------------------x~~+IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ-I-I-----------------------------------------------t~~+IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ-I-Il-------------------------------------------x~:~JIIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ-I-IIZ-----------------------------------------x;~~--I-IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ-I-IIZ-I-------------------------------------x~~+IIZ-=-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-IZ-Z-I-IZ-I-IZ-I-IZ_=_IIZ-=Z----------------------------------i;~~+IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-=-=Z-Z-I-IZ-=-IZ-I-IZ-I-IIZ-IZ-l-----------------------------x~:~~+IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-Z-=Z-Z-I-IZ-=-1Z-I-IZ-I-IIZ-IZ-ll--------------------------x~:~;-I-IIZ-I-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-Z-IZ-Z-I-IZ-=-IZ-I-IZ-I-IIZ-IZ-lll-----------------------

x~:~JIIZ·I·I;·=·IIZ·IIZ·IIZ·I·IZ·=·I=·=·=Z·Z·I·IZ·=·IZ·I·IZ·I·IIZ·IZ·IIIZ···_·············_·· x~:~JIIZ-=-I;-=-IIZ-IIZ-IIZ-I-IZ-=-I=-I-IZ-Z-I-IZ-=-IZ-1-IZ-I-IIZ-=Z-lllZ-:-----------------

ch01

29

28

=

-i;;---
-
lz-
-
;-
-
l
-
lz-
l;-
-
z-
-
l=-
--------------------------------------------------------------------------------------------

15:12

25

(Continued )

February 14, 2011

Table 31.

47

T*

8388607

=

T*

48

2097151

50

T

51

T*

52

T

= =

T*/41

X

=

255

=

T*/5

o

x~

I;·················································

; •••••••••••••••••••••••••••••••••••••••••••••••••

I•• x·················································· o

X3275

;;••••••••••••••••••••••••••••••••••••••••••••••••

120

53

T

=

T*/1266205

54

T

=

T*/19

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

55

T*

1048575

X·················································· --~L-Iliili.-.-i-ii.-i-ii.-i-ii.-.-i.-.-i.-.-i-ii-.-.-i.-. -i.-.-i.-.-i.-.-i_.-i-ii.-.-i.-.-i---

56

X·················································· ---~~:-+i-ii-i-ii-.-i-ii.-.-ii.-.-i.-.-i.-.-i.-.-.-ii-.-. -i.-.-i.-.-i.-.-i.-.-i_.-i-ii.-.-i.-.-i--X·················································· ---~>Iliiliili-ii-.-.-ii.-.-ii.-.-i.-.-i.-.-.-ii-.-.-i.-.i.-.-i.-.-i.-.-i_.-i-ii.-.-i.-.-i---

=

T*

56

=

57

T*

=

262143

58

T

=

T*/565

59

T*

=

288230376151711743

60 61

T*

T*

=

=

60

X·················································· ---~-~:-+i-ii-i-ii-i-i-ii-.-ii.-.-ii.-.-i.-.-i.-.-.-ii-..-i.-.-i.-.-i.-.-i.-.-i_.-i-ii.-.-i.-.-i---

1152921504606846975

ch01

T*

48

--x~~d-;;Z-;-I;-;-I-I;-;-Il;-;l=-IZ-l:-:-;-I-:-=-I-IZI-
z-;;z-;-
;-;-
;;-;-;;-;-
----------------i~;---llIZ-;-IZ-;-I-:-;-;-;Z-;-;Z-;-IZ-;-IZ-;-:-:-;-:-;Z-:-IZ-:-IZ-:-IZ-;-:;;-:-;;-:-;Z------------x~~~+;-IZ-;-IZ-:-I-:-Z-:-;Zl;Z-;-IZ-;-IZ-;-:Z-;-:-IZ-:-IZ-:-IZ-;-IZ-;-:;Z-:-;Z-;-IZ-;---------i~---I-;-IZ-;-IZ-;-I-;;-:-;Z-:-;Z-;-IZ-;-IZ-;-:-:-Z-:-IZ-:-IZ-:-IZ-;-IZ-;-:;;-:-;Z-;-;Z-;-I---

15:12

49

=

(Continued )

February 14, 2011

Table 31.

February 14, 2011

Table 31.

(Continued )

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

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

15:12

ch01

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

•••• •••• •• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •• •••• •••• •••• •••• •• •••• •••• •••• •••• •• •••• •••• •••• •••• •• •••• •••• •••• •••• •• •••• •••• •• •••• •••• •••• •••• •••• •••• •• •••• •••• •• •••• •••• •••• •••• •• •••• •••• •••• •• •••• •••• •••• :c

0

><

..:

N 'C

0

><

..:

~

0

0 ".

..:

..: V)

~

~

II -x-

II -x-

N

('1

'-D

N

M

'-D

.. on

".

0

..:

-., II

*N

"1"

'-D

0-

0

0 N

..:

>< \Q

0 "t-

"t0 <"l

II

II

*h

*h

or.

'-D

'-D '-D

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

~

"t-

\Q

0 <"l

00
"t0-

~ l'--..

0\Q

OO l'--..

~ II -le-

h

t'-D

•••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• ~

::

0

><

..:

~

0

..:

><

~

:!: ..:'"

0

..:


0

0

~ -.,


"t0-

0

O-

-.,

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

l'--.. \Q
V)
II

"t-

II

*h

II -x-

*h

II -x-

01

0

......

oo

121

'-D

t-

0

><

~

0-

00

'-D

'"on

0

..:

00

-.,

h

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

"t-


::on

0

..:

><

"t0

-., -.,

V)

lr)

II -x-

II -x-

h

h

h

t-

~

0

><

..:

or, or, <"'l

-.,

~ -x-

N II

h

N t-

M t-

"1" t-

February 14, 2011

15:12

ch01

Table 31.

(Continued )

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

•• •• •• •• •• •• •• •• ••

••• ••• •• •• •• •• •• •• ••••

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

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

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

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

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

<;;::,

~ -x-

0\ ......., 00

*h

<-

h

h

II

II

*h

h

00

r-

o

N

00

00

122

M

00

-.::I"

00

r-

OO

February 14, 2011

Table 31.

(Continued )

•• •• •• ••• •• •

15:12

ch01

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

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

•••••••••••••••••••••••••••• •••••••••••••••••••••••••••• •••••••••••••••••••••••••••• •••••••••••••••••••••••••••• •••••••••••••••••••••••••••• 00 00

~ -xh II h

00 00

•• ••

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

•••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• \0 0\

.......

00

II

II -xh

0\ 00

•• ••

-*

h

o

00 0\

0\

123

0\ 0\

0\ 0\

o o......

L

[sLeofEden

Data for generating and verifying the period of period-T orbits of 90 .

Attraetor

D

(x o,x T-l) g, (initial1tinal) bit strinf( on Period-T orbit

T* = 1 T* = 1

6

T* =2

7

T* = 7

8 9

T* = 1 T* = 7

--i~---f~-I;-I-Z-Z-I;-I--------------------------------------------------------------------------------------------------------------------------------------

10

T* = 6

11

T* = 31

12

T* = 4

13

T* = 63

14

T* = 14

15

T* = 15

--i:---f-Z-I;-I-Z-Z-I;-I-Z------------------------------------------------------------------------------------------------------------------------------------i;~+Z-I;-I-Z-Z-I;-I-Z-Z---------------------------------------------------------------------------------------------------------------------------------i;---f-Z-I;-I-Z-Z-I;-I-Z-Z-I-----------------------------------------------------------------------------------------------------------------------------i;,;--f-Z-I;-I-Z-Z-I;-I-Z-Z-I;--------------------------------------------------------------------------------------------------------------------------i;;--f-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-----------------------------------------------------------------------------------------------------------------------i;~+Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-------------------------------------------------------------------------------------------------------------------

16 17

T* = 1 T* = 15

--i;~+Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I------------------------------------------------------------------------------------------------------------

18

T* = 14

19

T* = 511

20

T* = 12

21

T* = 63

--i;;--f-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I;---------------------------------------------------------------------------------------------------------~~-~-f-Z-I;-I-Z-Z-I;-I-Z-Z-I;_=_Z-Z-I;-I-------------------------------------------------------------------------------------------------------i;;--f-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z----------------------------------------------------------------------------------------------------i;,;--f-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-------------------------------------------------------------------------------------------------i;,;--f-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I--------------------------------------------------------------------------------------------x~:~JZ-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I;-I-Z-Z-I;-----------------------------------------------------------------------------------------

T* = 3

--

22 I 23 I

I

T* = 62

-I T* = 2047

Xo I • • • •

--i;---f-z-
;-
-z--------------------------------------------------------------------------------------------------------------------------------------------------Xo Xl

I •••••• I ••••••

--i~---f-Z-I;-I-Z-Z-I-------------------------------------------------------------------------------------------------------------------------------------------Xo I • • • • • • • •

Xo I • • • • • • • • • • • • • • • •

ch01

--

Xo I • • •

15:12

124

3 4 5

February 14, 2011

Table 32.

February 14, 2011

15:12

ch01

Table 32.

(Continued )

•• •••• •••••• •••••••• ••••••••••

•••••••••••• •••••••••••••• •••••••••••••••• •••••••••••••••••• •••••••••••••••••••• •••••••••••••••••••••• •••••••••••••••••••••••• •••••••••••••••••••••••••• •••••••••••••••••••••••••••• ••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••• C"'l

-., -.,

\Q

00

II

*E-,

<;;::,

<;;::,

or,

00 C"'l

<'f)

<'f)

00 C"'l

II

II

II

II

II

*E-,

*E-,

-.,

*E-,

*E-,

00

o

N

*E-,

or,

~ -x-

0\

II

E-,

*E-,

II

II

<;;::, ~

*E-,

E-,

00 M

M

125

~

C"'l

II

*E-,

~

C"'l

-.,

'"

C"'l -.,

0\

II

II

II

*E-,

*E-,

*E-,

II

C"'l

or, C"'l

~

<;;::, ~

*E-,

47 I 4H 49

I T*

I

= 2

23

_1

T* = 16 T* = 2 21 _1

(Continued )

50

T* = 2046

51

T* = 255

52

T* = 252

53

T* = 2 26 _1

54

T* = 1022

x·················································· -~~-:-Ji-ii-i-i.-.-i-ii.-.-ii.-.-ii.-.-i.-.-i.-.-i-ii-.-iii.-.-i.-.-i.-.-i.-.-i_.-.-ii.-.-i.-.-i---

55

T* = 2 20 _1

X·················································· --~L-Iliili.-.-i-ii.-i-ii.-i-ii.-.-i.-.-i.-.-i-ii-.-i-i.-. -i.-.-i.-.-i.-.-._.-.-ii.-.-i.-.-i---

56

T* = 56

x·················································· ---~~:---I-i-ii-i-ii-.-i-ii.-.-i.-.-ii.-.-i.-.-i.-.-.-ii.-.-i.-.-i.-.-i.-.-i.-.-._.-.-ii.-.-i.-.-i---

57

T* = 511

X·················································· --~L-Iliiliil.-ii-.-.-ii.-.-ii.-.-i.-.-i.-.-.-ii-.-.-i.-.i.-.-i.-.-i.-.-._.-.-ii.-.-i.-.-i---

58

T* = 32766

59

T* = 2 29 _1

1=················································· = . I•• x·················································· == .

15:12

--x~~d-;;;-;-I=-;-I-I;-;-II;-I;-l=;-l:-:-;-;-:-=-;-I;I-
;-;;;-;-:1;-
;;-
-
;-;-
----------------i;~---I-;-:;-;-:;-;-:-:-;-;-I;ll;l:;-;-:;-;-:-:-;-;-1;-;-:;-;-:;-;-:;-;-;;;-;-1;-;-:;------------x~~;--I-;-:;-;-:;-;-:-:-;-;-I;ll;-;-:;-;-:;-;-;;-;-;-I;-;-:;-;-:;-;-:;-;-I;;-I-I;ll;l------x~~~-J;-:;-;-:;-;-:-I;-;-I;-;-I;-;-:;-;-:;-;-I-:-;-;-1;-;-:;-;-:;-;-:;-;-:;;-;-1;-;-:;-;-1---

February 14, 2011

Table 32.

X

T'

~ 60

T* = 2 30 _1

o

Xw

x

o

XT_l

I••• •••••••••••••••••••••••••••••••••••••••••••••••••• === .

x·················································· --~L-I-i-ii-i-ii-i-i-!-.-.-i.-.-i.-.-i.-.-i.-.••-.-.-i.-.-i.-.-i.-.-i.-.-._.-.-ii.-.-i.-.-i---

x·················································· ---~-~:---I-i-ii-i-ii-i-i-ii-.-ii.-.-ii.-.-i.-.-i.-.-.-ii -."lfi.-.-i.-.-i.-.-i.-.-._.-.-ii.-.-i.-.-i--x·················································· ··~~~:+i·ii·i·ii·i·i·ii·i·i.·.·i.·.·i.·.·i.·.·.·ii·.·.· i.·.·i.·.·i.·.·i.·.·i_.·.·ii.·.·i.·.·i···

ch01

126

j

61

I

o

x~

Table 32.

(Continued )

February 14, 2011

15:12

ch01

••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• • •••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• •••••• •• ••••• •••• •••• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••••• • •••• •••• •••• •••• •••• •••• •••• •••• •• •• •• •• •• •• •• •• •• •• •• •• •• ••••• • •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •• •• •• •• •• •• •• •• ••••• • •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• • •••• •• •• •• •• •• ••••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •• ••••• • •• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• ••••.... •• ••••.... •••• •••• •••• •••• ••••'" •••• •••• ••••=: •••• ~

0 ~

'"

lo
0

'"

lo
~

0

~

0

'"

lo
~

:c

0 ~

~

~

0 ~

lo
'"on

0 ~

lo
~

~

II

II

*h

*h

II -xh

('l

M

\.0

\.0

-.::t \.0

I

a

I

~

~

II

II

*h

*h

II -xh

*h

Ir)

\.0 \.0

t\.0

oo

\.0

'"'"C"'l

~

lo
........,

........,

........,

~

0

lo
\Q

II

\.0

127

"1

'"C"'l

:'!:

0

lo
QO ~

a

0\ ........,

~

0 ~

lo
'n

lo
........, I

~

0

on

lo
lo
........,

~

0 ~

lo
*h

II -xh

~ -xh II h

.......

('l

t-

t-

M

-.::t

or,

co

'"C"'l

II -xh

II -xh

II -xh

0\ \.0

0

t-

or,

0

~

\Q

I.r)

II

........,

I.r)

t-

t-

Table 32.

(Continued )

February 14, 2011

15:12

ch01

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••••••••••••••••••••••••••• •• ••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••• •• ••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••• •• ••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••• •• ••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••• •• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• -..,

-..,

<;;::,

I

00 "1-

0\

-..,

~ C""l

C""l

00

C""l

II

II

II

II

*h

*h

*h

*h

eo

r-

II

*h

I

"'"

\Q

"1-

o

-.., I

-.,

-,.

C""l

C""l

C""l

II

II

II

*h

*h

*h

N

M

o

00

00

128

eo

"1-

-.., "1-

o Ir)

Ir) Ir)

Ir)

C""l

C""l

II

II

II

*h

*h

*h

-.::I"

00

V)

eo

I

~ C""l

II

*h

r-

eo

February 14, 2011

15:12

ch01

Table 32.

(Continued )

•• •••••••••••••••••••••• ••••• ••••••••• •• •••••••••••••••••••• •••••••••••••••••• ••••••••••••• ••••••••••••••••• •• •••••••••••••••• •••••••••••••• ••••••••••••••••••••• ••••••••••••••••••••••••• •• •••••••••••• •••••••••• ••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••• •• •••••••• •••••• ••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••• •• •••• •• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• .,. .,. .,. '" ....

Q

..:

..<

00 ""t~

II

*h

co co

'<>

Q

Q

..:

..:'" l'-.-

""t-

e;::, <"'l

II

~

Q

..:

00

..: 0 0\

'".,.

Q

Q

..:

..:

or,

00

0\ 0 ""t-

II

II

-..,

*h

*h

0\ 00

o

*h

.... ~

Q

..:

00

..:

00 00

-..,

'" '"

Q

..:

Q

~ C")

<"'l e;::,

OO

-..,

II

II

*h

*h

~

Q

..:

..:

~

Q

..:

><

;;; ..:

Q

..:

~ I

'"""'~ <"'l

II

*h

~

~

><

-..,

~ -x-

h

II

h

<"'l

CV)

II

*h

...

I

'"<"'l II

*h

~ ..:

Q

..:

~ I

;::; ~ <"'l

II .;:-

h

co

0'1

0'1

129

~

Q

..:

><

-..,

.,.~

Q

Q

..:

..:

<"'l

<"'l 0\ 0 ""t-

II

II

*h

*h

I

~

o o,......,

L I Isle ofEden

Attraetor

3

T* = 2 T* =2

5

T* = 6

(x 0 ,x T-J ) ~ (initialJi,nal) bit string on Period- T orbit

I~ Xl rii-jj------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

---i~--t-==-;-=----------------------------------------------------------------------------------------------------------------------------------------------------------------------------i~--t-==-;-==-----------------------------------------------------------------------------------------------------------------------------------------------------------------------

I

I

6

D

T* = 2

Xo I • • • • • • Xl

I ••••••

T* = 14

--~~--t-==l==l;---------------------------------------------------------------------------------------------------------------------------------------------------------------

8

T* = 4

---i;---t-==-;-==-;-;=-----------------------------------------------------------------------------------------------------------------------------------------------------------T* = 14

--~~--t-==-;-==-;-==-;--------------------------------------------------------------------------------------------------------------------------------------------------------

10

T* = 6

---i~---t-==l==l;=l;----------------------------------------------------------------------------------------------------------------------------------------------------

11

T* = 62

--;:~+==-;-==-;-;=-;-;;------------------------------------------------------------------------------------------------------------------------------------------------

130

12

T* = 2

---i~+==-;-==-;-==-;-=;;---------------------------------------------------------------------------------------------------------------------------------------------

13

T = T*/3

--;~~+==l==-;-;=-;-;;;-;-----------------------------------------------------------------------------------------------------------------------------------------

14

T* = 14

--~~--t-==-;-==-;-==-;-==-;-;-=--------------------------------------------------------------------------------------------------------------------------------------;~;--t-==-;-==-;-;=-;-;;;-;-=;----------------------------------------------------------------------------------------------------------------------------------

15

T* = 30

16

T* =8

---i;---t-==-;-==-;-;=-;-;;;-;-=;-;-------------------------------------------------------------------------------------------------------------------------------

17

T* = 30

--i~;--t-==-;-==l;=l;;;-;-=;-;-=---------------------------------------------------------------------------------------------------------------------------

18

T* = 14

19

T* = 1022

20

T* = 12

21

T* = 126

22

T* = 62

23

T* = 4094

--:-I~--t-==-;-==-;-;=-;-=;;-;-=;-;-==------------------------------------------------------------------------------------------------------------------------x~~;;-t-==-;-==-;-;=-;-;;;-;-=;-;-==-;---------------------------------------------------------------------------------------------------------------------~~~--t-==-;-==-;-;=-;-;;;-;-=;-;-==-;-=----------------------------------------------------------------------------------------------------------------~;-;-t-==-;-==-;-;=-;-;;;-;-=;-;-==-;-==--------------------------------------------------------------------------------------------------------------i:~+==-;-==-;-;=-;-;;;-;-=;-;-==-;-==-;--------------------------------------------------------------------------------------------------------i~~;J==-;-==-;-;=-;-;;;-;-=;-;-==-;-==-;-=------------------------------------------------------------------------------------------------------

ch01

7

9

15:12

4

...

Data for generating and verifying the period of period-T orbits of 105 .

February 14, 2011

Table 33.

24 I T* = 2046

26

T= T*/6

27 28

T* = 1022 T* = 28 T* = 2(2

14

-1)

30

T* = 30

31

T* = 62

32

T* = 16

33 131

T* = 30

35

T* = 8190

36 T= T*/9

38

T* = 1022

39

T* = 8190

40

T* = 24

41

T* = 2046

42 I 43

T* = 254

44

T* = 124

45 46

I T* ~ /26 T* = 8190

T* = 4094

.

··i~~;+IZ·I·I=·;·==·;·==·;·I=·;·I·=;·I·=;·I·==·;·==·;·1=·;·11···············································

.

--i;~~+IZ-I-I=-;-I=-;-==-;-I=-;-I-=;-I-=;-I-==-;-==-;-I=-;-IIZ--------------------------------------------------------------x~~;;+-IZ-I-I=-;-I=-;-==-;-I=-;-I-=;-I-=;-I-==-;-==-;-I=-;-IIZ-;---------------------------------------------------------T* = 28

37

··i;~~+IZ·ll;·;·=;·;·==·;·I=·;·I·=;·I·=;·I·==·;·==·;·1=

--i~~;+IZ-I-I=-;-==-;-I=-;-I=-;-I-=;-I-=;-I-==-;-==-;-1=-;--------------------------------------------------------------------------i~~~+IZ-I-I=-;-I=-;-I=-;-I=-;-I-=;-I-=;-I-==-;-==-;-1=-;-1---------------------------------------------------------------------

I

T* = 62

34

--i;-+
z-
-
z-
-
z-
-==-;-
=-;-
-=;-
-=;-
-==---------------------------------------------------------------------------------------------------x;~~~JIZ-I-I=-I-I=-I-==-;-I=-;-I-=;-I-=;-I-==-I------------------------------------------------------------------------------------------------i~~;+IZ-I-I=-I-==-I-==-;-I=-;-I-=;-I-=;-I-==-I-=-------------------------------------------------------------------------------------------x~~~;JIZ-I-I=-I-==-I-==-;-I=-;-I-=;-I-=;-I-==-I-==-----------------------------------------------------------------------------------------i;~;+IZ-I-I=-I-I=-I-I=-;-I=-;-I-=;-I-=;-I-==-I-==-I------------------------------------------------------------------------------------~~-;+IZ-I-I=-I-==-I-==-;-I=-;-I-=;-I-=;-I-==-I-==-;-I--------------------------------------------------------------------------------

··i;~;+IZ·I·I=·;·==·;·==·;·I=·;·I·=;·I·=;·I·==·;·==·;·I=·;·IIZ·;·=·········································· . -~~-;+IZ-Il=-;-I=-;-I=-;-I=-;-I-=;-I-=;-I-==-;-==-;-I=-;-IIZ-;-=Z--------------------------------------------------l~~;JIZ-I-I=-;-==-;-==-;-I=-;-I-=;-I-=;-I-==-;-==-;-I=-;-IIZ-;-=Z-;-----------------------------------------------x~~;JIZ·I·I=·;·I=·;·I=·;·I=·;·I·=;·I·=;·I·==·;·==·;·I=111Z·;·=Z·;·=··········································· --i;~~--tlZ-I-I=-;-==-;-I=-;-I=-;-I-Z-;-I-=;-I-==-;-==-;-I=-;-IIZ-;-=Z-;-==----------------------------------------x;~~~JIZ-I-I=-;-I=-;-I=-;-I=-;-I-=;-I-=;-I-==-;-==-;-1=-;-IIZ-;-=Z-;-==-;-----------------------------------·t;~+IZ·I·I=·;·==·;·==·;·I=·;·I·=;·I·=;·I·==·;·==·;·I=·;·IIZ·;·==·;·==·;·I································ -i:~-tlZ-ll=-;-==-;-I=-;-I=-;-I-=;-I-I;-I-==-;-==-;-I=-;-IIZ-;-=Z-;-==-;-I=-----------------------------t;~+IZ-I-I=-;-I=-;-I=-;-I=-;-I-=;-I-=;-I-==-;-==-;-I=-;-IIZ-;-IZ-;-==-;-I=-;-------------------------x~~;;+IZ·I·I=·;·==·;·==·;·I=·;·I·=;·I·=;·I·==·;·==·;·1=·;·IIZ·;·I=·;·==·;·I=·;·I····················· -x~~~JIZ-I-I=-;-I=-;-==-;-I=-;-I-Z-;-I-=;-I-==-;-==-;-1=-;-IIZ-;-=Z-;-==-;-I=-;-I-Z------------------

ch01

29

T* = 4

15:12

25

I

(Continued )

February 14, 2011

Table 33.

47

T* = 2(2

23

49

T* T* = 2(2

21

8

---i;---t-==-;-==-;-==-;-;l;-;-I;-;-=;-I-=;-;-==-;-;=-;-1-1-;-1-=;-;-==-;-==-;-==-;-;-1;-;----------i~~-J=;-;-==-;-==-;-;=-;-;-I-;-;-=;-I-=;-;-==-;-;=-;- 1=-;-1-=;-;-==-;-==-;-==-;-;-1;-;-1------

-1)

T* = 2046

-x;14s-t-I;-;-=;-;-I;-=-ll;-;-I;-;-=;-;-=;-;-==-;-;=-;-1-1-;-1-1;-;-;;-;-;;-;-;;-;-;-1;-;-1;--x.o t=················································· = . X

T*

51

=

510

ch01

52

=

T = T*/6 :

53

x

T* = 2(2 26 -1)

o

XT_I

T* = 1022

54

t••• ===

.

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

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

132

55

T* = 2(2 20 -1)

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

56

T* = 56

X·················································· --~:~+ii-=-ii-=-ii-i-ii-i-i-i-i-i-ii-.-i.-.-iii-i-ii-i-i.-i -i-ii-.-ii-.-ii-.-i.-i-i-ii-i-ii--T* = 1022

57

58

x·················································· -~::~+ii-i-ii-i-ii-i-i.i-i-ii-i-ii-.-i.-.-iii-i-i.-.-ii-i-.-ii-.-ii-.-ii-i-i.-i-i-ii-i-ii--x·················································· --~-~~-:+ii-i-ii-i-ii-i-i.i-i-ii-i-ii-.-i.-.-iii-i-i.-i -i-i-i-i-ii-.-ii-.-ii-.-i.-i-i-ii-i-ii---

T* = 2 15 _2

59 I T* = 2(2 29 -1)

T* = 60

60

61

15:12

~

-i~~-;-tl=-Il;-I-I;-;-Il;-I-I;-I-I;-I-=;-;-==-;-==-;-I-1-;-
-=;1-==-;11;-==-1;-1;--------------

-1)

48

(Continued )

February 14, 2011

Table 33.

T

=

T*/63

x·················································· --~:~+ii-i-ii-=-ii-=-i.i-i-ii-i-ii-.-i.-.-ii.-i-i.-i-i-i-ii-ii-.-ii-.-ii-i-i.-i-i-ii-i-ii--X·················································· ·~~~·:+ii·i·ii·i·ii·i·ii·i·i·ii·i·ii·.·i.·.·ii.·i·i.·i· i·i·i·i·ii·.·ii·.·ii·.·i.·i·i·ii·i·ii···

February 14, 2011

15:12

ch01

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

Table 33.

(Continued )

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

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

•••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• ••••"' •••• •••• •••• •••• :;;

Q

~

~

'n

'"

Q

~

~

;;;

Q

~

~

on

'"

Q

~

~

\Q

:;;

Q

~

~

~

~

~

"'on

Q

~

~

~

Q

~

~

.......

'"

II -x-

II -x-

'"II

~

~

\Q


II -x-

II -x-

h

h

\Q

~

II

*h

~

a

h II h

II -x-

Q

~

~

Q

~

~

h

0\

....... 00

II

h

'"

Q

~

II -x-

c::::,

\Q

~

lr)

-.h r---

on 'n

Q

~

\Q

~

:r;--

.......

~

Q

~

~,

h

h

~

::>,

~

~

~ co

Q

~

*h

~ I

~ ~

~

....... II

II -x-

II

*h

h

M

"T

,'")

c::::,

"> ~

~ -xh

h

N \0

M

\0

"T \0

V)

\0

\0 \0

r---

\0

eo \0

133

0\

\0

0

r---

..-

r---

N

r---

r---

r---

February 14, 2011

15:12

ch01

•• I• •••••••••••••••••••••••• i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

··i·················································· ··i·················································· ··i·················································· • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Table 33.

(Continued )

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

• • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

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

l""l

V")

"-

'"~

l""l

II -x-

""II

h

*h ~ ,

~ ,

co

'"~

a......,

V")

V")

l""l

"t-

l""l

II

II

II

*h

*h

-x-

h eo

r-

o

N

00

00

134

M

eo

r-

eo

February 14, 2011

15:12

ch01

·· .. ·.............................................. .. .............................................. . .............................................. . ··············································i······ .............................................. . ··············································i······ .............................................. . .............................................. . ··············································i······ .............................................. . ··············································i······ .............................................. . ··············································i······ .............................................. . .............................................. . .............................................. .

Table 33.

(Continued )

•• • • • • • • • • • • • • • • • • • • • • • • I••••• • • • • • • • • • • • • • • • • • • • • • • •i• • • • • •

• • • • • • • • • • • • • • • • • • • • • • • • • •i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • : • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • i• • • • • •

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

135

D

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 • • •

Isle ofEden

5 6

T* = 3

7

T* = 7

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

8

T* = 4

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

T* = 1

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

T* =2

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

I

I

T* = 1

Xo I • • • • • •

_

.

---i~---I-=-=IIIZ-Z-Z-=-----------------------------------------------------------------------------------------------------------------------------------------------------------------

10

T* = 6

---i~---I-=-=-=IIZ-Z-Z-=-=-------------------------------------------------------------------------------------------------------------------------------------------------------------

11

T* = 31

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

12

T* = 2

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

136

13

T= T*/3 = 21

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

14

T* = 14

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

15

T* = 15

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

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

-x~~-~--I-=-;-=IIZ-Z-Z-=-;-=-=lZ-=-Z-Z-=-;--------------------------------------------------------------------------------------------------------------------------

20

T* = 12

··i;~·+=·;·=IIZ·Z·Z·=·;·=·=lZ·Z·Z·=·=·;I··············

_

.

22

T* = 62

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

23

T* = 2047

X";~~J=-=-;ll=-=-Z-=-=-;-=;=-=-Z-Z-=-=ll;Z-----------------------------------------------------------------------------------------------------------

21

T* = 63

ch01

T* = 7

9

15:12

L 3 4

February 14, 2011

Table 34.

24 I T* = 1023

26

T= T*/3 = 42

27

T* = 4

-x~~~;JIZ-I-IZ-I-IZ-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;------------------------------------------------------------------------------------------------i~~;+IZ-I-IZ-I-=Z-I-=Z-;-IZ-;-I-=;-I-=;-I-=Z-;-=------------------------------------------------------------------------------------------T* = 511

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

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

Table 34.

(Continued )

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

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

•••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• •••• ••••"' •••• •••• ••••:=: •••• :;;

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-


N \0

h

M

\0

"T \0

*'

\Q

h

II -x-

II

h

V)

\0

h

\0 \0

lr)

II -x-

h

h

~

C")

\Q

II -x-

h

C"l \Q

h

I

'"'"<""1

r---

\0

c::::,

a

0\

-..,

\Q

II -x-

h

eo \0

139

0\

\0

I

C"l

II

II -x-

0

r---

-.., -..,

,'")
00

*h

"t-

lr)

-..,

lr)

II

*h

h

..-

r---

C"l

00

lr)

II

S2;

*h

~

N

r---

M

r---

"T

r---

February 14, 2011

15:12

ch01

Table 34.

(Continued )

•• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• ••• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •• •••••••••••••••••••••• ••••• •••••••••••••••••••• ••••••••• •• •••••••••••••••••• ••••••••••••• •••••••••••••••• ••••••••••••••••• •• •••••••••••••• ••••••••••••••••••••• •••••••••••• ••••••••••••••••••••••••• •• •••••••••• ••••••••••••••••••••••••••••• •••••••• ••••••••••••••••••••••••••••••••• •• •••••• ••••••••••••••••••••••••••••••••••••• •••• ••••••••••••••••••••••••••••••••••••••••• •• •• ••••••••••••••••••••••••••••••••••••••••••••• ••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• •••••••••••••••••••••••••••••••••••••••••••••••••••• -...,

-...,

I

I

~

?'l

~

~

II -x-

II -x-

h

h

-...,

-...,

I

I

'"


~

~

II

II -x-

-x-

h

h

eo

r-

o

N

00

00

140

M

eo

lr) lr) ~

"t-

lr) ~

II

II

*h

*h r-

eo

Table 34.

(Continued )

February 14, 2011

15:12

ch01

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

141

February 14, 2011

15:12

ch01

* T 142

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

*

~.

"*

*

*

-/1.

--A~

* * * **

*

1

D

**

ll--~-~*:-------!r~--:*~*~~-- * **

10 ----*--~--------'*i----'---------.-------T-t~'-.--'

*

10

2

*

) for Fig. 47. Relationship between the length L of the bit strings and the maximum period T of the attractors (symbol rule 60 . In some cases (symbol ) the maximum theoretical value is too large to be verified experimentally, and we hypnotize that it is a multiple of the actual maximum period T .

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

143

*-

12

10 ----------------------~*_---,

T *-

10

10

***-

10

~

"

8

**

*-

*-

~

"

10

*-

6

**:

*-

~

"

10

4

*~

"

10

*-1.

*:

*-

~

"

**-

1

1

10

*:

* * ****

*-**

~

iI:t

~*-

*k

*-*- ***

*- *-

**-

** *-* *- *

**-

2

*-

D

*-

*-

*~*

*

*

L

10

2

"*

Fig. 48. Relationship between the length L of the bit strings and the maximum period T of the attractors (symbol rule 90 .

) for

February 14, 2011

15:12

ch01

* 10 144

12

TA Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science 1011

*

1010

10 9

*

*

*

*: *:

*

10 8

* \. * *: *

10 7

10 6

* * * * **"*" * * * ** *11k * *" * * *"* **" "**. * *"* ***~* *~ *"* 10 * *" * *** * * * ** * * * 1+-------......----------....................--~---------........------ .............. 1

]0

*

D

L

]02

*

Fig. 49. Relationship between the length L of the bit strings and the maximum period T of the attractors (symbol Isles of Eden (symbol ) for rule 105 .

) or

February 14, 2011

15:12

ch01

* *

10 12

T

Chapter 1: Quasi-Ergodicity

10 11

* *

1010

10 9

* 10

145

** *" *"

8

* *

10 7

* * *

10 6

*:

*"

* * * * * *** ** * * *** * ** * * * --M:- * * *"~* ~ ***'* *"* ~*:* * *-*** * *** *

*-

10

* * **

**

*" *: * * ........--.""'"T""""""*"--........------------......... 1+----------'llll'------~

__1

1

10

*

D

L

10

2

"*

Fig. 50. Relationship between the length L of the bit strings and the maximum period T of the attractors (symbol Isles of Eden (symbol ) for rule 150 .

) or

February 14, 2011

15:12

ch01

"* T 146

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

1m

10 22

10 20

*

10 18

"* -.A~

10 16

~

10 14

10 12

"fl'

4.

"* . *

10 10

-.A:

*

10 8

--' R

10 6

*-1- ,

....:r

~:

1-

~.

" 10 4

" ...L

10 2

*

~

1--! 1

"

...

.

*

"'"

*

*

~

,

~t~r~:' ~I'-

}>.:.

*.. '7'

,

"pk

~~ ~ ;r".'

'f-I:

)'~!' -,;::

""R ~

>of>'

10

____---rftL_____ L

D

Fig. 51.

**~:-\'l:

**

---*--~---

....

'* *

*

-~r -

Scale-free property for rule 60 .

,---------.-----_._,1'~"-'---'---"

,I

February 14, 2011

15:12

ch01

Chapter 1: Quasi-Ergodicity

10

147

*:

12

*:

T 10

10

*:

*: *:

10

*: *:

8

10

6

*:

*:

1 1

L

10

D

Fig. 52.

Scale-free property for rule 90 .

10

2

February 14, 2011

15:12

ch01

*

1012.--148

--.

TA Nonlinear Dynamics Perspective of Wolfram’s New Kind of Science 10 11

* * *

10 8

* \. * * *

10 7

10 6

*

10

*' *

* *"

1+-------......----------....................--~---------........------ .............. 1

L

]0

D

Fig. 53.

Scale-free property for rule 105 .

]02

February 14, 2011

15:12

ch01

* *

10 12

T

Chapter 1: Quasi-Ergodicity

10 11

*

10 10

10 9

* 10 8

**

*

10 7

* *

10 6

* *

*:

10

l+------~Ill'_------____,lIl~--""'"T""" .....---------........---.........__I 2 1 10 L 10

D

Fig. 54.

Scale-free property for rule 150 .

149

February 14, 2011

15:12

ch01

D D150

0

0

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

Therefore, we need to find all L and L
, with L

= L, for which ξL = ξL
. From now on, we will refer exclusively to rules 90 and 150 , for which TL∗ = 2s(L) − 1; however, the conclusions for rules 60 and 105 are very similar and can be achieved through a similar procedure. The following theorem represents an important advance towards our goal.

is TL∗ = 15 (see Table 30); therefore, ξL = TL∗ /L = 1 and o(ξL ) = 1. From Eq. (50) n = (o(L)/o(ξ)) − 1 = (4/1) − 1 = 3, and ((21 − 1)/1)1 + 21·1 + 21·2 + 21·3 ) = 15, as expected. In this case, we consider all values n > 1, i.e. n = 1, 2, 4, etc. to be sure of finding all L

= L such that ξL = ξL
= 1. For example,

Theorem 6.4 (Scale-free decomposition). Let L be the length of the bit strings, L odd, and let ξL be its scale-free order. If o(L) = s(L),23 then L can be uniquely expressed as

n : 1 L
=

21 − 1 (1 + 21 ) = 3 1

n : 2 L
=

21 − 1 (1 + 21 + 21·2 ) = 7 1

i

2o(ξL ) − 1
i·o(ξL ) 2 L= ξL i=0

(50)

where n = (o(L)/o(ξL )) − 1. This theorem is extremely powerful, because given any L for which o(L) = s(L) and its scale-free order ξL , we can easily find analytically all the L
for which ξL = ξL
by just varying the parameter n. Its proof requires only basic concepts from number theory, but we present it in Appendix B in order to illustrate here the application of the theorem through some examples. Example 6.1. For L = 21, the values of the multi-

plicative order and suborder are o(21) = s(21) = 6 (see Table 29), and the maximum period of the orbit is TL∗ = 63 (see Table 30); therefore, ξL = TL∗ /L = 3 and o(ξL ) = 2. From Eq. (50) n = (o(L)/o(ξ))−1 = (6/2) − 1 = 2, and ((22 − 1)/3)(1 + 21·2 + 22·2 ) = 21, as expected. Considering the following values for the parameter n, i.e. n = 3, 4, 5, etc., we can find an infinite number of L

= L such that ξL = ξL
. For example, n : 3 L
=

22 − 1 (1 + 21·2 + 22·2 ) = 21 3

n : 4 L
=

22 − 1 (1 + 21·2 + 22·2 + 23·2 ) = 85 3

n : 5 L
=

22 − 1 (1 + 21·2 + 22·2 + 23·2 + 24·2 ) = 341 3

.. . Example 6.2. For L = 15, the values of the multi-

plicative order and suborder are o(15) = s(15) = 4 (see Table 29), and the maximum period of the orbit

23

We assume that o(1) = 1, as illustrated in Example 6.2.

21 − 1 (1 + 21 + 21·2 + 21·2 + 21·3 + 21·4 ) 1 = 31 .. .  i (m+1) − 1, Note that in this example L = m i=0 2 = 2 ∗ (m+1) − 1, as and since ξL = 1 it follows that TL = 2 already stated in Theorem 4 of [Chua et al., 2007b]. Therefore, the scale-free decomposition gives an alternate proof for such theorem. n : 4 L
=

Example 6.3. For L = 73, the values of the multi-

plicative order and D suborder are o(73) = s(73) = 9 (see Table 29), and the maximum period of the orbit is TL∗ = 511 (see Table 30); therefore, ξL = TL∗ /L = 7 and o(ξL ) = 3. Table 35. Values of L < 105 having the same scale-free order ξL ≤ 15 for rule 90 , ξL = (2s(L) − 1)/L.

Scale-free order

ξL

Length L of the bit strings

1

3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, 4095, 8191, 16383, 32767, 65535,

3

21, 85, 341, 1365, 5461, 21845, 87381,

5

51, 819, 13107,

7

73, 585, 4681, 37449,

9

455, 29127,

11

93, 95325,

13

315,

15

273, 4369, 69905,

February 14, 2011

l09r

15:12

ch01

- - - - - - - - - - - - - - - - - -__

T ~L =2361.65 151 . Chapter 1: Quasi-Ergodicity ~L =564.93 .

~L=7 ~L =2.818 ...

~L=7

D

/ N (blue lines) include attractors of only one L odd; lines Fig. 55. For rule 90 , lines corresponding to a scale-free order ξL ∈ corresponding to scale-free orders ξL ∈ N (red lines) include attractors of infinitely many different odd values of L.

From Eq. (50) n = (o(L)/o(ξ))−1 = (9/3)−1 = 2, and ((23 − 1)/7)(1 + 21·3 + 22·3 ) = 73, as expected. If we “go backward” using n = 1, as already done as in Example 6.3, we apparently find a contradiction, because n=1:

L
=

27 − 1 (1 + 21·3 ) = 9 7

but Table 30 indicates that T9∗ = 31, and then ξ9 = 3.44
= ξ73 . However, this happens because o(9) = 6 and s(9) = 3, and since o(9)
= s(9), Theorem 6.4 cannot be applied to L = 9. The values of L corresponding to ξL ≤ 15 are presented in Table 35. The results of this section

February 14, 2011

152

15:12

ch01

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

can be summarized in the following theorem: Theorem 6.5 (Scale-free property of additive rules). Let L be the length of the bit strings and let TL∗ be the maximum period of the orbit of a nontrivial additive rule, then for any L
∈ N the scale-free property TL∗
/L
= TL∗ /L holds iff (a) both L and L
satisfy the conditions of Theorem 6.4, if L
is odd; (b) L
= 2n · L, with n ∈ N, if L
is even and L, L

= 2i ; Figure 55 gives a graphical interpretation of this theorem: red lines correspond to values of ξL ∈ N, which will include all odd L obtained from Theorem 6.4 (as stated in the first point of Theorem 6.5) plus their even multiples (as stated in the second point of Theorem 6.5); blue lines correspond to values of ξL ∈ / N, and then they include only one odd L (because L does not satisfy the conditions of Theorem 6.4) plus its even multiples (as stated in the second point of Theorem 6.5).

7. Concluding Remarks The first and most important result of this paper is the characterization of the quasi-ergodic phenomenon. Although it has been introduced empirically, quasi-ergodicity is tremendously useful to support our classification of local rules into six groups, since only complex and hyper Bernoulli-shift rules appear to be quasi-ergodic. For the first time, we have found an additional feature distinguishing the Bernoulli στ -shift rules (group 4) from the complex and hyper Bernoulli-shift rules (groups 5 and 6), besides the dependence of the two Bernoulli parameters σ and τ on the initial point and on the length L. We also included numerous figures and tables in order to show the qualitative meaning of quasi-ergodicity, and encourage readers to give a quantitative interpretation of it. In addition to quasi-ergodicity, the paper presents further results about a variety of topics. First of all, a complete explanation for the emergence of fractals in the time-1 characteristic functions was given. These results, which generalize those presented in [Chua et al., 2005b], allow us to classify each rule according to the number of fractal patterns it exhibits, and retrieve the rule number from the time-1 characteristic function. In our opinion, the number and the form of the fractal patterns may give information on the dynamical

behavior of the corresponding local rule; this aspect will be further analyzed in our next papers. A whole section has been devoted to introduce a new nomenclature for rules which harbor, or do not harbor, Isles of Eden. For the first time we were able to find that Isles of Eden are everywhere in 1-D cellular automata: in fact, only 28 rules out of 256 do not have any Isle of Eden! Furthermore, we show that at least for two rules of group 1, a string either belongs to the basin of a global attractor or is an Isle of Eden; hence, we can completely characterize these rules by analyzing their Isles of Eden. The Bernoulli στ -shift basin tree generation formula presented in Sec. 4 has a practical importance, because thanks to it we only need to analyze a fraction of the 2L possible bit strings to draw all of the DOD D forbasin-tree diagrams of a given local rule. This mula will be also used in our next papers, in which we will relate it with the “traveling waves” in cellular automata. Finally, the last section completes the work that began in [Chua et al., 2007a] and [Chua et al., 2007b] about additive rules 60 , 90 , 105 and 150 . Although some of the results were known, they were not easily accessible and we have therefore recast them using our notation for the reader’s convenience. Furthermore, we have included many tables, so that all experiments reported can be accurately reproduced and verified. In our opinion, the results presented in this paper shed new light on a number of phenomena, especially on the role of Isles of Eden and the behavior of Bernoulli στ -orbits. Most of our theorems and conjectures can be easily generalized to bi-infinite bit strings by taking advantage of the analogy between non-spatially periodic finite strings and spatially periodic bi-infinite strings. Furthermore, many rules behave like a left- or right-shift for certain subsets of strings, and this property can be used to define a completely new class of problems still unexplored.

Appendix A Theorem. Inequality (31) is given by

D

  I I 

  1− xi−1 2−(i+1) + xi 2−(i+1) − 1   i=0

≥ χ 60

i=0

  I I  

  ≥ xi−1 2−(i+1) − xi 2−(i+1)    i=0

i=0

February 14, 2011

15:12

ch01

D Chapter 1: Quasi-Ergodicity

where I


χ 60 =

|xi−1 − xi |2−(i+1)

(A.1)

i=0

The above formula can be split into two parts, one for the upper bound

Proof.

I
i=0

|xi−1 − xi |2−(i+1)   I I  

 −(i+1) −(i+1)  ≥ xi−1 2 − xi 2    i=0

(A.2)

i=0

≥

 But xi ∈ {0, 1} and max( Ii=0 xi 2−i ) = 1; this means that (A.7) holds, and consequently the inequality (A.3) holds.


Appendix B In order to prove Theorem 6.4 (scale-free decomposition), presented in Sec. 6 and recalled in the following, we need to introduce two lemmas from number theory, whose proof can be found in any good book on number theory. Lemma B.1. For n ∈ N, n = ab

2n − 1 = 2ab − 1

and one for the lower bound  I  I 

  1− xi−1 2−(i+1) + xi 2−(i+1) − 1   i=0

= (2a − 1)(1 + 2a + 22a + · · · + 2(b−1)a ) (B.1)

i=0

I


|xi−1 − xi |2−(i+1)

(A.3)

i=0

which can be analyzed separately. However, in both cases we will make use of the well-known triangle inequality |a1 | + |a2 | ≥ |a1 + a2 | ⇒ |a1 | + |a2 | + · · · + |aN |   N 
   ai  ≥ (A.4)  

Lemma B.2. Let o(n) be the multiplicative order of 2 (mod n), n = ab, then

o(n) = lcm(o(a), o(b))

The proof for inequality (A.2) follows directly from (A.4). As for (A.3), it is equivalent to   I I  

  xi−1 2−(i+1) + xi 2−(i+1) − 1 1≥   +

I


i=0

−(i+1)

|xi−1 − xi |2

(A.5)

i=0

Applying (A.4) to the second part of inequality (A.5), we find   I I  

  xi−1 2−(i+1) + xi 2−(i+1) − 1    i=0

i=0

+

I


−(i+1)

|xi−1 − xi |2

i=0

 I  
   ≥ xi 2−i − 1   i=0

Theorem (Scale-free decomposition). Let L be the

length of the bit strings, L odd, and let ξL be its scale-free order. If o(L) = s(L), then L can be uniquely expressed as n

2o(ξL ) − 1
i·0(ξL ) 2 L= ξL

i=0

(B.3)

i=0

where n = (o(L)/o(ξL )) − 1. Proof.

Since o(L) = s(L), the scale-factor order be-

comes 2o(L) − 1 2s(L) − 1 TL = (B.4) = L L L and, by the definition of multiplicative suborder, it follows that ξL =

o(ξL L) = o(2o(L) − 1) = o(L)

(B.5)

But, from Lemma B.2, o(ξL L) = lcm(o(ξL )o(L)) =

o(ξL )o(L) p

(B.6)

where p|o(L), which means that

(A.6) therefore the inequality (A.5) is equivalent to   I 
   1≥ xi 2−i − 1 (A.7)  

(B.2)

Now, we are ready to prove the following theorem.

i=1

i=0

153

o(L) = p · q

(B.7)

Then, combining Eqs. (B.5) and (B.6), we obtain o(L) =

o(ξL )o(L) ⇒ o(ξL ) = p p

(B.8)

February 14, 2011

154

15:12

ch01

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

and then we also find an expression for q, because q=

o(L) o(L) = p o(ξL )

(B.9)

By using Lemma B.1, we can easily find that 2o(L) − 1 = (2p − 1)(1 + 2p + · · · + 2(q−1) · p ) = (2o(ξ) − 1)(1 + 2o(ξ) + · · · + 2(q−1) · o(ξ) ) (B.10)

in Fig. 8(a), has the following firing/quenching patterns

2o(ξL ) − 1 1 = o(L) (1 + 2p + · · · + 2(q−1) · p ) ⇒ 2 −1   o(L) 2o(ξL ) − 1 ( o(ξ) −1) · p p 1 + 2 + ··· + 2 L= ξL

(B.11)

or, in other words,  2o(ξL ) − 1  1 + 2o(ξL ) + · · · + 2n · o(ξL ) L= ξL where n = (o(L)/o(ξL )) − 1.




D

Definition C.1. A local rule N is said to be LeftPermutive, iff for every x ˜0 , x˜1 ∈ {0, 1} the map ˜0 , x ˜1 ) is a permuting map of the alphabet f N (x−1 , x {0, 1} [Hedlund, 1969; Courbage, 1999; Courbage & Yasmineh, 2001; Pivato & Yassami, 2008]. Theorem C.1. The classical definition and defini-

tion based on Boolean cubes of left-permutive rules are equivalent. A left-permutive rule can be written in the form [Wuensche & Lesser, 1992, p. 30]

Proof.

(2) 010 ↓ X2

(3) 011 ↓ X3

(4) 100 ↓ X0

(5) 101 ↓ X1

(6) 110 ↓ X2

(0) 000 30 : ↓ D 0

(1) 001 ↓ 1

(2) 010 ↓ 1

(3) 011 ↓ 1

D(4)

100 ↓ 1

(5) 101 ↓ 0

(6) 110 ↓ 0

(7) 111 ↓ 0

then, according to (C.1), it is a left-permutive rule. (B.12)

Appendix C The D Classical Definitions and Definitions Based on Boolean Cubes of Permutive Rules are Equivalent

(1) 001 ↓ X1

D

Example. Rule 30 , whose Boolean cube is shown

where we used Eq. (B.8)

(0) 000 ↓ X0

X6 = X 2 = 1 − X2 , X7 = X 3 = 1 − X3 , where Xi ∈ {0, 1}. Straightforwardly, we can conclude that the definition of left-permutive rule given in Sec. 1, D based on the opposite colors of the vertex pairs {(0), (4)}, {(1), (5)}, {(2), (6)}, and {(3), (7)} of the Boolean cubes, is equivalent to the classical definition of expression (C.1).


(7) 111 ↓ X3 (C.1)

Numbers within brackets refer to the corresponding vertices of Boolean cubes in Sec. 1. It is evident that it is necessary to define only four outputs — namely X0 , X1 , X2 and X3 — and the remaining four outputs will be automatically defined, since X4 = X 0 = 1 − X0 , X5 = X 1 = 1 − X1 ,

Definition C.2. A local rule N is said to be Right-

Permutive, iff for every x ˜−1 , x ˜0 ∈ {0, 1} the map f N (˜ x−1 , x ˜0 , x1 ) is a permuting map of the alphabet {0, 1} [Hedlund, 1969; Courbage, 1999; Courbage & Yasmineh, 2001; Pivato & Yassami, 2008]. Theorem C.2. The classical definition and definition based on Boolean cubes of right-permutive rules are equivalent.

Following the same steps as in the previous proof, a right-permutive rule can be written in the form [Wuensche & Lesser, 1992, p. 30]

Proof.

(0) 000 ↓ X0

(1) 001 ↓ X0

(2) 010 ↓ X2

(3) 011 ↓ X2

(4) 100 ↓ X4

(5) 101 ↓ X4

(6) 110 ↓ X6

(7) 111 ↓ X6 (C.2)

Numbers within brackets refer to the corresponding vertices of Boolean cubes in Sec. 1. It is evident that it is necessary to define only four outputs — namely X0 , X2 , X4 and X6 — and the remaining four outputs will be automatically defined, since X1 = X 0 = 1 − X0 , X3 = X 2 = 1 − X2 , X5 = X 4 = 1 − X4 , X7 = X 6 = 1 − X6 , where Xi ∈ {0, 1}. Straightforwardly, we can conclude that the definition of left-permutive rule given in Sec. 1, based on the opposite colors of the vertex pairs {(0), (1)}, {(2), (3)}, {(4), (5)}, and {(6), (7)} of the Boolean cubes, is equivalent to the classical definition of expression (C.2).


February 14, 2011

15:12

ch01

D

D

D

Chapter 1: Quasi-Ergodicity

Example. Rule 86 = T † 30 has the following fir-

ing/quenching patterns (0) 000 86 : ↓ D 0

(1) 001 ↓ 1

(2) 010 ↓ 1

(3) 011 ↓ 0

D

(4) 100 D↓ 1

(5) 101 ↓ 0

(6) 110 ↓ 1

(7) 111 ↓ 0

then, according to (C.2), it is a right-permutive rule. Definition C.3. A local rule

N is said to be ˜0 , x ˜1 ) and Bi-Permutive, iff both maps f N (x−1 , x x−1 , x˜0 , x1 ) are permuting maps of the alphaf N (˜ ˜0 ∈ {0, 1} and x ˜0 , x ˜1 ∈ bet {0, 1} for every x ˜−1 , x {0, 1} [Courbage, 1999; Courbage & Yasmineh, 2001; Pivato & Yassami, 2008].

Theorem C.3. The classical definition and the defi-

nition based on Boolean cubes of bi-permutive rules are equivalent. Proof. From expressions (C.1) and (C.2), we find that bi-permutive rules have the form (0) 000 ↓ X0

(1) 001 ↓ X0

(2) 010 ↓ X2

(3) 011 ↓ X2

(4) 100 ↓ X0

(5) 101 ↓ X0

(6) 110 ↓ X2

(7) 111 (C.3) ↓ X2

Numbers within brackets refer to the corresponding vertices of Boolean cubes in Sec. 1. It is evident that it is necessary to define only two outputs — namely X0 and X2 — and the remaining six outputs will

155

be automatically defined, since X1 = X4 = X 0 = 1 − X0 , X3 = X6 = X 2 = 1 − X2 , X5 = X0 , X7 = X2 , where Xi ∈ {0, 1}. Straightforwardly, we can conclude that the definition of bi-permutive rule given in Sec. 1, based D on the fact that vertices of the Boolean cubes can 0 0 0 be divided into two subsets with opposite colors {(0), (3), (5), (6)} and {(1), (2), (4), (7)}, is equivalent to the classical definition of expression (C.3).

D




There are only four bi-permutive rules: 90 , 105 , 150 , and 105 .

o

90 :

(0) 000 ↓ 0

(1) 001 ↓ 1

(2) 010 ↓ 0

(3) 011 ↓ 1

(4) 100 ↓ 1

(5) 101 ↓ 0

(6) 110 ↓ 1

(7) 111 ↓ 0

105 :

(0) 000 ↓ 1

(1) 001 ↓ 0

(2) 010 ↓ 0

(3) 011 ↓ 1

(4) 100 ↓ 0

(5) 101 ↓ 1

(6) 110 ↓ 1

(7) 111 ↓ 0

150 :

(0) 000 ↓ 0

(1) 001 ↓ 1

(2) 010 ↓ 1

(3) 011 ↓ 0

(4) 100 ↓ 1

(5) 101 ↓ 0

(6) 110 ↓ 0

(7) 111 ↓ 1

165 :

(0) 000 ↓ 1

(1) 001 ↓ 0

(2) 010 ↓ 1

(3) 011 ↓ 0

(4) 100 ↓ 0

(5) 101 ↓ 1

(6) 110 ↓ 0

(7) 111 ↓ 1

o o

This page is intentionally left blank

February 14, 2011

14:49

ch02

♦

Chapter 2 PERIOD-1 RULES ♦

Ë D

This stage of our journey through the universe of one-dimensional binary Cellular Automata is devoted to period-1 rules, constituting the first of the six groups in which we systematized the 88 globally-independent CA rules. The first part of this article is mainly dedicated to reviewing the terminology and the empirical results found in the previous papers of our quest. We also introduce the concept of the ω-limit orbit with the purpose of linking our work to the classical theory of nonlinear dynamical systems. Moreover, we present the basin tree diagrams of all period-1 rules — except for rule 0 , which is trivial — along with their Boolean cubes and time-1 characteristic functions. In the second part, we prove a theorem demonstrating that all rules belonging to group 1 have robust period-1 rules for any finite, and infinite, bit-string length L. This is the first time we give analytical results on the behavior of CA local rules for large values of L and, consequently, for bi-infinite bit strings. The theoretical treatment is complemented by two remarkable practical results: an explicit formula for generating isomorphic basin trees, and an algorithm for creating new periodic orbits by concatenation. We also provide several examples of both of them, showing how they help to avoid tedious simulations. Keywords: Cellular automata; nonlinear dynamics; period-1 rules; group 1 rules; ω-limit orbits; attractors; Isles of Eden; orbit concatenation; seamless concatenation conditions; basin tree diagrams; isomorphic basin trees; genotype; phenotype; bi-infinite bit strings; null rules; Wolfram.

1. The Second Time Around

Our quest for an analytical holy grail is propelled by our firm conviction that there must exist an elementary yet rigorous analytical theory of cellular automata that underpins the empirical morass collected in Wolfram’s monumental tome [Wolfram, 2002]. In our current adventure, we return to Part IV [Chua et al., 2005a] armed with a new arsenal of tools, and tempered with three years of insights, to uncover the deeper mysteries that had eluded us on our maiden voyage.

We continue our odyssey on the universe of onedimensional cellular automata, as depicted in Fig. 1, and chronicled over the past nine episodes: Part I [Chua et al., 2002], Part II [Chua et al., 2003], Part III [Chua et al., 2004], Part IV [Chua et al., 2005a], Part V [Chua et al., 2005b], Part VI [Chua et al., 2006], Part VII [Chua et al., 2007a], Part VIII [Chua et al., 2007b], and Part IX [Chua et al., 2008]. 157

t

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

. Final bIt

-.l

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

C II e 2

(c)

-,..

[KJ

I

t X X~ ) = N( Xi-I' i' 1+1 t

t+l

X·I

Llnitialbit

• • • ~22

=4

INPUT at time t

••• ~23 =8

INPUT CODE

l

Xi - 1

l

l

Xi

158

!J

(b) ••• ~24 =16

rn D rn D

( (d)

ImI

D

ImI ImI

iii D W D D ImI rn D D D

(e) Universal Formula for all 256 Rules

l+l

X i +1

[]] ImI ImI ImI IT] ImI ImI D

rn ImI D rn ImI D

OUTPUT at time t+ 1 Xi

,...... ,...... ,......

D D D ,...... D ,...... D ,...... D ,...... D ,...... D

(f) Heaviside "step" Function Ii(w)

X;+l =c!{C +c [C +C I(C +C 8

Fig. 1.

7

6

S

4

3 X;_1

--~J ~w

+C +CjX;+l)IJ } 2 X;

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

D

ch02

(a)

Cell Cell 0 1

-

output E {O,l} t+l ,..... x.

14:49

•• 11

,.

Local Rule

,.

X'I - I t X.I t

February 14, 2011

input E {a, I}

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules

Table 1.

159

List of 256 local rules with their complexity index coded in red (κ = 1), blue (κ = 2), and green (κ = 3), respectively.

0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

133

134

135

136

137

138

139

140

141

142

143

144

145

146

147

148

149

150

151

152

153

154

155

156

157

158

159

160

161

162

163

164

165

166

167

168

169

170

171

172

173

174

175

176

177

178

179

180

181

182

183

184

185

186

187

188

189

190

191

192

193

194

195

196

197

198

199

200

201

202

203

204

205

206

207

208

209

210

211

212

213

214

215

216

217

218

219

220

221

222

223

224

225

226

227

228

229

230

231

232

233

234

235

236

237

238

239

240

241

242

243

244

245

252

253

254

255

256 Local Rules 246

247

248

249

250

251

67 ~~ .$'

Period-l 25 Period-2 Rules

Rules

..~~~

I"\~'" §' ",,,,

~v ~~ ~ ~""

~

.........• t=======1=-O--=8~~1~8 -C-om-p-Je-x I

Bernoulli

ernoulli-shift ules

O"1:-shift Rules

Fig. 2.

Partitioning of the 256 local rules into six classes.

February 14, 2011

160

14:49

ch02

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

There are 67 robust Period-1 rules.

February 14, 2011

Rule Number

N

14:49

Left - Right Transformation

Tt[N]

ch02

Global Complementation

T[N]

Left - Right Complementation

Rule Number

T*[N]

N

Left - Right Transformation

Tt[N]

Global Complementation

Left - Right Complementation

Chapter 2: Period-1 Rules

T[N]

T*[N]

Table 3. There are 67 globally-equivalent robust period-1 local rules. All local rules in each row are globally equiva200denote local rules with a complexity index κ = 1, lentoto each other. Rows with red, blue, or green background colors 2, or 3, respectively. 4 202

8

12 13

32 36 40 44

64 68 69 72 76 77

78 79 92

93 96 100 104 128 132

136 140 141 160 164 168 172 192 196 197

203 204 205 206 207 216 217 218 219 220 221 222 223 224 228 232 233 234 235 236 237 238 239 248 249 250 251 252 253 254 255

161

February 14, 2011

14:49

ch02

••• 162

•••

•••

•••

•••

•••

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

• • •••

•••

... •••

•••

•••

...

.•••

••• •••

•••

D . . . . .t---&.

•••

m

•••

••• •••

••• ,

•••

•••

.-

•••

•••

• •••

III

•••

I

•••

•

•••

lIm

•••

•••

•••

•••

lID

•••

Em]

•

•••

••

,

IDI

... _. .-

•••

•••

,

•

•••

•••

••• •••

•••

•••

•••

••

1m ••• •••

... •••

•••

•••

...

•••

•••

••

am

•••

... ... .-

•••

••

...

,

•••

iii

...

... •••

1m

•••

•••

•••

•••

III

1m

1:1

•••

•••

_.

•••

•••

•••

There are 25 globally-independent period-1 rules.

•••

•••

m

• •••

Table 4.

•••

•••

•••

1m ••• •••

••• •••

mI

.•••

February 14, 2011

14:49

ch02

88 Equivalence Classes of Local Rules Chapter 2: Period-1 Rules

163

25 Period-l 13 Period-2

Rules

Rules

..........

30

Bernoulli O',;-shift Rules

Fig. 3.

Partitioning of the 88 globally-independent rules into six classes.

1.1. Recap of Period-1 Rules Among 256 local rules, which we reproduced in Table 1 for the reader’s convenience, 67 belong1 to Group 1, dubbed Period-1 Rules since most random initial bit strings converge to a period-1 attractor.2 They constitute about 1/4 of the 256 local rules, as depicted in the CA rules subdivision pie in Fig. 2. The 67 Period-1 local rules are exhibited in Table 2 along with their defining Boolean cube and index of complexity (red for κ = 1, blue for κ = 2 and green for κ = 3). Some of these 67 local rules may also have attractors with a larger period T > 1, but with smaller basins of attraction, and hence are not as robust as the dominating period-1 attractors. Consequently, it makes sense to refer to Group 1 as Period-1 rules, mindful of the fact that it is not the only game in town. In view of the global equivalence theorem proved D in [Chua et al., 2005a], only 25 out of the 67 period-1 rules are globally independent in the sense that the dynamics of the remaining 42 rules can be trivially predicted and determined from one of the

D 1

D

corresponding equivalent rules listed in Table 3. For future reference, these 25 globally independent period-1 rules are extracted from Table 2 and exhibited in Table 4. The corresponding CA rules subdivision pie shown in Fig. 3 reveals that almost 1/3 of the 88 globally-independent rules are period-1 rules. The globally-equivalent rules associated with each of these 25 period-1 rules are listed in Table 5.

D

2. Basin Tree Diagrams

D

In Tables 6-1 to 6-24, we present a gallery of the basin tree diagrams3 for each of the 24 globallyindependent period-1 rules listed in Table 4, not including the trivial rule 0 . They are organized into 24 Galleries, each identified by a two digit number “N − m”, where the left digit N Dis identified with the local rule N , and the right digit m pertains to page m of Gallery N, m = 1, 2, . . . , 5, for all N (except 0 ) listed in Table 4. The first page (i.e. Gallery N − 1) of each Gallery N displays the following relevant

D

D

DD

Due to a software error, in our previous articles rules 94 and 133 were erroneously included in this group. The reader is cautioned that except for the 28 local rules listed in [Chua et al., 2008], all other local rules harbor some Isles of Eden. For Group-1 rules, only 4 out of 25 globally-independent local rules are deprived of such gems; namely, rules 0 , 8 , 36 and 78 . 3 The basin tree diagrams for all 256 CA local rules are also available on the webpage http://sztaki.hu/∼gpazienza/cellular automata 2

February 14, 2011

164

14:49

ch02

Rule Number

Left-Right Transformation

N

Tt[N]

Global Complementation

T[N]

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

o 4

8

12 13 32 36 40 44 72 76

77 78 104 128 132 136 140 160 164 168 172 200 204 232

Left- Right Complementation

T*[N]

Local rules which are globally equivalent to the 25 robust period-1 rules listed in Table 4.

Ivertex

23 = 8

I

X:_ 1

xI i

t

X r+i

Ix

t 1 + i

~·I·I·I·

-

I

1;1

Xlj] 14:49

=4

t

February 14, 2011

0.9


0.8

~·I·I·I·

0.5

ch02

- - .... .1.1.1. .... . 1.1.1. - !j·I·I·I· ~w 22

•

Basin Tree Diagrams of Rule 4 .

Table 6-1.

-

2 = 16

xf+1 165

IL

=


31

..

7 ••••

4

¢@ 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

PI =3 -=0.375 8

fl)C()

I

L4

1

5 P2=-=0.625 8

I Gallery 4-1 I

I

I

0.5

¢n-l

1

I

e:J8)C()OJ

········..···..···.·······D ..···..: 8:)

1 P2=2- =0.125 16

o. 0I'"

I .

;

,

1

PI=4

16=0.25

10 P3=-=0.625 16

4

L=5

ρ3 =

17 = 0.53125 32

2 = 0.3125 32

1 = 0.15625 32

Gallery 4-2

(Continued )

14:49

ρ2 = 5

ρ1 = 5

Table 6-1.

February 14, 2011 ch02

o.

166

4

ρ1 = 6

ρ4 = 6

ρ5 =

167

Gallery 4-3

29 = 0.453125 64

4 = 0.375 64

(Continued )

14:49

1 = 0.09375 64 1 ρ 2 = 3 = 0.046875 64 1 ρ3 = 2 = 0.03125 64

L=6

Table 6-1.

February 14, 2011 ch02

February 14, 2011

14:49

ch02

ρ2 = 7

I \

ρ5 =

1 = 0.0546875 128

51 = 0.3984375 128

•• a .-a-. •• ii.-i •.~2!. ·0· • G--. jie-it fI - .I. o • • • •-0 .0. o

1 = 0.0546875 128

00

0.:-

••

• • •-0 • • i i t-SJ .-~-. o

0

/ \

i ••~• • •

7 = 0.3828125 128

2 = 0.109375 128

ρ3 = 7

ρ4 = 7

L=7 4

- - - - - ~

168

Gallery 4-4

·/~e.-/·\--­

iio-

ρ1 = 7

Table 6-1.

(Continued )

00

4

ρ7 = 8

2 = 0.0625 256

4 = 0.125 256

1 = 0.03125 256

1 = 0.015625 256 1 = 0.03125 256

Gallery 4-5

ρ2 = 8

(Continued )

169

12 = 0.375 256

90 = 0.3515625 256

ρ5 = 8

1 = 0.0078125 256

ρ8 =

ρ4 = 2

14:49

ρ6 = 8

ρ3 = 8

L=8

ρ1 = 4

Table 6-1.

February 14, 2011 ch02

x =

L=3

ρ1 =

[− x

8 =1 8

+x +x

] 0

w

;;

o

~Q

170

0

e-----o-o I e

Gallery 8-1

L=4

~

1

( w)

C"' 01

3 − 2

0-::t

t i +1

V"'I ci

t i

~

t i −1

t': 0

t +1 i

~~~6CO OC! 0

5 25 = 32

0'1 ci

2 = 16

4

N

ci

ci

ρ1 =

φ8 '" ci

.., ":

o

16 =1 16

•

4

14:49

21 = 2

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

1

χ 18 -

27 = 128

0 1 2 3 4 5 6 7

xit+1 xit +1

-

0

7

3

2

xit−1 xit ~

6 20 = 1

2 =8

vertex

Basin Tree Diagrams of Rule 8 .

-

26 = 64

8 3

22 = 4

Table 6-2.

February 14, 2011 ch02

or,

-

_ ill: ...,

ci

o

'C>

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 • • • • -

,


r

5

2 = 32

11.5

5 •••• 6 •••• 7 • • • •

xJ+i = 4i[xJ-i -X: +xJ+i -~2 ] 185

1L 31

(W)

~ 1

•

~A

~

0.1

w

IL

!

•! •

(J ~

i

II

0.5

..

q)n-1

41

•

•t :.."

........•'"

L

~.:Z

8 8

/

0

/-

Pi = -=1

,

ch02

® ••••

~

24 = 16

14:49

23 = 8

•

Basin Tree Diagrams of Rule 32 .

- - ..... •••• ....

22 = 4

February 14, 2011

Table 6-5.

....••.

2 A=-=0.125

16

I

__I \ ..----

."

~

/

-~."Q. •

~:.----.:-

/4 '\ • ••.; - I ... ~

.~

/~

1Gallery 32-1 I·······

14 P2= -=0.875 16

I

32

L=5

ρ1 =

32 =1 32

(Continued )

Gallery 32-2

Table 6-5.

February 14, 2011 14:49 ch02

186

32

L=6

ρ2 =

2 = 0.03125 64

(Continued )

14:49

187

Gallery 32-3

62 = 0.96875 64

ρ1 =

Table 6-5.

February 14, 2011 ch02

32

L=7

ρ1 =

128 =1 128

(Continued )

Gallery 32-4

Table 6-5.

February 14, 2011 14:49 ch02

188

32

L=8

254 = 0.9921875 256

2 = 0.0078125 256

Gallery 32-5

(Continued )

14:49

ρ2 =

ρ1 =

Table 6-5.

February 14, 2011 ch02

189

23 = 8

Yertex

X i +1

X@§]


0.8

5 ~2 =32

\..4

•••• ••••

(5

••••

(6

•••• (W)

-=F. 1

,I'

~

r r ~.

/

~II

0.1

w

... .I

II

~

0.5

I)

q)n-I

0

190

IL 31

1L 41

! Pi = - = 0.25

,J

I G) • • • • 2

8

I

1

0.9

=d,[-l+l(x;_!-X;+X;+!-~)I]

2

X/+ 1

ch02

x;+!

Xi

/

14:49

-

X _ i 1

/

@ •••• ~3

2 4 = 16

/

February 14, 2011

r - •••• ••••

II 2 2 =4

•

Basin Tree Diagrams of Rule 36 .

Table 6-6.

! 2

Pi = 3 - = 0.75 8

!

•• •• ~~ ~ •~ .~ ~ :.--. '\ ~ I • 8J

OJ;.....

•

I,.....G-a-ll-ery~36--1-1 PI ~ 4 l~ ~.~~;

8

P2 = 16=0.5 .

I

36

L=5

22 = 0.6875 32

2 = 0.3125 32

Gallery 36-2

(Continued )

14:49

ρ2 =

ρ1 = 5

Table 6-6.

February 14, 2011 ch02

191

2 = 0.09375 64

2 = 0.1875 64

L=6

ρ3 =

(Continued )

------... ..

o

192

Gallery 36-3

46 = 0.71875 64

14:49

ρ2 = 3

ρ1 = 6

36

Table 6-6.

February 14, 2011 ch02

February 14, 2011

14:49

ch02

.ye

• • o/I·Y· j

Yi

t •

~. j

t

•

•

I-i

0

8-G

2 = 0.109375 128

4 = 0.21875 128

ρ1 = 7

ρ2 = 7

L=7

ρ3 =

86 = 0.671875 128

e~.~' 8-i e-i i t o • • e-i 8-G _ _

193

Gallery 36-4

0

36

(Continued )

./-

•/1 .",0./8. o-

• 0

Table 6-6.

-

2 = 0.03125 256

ρ2 = 4

10 = 0.3125 256

2 = 0.0625 256

ρ1 = 8

Gallery 36-5

(Continued )

ρ4 =

152 = 0.59375 256

14:49

ρ3 = 8

L=8

36

Table 6-6.

February 14, 2011 ch02

194

vertex I X _

i 1

1

Xi

\illi • • • • -

~

00 • • • •

= 16

2 =32

rro • • • • CW)

X:+ = dl!.-I (-2xLI - 2x: + X:+1 + 1) I] 1

2

195

1L 31 3 PI = -=0.375 8

-=F. 1

0

q)n

0.9

,.

•

0.7

•

fl.5

0.6 0.5

J 0.4

0.3 0.2

IJ ~

0.1

w

.

O.B

i

ch02

3)1 • • • •

5

X[1Q]

i 1

-

24

1

1X 1+ 1Xi1+1

14:49

- .... •••• ..

..,.~ " IIiMI Al.--

23 = 8

•

Basin Tree Diagrams of Rule 40 . 1

February 14, 2011

Table 6-7.

o

i

(J.)

I)

o

0.1 0.2 OJ 0.4 0.5 0.6 0.7 0.8 0.9

. 2. •

IL 41

It

PI = -=0.125 16

5 P2 = 8=0.625

•

,t\ 0/ n-l

I

• \

~

l

•

\

•

-.-o-.~.--o--. r o f •

I Gallery 40-1 I

•

f

14 P2 = -=0.875 16

I

40

L=5

ρ1 =

5 = 0.15625 32

ρ2 =

27 = 0.84375 32

(Continued )

14:49

196

Gallery 40-2

Table 6-7.

February 14, 2011 ch02

40

ρ3 =

59 = 0.921875 64

3 = 0.046875 64

2 = 0.03125 64

Gallery 40-3

(Continued )

14:49

ρ2 =

L=6

ρ1 =

Table 6-7.

February 14, 2011 ch02

197

40

e\,

;--..w a /~8

i

G

198

Gallery 40-4

121 = 0.9453125 128

14:49

ρ2 =

o--e~

I

7 = 0.0546875 128

(Continued )

..

ρ1 =

L=7

Table 6-7.

February 14, 2011 ch02

-.

0-"•

~ e~-I 246 = 0.9609375 256

A 8~. I

ρ3 =

fY' I

2 = 0.0078125 256

(Continued )

..

Gallery 40-5

14:49

ρ1 =

L=8

8 = 0.03125 256

I\

40

ρ2 =

Table 6-7.

February 14, 2011 ch02

199

x:+

200

IL

1

5

2 = 32

vertex

~ • • • •••• X i- 1

Xi

I

X i+ 1

X;+l

1

•

X@1l

1Q2 •

•••• ••••

0.9

....4 ) • • • •

0.6

(5

••••

0.5

""6

••••

rC7

•

•

• •

d(Wit-

]4------.

= d.[%-IC-4x:_1- 2x: +x:+ 1 +2) I

q)n

0.8 0.7

11,:"-1

.1

0.4 0.3

#

0.2 II

0.1

w

/

o

ch02

24 = 16

23 = 8

I

14:49

m- 2 2 =4

I

February 14, 2011

Basin Tree Diagrams of Rule 44 .

Table 6-8.

1.. i)

o

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

31

° 4)n-l

0,:"

1

1L 41

1 PI = 3 8=0.375

2 P2 = 8=0.25

3

I

P3 = 8=0.375

-

Gallery 44-1

I

~

~

~

~

~

~

P2= 1:=025

PI = 4

3

16 =0.75

I

44

L=5

6 = 0.9375 32

2 = 0.0625 32

Gallery 44-2

(Continued )

14:49

ρ2 = 5

ρ1 =

Table 6-8.

February 14, 2011 ch02

201

44

ρ4 =

4 = 0.0625 64

9 = 0.84375 64

202

1 = 0.046875 64

3 = 0.046875 64

Gallery 44-3

ρ2 =

ρ1 = 3

(Continued )

14:49

ρ3 = 6

L=6

Table 6-8.

February 14, 2011 ch02

ρ1 = 7

•

203

A

2 = 0.015625 128

15 = 0.8203125 128

ρ2 = 7

0 _··-G e.•

G-G-.

-w-·. 0

8-i»

.-I-i

.-.-it .-.-it

I-e-i

Gallery 44-4

3 = 0.1640625 128

(Continued )

14:49

ρ3 =

L=7

44

Table 6-8.

February 14, 2011 ch02

4 = 0.015625 256

L=8

ρ4 = 4

Gallery 44-5

7 = 0.109375 256

(Continued )

21 = 0.65625 256

7 = 0.21875 256

ρ3 = 8

ρ2 = 8

14:49

ρ1 =

44

Table 6-8.

February 14, 2011 ch02

204

I

I X IX+ l+l

.....

~l

5 ~2 = 32

1{7J I .

•

• •

i(Wi~

1]4-----. w

205

1 Pl=3-=0375 8 .

XJm

I

i>

0.9 ~.


0.8

~

0.7

0.6

liS

I'jIl

0.5

0.4 0.3

~

0.2

O.I~ oo

0.1

I

[I

02 . OJ Q4 Q5

~6

IL 41

i

0.)-

I)

091

O. 7 0.8

•

.

"x ~ /

'tIt) 11-1

fl)Cl)O)eJ

1 :························0··········· n. = 4 -=0.25 : ~l 16: .........................

I ~lr· 8---:.-J ~

5 P2 = 8 =0.625

ch02

-",

31

1-

•••• •••• 311 •••• •••• •••• • •••

xJ+l = 4[~-1 (-2xJ_l + X: - 2x:+1 +1)

IL

X1I X

14:49

-

24 = 16

vertex I

1

1

February 14, 2011

-

II~ 91--8

23 = 8

•

Basin Tree Diagrams of Rule 72 .

Table 6-9.

~/\

IGallery

e

72-1

I

p, = g=075 16

0

•

!"'o

I I

14:49

----..

----..

ch02

o

\

...

o

_0

"'..

...

I . /'_~I

• i..

.. ~.~ll/

.;!~. -

~

ρ1 = 5

72

L=5

2 = 0.3125 32

ρ2 =

22 = 0.6875 32

Table 6-9.

(Continued )

\

206

Gallery 72-2

February 14, 2011

72

o. o •• _. / 4(:\ ". '" . • w-e . ~/\ 1~~ o •••• . o .0 e£? 0

.-

.-.

207

Gallery 72-3

43 = 0.671875 64

1 = 0.046875 64

ρ1 = 3

ρ2 =

3 = 0.28125 64

(Continued )

14:49

ρ3 = 6

L=6

Table 6-9.

February 14, 2011 ch02

ρ1 = 7

1 = 0.0546875 128

6 = 0.328175 128

L=7

Gallery 72-4

(Continued )

ρ3 =

79 = 0.6171875 128

14:49

ρ2 = 7

72

Table 6-9.

February 14, 2011 ch02

208

2 256 = 0.0625

L=8

ρ1 = 4

1 = 0.015625 256

ρ3 = 8

Gallery 72-5

11 = 0.34375 256

(Continued )

ρ4 =

148 = 0.578125 256

14:49

ρ2 = 8

72

Table 6-9.

February 14, 2011 ch02

209

vertex I X:_ 1

23 = 8

tit Xi X i +1

I

I~I.I.I.I. 7 ••••

5

4

XJ+] =

210

IL

4

31

[

-x l_

11

+2x l -Xl

z+1

I

PI = 3

_.!.] 2

CWl

-=F. I

0

0.8 0.7

0.5

•

0.4 0.3 0.2

() -I

0.1

I)

w

8=0.375

I

fJ)C() P2 = 3

1

8

=0.375

L4 fJJ

I

I

OJ

4)n-l

~4 16 ~0.25

2

2

P3 =16=0.125

P3 = 8=0.25

.

~ 0)

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

Gallery 76-1

I

~8)OJOJ 05 05 05...•5

11

1 P2=4-=O.25 16

OJ

I

'iii

IJ 5

0.6

1

8)

/

q)n

0.9

ch02

.1.1.1. .1.1.1.

XI2§] I~

14:49

~·I·I·I·

-

u_ u2 = 16 2 =32

]

t+1 Xi

February 14, 2011

- - .... .1.1.1. .1.1.1.

22 =4

•

Basin Tree Diagrams of Rule 76 .

Table 6-10.

~

00:] t

IP4~2 136~0375 •

211

1 = 0.015625 32 1 = 0.015625 32

ρ2 = 5 ρ3 = 5

2 = 0.0625 32

1 = 0.015625 32

ρ5 =

L=5

Gallery 76-2

(Continued )

ρ4 = 5

3 = 0.46875 32

14:49

ρ1 = 5

76

Table 6-10.

February 14, 2011 ch02

76

212

1 = 0.09375 64

3 = 0.28125 64

ρ4 = 6

ρ6 = 6

ρ1 = 6

1 = 0.09375 64 1 ρ 2 = 6 = 0.09375 64 1 ρ3 = 6 = 0.09375 64

2 = 0.03125 64

Gallery 76-3

3 = 0.140625 64

(Continued )

ρ8 = 2

4 = 0.125 64

ρ5 = 3

1 = 0.046875 64

14:49

ρ9 =

L=6

ρ7 = 3

Table 6-10.

February 14, 2011 ch02

ρ3 ρ5

ρ1

L=7

2 ρ10 = = 0.015625 128

213

Gallery 76-4

4 = 0.21875 128

2 = 0.109375 128

ρ11 = 7

ρ9 = 7

ρ8 = 7

3 = 0.1640625 128

ρ2

ρ6

ρ4

14:49

3 ρ7 = 7 = 0.1640625 128

76

1 = 0.0546875 128

(Continued )

ρ1 = ρ 2 = ρ3 = ρ 4 = ρ5 = ρ6 = 7

Table 6-10.

February 14, 2011 ch02

214

ρ18 = 8

ρ7 ρ8

ρ6

Gallery 76-5

1 = 0.03125 256

3 = 0.09375 256

ρ1 = ρ 2 = ρ3 = ρ 4 = ρ5 = ρ6 = ρ 7 = ρ8 = 8

ρ4 ρ5

ρ2 ρ3

ρ1

4 = 0.125 256

2 = 0.0625 ρ 13 256

ρ10 = 8

8

ρ12 = ρ13 = ρ14 =

ρ12

3 = 0.09375 256

L=8

2 = 0.0078125 256

(Continued )

ρ9 = 4

4 = 0.125 256

1 = 0.015625 256

7 = 0.0546875 256

ρ15 = 8

3 = 0.046875 256

ρ11 = 2

ρ14

ρ19 = 4

14:49

ρ16 = 8

76

ρ17 =

Table 6-10.

February 14, 2011 ch02

Table 6-11.

Basin Tree Diagrams of Rule 77 .

February 14, 2011 14:49 ch02

215

77

ρ3 = 5

3 = 0.46875 32

2 = 0.0625 32

3 = 0.46875 32

Gallery 77-2

ρ2 = 5

(Continued )

14:49

ρ1 =

L=5

Table 6-11.

February 14, 2011 ch02

216

1 = 0.09375 64

1 = 0.09375 64

ρ1 = 6

ρ2 = 6

2 = 0.03125 64

L=6 ρ5 = 3

ρ4 = 3

3 = 0.140625 64

Gallery 77-3

3 = 0.140625 64

(Continued )

ρ6 = 2

16 = 0.5 64

14:49

ρ3 =

77

Table 6-11.

February 14, 2011 ch02

217

ρ5 = 7

8 = 0.4375 128

8 = 0.4375 128

L=7

Gallery 77-4

(Continued )

1 = 0.0546875 128

2 = 0.015625 128

ρ1 = ρ 2 = 7

ρ3 =

14:49

ρ4 = 7

77

Table 6-11.

February 14, 2011 ch02

218

ρ8 = 8

4 = 0.125 256

6 = 0.1875 256

3 = 0.09375 256

ρ3 = 8

Gallery 77-5

4 = 0.125 256

(Continued )

45 = 0.3515625 256

3 = 0.09375 256

1 = 0.015625 256

ρ4 = 8

ρ1 = 4

2 = 0.0078125 256

ρ6 = 2

ρ2 =

14:49

ρ7 = 8

L=8

77

ρ5 = 8

Table 6-11.

February 14, 2011 ch02

219

Table 6-12.

Basin Tree Diagrams of Rule 78 .

February 14, 2011 14:49 ch02

220

78

L=5

ρ2 = 5

(Continued )

ρ1 =

2 = 0.0625 32

14:49

221

Gallery 78-2

6 = 0.9375 32

Table 6-12.

February 14, 2011 ch02

6 = 0.28125 64

2 = 0.03125 64

ρ1 =

Gallery 78-3

(Continued )

ρ3 = 2

22 = 0.6875 64

14:49

ρ2 = 3

L=6

78

Table 6-12.

February 14, 2011 ch02

222

2 = 0.015625 128

L=7 18 = 0.984375 128

Gallery 78-4

ρ2 = 7

(Continued )

14:49

ρ1 =

78

Table 6-12.

February 14, 2011 ch02

223

ρ3 = 2

67 = 0.5234375 256

2 = 0.0078125 256

L=8

3

67 =2 = 0.5234375 Gallery 78-5 256

(Continued )

ρ2 = 8

15 = 0.46875 256

14:49

ρ1 =

78

Table 6-12.

February 14, 2011 ch02

224

Table 6-13.

Basin Tree Diagrams of Rule 104 .

February 14, 2011 14:49 ch02

225

104 L=5

ρ2 =

1 = 0.15625 32

(Continued )

14:49

226

Gallery 104-2

27 = 0.84375 32

ρ1 = 5

Table 6-13.

February 14, 2011 ch02

ρ3 =

2 = 0.03125 64

56 = 0.875 64

ρ1 = 6

Gallery 104-3

(Continued )

1 = 0.09375 64

14:49

ρ2 =

104 L=6

Table 6-13.

February 14, 2011 ch02

227

114 = 0.890625 128

Gallery 104-4

(Continued )

ρ1 = 7

2 = 0.109375 128

14:49

ρ2 =

104 L=7

Table 6-13.

February 14, 2011 ch02

228

1 = 0.015625 256 2 = 0.0078125 256 2 = 0.015625 256

ρ1 = 4 ρ2 = ρ3 = 2

7 = 0.21875 256

Gallery 104-5

(Continued )

ρ5 =

190 = 0.7421875 256

14:49

ρ4 = 8

104 L=8

Table 6-13.

February 14, 2011 ch02

229

Table 6-14.

Basin Tree Diagrams of Rule 128 .

February 14, 2011 14:49 ch02

230

128 L=5

231

Gallery 128-2

31 = 0.96875 32

1 = 0.03125 32

(Continued )

14:49

ρ1 =

ρ2 =

Table 6-14.

February 14, 2011 ch02

63 = 0.984375 64

1 = 0.015625 64

Gallery 128-3

(Continued )

14:49

ρ1 =

128 L=6

ρ2 =

Table 6-14.

February 14, 2011 ch02

232

ρ1 =

127 = 0.9921875 128

1 = 0.0078125 128

Gallery 128-4

(Continued )

14:49

ρ2 =

128 L=7

Table 6-14.

February 14, 2011 ch02

233

128 L=8

234

Gallery 128-5

255 = 0.99609375 256

1 = 0.00390625 256

(Continued )

14:49

ρ1 =

ρ2 =

Table 6-14.

February 14, 2011 ch02

Table 6-15.

Basin Tree Diagrams of Rule 132 .

February 14, 2011 14:49 ch02

235

3 = 0.46875 32

1 = 0.03125 32

Gallery 132-2

1 = 0.15625 32

(Continued )

ρ4 =

11 = 0.34375 32

14:49

ρ3 = 5

ρ2 =

132 L=5

ρ1 = 5

Table 6-15.

February 14, 2011 ch02

236

237

ρ2 = 2

1 = 0.03125 64 1 ρ3 = = 0.015625 64

1 = 0.09375 64

3 = 0.140625 64

Gallery 132-3

ρ4 = 3

5 = 0.46875 64

(Continued )

ρ6 =

16 = 0.25 64

14:49

ρ1 = 6

132 L=6

ρ5 = 6

Table 6-15.

February 14, 2011 ch02

132 L=7

ρ4 = 7

238

2 = 0.109375 128

1 = 0.0078125 128

1 = 0.0546875 128

ρ2 =

ρ1 = 7

ρ3 = 7

Gallery 132-4

8 = 0.4375 128

(Continued )

ρ6 =

29 = 0.2265625 128

14:49

3 = 0.1640625 128

ρ5 = 7

Table 6-15.

February 14, 2011 ch02

239

4 = 0.125 256 6 = 0.09375 256 13 = 0.40625 256

ρ6 = 8 ρ7 = 4 ρ8 = 8 45 = 0.17578125 256

3 = 0.09375 256

ρ5 = 8

ρ9 =

2 = 0.0625 256

1 = 0.00390625 256

ρ4 = 8

ρ3 =

ρ1 = 8

ρ6 ρ1

ρ3

ρ4

Gallery 132-5

ρ2

ρ5

ρ8

(Continued )

ρ7

ρ9

14:49

1 = 0.03125 256 1 ρ2 = 2 = 0.0078125 256

132 L=8

Table 6-15.

February 14, 2011 ch02

Table 6-16.

Basin Tree Diagrams of Rule 136 .

February 14, 2011 14:49 ch02

240

136 L=5 ρ2 =

241

Gallery 136-2

31 = 0.96875 32

(Continued )

14:49

ρ1 =

1 = 0.03125 32

Table 6-16.

February 14, 2011 ch02

136 L=6

ρ2 =

ρ1 =

1 = 0.015625 64

(Continued )

14:49

242

Gallery 136-3

63 = 0.984375 64

Table 6-16.

February 14, 2011 ch02

136 L=7

ρ1 =

1 = 0.0078125 128

(Continued )

14:49

243

Gallery 136-4

127 = 0.9921875 128

ρ2 =

Table 6-16.

February 14, 2011 ch02

136 L=8

ρ1 =

ρ2 =

244

Gallery 136-5

(Continued )

14:49

255 = 0.99609375 256

1 = 0.00390625 256

Table 6-16.

February 14, 2011 ch02

Table 6-17.

Basin Tree Diagrams of Rule 140 .

February 14, 2011 14:49 ch02

245

ρ1 =

1 = 0.03125 32

140 L=5

ρ3 = 5

1 = 0.03125 32

(Continued )

ρ4 = 5

4 = 0.625 32

14:49

246

Gallery 140-2

2 = 0.3125 32

ρ2 =

Table 6-17.

February 14, 2011 ch02

3 = 0.28125 64

1 = 0.015625 64

ρ3 =

ρ4 = 6

1 = 0.03125 64

Gallery 140-3

1 = 0.015625 64

(Continued )

ρ5 = 3

ρ6 = 6

5 = 0.46875 64

4 = 0.1875 64

14:49

ρ2 = 2

140 L=6

ρ1 =

Table 6-17.

February 14, 2011 ch02

247

ρ2 =

248

6 = 0.328125 128

2 = 0.109375 128

ρ5 = 7

ρ3 = 7

1 = 0.0078125 128

1 = 0.0078125 128

Gallery 140-4

(Continued )

4 = 0.21875 128

6 = 0.328125 128

ρ4 = 7

ρ6 = 7

14:49

ρ1 =

140 L=7

Table 6-17.

February 14, 2011 ch02

249

5 = 0.15625 256

4 = 0.125 256

ρ5 = 8

ρ6 = 8

3 = 0.09375 256

1 = 0.00390625 256

1 = 0.0078125 256

ρ4 = 8

ρ3 =

ρ2 = 2

1 = 0.00390625 256

ρ1

ρ2

Gallery 140-5

ρ3

ρ5

(Continued )

ρ4

ρ9 = 4

9 = 0.140625 256

ρ8 = 8

ρ7 = 8

7 = 0.21875 256

8 = 0.25 256

14:49

ρ1 =

140 L=8

Table 6-17.

February 14, 2011 ch02

Table 6-18.

Basin Tree Diagrams of Rule 160 .

February 14, 2011 14:49 ch02

250

1 = 0.03125 32

ρ1 =

Gallery 160-2

31 = 0.96875 32

(Continued )

14:49

ρ2 =

160 L=5

Table 6-18.

February 14, 2011 ch02

251

49 = 0.765625 64

14 = 0.21875 64

Gallery 160-3

ρ3 =

(Continued )

ρ2 =

1 = 0.015625 64

14:49

ρ1 =

160 L=6

Table 6-18.

February 14, 2011 ch02

252

160 L=7

1 = 0.0078125 128

127 = 0.9921875 128

Gallery 160-4

(Continued )

14:49

ρ2 =

ρ1 =

Table 6-18.

February 14, 2011 ch02

253

225 = 0.87890625 256

ρ3 =

Gallery 160-5

1 = 0.00390625 256

(Continued )

ρ2 =

30 = 0.1171875 256

14:49

ρ1 =

160 L=8

Table 6-18.

February 14, 2011 ch02

254

Table 6-19.

Basin Tree Diagrams of Rule 164 .

February 14, 2011 14:49 ch02

255

5 = 0.78125 32

ρ1 =

Gallery 164-2

1 = 0.03125 32

(Continued )

ρ3 =

6 = 0.1875 32

14:49

ρ2 = 5

164 L=5

Table 6-19.

February 14, 2011 ch02

256

ρ3 = 3

4 = 0.1875 64

3 = 0.28125 64

Gallery 164-3

(Continued )

ρ1 = 3

ρ4 =

ρ5 =

3 = 0.046875 64

25 = 0.390625 64

2 = 0.09375 64

14:49

ρ2 = 6

164 L=6

Table 6-19.

February 14, 2011 ch02

257

11 = 0.6015625 128

1 = 0.0546875 128 1 = 0.0078125 128

Gallery 164-4

ρ4 =

(Continued )

ρ3 =

43 = 0.3359375 128

14:49

ρ2 = 7

164 L=7

ρ1 = 7

Table 6-19.

February 14, 2011 ch02

258

1 = 0.03125 256

ρ3 = 8

12 = 0.375 256

7 = 0.109375 256

ρ4 =

259

Gallery 164-5

31 = 0.12109375 256

(Continued )

ρ5 =

93 = 0.36328125 256

14:49

ρ2 = 4

164 L=8

ρ1 = 8

Table 6-19.

February 14, 2011 ch02

Table 6-20.

Basin Tree Diagrams of Rule 168 .

February 14, 2011 14:49 ch02

260

1 = 0.03125 32

ρ4 =

261

Gallery 168-2

21 = 0.65625 32

5 = 0.15625 32

(Continued )

ρ3 =

5 = 0.15625 32

14:49

ρ1 =

168 L=5

ρ2 =

Table 6-20.

February 14, 2011 ch02

6 = 0.09375 64

6 = 0.09375 64

ρ6 =

262

ρ2 =

Gallery 168-3

46 = 0.71875 64

1 = 0.015625 64

(Continued )

2 = 0.03125 64

ρ3 =

3 = 0.046875 64

14:49

ρ5 =

ρ4 =

168 L=6

ρ1 =

Table 6-20.

February 14, 2011 ch02

263

7 = 0.0546875 128

7 = 0.0546875 128

ρ3 =

ρ2 =

99 = 0.7734375 128

ρ4 =

7 = 0.0546875 128

Gallery 168-4

1 = 0.0078125 128

(Continued )

ρ5 =

7 = 0.0546875 128

14:49

ρ6 =

168 L=7

ρ1 =

Table 6-20.

February 14, 2011 ch02

1 = 0.00390625 256

8 = 0.03125 256

4 = 0.015625 256

ρ5 =

264

2 = 0.0078125 256

Gallery 168-5

8 = 0.03125 256

ρ2 =

(Continued )

ρ6 =

8 = 0.03125 256

ρ9 =

ρ8 =

8 = 0.03125 256

8 = 0.03125 256

ρ7 =

209 = 0.81640625 256

14:49

ρ4 =

ρ3 =

168 L=8

ρ1 =

Table 6-20.

February 14, 2011 ch02

Table 6-21.

Basin Tree Diagrams of Rule 172 .

February 14, 2011 14:49 ch02

265

4 = 0.625 32

1 = 0.03125 32

ρ4 =

266

ρ3 =

Gallery 172-2

1 = 0.03125 32

(Continued )

10 = 0.3125 32

14:49

ρ2 = 5

172 L=5

ρ1 =

Table 6-21.

February 14, 2011 ch02

ρ5 = 6

7 = 0.65625 64

3 = 0.046875 64

Gallery 172-3

0

3 = 0.046875 64

ρ6 =

12 = 0.1875 64

14:49

ρ4 =

172 L=6

ρ3 =

(Continued )

1 1 ρ = = 0.015625 1 ρ 2 = 3 = 0.046875 64 64

Table 6-21.

February 14, 2011 ch02

267

1 = 0.0078125 128

12 = 0.65625 128

2 = 0.109375 128

ρ1

ρ6

7 = 0.0546875 128

Gallery 172-4

ρ3 =

(Continued )

ρ5 =

21 = 0.1640625 128

14:49

ρ4 = 7

ρ2 = 7

172 L=7

ρ1 = ρ 6 =

Table 6-21.

February 14, 2011 ch02

268

269

24 = 0.09375 256

20 = 0.625 256

ρ8 =

3 = 0.01171875 256

ρ6 = 8

4 = 0.125 256 16 ρ7 = = 0.0625 256

ρ5 =

4 = 0.0625 256

4 = 0.015625 256

ρ4 = 8

ρ3 =

ρ2 = 4

1 = 0.00390625 256

ρ6

ρ4

ρ5

Gallery 172-5

ρ7

ρ3

(Continued )

ρ8

14:49

ρ1 =

172 L=8

ρ1

2

Table 6-21.

February 14, 2011 ch02

Table 6-22.

Basin Tree Diagrams of Rule 200 .

February 14, 2011 14:49 ch02

270

ρ4 = 5

2 = 0.3125 32

1 = 0.03125 32

1 = 0.15625 32

ρ2 = 5 ρ3 =

1 = 0.15625 32

Gallery 200-2

(Continued )

ρ5 =

11 = 0.34375 32

14:49

ρ1 = 5

200 L=5

Table 6-22.

February 14, 2011 ch02

271

200 L=6

ρ7 =

ρ6 = 6

18 = 0.28125 64

Gallery 200-3

(Continued )

14:49

3 = 0.28125 64

1 = 0.09375 64 1 ρ 2 = 3 = 0.046875 64 1 ρ3 = 6 = 0.09375 64 1 ρ 4 = = 0.015625 64 2 ρ5 = 6 = 0.1875 64

ρ1 = 6

Table 6-22.

February 14, 2011 ch02

272

ρ1 = 7

273

5 = 0.2734375 128

Gallery 200-4

(Continued )

ρ6 = 7

ρ9 =

29 = 0.2265625 128

2 = 0.109375 128

14:49

ρ8 = 7

1 = 0.0546875 128 1 ρ2 = 7 = 0.0546875 128 1 ρ3 = 7 = 0.0546875 128 1 ρ4 = 7 = 0.0546875 128 1 ρ5 = = 0.0078125 128 3 ρ7 = 7 = 0.1640625 128

200 L=7

ρ1 ρ2 ρ3 ρ4 ρ5

Table 6-22.

February 14, 2011 ch02

274

ρ11 = 8

3 = 0.09375 256

1 = 0.00390625 256

ρ11

Gallery 200-5

5 = 0.15625 256

2 = 0.0625 256

ρ10 = 8

ρ12 = 8

2 = 0.0625 256

ρ8

ρ9 = 8

(Continued )

ρ14 =

ρ6

ρ2

8 = 0.25 256

47 = 0.18359375 256

ρ13 = 8

ρ4

14:49

ρ8 =

ρ1 = 4

1 = 0.015625 256 1 ρ2 = 8 = 0.03125 256 1 ρ3 = 8 = 0.03125 256 1 ρ4 = 8 = 0.03125 256 1 ρ5 = 8 = 0.03125 256 1 ρ6 = 4 = 0.015625 256 1 ρ7 = 8 = 0.03125 256

200 L=8

ρ1 ρ3 ρ5 ρ7

Table 6-22.

February 14, 2011 ch02

Table 6-23.

Basin Tree Diagrams of Rule 204 .

February 14, 2011 14:49 ch02

275

1 = 0.15625 32 1 = 0.15625 32

ρ6 = 5 ρ7 = 5

1 = 0.15625 32

ρ4 = 5 1 = 0.15625 32

1 = 0.15625 32

ρ3 = 5

ρ5 = 5

1 = 0.15625 32

1 = 0.03125 32

1 = 0.03125 32

Gallery 204-2

ρ8 =

(Continued )

14:49

ρ2 = 5

204 L=5

ρ1 =

Table 6-23.

February 14, 2011 ch02

276

1 = 0.015625 64

1 = 0.09375 64 1 = 0.09375 64 1 = 0.09375 64

ρ4 = 6 ρ5 = 6 ρ7 = 6

277

1 = 0.09375 64

1 = 0.09375 64

1 = 0.046875 64

ρ13 = 6

ρ12 = 3

ρ11 = 6

1 = 0.09375 64 1 ρ9 = 6 = 0.09375 64

ρ8 = 6

1 = 0.09375 64

Gallery 204-3

(Continued )

ρ6

ρ14 =

1 = 0.046875 64

ρ6 = 3

1 = 0.015625 64

1 = 0.09375 64

1 = 0.03125 64

ρ2 = 6

ρ10 = 2

14:49

ρ3 = 6

204 L=6

ρ1 =

Table 6-23.

February 14, 2011 ch02

278

1 = 0.0546875 128 1 = 0.0546875 128 1 = 0.0546875 128 1 = 0.0546875 128

ρ10 = 7 ρ12 = 7 ρ14 = 7 ρ16 = 7 1 = 0.0546875 128

1 = 0.0546875 128

ρ8 = 7

ρ18 = 7

1 = 0.0546875 128

1 = 0.0546875 128

1 = 0.0078125 128

1 = 0.0078125 128

Gallery 204-4

ρ 20 =

(Continued )

1 = 0.0546875 128

1 = 0.0546875 128

ρ19 = 7

ρ17 = 7

1 = 0.0546875 128

1 = 0.0546875 128

ρ13 = 7

ρ15 = 7

1 = 0.0546875 128

1 = 0.0546875 128

1 = 0.0546875 128

1 = 0.0546875 128

ρ11 = 7

ρ9 = 7

ρ7 = 7

ρ5 = 7

1 = 0.0546875 128 1 ρ3 = 7 = 0.0546875 128

ρ2 = 7

14:49

ρ6 = 7

ρ4 = 7

204 L=7

ρ1 =

Table 6-23.

February 14, 2011 ch02

ρ27

ρ10 = ρ 23 = ρ34 ρ10

279

1 =8 = 0.03125 256

ρ 2 = ρ3 = ρ 4 = ρ5 = ρ6 = ρ7 = ρ8 = ρ9 = ρ11 = ρ12 = ρ13 = ρ14 = ρ15 = ρ16 = ρ17 = ρ18

ρ18

ρ17

ρ16

=8

1 = 0.03125 256

ρ 27 = ρ 28 = ρ30 = ρ31 = ρ33 = ρ35

ρ19 = ρ 20 = ρ 21 = ρ 22 = ρ 24 = ρ 25 = ρ 26 =

ρ35

ρ34

ρ33

ρ32

ρ31

ρ14 ρ15

ρ30

1 ρ 29 = 2 = 0.0078125 256

ρ13

ρ12

Gallery 204-5

ρ26 ρ28

ρ25

ρ8 ρ9 ρ11

ρ24

ρ7

1 256 = 0.015625

ρ23

ρ6

14:49

=4

ρ22

ρ5

3

ρ19 ρ20 ρ21

ρ36 ρ1 = ρ36 = 1 = 0.00390625 256

(Continued )

ρ4

204 L=8

ρ2

ρ1

Table 6-23.

February 14, 2011 ch02

Table 6-24.

Basin Tree Diagrams of Rule 232 .

February 14, 2011 14:49 ch02

280

232 L=5

281

Gallery 232-2

11 = 0.34375 32

1 = 0.15625 32

(Continued )

ρ2 = 5

ρ4 =

11 = 0.34375 32

1 = 0.15625 32

14:49

ρ3 =

ρ1 = 5

Table 6-24.

February 14, 2011 ch02

1 = 0.09375 64 2 = 0.03125 64

ρ2 = 6

ρ3 =

Gallery 232-3

3 = 0.28125 64

(Continued )

ρ4 =

16 = 0.25 64

16 = 0.25 64

ρ5 =

14:49

1 ρ1 = 6 = 0.09375 64

232 L=6

ρ6 = 6

Table 6-24.

February 14, 2011 ch02

282

283

ρ3 = 7

2 = 0.109375 128

3 = 0.1640625 128

ρ1 = 7

29 = 0.2265625 128

2 = 0.109375 128

Gallery 232-4

ρ2 = 7

3 = 0.1640625 128

(Continued )

ρ6 =

29 = 0.2265625 128

14:49

ρ5 =

232 L=7

ρ4 = 7

Table 6-24.

February 14, 2011 ch02

284

45 = 0.17578125 256 45 = 0.17578125 256

ρ9 =

6 = 0.1875 256

ρ7 = 8 ρ8 =

4 = 0.125 256

ρ6 = 8

ρ1 = 4

ρ6

ρ4

ρ7

Gallery 232-5

ρ5

(Continued )

ρ8

ρ2

ρ9

ρ1

14:49

1 = 0.015625 256 2 ρ2 = = 0.0078125 256 3 ρ3 = 8 = 0.09375 256 3 ρ4 = 8 = 0.09375 256 4 ρ5 = 8 = 0.125 256

232 L=8

ρ3

Table 6-24.

February 14, 2011 ch02

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules

information: 1. Rule number N 2. Boolean cube of rule N 3. Explicit formula for generating the truth table of rule N , where xi−1 , xi , xi+1 ∈ {0, 1} 4. Truth table of rule N 5. Characteristic function χ1N of rule N 6. Sample set of time-1 return maps corresponding to different random bit strings φ0 ∈ [0, 1), where the transient components have been deleted to avoid clutter. These time-1 maps are all extracted from Table 2 of [Chua et al., 2005a]. Observe that each period-1 time-1 return map is represented by one point on the diagonal line, and only a sample subset of such points are shown to avoid clutter. In addition, for certain rules, some of the time-1 maps (printed in different colors) are period-T attractors (T > 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

February 14, 2011

302

14:49

ch02

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

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].

February 14, 2011

14:49

ch02

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.

February 14, 2011

304

14:49

ch02

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

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 .

February 14, 2011 14:49 ch02

305

14:49

ch02

Table 8-1.

(Continued )

February 14, 2011

306

14:49

ch02

Table 8-1.

(Continued )

February 14, 2011

307

Table 8-2.

Portraits of ω-Limit Orbits of Rule 8 .

February 14, 2011 14:49 ch02

308

Table 8-3.

Portraits of ω-Limit Orbits of Rule 12 .

February 14, 2011 14:49 ch02

309

14:49

ch02

Table 8-3.

(Continued )

February 14, 2011

310

14:49

ch02

Table 8-3.

(Continued )

February 14, 2011

311

Table 8-4.

Portraits of ω-Limit Orbits of Rule 13 .

February 14, 2011 14:49 ch02

312

ch02

(Continued )

14:49

Table 8-4.

February 14, 2011

313

Table 8-5.

Portraits of ω-Limit Orbits of Rule 32 .

February 14, 2011 14:49 ch02

314

Table 8-6.

Portraits of ω-Limit Orbits of Rule 36 .

February 14, 2011 14:49 ch02

315

ch02

(Continued )

14:49

Table 8-6.

February 14, 2011

316

Table 8-7.

Portraits of ω-Limit Orbits of Rule 40 .

February 14, 2011 14:49 ch02

317

ch02

(Continued )

14:49

Table 8-7.

February 14, 2011

318

Table 8-8.

Portraits of ω-Limit Orbits of Rule 44 .

February 14, 2011 14:49 ch02

319

14:49

ch02

Table 8-8.

(Continued )

February 14, 2011

320

Table 8-9.

Portraits of ω-Limit Orbits of Rule 72 .

February 14, 2011 14:49 ch02

321

ch02

(Continued )

14:49

Table 8-9.

February 14, 2011

322

Table 8-10.

Portraits of ω-Limit Orbits of Rule 76 .

February 14, 2011 14:49 ch02

323

14:49

ch02

Table 8-10.

(Continued )

February 14, 2011

324

14:49

ch02

Table 8-10.

(Continued )

February 14, 2011

325

14:49

ch02

Table 8-10.

(Continued )

February 14, 2011

326

ch02

(Continued )

14:49

Table 8-10.

February 14, 2011

327

Table 8-11.

Portraits of ω-Limit Orbits of Rule 77 .

February 14, 2011 14:49 ch02

328

14:49

ch02

Table 8-11.

(Continued )

February 14, 2011

329

14:49

ch02

Table 8-11.

(Continued )

February 14, 2011

330

ch02

(Continued )

14:49

Table 8-11.

February 14, 2011

331

Table 8-12.

Portraits of ω-Limit Orbits of Rule 78 .

February 14, 2011 14:49 ch02

332

ch02

(Continued )

14:49

Table 8-12.

February 14, 2011

333

Table 8-13.

Portraits of ω-Limit Orbits of Rule 104 .

February 14, 2011 14:49 ch02

334

14:49

ch02

Table 8-13.

(Continued )

February 14, 2011

335

Table 8-14.

Portraits of ω-Limit Orbits of Rule 128 .

February 14, 2011 14:49 ch02

336

Table 8-15.

Portraits of ω-Limit Orbits of Rule 132 .

February 14, 2011 14:49 ch02

337

14:49

ch02

Table 8-15.

(Continued )

February 14, 2011

338

14:49

ch02

Table 8-15.

(Continued )

February 14, 2011

339

ch02

(Continued )

14:49

Table 8-15.

February 14, 2011

340

Table 8-16.

Portraits of ω-Limit Orbits of Rule 136 .

February 14, 2011 14:49 ch02

341

Table 8-17.

Portraits of ω-Limit Orbits of Rule 140 .

February 14, 2011 14:49 ch02

342

14:49

ch02

Table 8-17.

(Continued )

February 14, 2011

343

14:49

ch02

Table 8-17.

(Continued )

February 14, 2011

344

ch02

(Continued )

14:49

Table 8-17.

February 14, 2011

345

Table 8-18.

Portraits of ω-Limit Orbits of Rule 160 .

February 14, 2011 14:49 ch02

346

Table 8-19.

Portraits of ω-Limit Orbits of Rule 164 .

February 14, 2011 14:49 ch02

347

14:49

ch02

Table 8-19.

(Continued )

February 14, 2011

348

ch02

(Continued )

14:49

Table 8-19.

February 14, 2011

349

Table 8-20.

Portraits of ω-Limit Orbits of Rule 168 .

February 14, 2011 14:49 ch02

350

14:49

ch02

Table 8-20.

(Continued )

February 14, 2011

351

ch02

(Continued )

14:49

Table 8-20.

February 14, 2011

352

Table 8-21.

Portraits of ω-Limit Orbits of Rule 172 .

February 14, 2011 14:49 ch02

353

14:49

ch02

Table 8-21.

(Continued )

February 14, 2011

354

14:49

ch02

Table 8-21.

(Continued )

February 14, 2011

355

ch02

(Continued )

14:49

Table 8-21.

February 14, 2011

356

Table 8-22.

Portraits of ω-Limit Orbits of Rule 200 .

February 14, 2011 14:49 ch02

357

14:49

ch02

Table 8-22.

(Continued )

February 14, 2011

358

ch02

(Continued )

14:49

Table 8-22.

February 14, 2011

359

14:49

ch02

Table 8-22.

(Continued )

February 14, 2011

360

ch02

(Continued )

14:49

Table 8-22.

February 14, 2011

361

Table 8-23.

Portraits of ω-Limit Orbits of Rule 204 .

February 14, 2011 14:49 ch02

362

Table 8-24.

Portraits of ω-Limit Orbits of Rule 232 .

February 14, 2011 14:49 ch02

363

14:49

ch02

Table 8-24.

(Continued )

February 14, 2011

364

14:49

ch02

Table 8-24.

(Continued )

February 14, 2011

365

ch02

(Continued )

14:49

Table 8-24.

February 14, 2011

366

February 14, 2011

14:49

ch02

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.

February 14, 2011

368

14:49

ch02

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

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)

February 14, 2011

14:49

ch02

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.

February 14, 2011

370

14:49

ch02

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

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.

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules

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

February 14, 2011

372

14:49

ch02

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

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

February 14, 2011

14:49

ch02

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].

February 14, 2011

374

14:49

ch02

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



· · ·
). 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.

February 14, 2011

14:49

ch02

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

February 14, 2011

376

14:49

ch02

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







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






:






→






→






→






→






→






→






.

February 14, 2011

14:49

ch02

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.

February 14, 2011

378

14:49

ch02

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

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”







.

February 14, 2011

14:49

ch02

Chapter 2: Period-1 Rules

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.

February 14, 2011

380

14:49

ch02

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

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.

February 14, 2011

14:49

ch02

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.

February 14, 2011

382

14:49

ch02

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

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.

February 14, 2011

14:49

ch02

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

February 14, 2011

384

14:49

ch02

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.

February 14, 2011

14:49

ch02

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.

February 14, 2011

386

14:49

ch02

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

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

11:3

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

October 23, 2010

388

11:3

References

References

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).

February 12, 2011

12:25

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

February 12, 2011

390

12:25

index

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

12:25

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

February 12, 2011

392

12:25

index

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

www.worldscientific.com 7835 hc

 ISSN 1793-1010