Computational Analysis of One-Dimensiona

COMPUTATIONAL ANALYSIS OF ONE-DIMENSIONAL CELLULAR AUTOMATA

WORLD SCIENTIFIC SERIES ON NONLINEAR SCIENCE Editor: Leon O. Chua University of California, Berkeley

Series A. MONOGRAPHS AND TREATISES Published Titles Volume 1:

From Order to Chaos L. P. Kadanoff

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Smooth Invariant Manifolds and Normal Forms 1. U. Bronstein and A. Ya. Kopanskii

Volume 8: Dynamical Chaos: Models , Experiments , and Applications V. S. Anishchenko

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A. M. Samoilenko and N. A. Perestyuk Volume 16: Turbulence, Strange Attractors and Chaos D. Ruelle Volume 17: The Analysis of Complex Nonlinear Mechanical Systems: A Computer Algebra Assisted Approach M. Lesser

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Series A Vol. 15

Series Editor: Leon 0. Chua

COMPUTATIONAL ANALYSIS OF ONE-DIMENSIONAL CELLULAR AUTOMOTH

Burton H. Voorhees Professor of Mathematics Athabasca University

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V

Preface

The growth of interest in the study of complex systems which has occurred over the past twenty five years is a prime example of the explosive development of an idea whose time has come. The data processing capacities of modern computers have made possible work that could not even be imagined before their invention. For the theoretician vast new possibilities have opened in the continual search for understanding. And from the practical direction, the rapid integration of societies, economies, and cultures in the shrinking world; and the problems concomitant on explosive economic growth, have made the understanding of complexity an essential criterion for survival. The goal of a united world and a sustainable economy will be achieved, but if we do not understand the processes in which we are involved, and the requirements which they place on us, the sustainable economy may be that of Ethiopia rather than Utopia. The problem with complex systems is captured in the name. Complex systems are--complex. This means that many of the usual tools which have been used to study simple systems may not apply. New methods of analysis, new conceptual approaches, and new synthesizing perspectives are required, and their development is the focus of much of the current research on "complexity." Cellular automata provide one of the most interesting avenues into the study of complex systems in general, as well as having an intrinsic interest of their own. Because of their mathematical simplicity and representational robustness they have been used to model economic, political, biological, ecological, chemical, and physical systems. Almost any system which can be treated in terms of a discrete representation space in which the dynamics is based on local interaction rules can be modelled by a cellular automata.

vi

The intent of this book is to give an introduction to the analysis of cellular automata ( CA) in terms of an approach in which CA rules are viewed as elements of a non -linear operator algebra, which can be expressed in component form much as ordinary vectors are in vector algebra. Although a variety of different topics are covered , this viewpoint provides the underlying theme. The actual mathematics used is not hard, and the material should be available to anyone with a junior level university background , and a certain degree of mathematical maturity. Each chapter concludes with an exercise section . The problems in these sections have been chosen to provide basic exercise in the kind of analysis, and the kinds of thinking involved in working with cellular automata of the kind studied in this book . The suggestion, for serious students , is to do each exercise by hand. In some cases it would be a trivial task to work a problem on a computer , but this is not the point. The idea is to develop expertise , and intuition in working with cellular automata , not with computers. That expertise can be found elsewhere. There are many loose ends in this book as well. In writing it, it was a constant struggle to stick to the immediate point rather than chase off after another tantalizing lead that might produce some further results. It is my hope that at least a few readers will be stimulated , or provoked, to take up these unfinished lines of work. Perhaps the best illustration of the way I view this book is given in the following tale:

One fine morning Nasrudin announced to his friends at the tea house that he was setting out to climb the high mountains which lay several days travel from his village . Having proclaimed his intention, he strode away from the village whistling a merry tune. Five days later he returned , somewhat worse for the wear, and made his way to the tea house. After drinking some tea to quench his thirst, he proceeded to dazzle those who were present with descriptions of the wonders he had seen while in the mountains. "Mighty waterfalls , and vast forests ! Cliffs that climbed beyond the sky! And animals, such wonders as you have never seen!"

VII

His audience was enchanted, and Nasrudin was warming to his topic when a young rascal seated in the front row interrupted. "Excuse me, Nasrudin" He said, "but please, tell me how you had time to see all of those wonderful things? It is a two day walk to the mountains , and two days back , and you were only gone five days." "Your are right," Nasrudin replied , "In the time I was gone, I only was able to reach the foothills . But from that vantage point I could easily see all of the things which I am describing." My hope is that in describing the little that I have been able to see, it will encourage others to go further. Vancouver, B.C. November 29, 1994

This page is intentionally left blank

ix

Table of Contents Preface Introduction Chapter 1 The Operator Algebra of Cellular Automata 1. Basic Definitions and Notation 2. Boolean Forms of CA Rules 3. The Canonical Basis Operators 4. Symmetry Transformations on CA Rules 5. CA Rules as Maps of the Interval 6. Exercises for Chapter 1 Chapter 2 Cellular Automata Arithmetic 1. Products of CA Rules 2. The Division Algorithm 3. Residue Arithmetic of Cellular Automata 4. Exercises for Chapter 2 Chapter 3 Fixed Points and Cycles 1. Fixed Points and Shift Cycles via Rule Decomposition 2. Jen's Invariance Matrix Method 3. Exercises for Chapter 3 Chapter 4 Commutation of CA Rules 1. Computation of Commutator Sets 2. Idempotence 3. Ito Relations 4. Some Interesting Sub-Groups 5. Exercises for Chapter 4

67 71 72 77 85

Chapter 5 Additive Rules: I. Basic Analysis 1. Conditions for Additivity

87

2. Cyclotomic Analysis 3. Injectivity 4. Another View of Injectivity 5. Exercises for Chapter 5 Chapter 6 Additive Rules: II. Cycle Structures and Entropy 1. State Transition Diagrams 2. Cycle Periods 3. Reachability 4. Conditions for States on Cycles 5. Entropy Reduction 6. Exercises for Chapter 6 Chapter 7 Additive Rules: III. Computation of Predecessors 1. Predecessors of 3-Site Rules 2. k-Site Rules 3. Examples 4. The Operator B 5. Exercises for Chapter 7

v 1 26 29 31 33 33 35 42 44 47 52 54 60 65

94 98 100 101 119 121 122 124 131 132 134 138 140 144 149

X

Chapter 8 The Binary Difference Rule 1. Basic Properties of D 2. The Graph of D:[0 , 1]-+[0,1] 3. Some Numerical Relations 4. A Density Result 5. Exercises for Chapter 6 Chapter 9 Computation of Pre-Images 1. Pre-Images and Predecessors 2. The Rule Graph and Basic Matrix 3. Computation of Pre - Images From the Basic Matrix 4. Pre-Images and the Jen Recurrence Relations 5. Exercises for Chapter 9 Chapter 10 The Garden of Eden 1. GE(X) and GE*(X) 2. Computation of GE*(X) 3. GE*(X) for 3-Site Rules 4. Classification With GE* 5. Exercises for Chapter 10 Chapter 11 Time Series Simulation 1. Cellular Automata Generating Time Series 2. Statistics of Time Series 3. Exercises for Chapter 11 Chapter 12 Surjectivity of Cellular Automata Rules 1. A Kitchen Sink Theorem 2. The de Bruijn Diagram 3. The Subset Diagram 4. The Semi-Group G(X) 5. The Subset Matrix and Some Replacement Diagrams 6. Exercises for Chapter 12 Appendix 1 Boolean Expressions for 2 and 3 -Site Rules Appendix 2 Canonical Forms and Decompositions of 3-Site Rules Appendix 3 Strongly Legal 3-Site Non- Generative Rules Appendix 4 The Mod( 2) Pascal Triangle Appendix 5 GE*(X) for 3-Site T-Equivalence Class Exemulars Appendix 6 U(X) for Selected Rules References

150 154 158 159 160 164 165 169 171 176 193 195 198 203 207 208 213 218 220 224 230 235 236 243 245 249 254 256 260 266 271

1

Introduction A.M. Turing, in a classic paper, "Systems of logic based on ordinals," names three abilities which are required for good mathematical reasoning. These are intuition, ingenuity, and, what he considered most important of all, the ability to distinguish between interesting and uninteresting Questions [1]. At the end of his introduction to the proceedings of the 1989 Los Alamos Workshop on Cellular Automata, H.A. Gutowitz quotes Turing, and then asks, "What makes a cellular automata interesting?" [2]. This question can be viewed in two ways. What is is that makes cellular automata an interesting class of mathematical objects, and what is it that distinguishes particular cellular automata for special attention? Of course, many different answers to these questions can be given. What is of interest to a particular individual, or research community, may not be so to another individual or community. For a pure mathematician, for example, cellular automata may have an intrinsic interest relating to formal aspects of their behavior. For a biologist, cellular automata may provide an interesting way to model biological phenomena such as cell sorting. There are many different individuals and groups, each finding cellular automata interesting from their own particular viewpoint, and these individuals and groups are often not in close communication with each other. Another way of saying this is to say that the boundaries around the field which includes cellular automata, the field of complex systems, are fuzzy. In this sense, the study of complex systems, and of cellular automata is in what Thomas Kuhn calls a pre-paradigmatic state. The threads are there, but the carpet has yet to be woven. What unites the various threads of interest, integrating them into the larger tapestry of science, is the goal of understanding. Both for the individual and for the community, science begins with questions, with the desire to understand. An eloquent expression of this underlying passion for understanding is found in a passage from Issac Newton's Optiks: "What is there in places almost empty of matter, and whence is it that the sun and planets gravitate towards one another, without dense matter between them? Whence is it that nature doth nothing in vain; and whence arises all that order and beauty which we see in the world? To what end are comets, and whence is it that planets move all one and the same way in

2 orbs concentric , while comets move all manner of ways in orbs very eccentric , and what hinders the fixed stars from falling upon one another? How came the bodies of animals to be contrived with so much art, and for what ends were their several parts? Was the eye contrived without skill in optics, or the ear without knowledge of sounds? How do the motions of the body follow from the will, and whence is the instinct in animals?" In this light, cellular automata are interesting because of the belief that they may contribute , perhaps substantially, to the more general goal of understanding ; that is, to the development of a view of the world in which that which is seen to occur --is seen to occur naturally . The reason for this belief is the fact that cellular automata are capable of generating complex patterns of global behavior on the basis of very simple local interaction rules. Because of this, cellular automata have the twin virtues of mathematical tractability, and representational robustness. The basic cellular automata modelling paradigm involves either an initially discrete representation space , or the course grain partition of a representation space into individual cells, each of which may display one of a finite number of possible values . A system configuration is an assignment of a value to each cell, and system evolution is described in discrete time via local interaction functions between cells . These local functions define the cellular automata rule. All cells are updated synchronically, and the value assigned to a cell at a given time step depends only on its value, and the values of certain neighboring cells at the preceeding time step. For example, in the Greenberg-Hastings [3,4] model for pattern formation in chemical reaction-diffusion systems, each cell in a 2-dimensional square lattice can be in one of three possible states: quiescent, active, or refractory. The model can be considered as strictly formal, or it can be supposed to have a physical analog in which cells are assumed small enough that chemical concentration dynamics within cells can be course grained into these three catagories , and all concentration gradients are zero, except at cell boundaries where discrete jumps may occur. The neighborhood of a cell, that is, the set of all other cells able to directly influence its evolution, is the von Neumann neighborhood, consisting of the cell itself together with the four cells directly above, below , to the right, and to the left, as indicated in Figure 1 below . Local interaction rules are

3 defined in terms of the influence of cells in the neighborhood of a cell on the updated value to be placed in that cell. i j+1 i-1 j

i+1j ij-1

Figure 1 von Neumann Neighborhood of Cell ij

The local interaction rules in the Greenberg-Hastings model implement the following dynamics: 1. If a cell is quiescent it remains so unless one or more of its neighbors is active. In this case, it becomes active at the next time step. 2. Active cells become refractory in the next time step. 3. Refractory cells return to quiescence in the next time step. Even though these evolution rules are exceedingly simple, they produce spiral wave patterns which mimic the behavior of such chemical reaction-diffusion systems as are found in the Belousov-Zhabotinsky reaction. An example, with only active cells highlighted, is shown in Figure 2.

Figure 2 Sample Pattern Generated by Greenberg- Hastings Model

4

The robustness of cellular automata as a modelling class is exhibited in the wide range of applications in which they have been employed since their introduction by J. von Neumann and S. Ulam [5,6] . They have been used in a variety of biological applications, including modelling heart fibrillation [71, aspects of development [8,9], and testing evolution theories [ 10]. In physics, cellular automata have been used to model non-linear chemical reaction systems [3 ,4], the evolution of spiral galaxies [11], dendritic crystal growth [ 12], and spin exchange systems [13,14], to list but a few cases. In computation theory , cellular automata can be considered as parallel computers [ 15,16, 171, and have been used as parallel multipliers [ 18,19], sorters [20], and prime number sieves [ 21]. They have also been extremely useful in image processing and pattern recognition [22,23,24]; and it is known that some cellular automata are universal computers , hence equivalent to a universal Turing machine [25,26]. Cellular automata have been studied under a variety of names: "cellular spaces," "cellular structures," "homogeneous structures," "tessellation structures ," "tessellation automata," and "iterative arrays." Over the past dozen years the term cellular automata has come to predominate , coincident with the major increase of interest in these systems stimulated by the ground breaking work of S. Wolfram [27 ,28,29,30,31,321. In terms of the question of interest, Wolfram shifted the focus from the possible relevence of cellular automata for modelling to the discovery of particular cellular automata with interesting properties. This question was posed with the term "complexity engineering " [32]; that is , the design of complex systems exhibiting predetermined properties. Within this overall direction,Gutowitz [2] distinguishes two lines of research , relating to what he terms the forward problem, and the inverse problem. The forward problem is to determine the characteristic properties of a given cellular automaton . In its basic form, the forward problem involves analysis of the behavior of particular cellular automata rules. This may involve such things as determination of cycles and /or fixed points; statistical analysis of space- time patterns generated; and computation of conditions for configurations to he on cycles. In its more general form, the forward problem involves development of analytic techniques which can be applied to such analysis . The forward problem has been a focus of much of the recent research on cellular automata, and many useful techniques have been

5

developed; e.g., the polynomial representations introduced by O. Martin, A.M. Odlyzko, and S. Wolfram [33] for studying cycle periods; the recurrence relations derived by E. Jen [34,35] for counting pre-images; the statistical and entropic measures introduced by Wolfram [27], and D.A. Lind [361; the mean field analysis emphasized by Gutowitz [37]; graph theoretic approaches [e.g., 38,39]; computation of fractal dimensions [40,41]; and the 2, and Z parameters [42,43]. These, and other methods of analysis provides a rich toolkit. The inverse problem, on the other hand, has received relatively little attention outside of the practical modelling literature. As already indicated, there have been specific applications of cellular automata in modelling; e.g., J.H. Conway's well known "Game of Life" [44] is an example of the construction of a cellular automaton to mimic some properties of living systems, but little has appeared on the abstract problem of model construction, or of determining classes of cellular automata which satisfy preestablished constraints. Work to date on these problems has been aimed at determination of global behavior in the space of all cellular automata rules; e.g., phase transitions and criticality [45,46,47]. From another, more purely mathematical direction, L.P. Hurd [48] has constructed cellular automata exhibiting various degrees of computational complexity. Nevertheless, the inverse problem remains less well researched than the forward problem. Part of the aim of this book is to provide a framework for further work on inverse questions by writing cellular automata rules in terms of variable coefficients which can become variables in constraint equations whose solution sets define the sets of rules having pre-determined properties.

Particular

Universal

Forward Problem Determination of characteristics and evolutionary behavior of given individual cellular automata Development of techniques and

Inverse Problem Design of cellular automata for specific computational or modelling tasks

measures for

satisfying general

Determination of cellular automata

analysis of sets of precellular automata specified constraints Table 1 Particular and Universal Aspects of Forward and Inverse Problems

6 A general overview of the main issues in the forward and inverse problems is shown in Table 1. Attention in this book is on 1-dimensional cellular automata with cells taking binary values. Although this is the simplest possible case, it one which has been the focus of much investigation. The configuration space is now a 1-dimensional lattice, either finite or infinite. If infinite it is either doubly infinite, or half-infinite; while if finite, boundary conditions are specified at the end points, or the lattice is taken as a circle. Neighborhoods are blocks of k cell sites. A 3-site rule, for example, has neighborhoods consisting of 3 cells. The 1-dimensional k-site rules are labeled by assigning them a binary number, which is generated by listing the 2k possible neighborhoods in ascending numerical order and treating the listing of 0's and 1's obtained by specifying which neighborhoods map to 0 and which to 1 (called the rule table) as a binary number. For example, for the 3-site rule 90 the neighborhood table is 000 001 010 011 100 101 110 111 0

1

0

1

1

0

1

0

and the binary number 01011010 equals the denary (base 10) number 90. The generality of the 1-dimensional case is shown by an early result of G.A. Hedlund [49], to the effect that 1-dimensional cellular automata are the shift commuting endomorphisms of the shift dynamical system. From a purely mathematical point of view, one of the interesting aspects of cellular automata is that they may point the way toward a reconceptualization of the idea of a function. If the configuration space is taken as the set E+ of all right half-infinite binary sequences then it maps to the interval [0,1] in a natural way by associating to each sequence µ1µ2µ3••• the point I µ; 2-'. Thus, cellular automata acting on E+ also define maps of [0,1]. Although there is a technical issue related to sequences which consist of all 1's after some index value, cellular automata are "almost" functions on [0,1], but they are not defined in the ordinary way, as point functions in which the value of the function at each point is specified. Instead,they are defined by their local action on finite sequence segments. This leads to selfsimilarity in their graphs. The graphs defined by cellular automata are generalized linear graphs, with self-similarity taken into account. The question is whether non-linear generalizations can be defined, perhaps by making alterations in the neighborhood structures.

7

Historically speaking, much of the original interest in 1-dimensional cellular automata was stimulated by the complexity of the spatio-temporal evolution patterns which they are able to generate. Patterns generated by some of the rules discussed in this book are shown in the figures at the end of this introduction. Inspection of these patterns leads to several observations, and raises a number of questions. Some rules, such as rules 46, 72, 172, and 116 produce very simple patterns, either identically reproducing themselves after the first few transient iterations, or emulate left or right shifts. Rules 184 and 226 are slightly more complicated in their behavior, giving the appearence of "particle" dislocations travelling either right or left against a fixed checkerboard background. Use of the canonical decompositions introduced in Chapter 3 shows that these two rules generate mirror image behaviors. Rule 184 emulates a right shift on configurations on which rule 226 a left shift, and vice versa. More interesting are rules such as rules 18, 22, 30, and 126 which produce space-time patterns characterized by an apparently random distribution of inverted triangle structures of varying sizes . One of the early problems in studying these and similar rules was to characterize the distribution of the inverted triangles, and determine the fractal dimension of the patterns generated. These rules fall into Wolfram 's class III, "chaotic" rules. Rule 30 has also been suggested as a possible random sequence generator [501. Rule 60 also shows a distribution of triangles. Initially the appearence of these inverted triangles was likened to a distribution induced by statistical fluctuations. Rule 60 is additive, however: if g and g' are given as distinct configurations then rule 60 acting on their sum is equal to the sum of rule 60 acting on each configuration independently. Further, it is known that starting from an initial configuration containing only a single 1, rule 60 will generate the mod(2) Pascal triangle. Thus the action of this rule on any configuration can only give an XOR superposition of mod(2) Pascal triangles. In the figure shown, this fact shows up in the eventual synchronization which occurs at the bottom, due to the use of periodic boundary conditions. Finally, the figure for rule 54 shown an apparent Brownian motion of particles against a background. Analysis of this rule [51] has verified this interpretation by identifying the background, and two distinct types of "particles."

8 The most widely studied rule is rule 90, which is just the square of rule 60. It is the simplest non-trivial 3-site additive rule (rule 60 is really a 2-site rule in disguise). The best known of the non-additive rules is rule 18, which has been shown to reduce to rule 90 on the invariant subset of configurations space consisting of all sequences containing only isolated 1's separated by an odd number of 0's [51,52]. Two distinct points of view can be taken with regard to the order generating properties of cellular automata, and these different viewpoints lead to different questions, and different choices of analytic tools. Many studies of cellular automata have concerned themselves with analysis of statistical features of their output, and have at least implicitely used an analogy to statistical and thermodynamical systems. Some times this is very explicit, as in Wolfram's paper "Statistical mechanics of cellular automata," [27] which contains the statement: "The two dimensional picture formed by the succession of configurations in time is characteristically peppered with triangle structures. These triangles are formed when a long sequence of sites which suddenly all attain the same value, as if by a fluctuation, is progressively reduced in length by 'ambient noise."' As already indicated, however, the appearence of order in these rules is known to be completely deterministic. Although statistical mechanical and thermodynamical analogies have been often used in studies of cellular automata, and have proved extremelt useful, the better analogy is perhaps to deterministic chaotic systems'. There is a similarity, for example, between the long term unpredictability of chaotic systems and the undecidability results of K. Culik, L.P. Hurd, & S. Yu [53] which show that membership in behavioral classes defined on the basis of long term behavior is undecidable. To be more precise, Culik & Yu [54] define four classes of cellular automata. For 1-dimensional binary cellular automata which map the neighborhood consisting of all O's to 0, these are defined by the four conditions: 1. All finite configurations evolve to the configuration of all O's. 2. All finite configurations evolve to an ultimately periodic configuration. lIn the second view, the order generated by cellular automata pre-exists in the particular evolution rule, and is a manifestation of symmetry breaking induced by the choice of initial configuration.

9 3. For any pair of configurations c l and c2 is is decidable whether c l evolves to c24. All other cellular automata. For an arbritrary given cellular automaton, however, membership in any of these classes is undecidable. Thus, the general long term behavior of cellular automata, starting from a given finite initial configuration, has about it an undecidability which is comparable to that found in chaotic systems, for which the long term evolution is unpredictable. This book deals with work which has been carried out over the past eight years on certain aspects of both the forward and inverse problems. As indicated in the title, the emphasis in this book is on the development of various analytic and computational tools and methods for analysis of cellular automata. Chapter 1 introduces the basic definitions and notation , discusses two different forms of representation, defines symmetry equivalence classes of cellular automata, and indicates the way that cellular automata rules define maps of the unit interval. Chapter 2 introduces a division algorithms for cellular automata, and studies their residue arithmetic. Chapter 3 presents a method for finding fixed points and shift cycles, and gives a review of an invariance matrix method developed by Jen [551 to count fixed points and cycles. In chapter 4 a method for computing commutator sets of specified rules is derived. This also allows an understanding of a result due to H. Ito [56], which in the formalism introduced in the chapter, generalizes to shed light on the question of reconstructability raised by Crutchfield [571. Chapters 5, 6, 7, and 8 deal with additive rules . In chapter 5 the conditions for additivity of a rule are formulated in terms of an "obstruction matrix." This is followed by a representation of additive cellular automata rules in terms of complex polynomials, a formalism which allows simple conditions to be written for the injectivity of additive rules . In chapter 6 results of Martin, Odlyzko, & Wolfram [33], and of Jen [35] relating to cycle periods and reachability are reviewed, and a method for finding conditions for states to lie on cycles is derived. The chapter concludes with a review of some results on entropy reduction. Chapter 7 is devoted to a method for computing predecessors for arbritrary configurations under arbritrary additive rules. Chapter 8 presents a detailed analysis of the binary difference rule, which is equivalent to a discrete derivative with respect to sequence index.

10 Chapter 9 deals with the question of computing pre-images for finite sequences when the evolution rule is not additive. A matrix technique is introduced which allows such computations in arbritrary cases, and connections are shown to Jen's work on recurrence relations and enumeration [34,35]. Chapter 10 considers the "Garden-of-Eden" of a cellular automaton, that is, the set of configurations which have no predecessors under the given rule. Particular results are derived for the set of all 3-site rules, and it is shown that this provides a means of classifying rules. Chapter 11 considers the use of cellular automata to generate time series. Methods are presented for determining if a given cellular automata could have generated a specified time series, and if so what are the possible initial configurations. This method is statistically unbiased, however, only in the case of rules having an empty Garden-of-Eden. Finally, Chapter 12 considers the question of surjectivity, which is equivalent to having an empty Garden-of-Eden. A number of other properties equivalent to surjectivity are investigated, and methods are introduced for determination of whether or not given rules are surjective. Acknowledgements The origin of this work can be traced to a 1987 conversation with Ralph Abraham at the University ofCalifornia, Santa Cruz. At that point I was looking for new research problems, and mentioned that neural networks seemed interesting. Ralph replied that if I wanted to study neural networks, it would be a good idea to first study cellular automata. I took that advice, became fascinated, and this book is the result. Portions of the work reported here have been published in various journals over the past eight years. The canonical basis operators of chapter 1, together with their use to determine fixed points and shift cycles as described in chapter 3, and the results reported in chapters 9 and 10, were published in Physica D. The work reported in chapter 7 was published in a paper in Communications in Mathematical Physics for 3-site rules , and a subsequent paper in Physica D for the general case . The division algorithm and residue arighmetic given in chapter 2; the commutation computations of chapter 4; the use of complex polynomials to determine injectivity of additive rules in chapter 5; and the analysis of the binary difference rule given in chapter 6 were originally reported in Complex Systems. The results on entropy

11

reduction given in chapter 6 were reported in the International Journal of Theoretical Physics . References to these papers are given in the biblography. Needless to say, the work reported here has benefited from conversations and interactions with many others . As already indicated, it was originally stimulated by Ralph Abraham , and further encouraged during four months of a sabbatical which were spent in Santa Cruz . The method given for computing predecessors in chapter 7 grew out of a conversation at the Santa Fe Institute with Wentian Li and Mats Nordahl. Much of the material in chapter 12 was first presented as a seminar at the Santa Fe Institute. The work in chapter 5 dealing with Ito relations , and much of the work in chapter 11 was stimulated by a number of conversations with Jim Crutchfield and Jim Hanson . The work in chapters 7, 9, and 10 was reported at a meeting in June , 1992 in Oberwolfach , Germany. Conversations there with Hans Otto Peitgen and Fritz von Haeseler were stimulating and the meeting itself was wonderful . Correspondences with Harold McIntosh have shed light on a number of issues relating to uses of the de Bruijn and subset diagrams discussed in chapter 12, and some of the results reported in chapter 4 were suggested in conversations with Erica Jen. This work was also assisted by a number of summer research assistants . Scott Bradshaw ( 1991 ), Winslow Lacesso ( 1992), Hong Nugyun and Marcus Molenda ( 1993), and Robert Newton ( 1994), all of whom made siginficant contributions . Assistance in programming was given by Ron Haukenfrers and John Ramsden of Athabasca University Computer Services. Completion of this book has been assisted by an Athabasca University Presidents Award for Research and Scholarly Excellence which provided the time to finish it close to the editorial deadline . Financial support has been provided by a 1989 grant from the Athabasca University Academic Research Committee , and by operating grant OGP - 0024871 from the National Science and Engineering Research Council of Canada.

12 Biblography 1. Voorhees , Burton Cellular automata , Pascal' s triangle, and generation of order. Physica D 31, E1988) 2. Voorhees , Burton . Predecessor states for certain cellular automata evolutions . Communications in Mathematical Physics 117 , ( 1988). 3. Voorhees , Burton . Entropy of additive cellular automata. International Journal of Theoretical Physics 28 , No. 11 , ( 1989). 4. Voorhees , Burton . Period multiplying operators on integer sequences modulo a prime . Complex Systems 3, (1989). 5. Voorhees, Burton. Nearest neighbor cellular automata over Z2 with periodic boundary conditions . Physica D 45, (1990). 6. Voorhees, Burton . Division algorithm for cellular automata rules. Complex Systems 4, (1990). 7. Voorhees , Burton . Geometry and arithmetic of a simple cellular automata . Complex Systems 5, (1991). 8. Voorhees , Burton . Symmetric group modelling of visual information. Il Nuovo Cimento 106B No . 10, (1991). 9. Voorhees, Burton . Determining fixed points and shift cycles for nearest neighbor cellular automata. Journal of Statistical Physics 66 Nos. 5/6 (1992). 10. Voorhees, Burton . Predecessors of cellular automata states : I. Additive automata. Physica D 68, (1993). 11. Voorhees, Burton. Predecessors of cellular automata states : II. Preimages of finite sequences . Physica D 73, (1994). 12. Voorhees , Burton & Bradshaw , Scott. Predecessors of cellular automata states : III. Garden- of-Eden classification of cellular automata rules. Physica D 73, (1994). 13. Voorhees , Burton . Commutation of cellular automata rules. To appear in Complex Systems. 14. Voorhees , Burton. A note on the injectivity of additive cellular automata . To appear in Complex Systems.

13 Graphs of Space-Time Output for Sample CA Rules Rules generating figures shown are nearest neighbor with periodic boundary conditions on a 128x128 lattice. Rule 46: Emulates left shift (Complimentary rule 116 emulates right shift). Rules 72 and 172: Emulate identity Rules 184: White "particles" travelling right, black "particles" travelling left against checkerboard background. Anihilation bitwise at collisions with "thickest" particle winning. Rule 226: Mirror image of rule 184. Rules 18, 22, 30, 126: Apparently random patterns of inverted triangles of varying sizes. Rule 60: Superposition of copies of mod(2) Pascal's triangle. Rule 54: Apparent "Brownian motion" of particles against a background.

14

Time Sequence Rule 46

15

Time Sequence Rule 116

16

Time Sequence Rule 72

17

R

vr

rr

IF

Time Sequence Rule 172

OF

OF

IF

V

r

18

Time Sequence Rule 184

19

Time Sequence Rule 226

20

Time Sequence Rule 18

21

Time Sequence Rule 22

22

J ,^ r= ^^-^^^T^^^^^Ji'^^^Y

J yr

=

J1

Jy M13V3'^j Yjit

r

1 -

J

^^=Y1^1- JIJS ^

J

7

I TYTJr1

4jy 'jJI7 ^i^PIT,

7Z -

? -

i

71i J

J

EN

r,

^;^_^; , -

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^r

^ r

^J7 ,^T JTr^7 T,_;rjJJy

r

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J

JT JJ;^■

JAY- T J ^^_ 7

7- J J J = J ., ,^^, 1Y{, ^T>^ ^^^"^^'_^J'1j ^'f77y-+ ^ 1 1 , Y ^ J ^ _ ^T>J

JJ J'J77>'^

^

r^ Tr

7j^t^ A^ 77:-7,7IJSY:rJ ^J7 7.7 r6 i^y 7 17 . - J 1rJ i l7 JI, ^T- 7 77- 7 T, ^jy7^ Jr7J 77?^ ar 77^^7^, 7 7'I7T

j 7-2

:t

i

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y

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J'ITT J { 1T^y JiT^1 -3 ,JJ

,'^

yJ

l.1

JNJ-^

,

J

, J JT J 7^ 7V 7r ^>1 J r7'^T 3

f +T^7^ ^ 7i J7

J 1J ' 7j r lr^ ^l,J^ Ji [J ^3; - _ ^.^ >' JiT T>' TJ J ;7T^^J yJJ7 . ^.y^TJ >'.7'^J S• ^S T,,f i==J,:= JT 1' ^ i J J7- J = -T ^'= J^J 1 yTr Jy-77 y,J ^^ J J77JY J'7^ r^^Yi ^ M J 'FJS- ^'^ yJ' ^yJ J JT. J7 ^ JIyT 7J 1'^J j^r^r

I I yTi M=Y^JIJ•17 ,FJTi_'-^rJ^^T^^,7

Time Sequence Rule 30

.^

'"

=T=om

lri,

w

9;

^7

-W

-

-Tyr=:.= 91F r

q

T_^ts_ Or-

±r

7TH

_:

7 7;-ter V T-T'T- TAT:: zT Tom: T-TT^ _ T 7=

TTY

lur

T

T==T

_

--

T-

T

T

T

IPR ^T

T--:-'-_==mow

T^ r7 94 IF

;r.

'T" Tom= - '--_-^=T-^'"T-

: ^:•:_-=7r: qpr

T_ :=^3-:==T===T err=:3:-T:: s.

Time Sequence Rule 126

24

Time Sequence Rule 60

25

Time Sequence Rule 54

26

Chapter 1 The Operator Algebra of Cellular Automata Cellular automata can be specified in a variety of ways: as local neighborhood maps, as global maps of a configuration space, as Boolean equations implementing truth conditions, as arithmetic recurrences, as finite replacement schemes, as maps of the interval, etc. This chapter provides an introduction to some of the standard forms in which cellular automata are presented. It also introduces the idea of representing the global map defined by a cellular automata in terms of abstract components defined with respect to a canonical basis. 1. Basic Definitions and Notation Cellular automata are discrete symbolic dynamical systems defined in terms of a lattice of sites, L; an alphabet of symbols, K; and an evolution rule X, which maps configurations at any given time t to new configurations at time t+1. A configuration, or state, is an assignment of a symbol from K to every site of the lattice L. The set of all possible configurations is the state, or configuration space, which will be denoted E in the generic case. Given a configuration µ(t) the evolution rule generates a new configuration µ(t+1) by synchronically assigning to every site of L a symbol choosen from the alphabet K on the basis of the symbols in a neighborhood of that site in the given configuration. In a fundamental paper, published in 1968, Hedlund [49] has shown that cellular automata are the shift commuting endomorphisms of the shift dynamical system. In all that follows, the lattice is taken as either a finite set of n sites located on the circumference of a circle, or as a right half-infinite one dimensional array of sites. The first case gives what Jen [55] calls a cylindrical cellular automata, because its evolution can be visualized as taking place on a cylinder. The configuration space in this case will be denoted En. It consists of all periodic sequences of symbols with period a divisor of n. In the second case, the configuration space is the set of all right half-infinite symbol sequences, and is denoted E+. When the configuration space is taken generically, it will be denoted by E. All considerations in this book will be restricted to binary cellular automata, for which the alphabet is the set K = 10, 1).

27 The view taken is that cellular automata are defined in terms of particular kinds of operators on configuration space. Configurations will generally be represented by lower case Greek letters , and operators by upper case Latin letters. If µe E and X:E->E then µi (1 0

'Wolfram lists neighborhoods in descending numerical order, so the listing of rules by components used in this book is the reverse of that used by Wolfram. For example, Wolfram gives rule 30 as 00011110 while in the notation of this book rule 30 is 01111000. The present convention is used as a means of emphasizing the connection between CAs and maps of [0,1] since each neighborhood can be viewed as equivalent to a sub-interval, as briefly discussed in section 4 of this chapter.

29 A k-site CA rule is said to be additive if it can be represented as a sum of shifts: k-1 X = a-r I asas (1.5) s=0

If neighborhoods are left justified in (1.5) then r=0. For symmetric rules r = (k-1)/2. A rule is linear if it can be written in the form k-1 X=a1+cr'laws (1.6) s=0 where 1 is the rule that maps all neighborhoods to 1. If a = 0 the rule is homogeneous. Otherwise it is inhomogeneous . Comparison with equation (1.5) shows that the k-site additive rules are just the k-site linear homogeneous rules.

Table 1.1 lists the additive 3-site rules in nearest neighbor form. Their expression in the form (1.5), and the symbols which will be used to denote these operators are also given. The rule that maps all neighborhoods to 0 is denoted 0, and the rule that maps all neighborhoods to 1 is denoted 1. This latter is not an additive rule since 1(µ+µ') =.1 while 1(µ)+1(µ') = 1+1= Q. The left justified form of these rules is obtained by multiplying each rule by the left shift a. Rule

Symbol

Components

Shift Form

0 240

0 a-1

00000000 00001111

0 0-1

right shift

204

I

00110011

I

identity

170 102

a

01010101

a

left shift

D*

01100110

I+0

60

D

00111100

I+c-1

90

S

01011010

c+0-1

150

A

01101001

I+a+a-1

Table 1.1 Nearest neighbor form of the 3-site Additive Rules

2. Boolean Forms of CA Rules Cellular automata can also be defined in terms of multinomial Boolean functions. If X:E-E is a k-site rule then Xis represented by a Boolean

30 function in the variables is (0:5s:5k-1) which is of degree at most k. Given the component expression for a rule, there is a simple proceedure for construction of its Boolean form in terms of the AND and XOR (exclusive or) operations. These are represented respectively by multiplication (ab = a AND b) and binary addition (a+b mod(2) = a XOR b). Given a one dimensional CA rule X:E-+E, written in its component form X = (xOxl...x2k_1), the Boolean expression for this rule is constructed as follows:

Algorithm 1.1: 1. For each iO...ik_1 in the neighborhood list write X(iO...ik-1) = A(iO...ik-2)ik-1 + Y(iO...ik-2) mod(2) with Y(iO...ik-2) = x2i, where i = iO...ik_2, and A(i0 ••ik-2) = 0 x2i = x2i+1 (1.7) 1 x2i # x2i+1

2. If the components of A are all ones, return to step 1 for the rule Y. If the components of A are not all ones , apply the decomposition of step 1 to both A and Y. 3. Continue to iterate this process until the rules A and Y found depend only on a single variable. At this point the expressions for A and Y can be easily determined. 4. Recombine the expressions found in this process to obtain the Boolean expression for the rule X. As an example, the Boolean form for the 3-site rule 18 will be computed. The rule table for rule 18 is 000 001 010 011 100 101 110 111 0 1 0 0 1 0 0 0 Taking X(iOili2) = A(i0il)i2 + Y(i0il) mod(2), application of step 1 of the algorithm yields the rule tables for A and Y as 11 10 00 01 A

1

0

1

0 = (1010)

1 0 = (0010) Y 0 0 Repeating the proceedure for Y gives Y(i0i1) = A'(iO)i1 + Y(i0) mod(2) with both A' and Yequal to (01). Hence X(i0) = Y'(ip) = i0. Applying the proceedure to A(i0il) = A"(iO)il + Y'(i0) mod(2) yields A" = Y" = (11) = 1. Hence the Boolean expression for rule 18 is given by X(i0ili2) = [A"(i0)il + Y"(iO)li2 + [A'(i0)il + Y'(i0)] mod(2)

31 = (1+i1)i2 + ioi1 + i0 =(1+il)(iO+i2) Boolean forms computed in this way for all two and three site rules are given in Appendix 1. Note: In order to simplify notation, in the future all sums of (0.1)-sequences. CA configurations. or CA rules will automatically be taken mod( 2) unless otherwise specified. 3. The Canonical Basis Operators Let {i0 ...ik-1} be the k-site neighborhood list with i the denary form of i0...ik- 1, and define a set { vk)) of 2k k-site rules by (i)

1 i = j (1.9) J 10 That is , vk) is the global rule having local form that maps the neighborhood i = i0...ik- 1 to 1 and all other neighborhoods to 0. The numerical label for vk) is 2i and it is clear that the global operator X for every k-site rule can be written as a linear combination of these basis operators: 2k_1 X = xivk) i=0 where xi is the i-th component of the rule X.

(1.10)

As a matter of convienience, a different notation will be used for the vk) when k = 3, as indicated in Table 1.2. 000

001

010

011

100

101

110

111

(0) v3

(1) v3

(2) v3

(3) v3

(4) v3

(5) v3

(6) v3

(7) v3

K

n+

t

R-

a

P+

X

il+ Table 1.2

Notation Used for 3-Site Basis Operators

The set { vk) } provides a basis for the non-linear algebra of k-site CA rules . The rule form given in equation (1.10) is just the operator form of Wolframs numerical labeling, as given in equation (1.2). It will be called the canonical form of a rule. Canonical forms are listed in Appendix 2 for all 3site rules.

32 The canonical form of a rule X can be determined either directly from the component form of the rule, or from its numerical label N(X) by expressing this as a sum of powers of 2. Canonical representations can be added directly, with coefficients reduced mod(2), and the component forms of rules can be added component-wise, mod(2). A k-site rule X = (xOxl...x2k-1) can always be extended to a k+m site rule. To do this, the generic k-site neighborhood iO...ik_1 is mapped to the set of k+m site neighborhoods Ni = fr0...rs-1i0...ik-IJO.- jm-s-1 } for arbritrary s and m-s digit blocks ro...rs_l and jO...jm_s_1. Every neighborhood in the k+m site neighborhood list which belongs to Ni is assigned the value X(i0...ik-1). This defines the (s,m-s) extension of X. The (0,m) and (m,0) extensions are called respectively the m-site right extension of X, and the m-site left extension of X. If m = 1 the terms right and left extension will be used. The mapping site remains fixed during extension. As an example, the right and left extensions of the 2-site binary difference rule (0110) are, respectively, (00111100) (rule 60) and (01100110) (rule 102). Since each of the canonical basis operators is associated with a single neighborhood, it is possible to consider the way in which the set of k-site basis operators maps to k+1 site basis operators under extension. This mapping is indicated in Figure 1.1.

Figure 1.1 Hierarchy of Canonical Basis Operators for 1 , 2, and 3-Site Neighborhoods This figure shows the form of a binary tree. At each level, each of the k-site basis operators is at the top of its own binary tree. Thus, if X is a k-site CA rule with left justified neighborhoods, each coefficient xi in equation (1.10) selects out the binary tree headed by vk). The (0,m) right extension of X to

33

k+m sites is then obtained by associating the coefficient xi to every basis operator in the intersection of the set { vk+m 10< _ r_ ni/2 set ai = 1 4. If a component ai is not contained in equations (2.6) set it to 0 5. Determine the rule R by R = Q+AX mod(2). This algorithm minimizes R in the sense that the components of R contain the fewest possible number of 1 's, although as will be seen later, there is an indeterminacy which arises resulting in the need to define larger equivalence classes than just residue classes. As an example of the division algorithm , the nearest neighbor rule 90 will be divided by the 2 -site rules (0110) and (0010 ). Rule 90 has been studied extensively [e.g., 33 ,59,60 ,61,62] as one of the simplest additive rules. For rule 90, q = (01011010)T, and for X = (0110)

46

(1 0 0 0) 0

1 0 0

0

0 0

1

0 0 1 0 X(2,1) = 0 0 1 0 0 0 0 1 0

1 0

(2.7)

0

1 0 0 0 so that equations (2.7) become 2a0=0 2a1=2 2a2

=

2

(2.8)

2a3 = 0 This indicates that A = (0110) as well. Note, however, that if rule 90 is taken in nearest neighbor form then the neighborhoods for X are {i 1i2} while for A they are {ioil}. The question of neighborhood choice shows up, for example, in the fact that the identity operator for {ili2} neighborhoods is (0011), while (0101) is the identity operator for {i0i1} neighborhoods , and the nearest neighbor identity, (00110011) is the product of (0101 ) and (0011). With this cavet, the 2-site rule D = (0110) is the square root of rule 90. (With left justified neighborhoods, the binary difference rule D is exactly the square root of rule 90 .) Note that D is the binary difference rule, with 3-site extension given by rule 60 for {ioil } neighborhoods, and by rule 102 for {ili2} neighborhoods (see Table 1 . 1 for expressions of these rules). If X = (0010) then 1 0 0 0 1 0 0 0 0 1 0 0 1 0 0 0 X(2,1) = 0 0 1 0

(2.9)

0 0 1 0 0 1 0 0 1 0 0 0 and equations (2.6) become 4a0 = 2 gal=1

(2.10)

47

2a2 = 1 Application of the division algorithm gives A = (1110). Computation now yields AX = (11111111), so there is a remainder, given by (10100101); i.e., rule 165. Thus rule 90 is the product of the rules (1110) and (0010) with a remainder given by rule 165. This second example serves to illustrate a significant point. In the division algorithm as given, ai is set to 1 if = ni/2. A remainder with an equal number of 1 components would be obtained, however, by taking ai equal to 0 in this case. When such a situation occurs, it will be called a case of equivocation. The choice made will be called positive equivocation, while setting ai to 0 for = ni/2 will be called negative equivocation. The possibility of equivocation has significant consequences for the arithmetic of residue classes. It should be clear that the division algorithm can be applied for any pair of rules X and Q so long as the neighborhood structure of Xis such that L(k,m;X) can be constructed. Even if Q is defined for neighborhoods of fewer sites than X, it can still be divided by X by first extending it to a rule defined for neighborhoods with more sites than X. The algorithm can also be generalized to rules defined over alphabets other than {0,1}, and to higher dimensional lattices. All that is required is the ability to construct the appropriate X(k,m) matrix, and choose the neighborhoods so that they overlap in the required way. 3. Residue Arithmetic of Cellular Automata If Q = AX+R, then by analogy with ordinary residue arithmetic, Q will be said to be congruent to R modulo X, written Q - R mod(X). Lemma 2.3 Congruence modulo X satisfies 1. Q Q mod(X) 2. Q = R mod(X) t= R Q mod(X) This asserts that congruence modulo X has the reflexive and symmetric properties. The possibility of equivocation, however, leads to cases in which transitivity does not hold. Consider, for example, rules Q, R, S, and X such that Q - R mod(X) and R = S mod(X). Then there are rules A and B such that Q = AX+R and R = BX+S. Thus Q = AX+BX+S = (A+B)X+S which formally looks as if Q - S mod(X).

48 Suppose, however, that Q = (00101110) while X = (0110). The matrix X(2,1) is given by equation (2.7) and application of the division algorithm yields A = (0111), R = (01010000). But it is easy to show that the rule R can be written as (01010000) = (0110)(0110)+(00001010). Thus, Q can be written as the combination Q = [(0111)+(0110)1(0110)+(00001010) = (0001)(0110)+(00001010) This equation is true , but it does not follow from the division algorithm unless negative equivocation is choosen. Further analysis indicates that each column of the matrix X(k,m) may or may not be equivocal for a given rule Q, and it is necessary to consider all possible combinations of equivocation. In order to deal with this kind of situation, the equivocation class of a rule Q is defined as the set of all rules congruent to Q under all possible choices of equivocation. Let c(X,m) be the number of columns of the matrix X(k,m) which do not contain all 0's. Let e(Q,X,m) be the number of columns of this matrix which are equivocal for Q = AX+R. Lemma 2.4 Let X be a given k-site rule. The number of k+m site rules which are congruent to 0 mod(X) is given by 2c(X,m) with 10, the number of states in En having only isolated 1's, or only isolated 0's, is given by Ln-1 where Ln is the n-th Lucas number. Proof By Lemma 3.6, Nn+1(1;1) = Nn(1;1)+Nn-1(1;1)+1. Addition of 1 to each side of this equality yields Nn+1(1;1)+1 = (Nn(1;1)+1) + (Nn- 0; 1)+ 1), and this is the Fibonacci recursion relation for the quantity Fn = Nn(1;1)+1. Further, N1(1;1) = 0 and N2(1;1) = 2, so the sequence generated for Fn is given by the Lucas sequence. I The technique of identifying configurations which are fixed points, or which lie on shift cycles can be applied to k-site rules as well as 3-site rules. All that is required is construction of tables similar to Tables 3.2 and 3.3, although this construction and its analysis becomes progressively more difficult as the neighborhood size increases. 2. Jen' s Invariance Matrix Method A way of counting fixed points and cycles for rules with periodic boundary conditions has been developed by Jeri [55]. For a symmetric k=2r+ 1 site rule X, a 2l -1x2k-1 invariance matrix, W(X) is defined such that the ij entry of this matrix is 1 if the binary expressions of i and j can be overlapped to form a k-site neighborhood which leaves the value at the mapping site invariant under X, and is 0 otherwise. To construct W(X), i and j are written in binary form i = i-ri-r+1 ...ir-1 and j = j-r+IJ-r+2... jr. W(X) is given by s - Js (-r + 1 (01010101) (00110011) (0011 )(00001111) (00100011) (00001110) (01010001) Figure 4.1 Two and Three Site Commutators of (0010)

2. Idem n otence The method used to compute commutator sets can also be used to determine rules which are idempotent. The question of which k-site rules are idempotent requires computation of the solution set to the equation X2 = X*, where X* is the extension of X to 2k-1 sites. The only technicality involved is to insure that the mapping site is preserved on both sides of the equation. If X is a k-site rule defined such that [X(g)]i = X(µi-r...µi+k-r-1) then X2 will be a 2k-1 site rule with [X2(µ)]i = X2(9i-2r•••9i+2k-2r-2)• Thus X must be extended

72 so that [X*( g)]i = X*(µi-2r..•µi+2k-2r-2). This requires the X neighborhoods to be extended as j1 •••jriO ...ik-1sl ... sk-r-1• Theorem 4.4 Let X be a k-site CA rule , and let s denote the mapping site of X2. Then X is idempotent if and only if 1 k 11 2k - n - llxr xis = x ^no...xlknl 2 J n=0...0 r=0 l r

J

(4.7)

As an example , consider the problem of determining all of the left justified 2-site idempotent rules. Expansion of equation (4.7) yields the set of independent equations 0 = x0(1+x3) 0 = xl+x0 ( 1+x1+x2 )+x0x1(x2+x3) 0 = x0(1+x1 )( 1+x2)+xl ( 1+x2x3) 0 = x0(1+xi )( 1+x3)+xl(1+x2+x3)+x1x2x3 (4.8) 0 = x0(xl+x2+xlx2+x2x3) 0 = x0(1+x1 )( 1+x2)+x1(1+x2x3) 0 = x0(1+x2 )( 1+x3)+x2(xl+x3)+x3(1+xlx2) After some computation , solution of equations ( 4.8) gives the complete set of left justified 2-site idempotent rules as 1(0000), (0010), (0011), (1011), (1111 )}. Similarly, the set of right justified 2-site idempotent rules is determined to be {(0000), (0100), (0101), ( 1101), (1111)}. 3. Ito Relations A comparative study of the 3-site rules 18 and 126 was carried out by Ito [56] , motivated by the remarkable similarity of the state transition diagrams for these two rules. He found that this similarity was due to the fact that rules 18 and 126 are related via the 3-site rule 252 . This relation is expressed by commutativity of the diagram of Figure 4.2. Relations of this type will be called Ito relations . In this diagram X stands for rule 18, Y for rule 126, and T for rule 252 . If the diagram commutes then YT = TX, and rules X and Y will be said to be Ito related via T. Theorem 4.5 shows that the Ito relation is determined in terms of the division algorithm of Chapter 2, and the commutation relations studied in this chapter.

73

E T {>E X

Y

0

V

-----> E E T Figure 4.2

Ito Relation of Rules 18 and 126

Theorem 4.5 For given CA rules X and T, there exists a rule Y such that X and Y are Ito related via T if and only if T I [X,T]. Proof If such a Y exists, then TX = YT. But TX = XT + [X,T], so this may be written as XT + YT = (X+Y)T = [X,T], and T does divide the commutator. On the other hand, if T I [X,T] there is an A such that [X,T] = XT+TX = AT, which may be written as TX = (X+A)T so Y is taken as X+A. I The same arguments leading to equation (4.6) prove the next theorem. Theorem 4.6 Let rules X and T be given. The set of rules A such that X and Y = A+X are Ito related via T is determined by the solution set of at(i) = xt(i)+tx(i) (4.9) For example, suppose that X = (01001000) (rule 18) and T = (0111) (note that rule 252 is just the 3-site right extension of T). For these rules, equations (4.9) become a0 = x0+t0 = 0 al=xl+tl=0 a3 = x3+t2 = 1 a4 = x4+t2 = 0 a5 = x5+t3 = 1 a6 = x6+t1 = 1 a7 = x7+tO = 0 The most general solution of these equations is A = (00a210110). In the case a2 = 1 this gives X+A = (01111110), which is rule 126, the case studied by Ito. If a2 = 0 then X+A = (01011110), indicating that rule 122 is also Ito related to rule 18.

74 The question of when , given two rules X and T, there is a third rule Y such that the diagram of Figure 4.2 commutes is important in connection with an analysis carried out by Crutchfield [57] on the possible use of cellular automata to infer underlying spatio -temporal dynamics from data series. In his approach the rule T is a "measurement function," which describes the action of the measuring instrument . The rule X is the actual underlying dynamics , and the rule Y is the observed dynamics. The conclusion reached was that in most cases cellular automata would misrepresent underlying dynamics . The reason, in terms of the present results, being that in most cases , for a given X and T, there is no rule Y which is Ito related to X via T. Thus it is quite important to be able to determine under just what sets of conditions it is possible to find rules for which the diagram of Figure 4.2 does commute. If X and T are given, then the possible rules Y, if any exist, can be found by solution of equation (4.9) and use of Y = A+X. On the other hand, if T and Y are given, which would be the more usual experimental situation, equation (4.9) is not much help. Instead, it is necessary to return to equation (4.5), in the form k

(2k-n-1 x1-no -1 1tr Yt(i) = 1 is. xl-nk-12 ik-1 r JI n=0...0 r=0

(4.10)

to obtain a set of equations which may be solved for the components xi of X. Another question which is of interest is to determine all possible rules X which satisfy T I [X,T] for a given rule T . In this case the equation YT = TX, written in the form yt (i) = tx(i), can yield a set of logical constraints on the components of X. If T is a k-site rule , while X and Y are r-site rules , then YT and TX will both be k+r- 1 site rules. The yt(i) are easily computed from the k+r-1 site neighborhood list, while the tx(i) will be given in terms of the list Li(k,r-1;X) defined in equations (2.1) and (2.2). The condition that yt(i) = Yt(j) whenever t(i) = t(j) then yields conditions on the possible values of the xi. For example, if T is the 2-site binary difference rule D = (0110), and X is assumed to be a 3-site rule , the initial step in solving for Xis indicated in Table 4.1. Inspection of this table indicates that there are eight constraint equations to be satisfied : txOx0 = tx7x7; ttOx1 = tx7x6; tx1x2 = tx6x5; tx1x3 = tx6x4; tx2x4 = tx5x3 , tx2x5 = tx5x2, tx3x6 = tx4x1 ; and tx3x7 = tx4x0.

75 In this notation, txrxs indicates the component of T labeled with the denary form of the binary number xrxs. i

vtf,

0000 0001 0010 0011 0100 0101 0110 0000

yO

i txOx0

yl Y3

tx0x1 txlx2

Y2 Y6

tx2x4

Y7

tx2x5

Y5

tx3x6

YO

i

1000 1001 1010 1011 1100 1101 1110 1000

txlx3

tx3x7

Yt(i)

Lx(i)

y4 Y5 y7 Y6 Y2

tx4x0 tx4x1 tx5x2 tx5x3 tx6x4

Y3

tx6x5

yl y4

tx7x7

tx7x6

Table 4.1 Yt(i) and tx (i) When T is the Binary Difference Rule

Working out the conditions which are imposed on the xi is simplified by use of Figure 4.3, which indicates the cases in which trs = tr's'.

Figure 4.3 Cases In Which trs = tij For T = D (Cases of equality shaded)

This figure, in conjunction with the constraint equations, indicates that txOx0 = tx7x7 and tx2x5 = tx5x2 are always satisfied, while the remainder of the constraints can be formulated in the form of the binary truth conditions [(xi=xj)n(xr=xs)]V[(xi=x j)n(xr=x's)] . In these expressions the indicies (i,j,r,s) take on the following sets of values el = (0,1,7,6), e2 = (1,2,6,5), e3 = (1,3,6,4), e4 = (2,4,5,3), e5 = (3,6,4,1), and e6 = (3,7,4,0). The rules X for which there exists a Y with YD = DX can now be computed via the flow chart of in Figure 4.4.

76

Figure 4.4 Computation of X Such That D I [X,D]

Table 4.2 lists the solutions for X having x0 = 0, together with the corresponding values of Y, which can be computed, once X is known. (Rules commuting with the binary difference rule D are indicated with a * in this table.)

0 24 36 60 66 90 102 126 142 150 170 178 204 212 232 240

X 00000000 00011000 00100100 00111100 01000010 01011010 01100110 01111110 01110001 01101001 01010101 01001101 00110011 00101011 00010111 00001111

0 116 72

Y 00000000* 00101110 00010010

60

00111100*

46 90 102 18 226 150 170 222 204 184

01110100 01011010* 01100110* 01001000 01000111 01101001* 01010101* 01111011 00110011* 00011101

132

00100001

240

00001111*

Table 4.2 Non-Generative 3-Site Rules Satisfying YD = DX (* Indicates Rules Commuting With D)

77

4. Some Interesting S ibgr,ps In Chapter 2 it was noted that the set of k-site CA rules formed an abelian group under component-wise binary addition. The division and commutation results which have been presented allow some interesting subgroups to be distinguished. This will be illustrated for the case in which both Q and T are 2-site rules. Table 4.3 lists the 3-site products of all 2-site rules. In this table, the row and column lables are the numerical labels for 2-site rules, indicated with an astrick to distinguish them from the 3-site numerical labels. The entry in the cell with row label ab* and column label cd* is the 3-site numerical label for the composite rule (ab*)(cd*). 9*_

1*

2-**

2

4*

E**

fL

Z

li*

9*

9_ *

0

0

0

0

0

0

0

0

0

0

1*

0

128

0

192 0

136 36

236 1

129 17

209 3

48

48

12

68

16

66

34

12

68

2*

4*

0

11*

12* =

14* IV

0

0

0 0

0

139 55 255

0

64

0

192 48

240

12

204 60

252 3

195 51

243

15

207 63 255

12

34

34

2

24

12

48

48

64 0

187

119 255

24

2

34

8 0

0

8

0

136 68

9*

0

72

116 60

7*

0

200

116 252 46

li*

255 55

139 3

255 183

139 195 209 153 165 237 237

165 153 209

195 139

183 255

255 119

187 51

102 170 34

204 68

136 0

68

204 34 48

66

16

68

170 102 238 17

153 85

102 90

90

18

18

238 126 254 19

209 17

221 85

129

153

1

17

102 46

60

219 119 255 63

236 36

238

221 51

136 0

116 72 0 255

192 0

11* 255 247

187 243 221 221 189 253 239 231 187 243 207 207

12* 255 63

207 240 243 51

195 3

252 60

204 12

127 255 128 0

191 255

240 48

192 0 1± 255 191 207 207 243 187 231 239 253 189 221 221 243 187 247 255 14* 255 127 255 63 255 119 219 19 254 126 238 46 252 116 200 0 ib± 255 255 255 255 255 255 255 255 255 255 255 255 255 255 255 255

Table 4.3 3-Site Produces of 2-Site Rules (ab* indicates numerical label of 2-site rule)

Referring to this table, consider the diagram of Figure 4.5, in which X and Y are 3-site rules; Q and T are 2-site rules; and Q is taken as a linear rule, excluding 0 and 1. Thus the rules considered for Q are the rules given by: rule 3*, (0011), which is the left justified identity I; rule 5*, (0101), which

78 is the left shift a; rule 6*, (0110), which is the binary difference rule D; rule 9*, (1001), 1+D; rule 10*, (1010), 1+a; and rule 12, (1100), 1+I. Also, in the diagram, X = QT and Y = TQ, so the commutator is [Q,T] = X+Y.

T ENE Y

V

T

DE

Figure 4.5 Commutator Diagram

If Q is taken as a, or 1+a then reading across the corresponding rows, and down the corresponding columns of Table 4.3 selects out the rules listed in Table 4.4a. Rule Co=onents

Cannonical Form

Additive Form

a+13-+x

34

(01000100)

TI++A

68 136 238 221 187

(00100010) (00010001) (01110111) (10111011) (11011101)

13++t

I+f3-+x

13-+x 13++13-+Tl++X+8+t 6++13-+r1-+X+t+x

0+13-+x D+13-+X 1+a+13-+X

13-+T1++Tl-+x+O+x

1+I+f3-+x

119 17 0 102 170 204 255 153 85 51

(11101110) (10001000) (00000000) (01100110) (01010101) (00110011) (11111111) (10011001) (10101010) (11001100)

13++11++T1-+8+t+x ^-+x

1+D+13-+

13++11++A+t 13-+T1++X+9

0 D a

13++13-+X+9

I

!3-+Tl-+X+x 13++T1-+t+x Tl++71-+9+x

1+f3-+x

1 1+D

1+a 1+I

Table 4.4a Composite 3-Site Rules With Factor a or 1+a

The shift a commutes with all rules, but 1+a does not. However, its commutators are already included among the rules in Table 4.4a. (If this were not the case , it would be necessary to include them, as will be the case when Q is taken as the binary difference rule D.)

79 Using the same notation as Table 4.3, the non-zero commutators of 1+0 are given by: [1+0,0*] = 255; [1+0,1*] = 102; [1+0,2*] = 153; [1+0,4*] = 153; [1+a,6*] = 255; [1+a,7*] = 102; [1+0,8*] = 102; [1+a,9*] = 255; [1+0,11*] = 153; [1+0,13*] = 153; [1+a,14*] = 102; [1+0,1] = 255. The remaining four commutators are 0. Under the operation of binary addition, the rules listed in Table 4.4a form a sub-group of the abelian group of all CA rules. This sub-group is defined by the group table given in Figure 4.6a. This table is given in compressed form in that the rules in the table could also be listed as indices for the table. This is unnecessary, however, since if rules X, Y, and Z satisfy X+Y = Z, then Y+Z = X and X+Z = Y.

34 68

34

68

136

238

221

187

119

17

0 102

102

204

255

153

0

170 204

170

153

255

85 51

51 85

136

170

204

0

102

85

51

255

153

238 221

204

170

102

0

51

153

255

153

85

51

0

85 102

170

255 204

187

153

255

51

85

102

0

204

170

119

85 51

51 85

255 153

153 255

170 204

204

17

0 102

0

170

102

Figure 4.6a Group Table for 3-Site Rules With Factor a or 1+a

If Q is taken as I, or 1+1, the rules to be considered are those listed in Table 4.4b. Similarity to the rules from Table 4.4a should be immediately obvious. In fact, the rules in these two tables are related to each other by the symmetry transformation Ti defined in equation (1.11) of Chapter 1, and thus correspond on En to two different orientations on the circle. The identity I commutes with all rules, but 1+I does not. As before, however, the commutators are among the rules in Table 4.4b. The non-zero commutators of 1+1 are given by: [1+1,0*] = 255; [1+I,1*] = 60; [1+I,2*] = 195; [1+1,3*] = 255; [1+1,4*] = 195; [1+1,6*] = 255; [1+1,7*] = 60; [1+1,8*] = 60; [1+1,9*] = 255; [1+1,11*] = 195; [1+1,13*] = 195; [1+1,14*] = 60; [1+1,1] = 255. Rule Components Cannonical Form Additive Form 12 (00110000) (3-+1 1+13++x

80

252 (00111111)

rl- +e a-1 + B+ + x B ++x o +B ++x 8+ +B-+il- +x +e +t D*+13+ + x

243 (11001111)

B++rl++rl- +x +e+x 1+I+B+ + x

207 (11110011)

B-+rl++rl- +x +t+x I+a-1 +13 + +x

63 (11111100)

B- +rl ++rl- +e +t+x 1+B+ + x

3 (11000000)

r1++K 1+D*+B++X

0 (00000000) 60 (00111100)

0 B-+q-+e+t D*

204 (00110011)

B++B-+x+e I

240 (00001111)

B++rl-+x+e a-1

255 (11111111) 195 (11000011)

1 B++rl++x+K 1+D*

51 (11001100) 15 (11110000)

Tl++rl-+e+K 1+I B-+rl++t+x 1+a-1

48 (00001100) 192 (00000011)

Table 4.4b Composite 3-Site Rules With Factor I or 1+I

As in the previous case, the rules listed in Table 4.4b form a sub-group of the abelian group of all CA rules. This sub-group is defined by the group table of Figure 4.6b. 12

48

192

252

243

207

63

3

255

195

187

15

12

0

60

204

240

68

60

0

240

204

195

255

15

187

204

240

0

60

187

15

255

195

252

240

204

60

0

15

187

195

243

255

195

187

15

0

60

204

255 240

207

195

255

15

187

60

0

240

204

240

0

60

204

60

0

192

63

187

15

255

195

204

3

15

187

195

255

240

Figure 4.6b Group Table for 3-Site Rules With Factor I or 1+I

Finally, for Q equal to D or 1+D, the rules which must be considered are listed in Table 4.4c. Since D, in contrast to a and I, does not commute with most other 2-site rules, this table is twice the size of Tables 4.4a and 4.4b.

81

Rule Components

Cannonical Form

Additive Form

18

(01001000)

A+x+t = 8+B++B-

46

(01110100)

71++71B++11++e+t

72

(00010010)

B++B-

a+x+t I +x +t

116

(00101110)

B++r1-+O+t

a-1+x+t

126

(01111110)

B++B-+r1++rI -+9+t e+p++p-+x+o =

8+0 +t

66

(01000010 )

13++r1+

0+(3++(3-+x+e

36

(00100100 )

A+t

I+(3++(3- +x +A

24

(00011000 )

237

(10110111 )

B--M8 + +B- +x +9+t+x

1+0 + x+t

209

(10001011) (11101101 )

B ++r1- +x +x

1+a + x+t

183

71++11- +x +8 +t +x

1+I +x +t

139

(11010001 )

B-+11+ +x +x

1+a-l+x+t

129

(10000001 )

x +x

1+0+ (3++(3-+x+e

189

(10111101 )

J3-+i1- +x +8+t+x

219

(11011011 )

13++i3-+T1++T1-+x+x 1+I+0++p-+x+O

231

(11100111 )

B++r ++X+8+t+x

0 60 90

(00000000) (00111100 ) (01011010 )

B-+r1-+9+t 13++13-+r1++i1-

102

(01100110)

B++11++O+t

D

255 195

(11111111) (11000011 )

B++r1+ +x+x

1 1+D*

165

(10100101 )

x +8+t+x

1+8

153

(10011001 )

B-+ri- +x +x

1+D

}++ + +

1+a+p++R-+x+8 1+a-1±+±a-±X±Q 0 D* 8

Table 4.4c Composite 3-Site Rules With Factor D or 1+D

The group table for these rules is given in Figure 4.6c. Unlike the group tables given in Figures 4.6a,b, this figure contains additional rules not listed in Table 4.4c. These are due to the non-commutativity of D. These additional rules are listed in Table 4.4d. It is interesting to note that the only rules which are not either commutators or products in this group table are rules 10, 80, 245, and 175.

82

116 72

46 46

0

116 90

18

90

102 60

0

60

72

102 60

18

60

0

102 90

24

36

126 209

139 183 237

189 231 219 129

108 54

165 153 195

147 201 245 175

66

10

80

255

102 54

108 80

10

165 255 195 153 201 147

175 245

90

10

80

108 54

195 255 165 345 175

147 201

0

80

10

54

66

108 54

10

80

0

90

102 60

24

54

108 80

10

90

0

60

36

10

80

108 54

126 80

10

54

102 60

108 60

0

102 90

147 201 245

153

108 195

153 165 255

147 201 245 175 255 165 147 175 245

165 255

90

245

175 147 201

153 195 255 165

0

175 245 201 147

153 195

139 165 255

195 153 201 147 175 245 90

175 0

90

102 60

0

60

183 153 195 255 165 245 175 147 201 102 60 165 255

175 245 201

147 60

0

102 90

195 153

195 153

165 255

108 54

10

80

108 80

10

102 54 90

10

80

108 54

0

80

10

54

108

10

80

0

90

102 60

153 54

108 80

10

90

0

60

102

165

10

80

108 54

0

90

195 153 165 255 80

10

54

189 147 201 245 175 255 165 153 231 201 147

175 245

165 255 195

219 245 175

147 201

153

129 175 245 201 147

153 195

102 201

209 255 165

237 195 153

175 245 201 147

195 255

195 108 54

102 60

108 60

102 90

0

Figure 4.6c Group Table Generated By 3-Site Rules With Factors D or 1+D

Rule Components Cannonical Form 54

(01101100) it ++1T-+0+t (00110110) f3++13-+O+t

= A+x+e

Comment 54 = [v,1+D], v = 1*,7*,11*,13*

10

(01010000) 13-+ii+

= I +x +A = a+x+9

80

(00001010) f3++rl-

= a-1+x+8

201 147

(10010011) 13++13- +x +1( (11001001) rl++il- +x +x

201 = [v,1+D], v = 2*,4*,8*,14* 147 = [v,1+D], v =8*,11*,13*,14*

245

(10101111) f3++Ty-+x+A+t+x

neither commutator nor product

175

(11110101) f3-+il+ +x +8+t+x

neither commutator nor product

108

108 = [v,1+D], v = 1*,2*,4*,7* neither commutator nor product neither commutator nor product

Table 4.4d Rules From Figure 4.6c Not Included in Table 4.4c

For each sub-group, inspection of the additive and cannonical forms shows that the elements separate into classes of four such that any three elements in a class sum to the fourth. For example, in Table 4.4c rules 18, 46, 72, and 116 form such a class, the additive forms of rules in this class being, respectively: 4+x+t, a+x+t, I+x+t, and a-l+x+t. The other such rule

83 classes in Table 4.4c are (24,36,66,126), (139,183,209,237), (129,189,219,231), (0,60,90,102), and (153,165,195,255). In addition, the rules in each sub-group are related by the symmetry transformations T1, T2, and T3 defined in Chapter 1. These relations are illustrated in Figures 4.7a,b.

Generating Rules Fzo n Identity Sub.Group

Generating Rules From Shift SubGroup

Figure 4.7a Symmetry Relations of Identity and Shift Sub-Group Rules (Full Sub-Groups Include Binary Compliments)

In Figure 4.7a only half of the sub-group elements are shown, the remaining half being the binary compliments of these, and sharing exactly the same symmetry relations. Figure 4.7b shows all of the sub-group elements. Rules 54, 108, 147, and 201 are commutators, but are not composite rules, while rules 10, 80, 175, and 245 are neither composite nor commutators, as indicated in Table 4.4d. The remaining rules in the subgroup are all composite. Of particular interest is the much discussed rule 18.

84

Figure 4.7 b Symmetry Relations Between Elements of D; 1+D Sub-Group

85 Rule 18 is the composition of the binary difference rule D on the 2-site rule (0111). Thus, the non-additive properties of rule 18 are determined by the rule (0111), which maps all 2-site neighborhoods except 00 to 1. Since [D(t)]i is 1 only when µi # µi+l this indicates that rule 18 yields 1's only at the boundaries of strings containing only isolated 0's. Examination of the additive decompositions of the rules in these different sub-groups suggests definition of domains of the state space on which the rules act as additive rules. For the rules of Tables 4.4a,b the nonlinear part of the rule is given by either B±+X, or 1+B+-+x, indicating that these rules reduce to their additive ((B±+x)(µ) = 0), or linear ((1+B±+x)(4) = 0) parts on all states which contain only isolated 1's. For the rules of Table 4.4c there are four domains which are defined: D1= (ge E I g contains only pairs of 1's) D2 = {µE E I g contains only pairs of 1's, and no isolated O's} D3 = (µE E I t contains isolated 1's or pairs of 1's, and no isolated 0's) D4 = (µE E I t contains only isolated 1's) D5 = {µE E 14 contains no isolated 0's or 1's) By inspection, D2cD1, D2cD5, and D4CD3. Also, the various nonlinear parts of the rules from Table 4.4c satisfy the conditions:

X+t I D1= X+t I DZ = B++13-+x+9 I DZ = x+6 I D3 = x+t I D1= B++B I D4 = A+t I D5 = 0. Domains D4 and D5 are those on which rules 18 and 126 respectively reduce to rule 90. It should be noted, however, that this reduction to an additive rule does not, in general, survive under more than a single iteration of the rule. For rule 18, for example, Hanson and Crutchfield [64] have shown that the domain on which this rule reduces to rule 90 regardless of the number of iterations is the set of states which consist of only isolated 1's separated by an odd number of 0's, and this domain is a proper subset of D4. The question of on what domain a rule reduces to an additive rule for all iterations is difficult, although an elegant empirical technique to search for such domains has been developed [64]. 5. Exercises for Chapter 4 1. Compute the commutator set for the 2-site rule (0111). 2. Compute the commutator set for the left and right extensions of (0111); e.g., for the rules (00111111) and (01110111).

86 3. Use equation (4.7) to determine whether or not the following 3-site nearest neighbor rules are idempotent: a) (00100000) b) (00010011) c) (01110100) d) (00111010) 4. Find the rules which are Ito relted to X via T = (0110) when X is: a) (00011000) b) (01001101) c) (01110001) d) (01000010) 5. Find all rules X such that T I [X,T] for T = (01111011) and (01011010). 6. Suppose that rules X, Y, and T in Figure 4.2 satisfy TX = YT+R. Derive a formula for TXn. 7. Show that rules 122 and 126 are Ito related to rule 18 via rule 238, as well as via rule 252. 8. Show that if T is rule 30 then the only possible rules X, defined for fewer than 4 sites, for which there is a rule Y such that the diagram of Figure 4.2 commutes are 0, a, a2, and rule 30. 9. Let X and Y be respectively k and r-site rules having the property that a) X(0il...ik-1) = 1+X(lil...ik-1) Y(Oil...ir_1) = 1+Y(lil...ir-1) b) X(Oil...ik-1) = 1+X(lil...ik-1) Y(iO...ir-20) = l+Y(iO...ir-2l) Show that in each case their commutator also has this property. 10. Rule 18 is equal to rule 90 plus rule 72, and is also equal to rule 150 plus rule 132. a) Show that rule 18 is identical to rule 90 for all iterations (i.e., for all n, Xn18 = 8n) on configurations containing only isolated l's separated by an odd number of 0's. b) Show that there is no configuration other then 0 on which rule 18 is identical to rule 150 for all iterations.

87

Chapter 5 Additive Rules: I. Basic Analysis In this chapter the conditions which a CA rule must satisfy in order to be additive are formulated in terms of a map from the Cartesian product of the configuration space with itself to the configuration space. This map is represented for k-site rules by a 2kx2k matrix. CA rules can be partitioned into equivalence classes on the basis of their additivity properties, as characterized in this matrix. An approach to the analysis of additive rules in terms of complex polynomials is also introduced, and used to derive a condition for the injective of additive rules. 1. Conditions for Additivity In Chapter 1, a cellular automata rule was defined as additive if it was linear and homogeneous; or equivalently, if it could be expressed as a sum of shifts, as in equation (1.5). The more common definition of additivity is that a rule Xis additive if X(µ+µ') = X(µ) + X(µ') V µ,µ'e E (5.1) In Theorem 5.2 it is shown that these two definitions of additivity are equivalent. As a preliminary, some results on the component form of additive rules are necessary. Equation (5.1) can be stated in terms of the components of X by using the fact that the neighborhood set for X is an additive group under site-wise addition mod(2). For example, Figure 5.1 gives the group table for 3-site neiehborhoods. 000

010

011

100

101

110

111

000

001

000

001

010

011

100

101

110

111

010

101

100

111

110

001

110 111

111 110

100 101

100

001

001

000

011

010 011

010 011

011

000 001

010

000

101

100

101

110

111

000

001

010

011

100

111

110

001

000

011

010

110

101 110

111

100

101

010

011

000

001

111

111

110

101

100

011

010

001

000

100 101

Figure 5.1 Group Table for 3-Site Neighborhoods

88 By definition, however, for a k-site rule X the neighborhood iO...ik-1 maps to xi under X. If X is to be additive, its action on the sum of two neighborhoods must give the same result as the sum of its action on each neighborhood separately. Hence the group table for k-site neighborhoods defines an addition table for the components of X. If X is additive then X(iO...ik-1) + X(jo...jk-1) = X(iO...ik-1 +iO...ik-1) for all neighborhoods. This condition will be written as Xi + Xj

+ Xi+j = 0

(5.2)

where the sum i+j is understood as the denary value of the site-wise addition of the binary forms of i and j. The table which this defines for 3-site rules is shown in Figure 5.2. X0 xp X1 x2 x3 x4 X5 xg x7

xl

x2

X3

x4

X5

x6

X7

0

0

0

0

0

0

0

0

o

0

xl +x2+x3 x1+x2+x3 x1 +x4+x5 X1+X4+X5 x1 +x6+x7 xl+x6+x7 0

xl +x2 +x3 x2+x4+x6 X2+X5+X7 x2+x4+ x6 X2+X5+X7

0

X1 +X2+X3

0

X1 +X2+X3 X1+X2+X3

0

+x4+x6 x3 +x4+x7 x1 +x4+x5 x2

0

x1 +x4+x5 X2+X5 +X7 x3+x5+x6 x1+x4+x5

0

+x5+x6 +x4+x6 x3 +x5+x6 x 2+x4+x6 x3 x1 +x6+x7 x2

p

xl +x6 +x7 x2+x5+x7 x3+x4+x7 x3+x4+x7 x2+x5+x7 xl+x6+x7

0

X3+X4 +X7 x3 +x5 +x6 x3 +x5+x6 x3+x4+x7 0

+x 4+x +x4+x6 x3 x1 +x4+x5 x2 7 0

x3+x5+x6 X2+X5+X7 0

x1 +x 6 +x 7 0

Figure 5.2 Component Sums Defined by Neighborhood Table of Figure 5.1 Continuing the 3-site example, equation (5.2) yields the set of conditons

x0 = xl+x2+x3 = xl+x4+x5 = xl+x6+x7 =x2+x4+x6 = x2+x5+x7 = x3+x4+x7 = x3+x5+x6 =0

(5.3)

and the independent equations in this set may be taken as x0=0 xl+x2+x3 = 0 xl+x4+x5 = 0

(5.4)

xl+x6+x7 = 0 x2+x4+x6 = 0 giving the most general 3-site rule satisfying equation (5.1) as X = (0,x1,x2,x1+x2,x4,xl+x4,x2+x4,xl+x2+x4) The next theorem generalizes this form to general k-site rules. Theorem 5.1

(5.5)

89 A k-site rule X satisfies equation (5.2) if and only if k-1 Xi = Xik -s-1x23 `d i = i0...ik-1 (5.6) s=0

Proof By equation (5.5) the claim is true for k = 3, and it is easily verified for k = 2. Suppose that the claim is true for k-site rules. Let G(k) denote the ksite neighborhood group table; and G(0)(k), G(1)(k) denote this table with all elements extended on the left by the addition of a 0 or a 1 respectively. Then G(k+ 1) = G(o)(k) G(1)(k)

G(1)(k) G(o)(k)) Equations (5.2), as applied to G(k+1), can be divided into two sets: a) Those arising from terms of the form OiO...ik_1 + G(0)(k). These are the equations which determine the components of the k-site additive rules. Thus, they will determine the first 2k components of the k+1 site additive rules , which will have the same form as for the k-site additive rules. These will be denoted xi, with 0 2n.

144

Proof If g is on a cycle of D with period a divisor of 2n then by equation (7.23), (I+D2n)(g) = 0 (7.24) Since D = I+a, by Lemma 7. 11, D2n = I+a2n , hence equation (7.24) becomes a2n(µ) = 0, which completes the proof. I

4. The Operator B A major role in computing solutions to equations of the form X(µ) =13 is played by the operator B:E+->E+ defined in equation (7.1). As indicated at the beginning of section 2 of this chapter, operating on a configuration with B can be seen as integration with respect to sequence index. Lemma 7.2 gives a formula for [Bn(g)h, and Lemma 7.3 indicates that B commutes with the right shift. It is not true, however, that B commutes with the left shift. Lemma 7.13 r-1 [B,ar] = BPJ 1as (7.25) s=0

where P1 is the projection defined by P1(µ) = µ1a(1). Proof The claim is easily demonstrated for r = 1, and the remainder of the proof follows by induction. Assume that equation (7.25) is true for r. Then [Bar+l] = Bar+1 + ar+1B = Bara + ar+1B but r-1 Bar = arB + BPl as s=0

so that [B,ar+l] becomes r-1 arBa + BP 11 as+1+ ar+1B s=0 and on substitution of Ba = aB + BP1 this yields equation (7.25) with the summation from 0 to r. I

In the examples of section 3, it often occured that the predecessor of a periodic configuration µ had a period greater than that of its image. This is a consequence of the period doubling property of the operator B, described in the next two theorems. Theorem 7.14 Let µE E+ have period n. Then:

145

1. If [B(9)1n = 0 then B(µ) also has period n. 2. If [B(µ)]n = 1 then B(µ) has period 2n. Proof Let Itr be the first non-zero entry in the configuration g. By definition, [B(4)]n+1 = [B(9)1n + 9n+1• If g has period n then this equals [B(9)1n + µl. Hence, if [B(9)1n = 0 it follows that [B(g)]n+r = [B(µ)]r for 15 r< n and thus B(µ) has period n. Suppose that [B(µ)ln = 1. Then [B(µ)]n+r = 1+[B(µ)]n for 1- 1 . But these coefficients come from the s-th diagonal of the

mod(2) Pascal triangle , and no diagonal of this triangle consists of a leading 1 followed by only 0 's. Hence equation ( 7.27) can never be satisfied for µ * 0. The proof of Theorem 7.16 follows from the non -existence of a diagonal consisting only of 0 's following the leading 1 in the mod(2) Pascal triangle. This triangle does, however, contain diagonals which have arbritrairly large numbers of 0's following the leading 1, and this fact can be used to show that B is almost periodic , in the sense of being self-accumulating. That is, every iterate of B is an accumulation point for further iterates. Theorem 7.17 For all n,s >- 0, and for all µE E+ g(Bn(µ),Bn+2s (µ)) < [2(22s - 1)l-1 (7.29) where g( p,p') is the metric on [0,1] defined in section 4 of Chapter 1.

Proof It is sufficient to prove the theorem for n = 0. For this case, consider the sequence 4 = µ - B25 (µ), which can be shown to have components

147

i -1

2s +.1 Ij E j=1 i-j+1 - ) The coefficients in this sum are the first i entries of the 2s diagonal of the mod(2) Pascal triangle, with the first entry excluded since the sum is only up to i- 1. By Lemma A.5 of appendix 4, these coefficients are all 0 for i 1. But l g(µ,B 2s (µ) I ^_^' -2 - 21 i122i+1 2 i=1

The final sum in this expression is just (22s-1 )- 1, finishing the proof. I The content of this theorem is summarized by saying that the operator B is self-accumulating; i.e., with respect to the topology induced on E+ by the metric g, the sequence Bn(g) eventually returns to arbritrairly small neighborhoods of its previous iterates. It is also possible to show that B has no dense orbits . Define a partition of E+ by E+ = U ET where Er = WE E+ I µi=0, i 2-s-2-r. Theorem 7.18 No orbit of B: Es -* ES is dense in Eg for any s. Proof It is sufficient to prove the theorem for the case s = 1. Suppose that µE Ei is such that the orbit of B iterated on g is dense in E. Then, for any given 13,13'E Et, and for all N < oo there will be integers n and m (which may depend on N) such that g(13,Bn(g)) < 2-N and g(B',Bm(g)) < 2-N (7.30) Without loss of generality, assume that n and m are the smallest integers for which these inequalities hold. In terms of the components of g, 13, and B' equation ( 7.30) requires that 13i = [Bn(g)]i and 13'i = [Bm(g)]i, i 2r-1( 2r-1). I 5. Exercises for Chapter 7 1. Use equations (7.10), (7 . 11), and Theorem 7.7 to find the general solution to A(µ) = 01001. 2. Give a proof of Theorem 7.10. 3. Find the general solution to the homogeneous equation X(µ) = 0 for the following operators X: a) a+a2 b) I+a3 c) I+a+a3 d) I+a2+a4 4. Find the general solution to X(µ) = 10010 for each of the operators in exercise 3. 5. Use equation (7.23), Theorem 7.5, and Lemma 7.11 to prove that the configurations on cycles of rule 150 which have period a divisor of 2n are given by 2n 2° l bsa(s ) + X asBsa(s+2° ) s=1 s=1

6. Use the result of exercise 5 to find all configurations on period 4 cycles for rule 150. 7. Find the spatial period of configurations on period 5 cycles of D. 8. Use Lemma 7.13 to compute [Br,as].

150

Chapter 8 The Binary Difference Rule This chapter presents an analysis of the geometrical and arithmatical properties of the binary difference rule D. The first part of the chapter contains derivations of some of the basic characteristics of this rule, and then gives an analysis of the graph of D:[0,1]- [0,1]. This is followed by some number theoretic relations contained in the graph, and a theorem which demonstrates that the set of all predecessors of all orders of any point in [0,1] is uniformly distributed in [0,1]. 1. Basic Properties of D After the right and left shifts, perhaps the most ubiquitous cellular automata rule is the binary difference rule. This is the rule D:E-*E defined by [D(i)]i = µi+µi+1 (8.1) which has frequently been mentioned to in previous chapters. This rule was particularly useful in Chapter 7, where it provided the basis for solution of the predecessor equation for additive rules in general. As indicated in Chapter 7, equation (8.1) can be seen as an analogue of Taylor's theorem, indicating that D can be viewed as a discrete derivative with respect to sequence index. Since D(µ) is uniquely defined for all ge E+, the map D:E+->E+ is a function. If D is taken as a map D:[0,1]-*[0,1], however, it is not a function. This is easily seen by consideration of the configurations g and g' defined by gi i < n µi i < n µi= 1 i=n µ;= 0 i=n 0 i>n 1 i>n Both of these configurations define the same point of [0,1], but _ [D(µ)]i i #n -1 [D(µ')], 1+[D(g)]i i=n-1 Every configuration of the specified form corresponds to a rational point of [0,1] having as denominator a power of 2. Hence D:[0,1]-4[0,1] is double valued at all points of the form I(/2s. Despite this, however, D is sufficiently function like that it can be treated as such in many cases.

151

This chapter presents some of the more elementary properties of the binary difference rule. The following lemma follows immediatly by iteration of equation (8.1). Lemma 8.1 [Dn(g)h = n+1(n+1 ) j=1 J

where the coefficients are drawn from the n+1 row of the mod(2) Pascal triangle. One question is whether or not D has cycles. Clearly it must when acting on the finite state space En . The next lemma, a version of Theorem 6.10, details the maximum number of iterations that are necessary to reach a cycle. Lemma 8.2 Let n = 2mn0 with no odd. Under iteration of D, every element of the state space En maps to a cycle or fixed point in at most 2m iterations. Proof If no = 1 then by equation (8.1) and the property of the mod (2) Pascal triangle [D2m(µ)]i = µi + 9i+2m = 0 since i+2m = i mod(n). If no > 1, let k be the smallest integer such that 2k = 2m mod(n). Then [D2k(g)]i = gi + g42k = 11i+2m = [D2m(g)]i which may be written as D2k-2m(D2m(µ)) = D2m(g)]i indicating that D2m(g) is on a cycle. I Returning considerations to the space E+, a configuration ge E+ will be said to terminate at n if there is an n > 0 such that gn = 1, while gi = 0 for all i > n. Likewise, a configuration will be called eventually periodic if there are integers n and m such that for all i, gi+n+m = gi+n• If m is the smallest integer such that this is true then g will be said to be eventually periodic with period m. If no such m exists, then g is non-periodic. If D is treated as a map of the interval [0,11 then eventually periodic sequences correspond to rationals while non-periodic sequences correspond to irrationals. The next few lemmas and theorems establish the invariance of the sets of rationals, and of irrationals, under the mapping defined by D. This, in turn, relates to the fact, mentioned at the end of Chapter 6, that D acting on En reduces entropy, but acting on E+ does not do so. This topic will be taken up further in Chapter 12.

152

Lemma 8.3 Let gE E+ terminate at n. Then µ lies on a cycle of D having period 2r where r is the smallest integer such that n:5 2r. Proof Since the 2r-th row of the mod(2) Pascal triangle contains all 0's, excepting the initial and final 1, application of equation (8.2) yields the following expression [1)2r(g)]i = µi + µi+2r Since g terminates at n:5 2r this means that [D2r(µ)]i = µi for all i. Suppose that there is an s > 0 such that D21-s(g) = g. By equation (8.2) this can be written out as 0=(2' -s+1) 2 1) µ2+ +(2r s+1 µi

Il J 0 = (2r -2s+ 1

)µ3+...+ 1 2r, -s+1 lµj + 2r + 1 µj+1

J

0 = [2r -s + 11µn-1+...+(2r 3 + 1 2 0=

[

)

un

(8.3)

2r -s+ll 2 )

un

where J n 2r-s+1- n = 2r-s+l 2r-s+1- n, and in addition

153

[

2r_ s^1J_o 2

:5 jn

Additivity of D, combined with Lemmas 8.2 and 8.3 then yields a theorem on the action of D on eventually periodic sequences. Theorem 8.4 Let g be as above , let nm(B) be the period of the cycle to which B goes under iteration of D, and let r be the smallest integer such that n n, µr is determined in terms of 92 ,...,µn by a finite recurrence relation, and g must therefore be eventually periodic. I Theorem 8.6 Let µe E+ be non-periodic . Then D (µ) is non-periodic. Proof

154

The proof is by contrapositive, making use of the formula given in Theorem 7.4 for the solution of the equation D(µ) =13. Suppose that D(µ) is eventually periodic with period n. By Theorem 7.4, and the period doubling property of the operator B, µ must also be eventually periodic, with period either n or 2n. I From the above theorems and lemmas it is apparent that D maps eventually periodic configurations to eventually periodic configurations, and non-periodic configurations to non-periodic configurations. Thus, as a map of the interval , D maps rationals to rationals and irrationals to irrationals. In Chapter 12 this property will be shown to characterize all surjective rules. 2. The Graph of D:10.11-40.11 The graph of D:[0,1]-+[0,1] is shown in Figure 8.1 at the end of this chapter. Some of the general properties of this graph are derived in this section. Lemma 8.7 The graph of D:[0,1]-*[0,1] is reflection symmetric about 1/2. Proof D(1/2) = D(1Q) = D(01) = 1/2. Hence 1/2 is a fixed point. Suppose that XE [0,1 /2) and take two configurations µ, µ'E E+ such that µ = 2 - x and µ' = 2 + x. Then µ+µ' = 1 so that D( µ+µ') = 0. Since D is additive this means that D(µ) = D(µ'). I Without loss of generality, further considerations of the graph in Figure 8 . 1 will be restricted to the interval [0,1/2]. Coordinates can be introduced labeling points on this graph, based on the sequence {S(al)} = {(2

-al , 2-ai I 101Q while x(1,1,1)-0101. 2. For 1jO...jn-lE GE*(T2X)

3. jO...jn-lE GE*(X) a 1...1 + jO...jn-l€ GE*(T3X) Proof The first part of the theorem follows directly from left/right symmetry. In order to prove the second suppose that jo..in-lE GE*(X), but there is a sequence iO...in_1 such that T2X(iO...in-1sl...sk) =jO..jn-1 for some sl...sk. Taking in+k-1 = sk for notational convenience, this means that for all r< n, T2X(ir...ir+k-1) =Jr. Since jO...jn-le GE*(X), however, there must be at least one r such that X(ir...ir+k-1) #Jr. But by the definition of the transformation T2, T2X(ir...ir+k-1) = X(ir...ir+k-1 + 1...1) for all r. Hence T2X(ir...ir+k-1) = jr implies that X(ir...ir+k-1 + 1...1) = jr as well, contradicting the assumption that jO...j n- lE GE *(X). Finally, suppose thatjO...jn-lE GE*(X) but 1...1 +jO...jn-1 GE*(T3X). Then there is an iO...in-1 and an s1...sk such that T3X(iO...in-1sl...sk) = 1...1+ JO.-in-1• But T3X(9) = (1 + X)(µ) = 1(µ) + X(µ), and i(µ) = 1 for all R. Therefore, T3X(iO...in-lsl...sk) = 1...1+ X(iO...in-lsl...sk) = 1...1+ JO...Jn-1, so that X(i0...in-lsl...sk) =JO•••Jn-1, contradicting the initial assumption. I

198

3. GE*(X) for 3-Site Rules The space of 256 3-site rules is partitioned into 47 equivalence classes by the transformations T1, T2, and T3. Appendix 5 gives either GE*(X), if it is finite, or the initial portions of GE*(X) otherwise, for particular exemplars of each of these equivalence classes . The T-invariant equivalence classes are listed in Table 10.1. (0,255) (1,127,128,254)

(2,8,16,64,191,239,247,253)

(4,32,223,251)

(3,17,63,119,136,192,238,252)

(5,95,160,250)

(6,20,40,96,159,215,235,249)

(10,80,175,245)

(7,21,31,87,168,224,234,248)

(15,85,170,240)

(9,65,111,125,130,144,190,246)

(18,72,183,237)

(11,47,81,117,138,174,208,244)

(19,55,200,236) (22,104,151,233)

(12,34,48,68,187,207,221,243)

(23,232)

(14,42,84,112,143,171,213,241)

(24,66,189,231)

(25,61,67,103,152,188,194,230)

(29,71,184,226)

(26,74,82,88,167,173,181,229)

(33,123,132,222)

(27,39,53,83,172,202,216,228)

(36,126,129,219)

(28,56, 70,98,157,185,199,227)

(37,91,164,218) (43,113,142,212)

(30,86,106,120,135,149,169,225)

(46,116,139,209)

(38,44,52,100,155,203,211,217)

(50,76,179,205)

(41,97,107,121,134,148,158,214)

(51,204)

(45,75,89,101,154,166,180,210)

(54,108,147,201)

(58, 78,92,114,141,163,177,197)

(57,99,156,198)

(62,110,118,124,131,137,145,193)

(13,69,79,93,162,176,186,242)

(35,49,59,115,140,196,206,220)

(60,102,153,195) (73,109,146,182) (77,178) (90,165) (94,122,133,161) (105,150) Table 10.1 T-Equivalence Classes of 3-Site Rules

199

If the elements of GE*(X) are computed out to sequences of length eighteen. Four first order catagories are obtained. These are defined as: EM: For rules in this catagory GE*(X) is empty. The rules in EM are listed in Table 10.2 FI: For rules in this catagory GE*(X) is finite. SI: For rules in this catagory, the number of elements in GE*(X) of length 7 there are two seeds, having the forms 010...0100, and 010...0111 with the central block of 0's having length n-5 in each case. The number of seeds scales with n according to N(n) = 2n-11 (10.6) Rule 172 1010 is a seed. For all n >_ 7 there is a single seed with the form 001...1001 if n is odd, and 101...1..1 if n is even. The central block of 1's has length n-5 and the number of seeds scales with n according to N(n) = n-5

(10.7)

3. The LI catagory includes several rules which have been extensively studied, in particular rules 18 [52,64], 22, and 110 [74,75]. For a rule X denote the number of seeds of length n by K(n,X). Table 10.3 gives the value of K(n,X), n_ 3 (10.8) Proof The set Sr + 1(1;1) of length r+1 sequences containing only isolated 1's can be constructed as follows: a) Add a 0 to the right end point of each sequence in Sr(1;1) to obtain a set Sr+1 ( 0). The elements of this set have only isolated 1 's, and in addition, they terminate in 0; b) Add a 1 to the right end point of each sequence in the set Sr (0) of elements of Sr( 1;1) which terminate in 0. Call this new set Sr + 1(1). Clearly this set consists of length r+1 sequences containing only isolated 1's, and terminating in 1. Then, Sr+1(1 ; 1) = Sr+l (0)uSr+1(1),){0...011 since the final sequence accounts for the only remaining element of Sr+1(1;1). By construction , the elements of Sr+1 ( 0) will have an odd or even number of 1 's if the corresponding elements of Sr(1;1) do, hence the number of sequences with an odd number of isolated 1's contained in Sr+l ( 0) is given by Kr. The elements of Sr+l(1), on the other hand, will have an odd number of l's if the corresponding elements of Sr(O) have an even number of l's. But

202 Sr(0) is just Sr-1(1;1) with a 0 appended to the right end point of each member. Hence the number of sequences in Sr+1 ( 1) having an odd number of isolated 1 's is just the number of sequences in Sr-1(1;1) with an even number of 1's. But this is just the number of length r-1 sequences with only isolated 1's minus the number of length r- 1 sequences with an odd number of isolated 1's. Finally, the term { 0...01} is counted by adding a final 1. Thus, recalling Corollary 3.7, Kr+1 (1;1) = Sr(1;1) + (Lr-1 - 1 - Kr-1) + 1, and this proves the lemma. Theorem 10.8 Excluding the anamolous seed 111 , K(n,18) satisfies 0 n7 (10.9) K(n-4, 8 )+ Kn_6 n Proof Excepting the seed 111, all elements of GE *( 18) are of the form 110s3 ... sn-4011 where s3...sn -4 has an odd number of isolated 1's. This sequence has length n-6, so if n < 7 no such sequences can exist . The number of such sequences will be Kn-6 plus a term which adds in those sequences which are allowed by open boundary conditions , but forbidden by periodic boundary conditions . All such additional sequences will have the form 10s5...sn-701 where the n- 10 digit sequence s5...sn-7 contains an odd number of isolated l's. But this is exactly the condition on the interior sequence contained in K(n-4,18).1 Equation ( 10.9) can be expanded to yield In461

K(n,18 ) = JK(n-6) mod(4)+4j j=0

(10.10)

Rule 110 This rule has been studied as a possible example of a rule which carries out computations at the edge of chaos [75,76]. Examination of the pattern of seeds suggests that a sequences is in GE *(110) if and only if it has the form 010s3... sn-4010 where s3...sn -4 consists either of all O's, or of blocks of 111 separated by one or more 0 's. The values listed in Table 10.3 follow the recursion relation K(n,110) = 0 n- 5 (10.11) The number of sequences consisting of blocks of 111's separated by one or more 0's can be computed from Lemmas 3.5 and 3.6. Using these lemmas, the number of seeds of length n is given by K(n,110) = Kn-6(1;3) + 1 in which the final 1 accounts for the s3...sn-4 = 0... 0 sequence. Rule 152 Explicit conditions for membership in GE*(152) appear to be rather complex. Inspection of the list of seeds for n 6, seeds have the form 11s2...sn-311 with the internal sequence having only isolated 1's. For rule 22 the seed form is 10s2...sn-301. 4. Classification With GE* Four catagories of 3-site rules have been distinguished, based on the behavior of the set GE*: either it is empty (EM), finite (FI), grows linearly with n (SI), or grows at a rate faster than linear with n (LI). Since, roughly speaking, the elements of GE* define how many "holes" are cut out of [0,1] to form the Cantor set X([0,1]), it is to be expected that the fractal dimension of this set will satisfy d(LI) < d(SI) < d(FI) < d(EM) = 1. Thus, the greatest entropy reduction is expected for rules in the LI catagory. The best known classification of CA rules has been made by Wolfram based on limiting behavior. The four classes distinguished are defined as [28]: Class I: rules which evolve after a finite number of steps, from almost all

204

initial states , to a unique homogeneous state in which all sites have the same value ; Class II : Rules which evolve to simple structures , either fixed points or cycles . The set of these structures corresponds to the words generated by a regular grammer; Class III: These rules evolve to aperiodic , chaotic patterns. These patterns appear as asymtotically self-similar fractals ; Class IV: Rules for which the persisting structures exhibit no simple pattern, and appear to be essentially unpredictable. This classification has been refined by Li and Packard [58], who distinguish five catagories: Null Rules (Wolfram Class I); Fixed Point Rules (Wolfram Class Ila); Periodic Rules (Wolfram Class IIb); Locally Chaotic Rules (chaotic dynamics confined by domain walls); and Globally Chaotic Rules (Wolfram Class III). Wolframs Class IV is difficult to fit into the Li & Packard scheme, falling under either the chaotic heading (based on spatial response to perturbations ), or the periodic heading (based on possible periodicity of limit configurations). The classification on the basis of GE * does not respect the Wolfram, or Li & Packard classes since , although the transformation T2 leaves GE* invariant, it crosses the Wolfram/Li&Packard boundaries . For example, both rule 18 and rule 72 are in the LI catagory, yet in terms of asymtotic behavior rule 18 is Class 3 (or Chaotic in the Li & Packard scheme) while rule 72 is Class I (Null). Rules 18 and 72 are related by T2, however , so by Theorem 10.6, GE *( 18) = GE *( 72). Table 10 .4 lists the members of the Li & Packard classes which belong to each of the GE * catagories . Note that exemplars are shown only for the 88 equivalence classes distinguished under the transformations Ti and T2T3. Several points are worthy of note: 1. GE* is finite for all null rules; 2. With two exceptions , all rules for which GE* is finite are either null, fixed point, or periodic. That is, in Wolfram Classes I and II. 3. With the same two exceptions , chaotic rules are found either in the EM or LI catagories. 4. The only rules in EM which are not chaotic are the identity, powers of the shift, and 1 plus these rules. The exceptions mentioned in points (2) and (3) are rules 54 and 129. These rules are in Wolfram Class IV, and are assigned to the chaotic class by Li & Packard , although they suggest that other considerations might well place them in the periodic class. Rule 54 is related to rule 108 by both T 1 and

205 T2, and rule 108 is in the periodic class, so it seems reasonable to place rule 54 in this class as well. Likewise, rule 129 is related to rule 126 by the transformation T2T3 so an argument could also be made for placing this rule in the periodic class.

Null Fixed Point

Periodic

Locally Chaotic Chaotic

EM FI SI 0,8,32,40,128, 136,160,168 44,46,58,78, 2,4,10,12,13, 170,204 172 24,34,36,42, 56,57,76,77, 130,132,138, 140,162,184, 200,232 1,3,5,6,7,9,11, 27,38 15,51 14,19,23,28, 29,33,35,43, 108,142,156, 178 154 30,45,60,90, 105 106,150

54,129

LI

72,104,152, 164

25,37,41,74, 131,133,134

26,73 18,22,137, 146,164

Table 10.4 GE* Catagories for Li & Packard Classes

One final point is that membership in the Wolfram catagories has been shown to be formally undecidable [53], and results have been obtained which indicate that class membership may be undecidable in any scheme based on infinite time behvior [77]. The GE* catagories, on the other hand, are decidable, and the classes which they define are invariant under the transformations Ti, T2, and T3. The conceptual difficulty with GE* classification is that members of the same catagory may have very different long term behaviors, as indicated already in the example of rules 18 and 72. What is interesting about these two rules in particular is that they are defined on exactly the same Cantor set, yet have very different behaviors on this set. Rule 72 has the left justified decomposition a+x+t (or, the nearest neighbor decomposition I+x+t), while rule 18 has the decompositions 6+B++Band 4+x+t. Under rule 72, all states map to states which contain only isolated pairs of 1's, and these shift to the right with each iteration (left justified) or remain fixed (nearest neighbor). Rule 18, on the other hand, imitates rule 150 on these configurations, and imitates rule 90 on domains

206 which consist of regions of isolated l's separated by an odd number of 0's, sandwiched between domain walls consisting of 11 's, which preform anihilating random walks [64]. This suggests that GE* classification is useful when the interest is in the nature of the Cantor set which is the domain of a rule, and that a classification based on long term behavior is more of interest from a dynamical perspective. Another classification of 3-site rules has been given by Jen [35], based on the recurrence relations for counting pre-images. Since GE* is defined as the minimal set of sequences having no pre-images , it is the generator of the null set of these relations. That is, for a given rule X, the elements of GE*(X) are generators of the set of sequences which yield 0 when substituted into the recurrence relation for X. The catagories distinguished by Jen are defined in terms of the number of pre -images of a sequence s as: A. For all s the number of pre -images is constant; B. The number of pre-images of s is a product of integers representing block lengths (of 0's and 1' s) in s; C. The number of pre-images of s is a product of integers representing specific terms in s; D. The number of pre-images of s satisfies a telescoping recurrence relation; E. The number of pre -images of s is given by terms in a sequence whose values vary periodically; F. Linear recurrence relations with non-constant coefficients which do not obviously simplify (default catagory). Table 10 . 5 indicates the GE * catagories occupied by the rules in the six different classes defined by Jen. With the exception of rule 0, Jen's Class A corresponds exactly to the catagory EM, corresponding to an empty Garden-of-Eden. GE* is finite for rules in Classes B, C, and F with the exception of rules 38, 44, and 46 in Class B; rules 18 and 72 in Class C; and rules 58, 74, and 78 in Class F. All rules in Class E are in the LI catagory except rules 27 and 172 which are in SI. No rules in SI are in Classes C or D, and no rules in LI are in Class B. In addition, almost all of Jen's classes are preserved under T-transformations, the exceptions falling into the default class F. With further analysis it might be possible to show that rules in her class F can fit into other classes.

207 EM FI SI A 15,30,45,51, 0,255 60,90,105, 150,170,204 B 11,12,14,24, 38,44,46 34,35,42,138, 140 C 1,2,3,4,7,8,16, 17,19,28,32, 36,50,56,57, 76,126,128, 136,156,162, 168,184,200 D 5,6,9,23,33, 40,54,108, 130,132,160, 232 E 27,172 F

10,13,29,43, 77,142,178

58,78

LI

18,72

37,62,73,94, 110,122,146

22,25,26,41, 104,134,152, 164 74

Table 10.5 GE* Catagories for Jen Classes

5. Exercises for Chapter 10 1. In section 1 of this chapter, a counter-example is given to the conjecture that GE*(X) = 0 implies that X has the intermediate value property. Find another counter-example to this conjecture. 2. Find all 3-site rules X for which the specified sets of sequences are in GE*(X): a) 101 b) 101, 111 c) 1100 d) 101, 1001 e) 01110 3. Suppose that Xis a rule for which GE*(X) = {111}. Use Lemmas 3.5 and 3.6 to compute a formula for the fractal dimension of the Cantor set which is the image of X on [0,1].

208

Chapter 11 Time Series Simulation In this chapter the possibility of finding CA rules and initial conditions which generate specified time series is considered. The strongest general result is that no k-site rule can generate arbritrary time series with entries of k or more digits, while a k- site rule can generate arbritrary time series with k-1 digit entries if and only if it is linear in the final variable. A matrix technique is derived which allows determination of whether or not a given rule could have generated a specified time series , and if so what are the possible initial conditions. This allows estimation of probabilities for specified series , but these estimates are unbiased only for rules with empty Gardens-of-Eden. 1. Cellular Automata Generating Time Series Measurements carried out on a dynamical system yield discrete time series of finite accuracy. The theoretical problem is to reconstruct the underlying dynamics from this data set. This reconstruction is generally carried out within a particular theoretical framework which specifies the kinds of modelling paradigms which are to be used. For example, in classical physics it is assumed that modelling will be carried out in terms of either ordinary or partial differential equations. Cellular automata form one of the simplest model classes for systems which exhibit spatially and temporally discrete pattern generation. Several researchers have proposed them as models of pattern formation [78,791, and they have been employed in a variety of applications for theoretical modelling. The suggestion has also been made that CA equations of motion could be infered from time series data [80,811. That is, given a time series, that it would be possible to determine a CA rule which generated that series from an initial condition. This last claim has been questioned on the basis that if distortions introduced by the measuring apparatus are included, CAs do not form a wide enough modelling class to accurately reproduce underlying dynamics [571. Despite this negative result, the question of when a CA dqg serve as a good model for the dynamics which generate a time series is still of interest. In fact, if it turns out that the output time series of a particular measuring instrument can be represented by a CA, this conveys more information than if it were the case that all time series could be so simulated.

209

If the effect of the measuring instrument itself can be modelled by a CA rule, say T, then the condition that the dynamics be modelled by a CA rule is that there exist rules X and Y such that Y generates the output series, and X is Ito related to Y via T. In this case, the series observed is YnT(µ(0)) where µ(0) is the initial configuration, so that the rule sought is Y acting on an initial state T(µ(0)). The methods discussed in Chapter 4 can then be used to compute X. Let S(n,t) be a time series of n digit binary numbers containing t+1 elements: S(n,t) = µ1(0)...µn(0) µ1(1)...µn(1) 41(t)...9n(t) The two questions to be considered are: 1) Could a given CA rule X:E+-*E+ have generated this series, and if so, what are the possible initial states? 2) What CA rules, if any, could have possible generated this series? The second of these questions is the more difficult, although a partial answer is easy to obtain. Theorem 11.1 Let X:E+_E+ be a k- site rule . Then X cannot generate an arbritrary series S(n,t) for any n > k. Proof By definition , if i = io ...ik-1 then X cannot generate any series which contains the sequence i0...ik-1 x'i since X(i0...ik_ 1) = xi. I One approach to finding an answer to the first question is through the process of backward reconstruction . This method begins with the last entry in the time series and looks for a sequence s1...sk-1 such that when this sequence is appended to the penultimate entry in the time series , the final entry is generated by the given rule. That is X4t1(t-1)...9n(t- 1)s1...sk-1) = µl(t)...µn(t) (11.1) If this is possible , then the process is repeated with the new sequences µ1(t-l) ...µn(t-l)sl...sk-1, for all sl...sk-1 such that equation (11.1) is satisfied, now moved to the right side of this equation , while the argument on the left

210

side is now µ1(t-2)...gn(t-2)rl...r2k-2. An algorithmic proceedure for carrying out these computations has been given by Wuensche & Lesser[43]. So long as n < k there is no a priori restriction on the possibility of success in this process, but it is computationally intensive, and there are often cases in which it will fail, although this failure may not become apparent until the process has continued for some time. For example, Rule 22 cannot generate the time series 01,10,11,01,11, as indicated by the following attempted reconstruction: 01011 # 1 100111 0 01 110000 0100 11 There is one case, however, in which the backward reconstruction process will always succeed. Theorem 11.2 Let X:E+- *E+ be a k-site rule which is linear in the final variable. Then X can generate arbritrary sequences S(n,t) for all n < k. Proof Since Xis LFV, X(iO...ik-20) = 1 + X(iO...ik-21). Since n < k this means that there will always be the freedom to flip the final bit from 0 to 1 or vice versa, in order to escape an impass. I The next theorem shows that this result is the best possible. Theorem 11.3 There exist time series S(k-1,t) which cannot be generated by any ksite rule which is not LFV. Proof If a rule X is not LFV then there will be at least two neighborhoods iO...ik-20 and iO...ik-21 which map to the same value , say 1+a = a', under X. Let S(k-1,t) be any time series which contains the conbinations iO...ik-2 a.......... i0...ik-2 a..........

211

for all possible choices of iO...ik-2. Clearly, no X which is not LFV can generate such a series. I To summarize the conclusions of the preceeding paragraphs, if X:E+_*E+ is linear in the final variable then X can generate all time series S(n,t) for which n < k, but there will be time series with n > k which X cannot generate. If X is not LFV, then there will be time series S(k-l,t) which cannot be generated. The more general results for k-site rules are best introduced through the example of single digit time series S(1,t) and 3-site rules. The question is now, given a 3-site rule X, and a time series S(l,t), is it possible that the rule X could have generated this series, and if so, what are the initial conditions? It would also be useful to know the probabilities that one rule X generated the series as compared to some other rule X'. If a 3-site rule X can generate a time series S(1,t) = {41(0),...,µ1(t)} then a backward reconstruction as indicated below is possible. µ1(0)42(0).93(0)... µ 1(t-3)µ2(t-3)µ3(t-3)µ4(t-3)µ5(t-3)µ6(t-3)µ7(t-3) µ 1(t-2)µ2(t-2)µ3(t-2)µ4(t-2)µ5(t-2) µ1(t-1)µ2(t-1)93(t-1) µ1(t) That is, there must be at least one neighborhood starting with µ1(t-1) which maps to µ1(t); there must be at least one five digit string starting with µ1(t-2) which maps to one of these neighborhoods, and etc. This situation can be expressed in terms of the matricies X(k,m) defined in Chapter 2. Recall that X(k,m) is a 2k+mx2m+1 matrix with ij element given by the j-th element of the complete Boolean product of Li(k,m;X). To formalize the backward reconstruction process for one digit time series, these matricies are partitioned into quadrants: X(k,m) = (PQO(m+ 1) P0,1(m+ l)1 p10(m+ 1) P1,1(m+ 1)) (11.2) In the case under consideration, the reconstruction 91(t-1412(t-1)43(t-1) µ1(t) will be possible for arbritrary µ1(t) and µ1(t-1) if and only if the 8x2 matrix

21 2

P0,0(1) P0,1(1) (p10(1) P1,1(1)

X(3,0) =

(11.3)

xg X6 X'7 X7

has non -zero entries in each quadrant pb,c(1). Likewise, the reconstruction µ 1(t-2)42(t-2)µ3(t-2)94(t-2)95(t-2) 91(t-1)µ2(t-1)µ3(t-1) will be possible for all µ1(t-1) and µ1(t-2) if and only if the 32x8 matrix X(3,2) = po,0(3) P0,1(3) (P1,0(3) p1,1(3) has at least one non-zero entry in each quadrant. By construction, if the 16x1 column matrix pa,b(3)Pb,c(1) has any entry equal to 1 then there is a five digit sequence aµ2(t-2)93(t-2)94(t-2)µ5(t-2) which maps to the three digit sequence bµ1(t-1)92(t-1), which in turn maps to c under the rule X. That is, the rule X can generate the time sequence a, b, c if and only if pa,b(3)Pb,c(1) # 0. The k-site generalization of this is given in the next theorem. Theorem 11.4 Let X:E+-4E+ be a k-site rule and let S(1,t) = {µ1(0),...,µ1(t)} be a single digit time series. Then X can generate S(1,t) if and only if the column matrix defined by the product Pµ(0),µ(1) ((k - 1)t - k + 2)Pµ(1),µ(2) ((k - 1)t - 2k + 3)... P4(t-1),µ(t) (1) 0 _ [IPµ(t-s-1),µ(t-s)((k-1)s+ 1) (11.4) s=t-1

has at least one non-zero entry. The binary form of the row index for nonzero entries of this matrix are the initial conditions for the time series. Since the above theorems give not only the possibility of a rule X generating a given time series, but also the initial conditions from which this series might have been generated, it allows comparison between different rules . That is, if X and X' are two rules both of which could have generated a time series , then the relative probabilities of these two rules as the actual

213

generators of the series can be taken as related to the relative numbers of initial conditions from which each could in fact have generated the series. For example , if X generates the series from a single initial condition, while X' could have generated the series from four distinct initial conditions , then all other things being equal the probabilities for X and X ' to have generated the series would be 20% and 80% respectively. For a time series S (n,t) with n < k, a similar result holds. Let a(s,n) represent the sequence 91(t-s )...µn(t-s), and define pa(s-1 ,n),s(s,n)( r) by a partition of the matrix X(k,r-1 ) into 22n blocks of equal size. Theorem 11.5 Let S(n,t) be as above , with terms denoted a(s,n). A k- site rule X will generate this series if and only if 0 X7 1 1Pa(s-1,n),a(s,n ) (( k - 1)s + n) # 0 (11.5) s=t-1 Further, the binary forms of the indices for rows having non -zero entries are

the initial conditions for the series. Since the p-matricies are defined in terms of the components of X, this gives an immediate test for whether or not a given rule can generate a given time series . If a rule is not given , on the other hand, it gives a set of necessary and sufficient constraints on the components of any rule which could possible generate a given time series. A simple argument utilizing the properties of matrix multiplication yields a count of the number of distinct time series which a given rule could generate. Theorem 11.6 The number of distinct time series S(n,t) beginning with a (t,n) and ending with a(0,n) is given by the number of 1's in the block of the product 0 11 X(k, (k - 1)s + n - 1) S=t-1

which has row indices with binary form starting with a(t,n). 2. Statistics of Time Series In analogue with the spatial measure entropy defined in equation (10.1) there is a temporal measure entropy defined for time series S(1,t) of length n by

214

Sn(X)=- 1 lim I pn(s) log2P. ( s) n t-*°° SEE n

(11.6)

where pn( s) is the probability of the length n sequence s occuring in the temporal output series S(1,t). An equivalent entropy can be defined for series S(n,t). In the following, however , only sequences S(1,t) will be considered. For these series, the sum of the column matricies defined in equation (11.4) gives the total number of initial configurations from which a rule X can generate the specified series. Division of this number by the total number of possible initial configurations gives the probability that a particular series will appear. The catch in this argument , however, is that some configurations may be forbidden for a rule X because they contain sequences from GE*(X). These configurations may appear only once , at the beginning , and thereafter will never appear in the later evolution of the rule . This can act to skew the naive probabilities computed from counts of the column entries in equation (11.4), and corrections will need to be made. An infinite sequence p is called infinitely distributed if for all n, every possible combination of n digits appears with equal probability 2-n. The probabilities computed from the products of equation ( 11.4) are based on the assumption that all spatial sequences are infinitely distributed at each step in the iteration of the rule X. As indicated , this assumption will be false if there are Garden-of-Eden configurations . On the other hand, if GE*(X) = 0 then a theorem of Hedlund [49] states that for all n, every n digit sequence has an equal number of pre -images. Since each pre-image will be an n+k-1 digit sequence , each such possible sequence must occur with equal frequency so if a sequence is infinitely distributed then its successor sequences, under a rule for which GE*(X) = 0 , will also be infinitely distributed. Theorem 11.7

Let µe E+ be infinitely distributed and let X:E+-*E+ be a k - site rule. Then for r > 0, Xr(t) is infinitely distributed if and only if GE *(X) = 0. Table 11 . 1 shows probabilities for 1 , 2, and 3 digit sequences in S(1,t) for several representative 3-site rules, computed from the matrix product introduced in Theorem 11.4. These are estimates of probabilities for the appearence of the given 1, 2, and 3 digit sequences as the specified rules evolve from an infinitely distributed initial condition.

215

Rule 18 .75 0 1 .25

22 .625 .375

30 .5 .5

54 .5 .5

90 .5 .5

108 .5 .5

110 .375 .625

114 .5 .5

120 .5 .5

150 .5 .5

178 .5 .5

.25 .25 .25 .25

.125 .375 .25 .25

.375 .125 .125 .375

.375 .125 .125 .375

.25 .25 .25 .25

.375 .125 .125 .375

.0312 .0938 .2188 .1562 .0625 .1875 .1562 .0938

.3125 .0625 0 .125 .0312 .0312 .0312 .3438

.3125 .0625 .0625 .0625 .0625 .0625 .0625 .3125

.125 .125 .125 .125 .125 .125 .125 .125

.2812 .0938 0 .125 .125 0 .0938 .2812

00 01 10 11

.375 .125 .375 .125

.25 .25 .375 .125

.125 .375 .375 .125

.25 .25 .25 .25

.25 .25 .25 .25

000 001 010 011 100 101 110 111

.3125 .0625 .0312 .0938 .3125 .0625 .0938 .0312

.1562 .0938 .1562 .0938 .25 .125 .0938 .0312

.0312 .0938 .2812 .0938 .0938 .2812 .0938 .0312

.0938 .1562 .1875 .0625 .125 .125 .125 .125

.125 .125 .125 .125 .125 .1875 .125 .0625 .125 .1875 .125 .0625 .125 .1875 .125 .0625 Table 11.1

Predicted Probabilities for 1, 2, and 3 digit Time Series for Some 3-Site Rules Based on Preservation of Randomness in Automata Evolution (Although listed horizontally, sequences are temporal outputs)

Table 11.2 gives probabilities computed empirically by evolution of sixteen different 1023 digit pseudo-random sequences for 511 iterations and averaged over the sixteen outputs. In both of these tables, the rules are taken as left justified. There are differences which arise if the mapping site is changed which will be discussed later in this chapter. Rule 0 1

18 .751 .249

22 .648 .352

30 .501 .499

54 .530 .470

90 .513 .487

108 .689 .311

110 .427 .573

114 120 .438* .511 .562* .489

150 .501 .499

178 .511 .489

00 01 10 11

.627 .126 .126 .122

.429 .219 .219 .133

.118 .383 .383 .116

.292 .239 .239 .229

.265 .247 .247 .241

.454 .236 .236 .075

.090 .337 .337 .235

.439* .000 .000 .561*

.390 .120 .120 .371

.252 .247 .247 .254

.425 .084 .084 .406

000 001 010 011 100 101 110 111

.560 .065 .063 .063 .065 .061 .063 .059

.295 .133 .155 .065 .133 .086 .063 .070

.027 .092 .294 .088 .092 .290 .088 .027

.239 .057 .208 .031 .055 .183 .031 .198

.136 .248 .010 .130 .206 .081 .120 .191 .187 .126 .045 .152 .130 .206 .079 .118 .029 .258 .126 .045 .152 .114 .029 .083 Table 11.2

.439* .000 .000 .000 .000 .000 .000 .561*

.332 .059 .059 .061 .059 .061 .059 .310

.128 .126 .120 .126 .126 .122 .126 .130

.341 .085 0 .085 .085 0 .085 .321

Empirical Probabilities for 1 , 2, and 3 digit Time Series for Some 3-Site Rules (Although listed horizontally, sequences are temporal outputs)

216 Comparison of these two tables gives some indication of the degree to which a rule fails to satisfy the criterion that each successive iteration preserve the random characteristic of the initial condition. As expected, empirical results for the two additive rules, 90 and 150, match the computed values of Table 11.1 closely, while the results for the non-linear rules do not. For the additive rules the expected temporal entropy for the three values of n used is equal to 1, which is maximal. For other rules it will be less than this. Table 11.3 gives the p-matrix estimates of entropies, and the entropies computed from the empirical probabilities of Table 11.2, together with the difference dS between these two numbers. 90 1.000 .998 .002

108 1.000 .894 .106

110 .955 .984 -.029

178 114 120 150 1.000 1.000 1.000 1.000 .989 1.000 1.000 1.000 0 0 0 .011

Rule 18 Si(p) .812 Si(e) .811 dS .001

22 .955 .936 .019

30 1.000 1.000 0

54 1.000 .997 .003

S2 (p) .906 S2 (e) .773 dS .133

.953 .936 .017

.906 .893 .013

1.000 1.000 1.000 .953 .996 .999 .890 .931 .004 .001 .110 .022

.906 .495 .411

.906 .897 .009

1.000 .906 1.000 .826 0 .080

S3 (p) .834 S3(e) .743 dS .101

.943 .930 .013

.874 .856 .018

.980 .883 .097

.716 .330 .386

.850 .836 .014

1.000 .807 1.000 .755 0 .052

1.000 .953 .938 .999 .864 .911 .001 .089 .027 Table 11.3

Sn(X) Entropies for Time Series for n = 1,2,3 (Sn(p) is from p-matricies, Sn(e) is empirical)

The values computed from the p-matrix method can be adjusted for those rules with GE* # 0 by excluding contributions from the row index in equation (11.4) which contain a sequence from GE*. For rule 18, for example, the first element of GE*(18) is 111. There are 24 of the row indices in the product Pa,b(3)Pb,c(l) which do not contain this sequence. When all other indicies are excluded the new probability estimates for 2 and 3 digit series for rule 18 are: 00 01 10 11 000 001 010 011 100 101 110 111 .542 .177 .167 .125 .292 .125 .042 .083 .25 .042 .125 .083 For greater accuracy it would be necessary to compute products involving longer initial sequences, excluding those containing a 111 block, and computing statistics over those which remain. Wolfram [281 has pointed out that for n digit sequences, substantially more than 2n samples need to be considered to obtain accurate estimates for length n block probabilities, those

217 estimates based on small samples will systematically underestimate the entropies computed.

The value S1(p) is just the percentage of l's in the rule table. This quantity is Langton' s ? parameter. Rule classes can be roughly defined by choosing values of ? between 0 and 1 /2. Maximum chaos in the space-time patterns generated tends to occur at A. = 1 /2. Class IV behavior in Wolfram's classification occurs as a phase transition between periodic class II and chaotic class III behaviors . In terms of temporal series, A. is just the probability that the symbol at time t in the series is a 1 , given an equal probability of k-site neighborhoods occuring at time t-1. The entries for rule 114 in Table 11 . 2 are starred because the empirical output from this rule is not well represented by the average values given in the table. In the actual output from this rule, either 0 , 00, and 000; or 1, 11, and 111 showed up almost exclusively . There were seven cases in which the value 0 appeared an average of 510 of 511 . times, the series 00 appeared an average of 508 of 510 times, and the series 000 appeared an average of 507 of 509 times, all other values being essentially 0. In the remaining 9 cases the value 1 appeared an average of 508 of 511 times, the series 11 appeared an average of 507 of 510 times, and the series 111 appeared an average of 506 of 509 times. In seven of the sixteen runs there were flips from 0 ' s to l 's. This gives an example of a bifurcation between two possible behaviors, with the evolution rule being very sensitive to the initial conditions choosen. This can be seen by consideration of the rule table. The rule table for rule 114 is 000 001 010 011 100 101 110 111 0 1 0 0 1 1 1 0 Thus, if the initial condition is a sequence starting with a 0 the only way that a 1 can appear in future iterations is if the neighborhood 001 appears , while if the initial sequence starts with a 1 then the only way that a 0 can appear in later iterations is if the neighborhood 111 appears first. Rule 178 gives an example of another behavior. Here the series 010 and 101 can never occur, as can be seen from the attempted backward reconstructions . The rule table for rule 178 is 000 001 010 011 100 101 110 111 0 1 0 0 1 1 0 1 so the attempted reconstruction goes as indicated below.

218 001#

110#

110

001

0 1 One point of interest relates to rule 30. The empirical and p-matrix generated probabilities for series match within accepted errors, which is to be expected since the Garden-of-Eden for rule 30 is empty. Examination of Tables 11.1 and 11.2 indicates that the probabilities for sequences to appear in S(l ,t) peaks about sequences of alternating 0's and 1's. It has been suggested, however, that this rule can serve as a random sequence generator [50]. Wolfram claims that the sequence obtained by taking successive time values at a fixed site, or at sites lying along any diagonal (i.e., gi(t) or Ri±t(t)) through the spacetime pattern generated by iteration of this rule will be random . The difference is that Wolfram uses a nearest neighbor realization of rule 30 while that used here is left justified. The rule table for rule 30 is 000 001 010 011 100 101 110 ill 0 1 1 1 1 0 0 0 In the left justified form of the rule, the probability that a 0 value of gl (t) will be followed by a 1 value of µi(t+1) is .75, as is the probability that a 1 value will be followed by a 0 value . In nearest neighbor form, however, each of these probabilities is .5. Likewise, computation of the products pa,b(3 )Pb,c(1) indicates that in nearest neighbor form the probabilities of all three digit series are equal to .125. Since these two forms of the rule are related by a shift, the sequence gl(t) in this paper becomes the diagonal sequence gl+t(t) in nearest neighbor form. Right diagonal sequences in the nearest neighbor form, then, will have sequence distributions peaked about alternating sequences of 0's and l's, while nearest neighbor vertical series can be expected to be random. 3. Exercises for Chapter 11 1. Use backward reconstruction to determine which of the following rules could have generated the time series 10, 01, 00 , 10, 00: a) Rule 18 b) Rule 22 c) Rule 54 d) Rule 110

219

2. Find a 2 digit time sequence which could not have been generated by rule 30. 3. Use Theorem 11.5 to prove that no 3-site rule having xg = x7 = 0 can generate a 2 digit time series containing the sequence 11, 11. 4. Find the initial conditions from which the following rules generate the single digit time series 1, 0, 0, 1. a) Rule 54 b) Rule 18 c) Rule 22 d) Rule 110 5. Let X be a k-site rule which is linear in the k-1 variable. CanX generate all k-2 digit time series? Why?

220

Chapter 12 Surjectivity of Cellular Automata Rules A CA rule is surjective if every configuration has a predecessor. There are a number of important properties of CAs which are equivalent to surjectivity, a number of which are listed in a "kitchen sink" theorem in section 1. There are also graph theoretic techniques for the study of surjectivity, centered on the de Bruijn diagram which is defined in section 2. Another diagram, derived from the de Bruijn diagram, is the subset diagram. Both diagrams provide surjectivity criteria. Analysis of the adjacency matrix for the de Bruijn diagram allows definition of a semi-group associated to each CA rule, and several theorems on the structure of this semi-group are proved. Finally, a replacement diagram is associated to each CA rule. 1. A Kitchen Sink Theorem As defined in Chapter 10, a CA rule is surjective if every configuration has a predecessor, and injective if this predecessor is unique. The interest in surjectivity was first stimulated by Moore [82], who introduced the idea of Garden-of-Eden configurations. As indicated already, a rule is surjective if and only if it has an empty Garden-of-Eden. In Chapters 9 and 10 the Garden-of-Eden set GE(X) was defined in terms of the set of seeds GE*(X), and lemma 9.1 showed that a configuration is in GE(X) if and only if it contains at least one sequence from GE*(X). Whether or not specific rules are surjective is a major question. No general test for surjectivity is known, and for dimensions greater than 1, no such test can exist. This is a consequence of the following theorem proved by Kari [68] on the basis of the undecidability of tilings of the plane. Theorem 12.1 (Kari, [681) For any dimension d >_ 2 it is undecidable whether or not an arbritrary d-dimensional CA rule is injective, or surjective. For dimension 1 there are proceedures for determining whether or not specified rules are surjective. Finding an effective proceedure for all 1 dimensional rules, however, remains an open problem, and forms the focus of interest in this chapter. Theorem 12.2 (Amoroso & Patt, [831) Let X be a 1-dimensional k-site CA rule. Then the injectivity and surjectivity of X are decidable.

221

The classic study of surjectivity is contained in the paper by Hedlund [49]. Several of the results of this paper are synthesized in the following three theorems: Theorem 12.3 Let X be a k-site CA rule . Xis surjective if and only if every finite sequence sl...sn has exactly 2k-1 pre-images, and every infinite configuration has at most 2k-1 predecessors. Corollary 12.4 If a rule Xis surjective then it has an equal number of 0's and 1 's in its rule table. Proof Every finite sequence has exactly 2k-1 pre -images. In particular, therefore, 0 and 1 each have 2k- 1 pre-images, and these are the k-site neighborhoods. I Theorem 12.5 Let X be a k-site surjective CA rule . If gE E+ is periodic or eventually periodic, then all predecessors of.t under X are also periodic or eventually periodic. Proof Suppose to the contrary that X is surjective, and that B has period n in E+, but that B has a non-periodic predecessor g. But arn(B) = 13 for all r >_ 1. Since g is not periodic, however, arn(µ) * asn(g) for any r,s > 0, r,*s , otherwise it would be the case that a(r-s)n ( g) = g and g would be periodic . For all r >_ 1, however, X(arn(g)) = arnX(g) = am(13) = B . Thus B must have at least a countable infinity of predecessors , in contradiction to Theorem 12.3, and the claim is demonstrated. I Theorem 12.6 _ 1 and suppose that X:E+_3E+ is not surjective. Then there Let n > exists a finite sequence sl...sr having more than n pre -images . There also exist spatially periodic configurations having an uncountable number of predecessors. In terms of the map X :[0,1]-9[0,1] periodic and eventually periodic configurations correspond to rationals . Since the rationals are countable, Theorem 12 . 6 implies that if X is not surjective then there will be rationals in [0,1] having uncountably many irrational predecessors . If X is surjective, on the other hand, Theorem 12.5 requires that all rationals have only a finite

222 number of rational predecessors. Use of this in a contrapositive argument proves the next theorem. Theorem 12.7 X:E+-E+ is surjective if and only if X:[0,1]-[0,1] maps rationals to rationals and irrationals to irrationals. The next lemma shows that surjectivity is preserved under composition of rules. Lemma 12.8 Let X and Y be CA rules defined on E+. The composition XY is surjective if and only if X and Y are each surjective. Proof Clearly XY will be surjective if X and Y are surjective. Suppose then that XY is surjective, but X is not. Then there will be a configuration B which does not have a predecessor under X. But it does have one under XY, say g. Thus XY(µ) = B, hence Y(µ) is a predecessor to B under X, contradicting the assumption. Suppose that Y is not surjective. By theorem 12.5 there will be a periodic configuration, say B, with an uncountable number of predecessors under Y. Let these predecessors be denoted µ(s). Then XY(t(s)) = X(B), hence the configuration X(B) has uncountably many predecessors under XY, but this cannot be since by theorem 12.3 and the surjectivity of XY this rule can have at most 2k+r-1 predecessors where Xis a k-site rule and Y is an r-site rule. I The spatial measure entropy of a rule X was defined for sequences of length n in equation (10.1), and was related to the fractal dimension of the Cantor set d(X) defined by X in equation (10.2). The topological entropies for sequences g are defined by [28]: 1 2n St (n; t)=-nlog2 H(pn(i;µ)) i-1

(12.1)

where pn(i;µ) is the probability of occurence in g of the i-th length n sequence and O H(p)={0 Pp=0 Comparison of equations (10.1) and (12.1) indicate that the topological and measure entropies satisfy the inequality Sn(X) S St(n;X(-t)) < 1 (12.2)

223 Since elements of GE*(X) have length greater than or equal to some critical length nc the topological entropy will be 1 for values of n less than this critical length. If a configuration µ is infinitely distributed and Xis a k-site surjective rule then, as indicated by Theorem 11.7, the configuration X(µ) will also be infinitely distributed since each n digit sequence in X(µ) has 2k-1 pre-images, and each of the 2n+k-1 digit pre-images occurs in g with equal probability. Thus Sn(X) = 1 if X is surjective. In fact, if g is infinitely distributed and Sn(X(4)) = 1 then X is surjective since maximum spatial measure entropy requires that all probabilities be equal, implying that for all n, all length n sequences have the same number of pre-images. In addition, if Sn(X(g)) = 1 for all n, then by equation (10.2) the image of X has dimension d(X) = 1. The temporal probabilities for occurence of length n series were discussed in Chapter 11, and their temporal measure entropy is defined in equation (11.6). It was pointed out that the value of this entropy for a rule X, as estimated via the p-matrix method, would be expected to correspond to the empirically determined entropy if and only if GE*(X) = 0. This follows since the p-matrix technique assumes that the property of infinite distribution is preserved under interation of X, and this is true if and only if X is surjective. The various properties equivalent to surjectivity which have been discussed can be combined into a "kitchen sink" theorem, although it must be understood that this theorem makes no claim to be exhaustive. Theorem 12.9 Let X be a k-site CA rule. Then the following statements are equivalent: 1. X is surjective. 2. GE*(X) = GE(X) = 0. 3. Every finite sequence sl...sn has exactly 2k-1 pre-images. 4. Every element of E+ has at most 2k-1 predecessors. 5. X maps eventually periodic configurations to eventually periodic configurations, and non-periodic configurations to non-periodic configurations. 6. X:[0,1]-x[0,1] maps rationals to rationale and irrationals to irrationals. 7. X has maximal spatial measure entropy (Sn(X) = 1 for all n). 8. The Cantor set defined by X has dimension d(X) = 1.

224 9. X preserves the infinite distribution property of configurations. 10. The p-matrix method is an accurate estimator for temporal measure entropies. In Chapter 6 it was pointed out that while all non -injective rules defined on En led to a decrease in entropy , a similar result was not true for some of these rules when defined on E+. Indeed, the question might arise as to why any rule can have maximum entropy since it is known that periodic configurations map to fixed points or cycles . For a finite configuration space this will necessairly lead to a reduction in entropy unless a rule is injective. For the configuration space E +, however, the situation is different . Property 6 in theorem 12.9 provides the explanation for the entropy reducing properties of CA rules . The set of eventually periodic configurations, corresponding to the set of rationals in [0,1], has measure 0 with respect to the full configuration space . Hence, if a rule is surjective , it has an entropy reducing effect only on a set of measure 0 . On the otherhand, if a rule is not surjective, then by Theorem 12.6 there will be rational elements of [0,11 having uncountably many irrational predecessors . In these cases, entropy reduction is only to be expected. 2. The de Bruijn Diagram Applications of graph theory have played a major role in much of the research on cellular automata . For present purposes the two graphs which are important are the de Bruijn diagram and the subset diagram . Reviews detailing many of the possible applications of these, and other graphs to the study of cellular automata have been published by McIntosh [38,39,84]. The de Bruijn diagram plays a central role in Jen 's work on computation of preimages [34,85], and in Wolframs studies of the relation between cellular automata and formal languages [29]. These diagrams can be considered one of the essential tools of the trade. For a k- site CA rule X, the de Bruijn diagram is defined as a labeled directed graph with 2k- 1 verticies and 2k edges . The verticies are labeled with the binary digits of the integers ranging from 0 to 2k-1-1. An edge is directed from vertex i = i0...ik-2 to vertex j = j0•••jk - 2 if and only ifjs = is+1 for all s such that 0:5 s:5 k-3 . That is , if and only if these two digit strings can be overlapped to form a k-site neighborhood, which will be denoted i•j. When

225 this is the case, the edge connecting vertex i to vertex j is labeled with the value X(i *j). The de Bruijn diagram without edge labels tells which k-1 digit strings can be overlapped to form a k-site neighborhood. When the edge labels are included it gives the same information about a rule as does the rule table. As will be seen, however, it also provides a means of continuing the analysis of a rule, unpacking the information in the rule table. Figure 12.1 shows the de Bruijn diagram for the generic 3-site rule, labeled in terms of components. An immediate property of this diagram is that it allows determination of pre- images. KO

00

7 Figure 12.1 Generic de Bruijn Diagram for 3-Site Rules

If sl...sn is any binary sequence, the pre-images for this sequence under a rule X are found by finding all length n paths in the de Bruijn diagram for X which have edge labels sl...sn, and then constructing the length n+k-1 sequences generated by overlapping the k-1 site blocks which label the verticies of these paths. If no such path exists, then sl...sn has no pre-image [38,391. For example, if X is the 3-site rule 30 and the sequence 10011 is given, the pre-images of this sequence can be read off from the de Bruijn diagram of Figure 12.2.

226 Examination of this figure shows that there are four paths with the label 10011. These connect the verticies listed, together with the pre- images constructed from their overlap, in Table 12.1. Connected Verticies

Pre-Image

10, 00, 00, 00, 01, 10 10, 00, 00, 00, 01, 11

1000011

01, 11, 11, 10, 00, 01

0111001

01, 11, 10, 01, 10, 00

0110100

1000010

Table 12.1 Pre-Images for 10011 Under Rule 30

0

11

0 Figure 12.2 de Bruijn Diagram for Rule 30

In Definition 9.3 linearity in the r-th variable was introduced, and the particular cases of linearity in the initial variable (LW), and linearity in the final variable (LFV) were distinguished, and a rule possessing one or the other of these properties was called linear in an extreme variable (LEV). It will turn out that these properties are closely related to surjectivity. Lemma 12.10 Let X be a k-site CA rule. Then:

227

1. If X is LIV the incoming edges at each vertex in the de Bruijn diagram for X have distinct labels. 2. If X is LFV the out going edges at each vertex of the de Bruijn diagram for X have distinct labels. Proof If X is LIV then, by Lemma 9.4 (equation (9.3)) xi+2k_1 = 1 + xi for all i between 0 and 2k- 1-1. But each vertex il...ik-2 of the de Bruijn diagram for X has incoming edges from verticies OiO...ik -3 and 1i0 ...ik-3, and hence overlaps with these verticies to form the neighborhoods 0i0...ik -2 and 1iO ...ik-2 which map to distinct symbols under X. Likewise, if X is LFV then x2i+1 = 1 + x2i for 0 n, but are not the global map of any CA rule with k = n. Proof Let jr denote a block of r 1's. Then , for r _> 2, let s = 1r00 , s' = 0jr-k00, and define an r+2 site rule X such that for i = i0...ir+1 0 i=s X(i) = 1 i = s'

ip otherwise

233 With this definition, for all µe E+, X2(µ) = g, hence X is injective. If X is defined for more than 3-sites, it need not be LEV. Further, if X is defined for 4 sites then, if it were composed it would have to be a composition of a 2site and a 3-site rule, but it is easy to check that this is not the case. I The proof of this theorem is accomplished by exhibiting rules which satisfy the desired conditions. The way in which these rules were constructed, however, leads to the possiblity of other constructions as well. For example, in the proof given, the rules produced were taken as left justified. They could equally well be taken as defined for other mapping sites , and in fact Amoroso & Patt defined them for i 1 as the mapping site. Of the eight injective 4-site rules which are prime but not LEV, four are non-generative. The component expressions for these rules are given in Table 12.2. (0010110100001111) (0011100100110011) (0011001101100011) (0000111101001011) Table 12.2 Component Forms for Prime 4-Site non-LEV Injective Rules

The determination of exactly what conditions the components of a CA rule must satisfy in order for the rule to be surjective remains an outstanding problem. The matricies d0 and d1 can also be used to construct the subset diagram, a construction which McIntosh calls the vector subset diagram [38,391. As in Figure 12.3, let each of the verticies of the de Bruijn diagram for a rule X be taken as a coordinate for a binary vector in 2k-1 dimensions. Then each of the possible 22k-1 vectors c in this space corresponds uniquely to a subset C of the full subset via (0 vertex i e C

C: =

Sl 1 vertex i 4< C The pruned subset diagram is then generated by the following algorithm:

Aleorithm 12.20 1. Let c(1) be the vector with all 1 components. Take c(1) as the initial vertex.

234 2. Compute c(1)d0 and c ( 1)dl. Set all terms greater than 1 equal to 1. Take these products as the next two verticies , with an edge labeled 0 directed from c(1) to c(1)d0 and an edge labeled 1 directed from c( 1) to c(1)dl. 3. Repeat this proceedure , using c( 1)d0 and c( 1)dl as the new initial vectors . Continue until no new vectors are generated. 4. The vector c(0) consisting of all 0 entries corresponds to the empty subset. GE*(X) is empty if and only if the vector c(0) is not generated in this proceedure. This is the method used to generate the subset diagram for rule 54 given in Figure 12 . 3. Note that if a rule Xis LIV then by Lemma 12.17 c(1)d0 (X) = c(1)dl(X) = c(1), hence the pruned subset diagram consists of only a single vertex, c(1), with two loops. If a rule is not surjective then the subset diagram provides yet another means of computing GE*. If M is the adjacency matrix of the subset diagram (either the full diagram , or the pruned diagram will do) then the ij entry of Mn counts the number of length n paths which start at vertex i and end at vertex j . In particular, the entry with the label c(1),c(0 ) will count the number of length n paths which begin at c(1 ) and terminate at c(0), and the labeling of these paths gives all length n sequences without pre -images. The edges in the subset diagram as defined are labeled with either 0 or 1, but they could equally well be labeled with d0 and dl . With this labeling, the entries in the adjacency matrix of this diagram will be 0, d0, and dl. For example , for rule 54 this form of the matrix M(54) is given in Figure 12.4. do + d1

0

0

0

0

0

0

0

0

0 0

d1

0

0

do

0

0

0

0

0

0 0

do

0

0

0

0

0

0

0

d1

0 0

0

0

0

do

0

0

0

0

d1

0 0

0

do

d1

0

0

0

0

0

0

0 0

0

do

0

0

0

0

0

0

0

d1 0

0

0

0

0

d1

0

0

do

0

0 0 0 0

0

0

0

0

0

0

0

do

d1

0

0

0

0

0

d1

do

0

0

0 0

0

0

0

0

0

0

do

0

0

d1 0

0

0

0

0

0

0

0

do

0

d1 0

235 In this form the c( 1),c(0) entry of Mn not only counts the number of length n sequences without pre -images, but lists them, as products of d0 and dl. In this way, the difference between the cases in which GE* is finite or infinite is seen to depend on the way in which the matrix M behaves when raised to powers . If after some finite n the c(1),c(0 ) element always contains d0 and d1 products which have already appeared for lower values of n, then GE* will be finite. In the rule 54 example, computation yields the first non-zero [Mn(54 )lc(1),c(0) as

[M5(54) lc(1),c(0 ) = dldpoldpd1 + d1dpdidp +dpdidpd1 indicating that the first entries in GE* (54) are the five digit sequences 10101, 10110, and 01101 . No new seeds appear until n = 8, where the sequence 10111101 shows up, after which no further seeds appear. 4. The Semi-Group OX) The free semi-group generated by d0(X) and di(X) was introduced in Theorem 12.15. In this section some of the properties of this semi -group are determined. Theorem 12.21 G(X) is a group if and only if Xis both LIV and LFV. Proof If X is both LIV and LFV then by Theorem 12.12(1) G(X) is a permutation group. The proof in the other direction will proceed by contrapositive. Suppose that Xis not LW. Then, by Theorem 12.12(2) both d0 and di will have at least one row of zeros, say r0 and rl . Further, these cannot be the same row since d0+dl = d. Assume now that G(X) is a group. Then it will contain an identity element e, and all elements will have inverses . In particular, d0 and dl will have inverses . Let v0 be a row vector which has a 1 in the r0 position only, and vl a row vector with a 1 only in the rl position . Then v0 (d0d-10) =0 since vodo = 0 . Likewise , vl(dld- 1l) = 0. Thus, v0(edl ) = vl(ed0 ) = 0. But these last products cannot be zero since d 1 must have a pair of l 's in row r0, and d0 must have a pair of l 's in row 1. Hence not all elements of G(X) can have inverses , and G(X) cannot be a group. Likewise , if X is not LFV then each of

236 d0 and dl will have a column of 0's, and these cannot be the same column, so a similar argument applies to show again that G(X) cannot be a group. I As might be expected, the semi-groups defined by rules which are related by the transformations Ti, T2, and T3 are also closely related. The action of these transformations on a rule X can be expressed in terms of their action on the components of X--through their actions on neighborhoods in the case of Ti and T2, and in terms of complimentation in the case of T3. Thus [Tl(X)]j = xT10), [T2(X)]j = xT2(j), [T3(X)]j = 1+xj. Proof of the next theorem follows from the observation that if j and m are k-1 site partial neighborhoods which can be overlapped to form a k-site neighborhood then T1(j ♦ m) = T1(m),T1(j) and T2(j ♦ m) = T2(j),T2(m). Theorem 12.22 Let X be a k-site CA rule. The groups G(X), G(Tl(X)), G(T2(X)), and G(T3(X)) are isomorphic. The isomorphisms relating these groups are defined in terms of their generators by: [ds(Tl(X)]jm = [ds(X)1T1(m),T1(j) [ds(T2(X)]jm = [ds(X)1T2(j),T2(m) [ds(T3(X)Ijm = [ds'(X))]jm s' = 1+s mod(2) Appendix 6 lists the semi-group G(X) for some rules of interest. 5. The Subset Matrix and Some Replacement Diagrams In construction of the subset diagram via Algorithm 12.20, the d0 and dl matricies were multiplied on the left by vectors c which described subsets of the full subset of de Bruijn verticies. In addition, the "topological set" condition that terms greater than 1 be set to 1 was imposed. If the i-th component of the product cds is non-zero it means that there is an edge labeled s from one of the verticies in the subset characterized by c to vertex i. Similarly, if the i-th component of n

c[Tdsj j=1

is non-zero it means that there is a path labeled sl...sn from at least one of the verticies in the subset characterized by c to vertex i. The matricies d0 and dl can also be used to multiply column vectors on the right. If c is defined in the same way as before, but now as a column

237 vector, then if the i-th component of dsc is non-zero it means that there is an edge in the de Bruijn diagram labeled s from vertex i to one of the verticies in the subset characterized by c. Without the topological set condition, the i-th component of n r1dsj j=1

c

(12.4)

counts the number of length n paths labeled sn...sl which originate at vertex i and terminate at a vertex in the subset characterized by c. (Note the reverse order of indices!) Thus, if c is taken as a unit vector with r component equal to 1 and all other components equal to 0, then the product in (12.4) counts the number of length n paths labeled sn...s1 which originate at vertex i and terminate at vertex r. In addition, by a process similar to that of Algorithm 12.20, a diagram is generated which in general will be infinite, but which can, at least in some cases , be characterized by a relatively simple recurrence diagram, which shown how CA rules can generate specific number sequences such as the Fibonacci sequence. The following examples will serve to illustrate the idea: 1. For the 2-site binary difference rule D = (01110) do(D)=I0 1 d1(D)=C1 Ol 0) 0 1 The diagram generated, taking c = 10 as the starting point, is shown in Figure 12.4.

d 10 01 d1 Figure 12.4 2. For rule 90 , S = (01011010) 1 0 0 0 0 1 0 0 do(8) = 0 0 O1 00 d1(8) = 0 1 00 0 0 0 0 1 0 0 1 0 and the diagram obtained is shown in Figure 12.5.

238

0010

d1

1000

Figure 12.5 In both of these first two examples the diagrams are compact. There are no positive feedback loops and no numbers larger than 1 appear in any of the vertex nodes . Thus, the number of paths connecting pairs of verticies will be a constant, independent of the path length. Note also that only unit vectors appear as vertex labels in these diagrams, a consequence of the fact that the d0 and dl matricies are permutation matricies. The next set of examples consider more general cases, in which the rules considered are not additive, and GE* is not empty. In these examples the diagrams will be infinite, but it will be seen that they can be easily represented by finite replacement diagrams. That is, diagrams which start with the vector c = (abcd), with variable components, and indicate how these components map at each multiplication by d0 or dl. The replacement diagrams for the rules given in the two previous examples represent simple permutations. For the rules D and 8 respectively, these are a-^b b->d

a-a a b^c c->b

C->a C1-4C

d-,d

D

S

3. Let X be the 2-site rule (0111). Then

do (X) _ (0 0) d1(X) _

(0

1

These matricies generate the infinite diagram which is shown in Figure 12.6.

239

d0 01 d^ 00 'd1l 11 d

d 23 -20

.d1 35 >30 d1i

Figure 12.6 Although infinite, inspection shows that this diagram can be represented by the finite recurrence diagram with replacements shown below. C11

4. For the 3-site rule 18 (1 0 0 0 0 0 1 11

dp(18)=

I0

1 00

0 0

1 1

d1(18) =

0

1 0 0

0

0

0 0

1 0 0 0 0

0 0 0

240 The diagram generated with these matricies from an initial 0001 is shown in Figure 12.7.

d1 10000

d0

0101

d1

d1 ->1000

0010

d0 J d1- >220000 d j; 0020 d0`J 0323

dl 3000 d

d0 0535

d0^

0030

d0o 5000 d

d

0050

d Figure 12.7

Again the diagram is non-compact, and can be seen to be generating Fibonacci sequences in its output. The numerical relations which it contains are generated by the recurrence diagram

d0 b-+s

241 In reading off pre-image numbers from this diagram the additional information dO(000a) = (OaOa), dl(000a) = (0000), dO(Oa00) = (00a0), and d1(OaO0) = (a000) is useful. As an example, suppose that the sequence 000110001 is given. The number of paths with this label in the de Bruijn diagram, which terminate at 00 is determined by starting at a000 with the initial condition a = 1, and following the arrows labeled in the order 100011000 (recall the reverse ordering in equation (12.4)), making the indicated substitutions along the way. This gives 1000-0010--)0101->0111-0212---)2000->0020-)0202->0222-->0424 indicating that there are four paths labeled 000110001 which start at 01 and terminate at 00, four which start at 11, and two which start at 10 and terminate at 00. The replacement diagrams can most compactly be specified in terms of the mappings of the full subset vector, represented abstractly for 3- site rules as abcd. For rule 18, the diagram becomes b -^cf dd0 c-,b d-+ cid

abcd

d1a-4b ^b-40 C -4 a

d-40

Likewise, for the 2-site rule of example 3, the diagram is d0_^, A-d1 a-4b b -^ O b-a+b With c( 1) the vector consisting of all 1 entries, the i-th component of the product n

fj dsj c(1) j=1

counts the number of length n paths labeled sn...sl in the de Bruijn diagram which start at vertex i = iO...ik-2. This counts the number of pre-images of the sequence sn...sl which start with iO...ik-2. This is the number LS (n; X) used in the Jen recurrence relations for numbers of pre-images [33,34]. Thus the total number of pre-images of sn...sl is given by the expression 2k-1 -1

N(sn...si ) =

E

[[ ft ds.]c(1)]

i=0 j=1

(12.5)

242 which gives a connection of the replacement diagram to Jens recurrence relations. 5. In the rule 18 example, the number series which were generated satisfied Fibonacci recurrence relations . Rule 12 provides an example in which the number series is a simple counting relation. For rule 12

(1 1 0 0' (0 0 0 0' d1(12) =

d0(12)= 0 0 0 0 0 0 1 1

0 0 1 1 0 0 0 0 0 0 0 0

generating the diagram of Figure 12.8.

1000 d1 0000 ----

1010^•'0100,• d1.

10 1 d 1 ..

d1

0200 ,.

d>2020 d-1 } 0400

0300 , •"0d>3030 -}>

2022

Figure 12.8 The replacement diagram for this rule is CIO dl b-+0 abcdb->cid c -* a+b c -00 C1-+ c 4d d ->0 In the first two of these examples, the number of paths determined from the diagrams generated remained constant. In the remaining examples this number grew with path length as the result of the appearence of a positive feedback loop in the diagram. The disadvantage of the compact form of the replacement diagram is that it does not make the presence of feedback loops manifestly obvious, while the presence of such loops is directly related to surjectivity since all finite sequences have the same number of pre -images for surjective rules . On the other hand, the compact form does present an

243

iterated replacement scheme on 2k-1 letters which is easy to implement on a computer. Theorem 12.23 Let X be a k-site CA rule . Xis surjective if and only if a positive feedback loop does not appear in the replacement diagram for X. Proof If a positive feedback loop appears then the number of paths with a given label will grow with increasing path length. It will always be possible to choose a path long enough that this number exceeds 2k-1 so by Theorem 12.3, X cannot be surjective. Suppose that Xis not surjective. Then , by Theorem 12.6, for any given n there will be a finite sequence having more than n pre -images . This can only be the case if there are more than n paths with this sequence as label in the de Bruijn diagram for X and hence the number of paths increases with path length for at least some paths . But this can only occur if there is a positive feedback loop in the dsc diagram. I 6. Exercises for Chapter 12 1. Prove that LIV and LFV are sufficient conditions for the matrix products VT(X)V(X), and V(X)VT(X) respectively, to be diagonal. 2. Find a prime 5 -site injective rule which is not LEV. 3. Construct the de Bruijn diagrams for the following rules: a) (01011000) b) (01111100) c) (0010110100001111) d) (01001000) e) (01011010) 4. Use the de Bruijn diagrams to find all pre-images for the following sequences for each rule in exercise 3: a) 0110 b) 011100 c) 1001011 d) 010101 e) 101001000 5. Use Theorem 12.13 to compute the number of pre -images for each of the sequences in exercise 4, under each rule from exercise 3.

244 6. Construct the pruned subset diagram for each rule in exercise 3. 7. Refering to Figure 12.3, find all 6 digit sequences without pre-image under rule 54. 8. Construct the replacement diagrams for the following 3-site rules: a) Rule 36 b) Rule 164 c) Rule 108 d) Rule 22 e) Rule 178 9. Compute the number of pre-images of each of the following sequences for each rule in exercise 8: a) 10 b) 11111 c) 011011 d) 01010111 e) 10110110101 10. For rule 18 prove that ldo+1(18)d1(18 )]0 = Fn+2 where Fn is the n-th Fibonacci number (FO = F1 = 0, F2 = 1).

245

Appendix 1 Boolean Expressions for 2 and 3-Site Rules Expressions are given only for rules with x0 = 0. The Boolean expression for a rule (1xl...x2k_1) is obtained by adding 1 to the expression for the rule (0x1 ...x2k_1). The rule is applied to variables x,y (2-site) or x,y,z (3site ). All sums are mod(2). The Boolean expressions in these tables are given in terms of the AND and XOR (exclusive or) operations. They differ from the Boolean entries given by Wolfram in his Table 1, pages 516-521 of Cellular Automata and Complexity since he uses the AND and OR operations. 1. Two Site Rules: Boolean Expression 0 xy xy X,

Rule 0000 0001 0010 0011 0100 0101 0110 0111

xy

y x+y x+ +x

2. Three Site Rules: Rule

A

Y

00000000 00000001 00000010 00000011 00000100 00000101 00000110 00000111 00001000 00001001 00001010 00001011 00001100 00001101 00001110 00001111 00010000 00010001 00010010 00010011 00010100

0000 0001 0001 0000 0010 0011 0011 0010 0010 0011 0011 0010 0000 0001 0001 0000 0100 0101 0101 0100 0110

0000 0000 0001 0001 0000 0000 0001 0001 0010 0010 0011 0011 0010 0010 0011 0011 0000 0000 0001 0001 0000

Boolean Expression 0 xyz xyz' xy xy'z xz x(y+z) x(y+z+yz) xyz' x(1+y+z) xz' x(1+y'z) xy' x(1+yz') x(l+yz) x x'yz yz y(x+z) y(x+z+xz) z(x+ )

z( ,x+T) z A Ax+z,Ax Ax+z(A+x+T) z(A+x+T) zA x xx+^+x zAx+Ax+A+x zAx+A+x

A+x zAx+,zx+A,x ,zx+Ax A+,zx zAx+A+x z,Ax+A zx+A zx+Ax z,Ax+, x A zAx+A

zAx+A,x Ax zAx+,zA+A ,zA+Ax zA+x zd x+A+x Ax+,z(A+x) z(Ax+A+x) zAx+,z(A+x) z(A+x) zA+zx z,Ax+zA zx+z^X

zA+zx+Ax A + (z x+T )A zA (z+x)A

,z,^x zA,x+x zA+x zA+Ax ;fx+zAx

zA+ zx zAx+ ,Ax z^fx+(z+^f+T)x zA+(z+X+T )x zA+zx+Ax zAx+zA+zx+Ax (Ax+A+x)z

TTTTT000 01111000 TOT1T000 00111000 11011000 OTOTT000 TOOTT000 OOOTT000 TTTOT000 OTT01000 TOTOT000

0010 TOT0 1010 OOTO OTTO TITO TITO OTTO OTTO TITO TTTO

TT00 TT00 OT00 OT00 TT00 TT00 OT00 0100 T000 T000 0000 ,

00000100

OOTO

OOTO

TOTOOOTO OOTOOOTO 110000TO 010000TO 100000TO 000000TO TTTTTTOO 01111100 TOT1T100 OOTTTTOO TTOTTTOO 0101TT00 TOOI1100 OOOITT00 TTTOTT00 01101100 10101100 OOTOTT00 TTOOTT00 OT001T00 10001100 OOOOTT00 TTTTOT00 OTTIOT00 10110100 OOTTOT00 11010100 OTOTOT00 TOOTOT00 00010100 11100100 OT100T00 TOTOOT00 00100100 11000100 OTOOOT00 TOOOOT00

TTOT OTOT OOOT TOOT TOOT OOOT 0000 T 000 T000 0000 OT00 TTOO TT00 OT00 OT00 TT00 TT00 OT00 0000 T000 1 000 0000 OOTO OOTO TOT0 OOTO 0110 TITO TI10 OTTO OTTO T1TO TITO OTT O OOT O T010 TOT0

0000 0000 T000 T000 0000 0000 TTTO TTTO OTTO OTTO TITO TTTO OTTO OTTO TOT0 TOT0 OOTO OOTO TOT0 TOT0 0010 OOTO TITO TITO OTTO OTTO 1110 TITO OTTO OTTO TOM TOT0 OOTO OOT 0 TOT O TOT0 0010

9bZ

247 (1+x'y)z+xy y'z+xy (x+z)y' xy'+(l+x'y)z x+(1+x'y)z x+y'z xy'+x'y'z xy'+(1+x+y)z x+(1+x+y)z x+x'y'z x'z ( 1+xy')z xy+(l+xy')z xy+x'z ( 1+xy)z z xy+z z+xyz' xy'+(l+xy)z xy'+z x+z x+z+xyz xy'+x'z z+xy'z' x+(1+xy')z x+z+xz x'(y+z) x y+(l+xy')z y+(1+xy')z y+x'z x y+(l+xy)z x'y+z y+z y+(1+xy)z x+y+(1+xy)z x+y+z x+y+z+xy x+y+z+xyz' x+y+z+xz x+y+z+xy'z x+y+z+xz+xyz' x+y+z+xy+xz x'y+x'y'z x y+(l+x+y)z y+(1+x+y)z y+x'y'z x y+y'z

01000110 01000111 01001000 01001001 01001010 01001011 01001100 01001101 01001110 01001111 01010000 01010001 01010010 01010011 01010100 01010101 01010110 01010111 01011000 01011001 01011010 01011011 01011100 01011101 01011110 01011111 01100000 01100001 01100010 01100011 01100100 01100101 01100110 01100111 01101000 01101001 01101010 01101011 01101100 01101101 01101110 01101111 01110000 01110001 01110010 01110011 01110100

1011 1010 1010 1011 1011 1010 1000 1001 1001 1000 1100 1101 1101 1100 1110 1111 1111 1110 1110 1111 1111 1110 1100 1101 1101 1100 1100 1101 1101 1100 1110 1111 1111 1110 1110 1111 1111 1110 1100 1101 1101 1100 1000 1001 1001 1000 1010

0001 0001 0010 0010 0011 0011 0010 0010 0011 0011 0000 0000 0001 0001 0000 0000 0001 0001 0010 0010 0011 0011 0010 0010 0011 0011 0100 0100 0101 0101 0100 0100 0101 0101 0110 0110 0111 0111 0110 0110 0111 0111 0100 0100 0101 0101 0100

01110101

1011

0100

z+x'yz'

01110110

1011

0101

y+(1+x' )z

248 01110111 01111000 01111001 01111010 01111011 01111100 01111101 01111110 01111111

1010 1010 1011 1011 1010 1000 1001 1001 1000

0101 0110 0110 0111 0111 0110 0110 0111 0111

y+y'z x+y+y'z x+y+(1+x'y)z y+(x+z)y'+xyz y+(x+z)y' x+y+x'y'z z+(x+y)z' x+y+z+xy+xz+yz x+y+z+xy+xz+yz+ xyz

249

Appendix 2 Canonical Forms and Decompositions of 3-Site Rules The first column of this table lists the decimal label of the rule. The second column gives the label of its conjugate rule under the transformation T1. If this column is empty then the rule is self-conjugate. The third column gives the expression of the rule in terms of the canonical basis operators, and the fourth column lists the strongly legal decompositions of the rule. Recall that if a rule Xis decomposed as X = A+F where A is an additive rule then this decomposition is strongly legal if the kernel of F in En-{Q,j is large. What this requires is that F be one of the following combinations of basis operators, listed togeather with the configurations which they map to Q, in Table A2.1. The full listing of conditions for a configuration to map to Q under various combinations of basis operators is given in Table 3.3. F 13+,13±+X T1+, T1±+K

Configurations Manned to 0 Contain only isolated 1's Contain only isolated 0's

13++8,13±+X+8

Isolated 1's separated by two or more 0's

11±+t, Tl±+t+K x 8 t K

Isolated 0's separated by two or more 1's Contains no 111 blocks Contains no isolated 0's Contains no isolated 1's Contains no 000 blocks

Table A2.1 Note that this table lists forms involving K. If this basis operator is involved in the canonical form of a rule, then the rule is generative. In terms of the listing given below, canonical forms for generative rules with numerical label 2m+ 1 are obtained by adding the basis operator K to the form for the non-generative rule with label 2m. Rule 0 2 16 4 20 6 8 64 10 80

Canonical Form

Strongly Legal Decompositions

11+ t it++t 13-+T1+

a2+13-+X+B a+13++13-+x aD+13++8 none a2+X+8

12 14

13-+t 13-+Tl++t

a+f3++X a2+X+8+t, aD+13++f3-+8

68 84

3-

250 16 18 20 22 24 26 28 30 32 34

2

I+B+ +x +9

66 82 70 86

iiTl++rl71-+t rl++tj-+t 13-+,nB-+il++iiB-+i1 +t B-+il++Tl-+t

48

il++9

none a2+B- +x

9+t

a+B++B-+x+9

6

8

36

5+13++B-, A+X+t D+13-+9

e+x I+B++B-+x+9, D+9+t S+B+ D+9

e +B- + x

38 40 42

52 96 112

t1++9+t B-+9 B- +Tl ++8

aD+B+ D+il-+t a2+x, D+il++il-+t

44

100

B-+0+t

a+B+ +x+9, D +T--

46 48 50 52 54 56 58 60

116 34

B-+Tl++9+t Ty-+9 Tl++rl-+9 iI-+8+t

a2+x+t, aD+B++B-, D+tl++,qI+B++x

8+B++B-+9 , 4+x+9+t D+B-

Ti++il-+8+t

o+x+9

98 114 102

B-+il-+9

I+B++B- +x, D+t 8+B++9, D+il++t D

62 64 66 68

118 8 24 12

8- +Tl ++Ti- +9+t 13+

70 72

28

B++il++t B++B-

74

88

38

76 78 80 82 84 86 88

90

92 10 26 14 30 74

B-+,q++tl-+9 B-+Tl-+8+t

B++fl+ B++t

D+,q+, 4+B-+x+9 none a2+B++B-+x+9, aD+9+t

a+B- +x (;D+9

a+x +t, S+ti + + l-

B++B-+il+

a2+B++x+9, S+,q-

13++B-+t

a+x, S+tl++ij-+t

B++B-+71++t B++qB++tl++rlB++rl-+t

aD+B-+9 , S+il -+t

B++1l++Tl-+t

B++B-+rl-

B++B-+ij++rl-

I+x+9

S+BI+x+9+t , D+B++B-+9

4+B+ +x I+B- +x +9, S +Ti+ S

251

92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154 156 158 160 162 164 166

78

6++B-+Tl-+t

D+B++O, S+11++t

B++B-+T1++T1-+t

S+t, a+B++B-+x

40 56 44 60

B++e

aD+Tl++t

8++T1++e B+ +e +t B++T1+ +e +t

a+B- +x +e, aD+il+ aD

B++B-+e B++B-+Tl++e

a+x+e+t

120

B++B-+e+t

a+ x+e aD+BI+xaD+il++r1-+t aD+Tl-+t, 8+B-+O I+x+t, aD+Tl++Tl -, D+B++BaD+TI-, A+B+ +x+e

124 42 58 46 62 106 110

144

B++B-+T1++e+t B++T1- +e B++11++Tl- +e B++Ti- +e +t B++Tl++TI-+e+t

134 194 210 198 214 176 180

a2+B+ +x

B++B-+TI-+e

I+B-+x

B++B-+Tl++Tl-+e

8+e

B++B-+T1-+e+t B++B-+TI++Tl-+e+t

D+B+

x 11++x

A+Tl++TI-+t

x+t

148 192 208 196 212 130

a2+B++B-+x, aD+t

8+e+t, a+B++B-+x+8

a2+B-+e , A+Tl-+t

B-+Tl+ + x

a+B++B-, 4+Tl++ilaD+B++x+e, a+rlnone a2+e

B-+x+t

a+B+

B-+T1++x+t n-+x Tl++T1-+x

a2+e+t, aD+B++B-+x+e I+B+ +e, 4 +T1 ++t

Tl++x+t B-+x

T1-+x+t Tl ++T1-+x+t

B-+Tl-+x B-+Tl++,n-+x

S+B++B-+x, A+t D+B- +x +e, A +Tl+

A I+B++B- +e, D + x + e +t S+B+ + x

B-+rl-+x+t

D + x+e

8-+Tl++T1-+x+t

4+B-

x+e Tl++x+e

none a2+B-

x+e +t

a+B++B-+e aD+B++x

T1+ + x+ e +t

252 168 170 172 174 176 178 180

224 240 228 244 162 166

182 184 186 188 190 192

226 242 230 246 136

194 196 198 200 202 204

152 140 156

206 208 210 212 214 216 218 220 222 224 226 228 230

216 220 138 154 142 158 202 206

a2+11+

13- +Tl+ +x + e

a2

B-+x+e+t 13- +il + +x + e +t

a2 +1+ +t,

rl- + x +e Tl++rl- + x +9 rl - + x+e +t

I+13+

Tl++TI -+x+9+t 13-+r1-+x +e 13-+T1 ++rl -+x+e B-+Tl- +x + e +t 13-+Tl++TI-+x+9+t 13++x

168 184 172 188

252 170

a+13++e aD+13++13- + x

8+I3+ +B- +x +e, A +e+t D+B- + x A+e a2+11++rl-, I+B++13-, D+x+t a2+T1a2+rl++T1-+t, D+x a2+rl-+t, M+13-+e none a2+B++B-+9 , aD+x+e+t a+B-

B++B-+T1+ +x +t B++T1-+x B++71++71-+x B++T1-+x+t B++Tl ++rl-+x+t 8++B-+T1- +x 8++B-+Tl++T1-+x B++B-+r1- +x +t

a+rl+, aD+B-+x+9 I+9

B++x+9 B++r1++x+9 B+ +x +e +t B++r1 + +x +9+t

8++B- +x+e 248

a2 +t,

B++Tl++x B++x+t B++71+ +x +t B++B-+x B++B-+r1++x B++B- +x +t

13++13-+r1++T1-+x+t

232 234 236 238 240

B - +x +e

8++B-+il++x+e B++B-+x+9 +t B++B-+T1+ +x+e +t B++il-+x+e

aD+x+9 a+t a2+6++9 , a+71++1 a

8+B-+x I+e +1, D+B++B-+x +e O+B+ I+B-+e , a +Tl -+t a+Tl++T1-+t,

8+x

a+T1-, D+B++x+9 a+T1++r1-, 8+4+t, O+B++BI+11a2+B++B-, I+rl++rl-, aD +x +t I+11- +4 a+B- +e I+r1++rl-+t, aD+x a+e+t

a2 +B+

a+e aD+B- + x I

253 242 244 246 248 250 252 254

186 174 190 234 238

B++il++il-+X+8 B++qq -+X+0+t

I+11+ I+t, D+B++B-+X

8++ Ti++1-+X+8+t

B++B-+il-+X+8

I+tl++t, A+B++O I+6-

B++B-+tI++,I-+x+0

8+x+0

B++B-+11-+X+0+t B++B-+rl++tl-+x+0+t

D+B++x

S+x+O+t, a+B++B-+0

254

Appendix 3 Strongly Legal 3-Site Non-Generative Rules This appendix lists the strongly legal non-generative 3-site rules which reduce to the identity or to shifts. The forms given are for nearest neighbor rules. The expressions given can be changed to their left justified form by the substitutions a-1-I, I->a, and a->a2. Table A3.1 lists rules which are have strongly legal fixed points. Table A3.2 lists rules with strongly legal shift cycles. Note that the same rule may have both strongly legal fixed points, and strongly legal shift cycles. 1. Configurations with only isolated 1's are fixed

140

1+13+

196 1+13- 132 I+B++13-

12 I +B + +x 68 I+B-+x 4 I+B++B-+x 2. Configurations with only isolated 0's are fixed 206 I+il+ 220 I+rl- 222 I+il++il3. Configurations with isolated 1's separated by two or more 0's are fixed 172 I+13++0 228 I+B-+O 164 I+B++13-+0 44 I+B+ +x +O 100 I+B-+x+O 36 I+B++B-+x+9 4. Configurations with isolated 0' separated by two or more 1's are fixed 202 I+rl++t 216 I+rl-+t 218 I+il++il-+t 5. Other cases 236 1+0 Fixed points have no isolated 0's 200 I+t Fixed points have no isolated 1's 76 I+x Fixed points have only 1 and 11 blocks 108 I +x +O Fixed points have 1 and 11 blocks, no isolated 0's 232 I+O+t Fixed points have no isolated 0's or 1's 72 I +x +t Fixed points have only 11 blocks 104 I +X +O+t Fixed points have 11 blocks and no isolated 0's Table AM Strongly Legal Fixed Point Rules Complimentary rules are indicated by pairing them within parentheses. In nearest neighbor form, the right and left shifts are complimentary. Hence if X and Y are complimentary rules, and X has a strongly legal right shift decomposition then Y will have a strongly legal left shift decomposition, and vice versa.

The comments made in section 1 of Chapter 3 describe some cases of rules which have both strongly legal right and left shift decompositions, acting on different configuration subsets. Decompositions in which both a shift and the identity occur are also possible.

255 1. Configurations with isolated l's on shift cycles (234,248) (a+B+,^l+B ) (162,176) ((rl+B+,a+B-) (106,120) (a+13++x,(r1+13-+x) (a+B'+x,a+B-+x) (34,48) (98,56) (a+B++B-+x,(rl+B++B'+x) 2. Configurations with isolated 0's on shift cycles (168,224) (a+il+,(rl+rl-) (186,242) ((rl+rl-,a+rl+) 3. Configurations with isolated 1's and 0's on shift cycles (226,184) (a+B++B' = a-1+71++il-,(Y'1+B++B' = a+rl++ij-) Rule 226 reduces to a left shift on configurations with only isolated 1's, and a right shift on configurations with only isolated 0's, and vice versa for rule 184. 4. Configurations with isolated l's separated by two or more 0's (202,216) (a+B++e,a-l+B-+8) (130,144) (a+B-+O,a-1+B++9) (74,88) (a+B++x+9,(Y'l+B-+x+8) (2,16) (a+B'+x+8,(rl+B++x+9) (66,24) (a+B++B'+x+A,(rl+B++B'+x+9) 5. Configurations with isolated 0's separated by two or more l's (168,224) (a+Tl++t,a'l+Tl-+t) (186,242) (a+rl-+t,a-l+Tl++t) (230,188) (a+rl++rl-+t,a-1+Tl++rl-+t) 6. Other Cases (138,208) (a+9,(rl+8) No isolated 0's Only 1 and 11 blocks (42,112) (a+x,(-l+x) (174,244) (a+t,a-1+t) No isolated 1's (10,80) (a+x+9,a-1+x+8) No isolated 0's, only 1's and 11's (142,212) (a+0+t,a'1+0+t) No isolated 0's or isolated 1's (46,116) (a+x+t,(rl+x+t) Only 11 blocks (14,84) (a+x+O+t,a-l+x+O+t) Only 11 blocks, no isolated 0's Table A3.2 Rules With Strongly Legal Shift Cycles

256

Appendix 4 The Mod (2) Pascal Triangle Many of the properties of the additive cellular automata depend on properties of Pascal's triangle taken modulo 2. In this appendix some of the more important properties of this triangle are given. Most of the results presented here are due to Calvin Long [891, who proved them for the more general case of Pascal's triangle taken modulo any prime p. Figure A. 1 shows the first 33 rows of this triangle. 1 11 101 1111 10001 110011 1010101 11111111 100000001 1100000011 10100000101 111100001111 1000100010001 11001100110011 101010101010101 1111111111111111 10000000000000001 110000000000000011 1010000000000000101 11110000000000001111 100010000000000010001 1100110000000000110011 10101010000000001010101 111111110000000011111111 1000000010000000100000001 11000000110000001100000011 101000001010000010100000101 1111000011110000111100001111 10001000100010001000100010001 110011001100110011001100110011 1010101010101010101010101010101 11111111111111111111111111111111 100000000000000000000000000000001 , Figure Al First 33 Rows of Mod (2) Pascal Triangle

257 This triangle is generated by the binary difference operator D acting on the space of double infinite binary sequences with an initial sequence containing only a single 1. Two immediately obvious properties are the presence of self similar copies of the Pascal triangle at different size scales, and the separation of these copies by inverted triangles of 0's. This structure is explained by an elegant structural theorem. Let An,k denote the triangle 2m n 2m k

2mn+2m -1l 2°'n+2m-1 ( 2mk ) ..................(2mk+2m - 1 / Theorem A4.1 (Long, [ 89]) above is the triangle An,k (0