Elementary Cellular Automata with Memory - Semantic Scholar

32 downloads 0 Views 881KB Size Report
Table 1(a). Rule 254 (11111110) starting with a single-site live cell until T. 8. ... outer live cells. As long as the local rules of both historic and ahistoric automata.
Elementary Cellular Automata with Memory ´ Alonso-Sanz Ramon

´ ETSI Agronomos (Estad´ıstica), C. Universitaria. 28040, Madrid, Spain Margarita Mart´ın

F.Veterinaria (Bioqu´ımica y Biolog´ıa Molecular IV, UCM), C. Universitaria. 28040, Madrid, Spain Standard cellular automata (CA) are ahistoric (memoryless), that is, the new state of a cell depends on the neighborhood configuration of only the preceding time step. This article introduces an extension to the standard framework of CA by considering automata with memory capabilities. While the update rules of the CA remain unaltered, a history of all past iterations is incorporated into each cell by a weighted mean of all its past states. The historic weighting is defined by a geometric series of coefficients based on a memory factor (Α). A study is made of the effect of historic memory on the spatio-temporal and difference patterns of elementary (one-dimensional, two states, nearest neighbors) CA starting with states assigned at random.

1. Introduction

Cellular automata (CA) are discrete, spatially explicit extended dynamical systems. A CA system is composed of adjacent cells1 or sites arranged as a regular d-dimensional lattice (d is in most cases 1 or 2), which evolves in discrete time steps. Each cell is characterized by an internal state whose value belongs to a finite set. The updating of these states is made simultaneously according to a common local rule involving only a neighborhood of each cell (e.g., [1] for a definition of CA). CA are discrete par excellence, all three of their dimensions (space, time, and state) are discrete. This perfectly fits the features of digital computers, enabling exact computation. The synchronicity of the updating mechanism, the regular topologies, and the locality of interactions2 make the CA paradigm ideally suited for parallel computers.  Electronic

mail address: [email protected]. bricks of an oversimplified microworld which do not try to emulate real particles as in Molecular Dynamics. 2 A feature that relates CA with Markov Random Field models, used in spatial statistical modeling and analysis. 1 The

Complex Systems, 14 (2003) 99–126;  2003 Complex Systems Publications, Inc.

100

R. Alonso-Sanz and M. Mart´ın

CA are often described as a counterpart to partial differential equations [2], capable of describing continuous dynamical systems. Roughly speaking, two main levels are studied in a natural system, corresponding to the scale of observation: the microscopic and the macroscopic. Often, the complexity of the macroscopic world is apparently disconnected from that of the microscopic world, although the former is driven by the latter: the microscopic details are lost when the whole system is seen through a macroscopic filter. CA work at the microscopic level which drives the macroscopic behavior: the idea is not to try to describe a complex system from above—to describe it using difficult equations— but to simulate it by the interaction of cells following easy rules. These rules are often formulated in a natural language (with minimal or soft mathematical demands) and are easily translated into a computer programming language, as postulated by the Artificial Intelligence community (particularly when designing Expert Systems). In other words: not to describe a complex system with complex equations, but to let the complexity emerge by interaction of simple individuals following simple rules. From the theoretical point of view, CA were introduced in the late 1940s by John von Neumann and Stanislaw Ulam. One can say that the cellular part comes from Ulam, and the automata part from von Neumann. But CA were shunted onto a dead track for a couple of decades [3] and did not reach the general public until 1970, when Martin Gardner published an account of John Conway’s Game of “Life” in Scientific American [4, 5]. Life was destined to become the most famous CA and an inspiration to a generation of Artificial Life researchers. A number of people at MIT began studying CA beyond Life during the 1970s. Probably the most influential figure there was Edward Fredkin, who around 1980 formed the Information Mechanics Group at MIT along with Tommaso Toffoli, Norman Margolus, and G´erard Vichniac. By 1984, Toffoli and Margolus had nearly perfected the CAM-6 CA machine, a special computer designed for the lightningquick execution of CA, and were generating some publicity about it. In addition, in the middle 1980s (perhaps the golden age of CA), Stephen Wolfram was publishing numerous articles about CA [6] which hooked a number of researchers. In these articles, Wolfram suggested that many physical processes that seem random are in fact the deterministic outcome of computations that are simply so convoluted that they cannot be compressed into a shorter form and predicted in advance (he spoke of these computations as incompressible). During the following years, CA were developed and used in many different fields. The requirements for the application of the CA approach to real problems (connecting different levels of detail) enlarged the basic paradigm, leading to systems related to CA (mere extensions in Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

101

some cases) such as: inhomogeneous,3 asynchronous4 continuous state spaces,5 Lattice-gas6 or macroscopic7 automata. Today, some authors use the more comprehensive term grid-based models [15] in order to be freed of the restrictions that the CA paradigm imposes. A vast body of literature has been produced on these topics.8 To a great extent, the papers on CA relate to many different, unrelated areas of physics.9 But the aim of this contribution is not to present a comprehensive review of the avatars of the CA approach but rather to enlarge the paradigm in a relatively unexplored direction: the consideration of all past states (history) in the application of the CA rules, as explained in section 2. To materialize this, we start from the simplest scenario: onedimensional (d  1) CA with two possible values (k  2) at each site: a  0, 1, with rules operating on nearest neighbors (r  1). This is the scenario of [28], a landmark review paper largely responsible for the resurgence of interest in CA in the 1980s. The recent book by Wolfram [29] will receive special attention in the CA community (see two early comments by Casti [30] and Giles [31]). Thus, this paper deals with elementary rules (d  1, k  2, r  1), which following Wolfram’s [28] notation, are characterized by a sequence of binary values (Β) associated with each of the eight possible 3 Or nonuniform: where different cells may be ruled by different transition functions. Nonuniform CA were investigated by Vichniac et al. [7] and Sipper [8]. Kauffman’s [9] Random Boolean Networks allow arbitrary rules and connections which may be different at each site. Cellular Neural Networks [10] include CA as a special case. 4 Or lattice Monte Carlo simulations: cells are not updated simultaneously [11]. 5 Such as Coupled Map Lattices (CML), in which time and space are discrete but state is continuous [12]. 6 In which the update is split into two parts: collision and propagation, intended to guarantee propagation of quantities while keeping the proper updating rules (collision) simple [13]. 7 State variables refer to macroscopic quantities; cell dimension is larger [14]. 8 The biennial Conference on CA for Research and Industry (ACRI, from its original Italian acronym) aims to present an international forum for researchers who are active in the CA field, as well as for those interested in evaluating the possibility of applying them in their own fields [16, 19]. Ilachanski [20] has recently reviewed the subject. His book contains an extensive bibliography and provides a listing of CA resources on the World Wide Web. Outstanding, the freeware DDLab [21] is a useful tool, in permanent upgrading, for studying both CA and Random Boolean Networks. 9 The book by Chopard and Droz [22] might serve as an excellent textbook for physicists. The updated book by Ilachanski [20] is also intended mainly for a physicist audience. Some modern books on statistical mechanics; for example, Wilde and Singh [23], incorporate CA as a tool to study systems that are far from equilibrium. It has been argued that CA , intimately related to discrete statistical models, will play an important part elucidating basic ideas and general principles of statistical mechanics. Conversely, Rujan [24] also studies the usefulness of statistical physics methods to describe the properties of probabilistic CA. The statistical mechanics of probabilistic CA was studied by Lebowitz, et al. [25] and Alexander, et al. [26]. The book edited by Dieckmann, et al. [27] frames CA in the context of (ecological) spatial analysis.

Complex Systems, 14 (2003) 99–126

R. Alonso-Sanz and M. Mart´ın

102

(T

(T

triplets: (ai 1 , a(T i , ai 1 ) 111 110 101 100 011 010 001 000 8

Β1 ,

Β2 ,

Β3 ,

Β4 ,

Β5 ,

Β6 ,

Β7 ,

Β8 binary  Βi 28 i i1

decimal.

The rules are conveniently specified by a decimal integer, to be referred to as their rule number, R. The rule numbers for elementary CA range from 0 to 255. We will review in passing the recent literature invoking elementary rules. Special attention will be paid to legal rules [28]: rules that are reflection symmetric (so that 100 and 001 as well as 110 and 011 yield identical values), and quiescent, that is, they do not transform a dead cell with dead neighbors into a live cell (their binary specification ends in 0). These restrictions leave 32 possible legal rules of the form: Β1 Β2 Β3 Β4 Β2 Β6 Β4 0. Another set of 32 categorized rules are those reflection symmetric but nonquiescent, with binary codes of the form: Β1 Β2 Β3 Β4 Β2 Β5 Β4 1 (the binary specification of any nonquiescent rule ends in a 1). There are 128 semi-asymmetric (either Β2 Β5 or Β4 Β7 , but not both) rules and 64 fully asymmetric (Β2 Β5 and Β4 Β7 ) rules. There are 88 equivalence classes, that is, 88 fundamentally inequivalent rules, formed under the negative, reflection, and negative + reflection transformations [32]. 2. Memory

Standard CA are ahistoric (memoryless); that is, memory of only the previous iteration is taken into account to decide the next one. Thus, if a(T is taken to denote the value of cell i at time step T, the site i values evolve by iteration of the mapping: a(T 1  Φ((a(T i i )), where Φ is an arbitrary function which specifies the CA rule operating on the neighborhood  of the cell i. In this paper, a variation of the conventional one-dimensional CA is considered, featuring cells by their mean state in all the previous time steps. Thus, what is here proposed is to maintain the rules Φ unaltered, but make them actuate over cells featured by their mean state: a(T 1  Φ((fi(T )), with fi(T being the mean state of cell i after i iteration T. We refer to these automata considering historic memory as historic (or  with Toffoli and Margolus’ permission), and to the standard ones as ahistoric. It should be emphasized that the memory mechanism considered here is different from that of other CA with memory reported in the literature. Typically, they determine the configuration at time T 1 in terms of the configurations at both time T and time T 1. Particularly interesting Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

103

are the second order in time (memory of capacity two) reversible10 rules due to Fredkin. These rules (referred to by Adamatzky in [41] as the class ), are of the form: (T 1

a(T 1  Φ(a(T i i )  ai

,

with  denoting the exclusive OR (XOR) operation. As a natural extension of the memory mechanism here proposed, history can be incorporated into these rules in the form [42, 43]: (T 1

a(T 1  Φ(fi(T )  f i i

.

But to preserve the reversibility inherent in the XOR operation, the reversible mechanism incorporating memory must be of the form (see [44]): (T 1

a(T 1  Φ(fi(T )  ai i

.

Some authors; for example, Wolf-Gladrow [13], define rules with memory as those with dependence in Φ on the state of the cell to be updated. So in the r  1 scenario, rules with no memory take the form: (T

(T

(T

(T

a(T 1  Φ(ai 1 , ai 1 ). Rule 90: a(T 1  ai 1 ai 1 mod 2, would be an i i example of a rule with no memory. Our use of the term “memory” is not this. The effect of memory starting with a single-site seed was reported in [42] regarding one-dimensional CA and in [43] regarding totalistic twodimensional CA. In an earlier work we analyzed the effect of memory in other CA scenarios: (i) Conway’s Game of Life [45] and (ii) the spatial formulation of the Prisoner’s Dilemma [46–50]. These studies become a starting point for the motivation of the present, projected in a more general scenario. 3. Full history at work: Some simple examples

The simplest way to take history into account is to feature cells by their most frequent state, thus not necessarily the last one. In case of a tie, the cell will be featured by its last state. Rule 254 (1111110) progresses as fast as possible (i.e., at the speed of light): it assigns a live state to any cell in whose neighborhood there would be at least one live cell. Thus, in the ahistoric model, a single-site live cell will grow 10 Reversible systems are of interest since they preserve information and energy and allow unambiguous backtracking [33]. They are studied in computer science in order to design computers which would consume less energy [34]. Reversibility is also an important issue in physics [35–38]. Gerald ’t Hoof in a speculative paper [39], suggests that a suitably defined deterministic, local reversible CA might provide a viable formalism for constructing field theories at Planck scale. Svozil [40] also asks for changes in the underlying assumptions of current field theories to make their discretizations appear more CA-like.

Complex Systems, 14 (2003) 99–126

104

R. Alonso-Sanz and M. Mart´ın

Table 1(a). Rule 254 (11111110) starting with a single-site live cell until T  8.

Historic and ahistoric models. Live cells:

 last,  most frequent.

Table 1(b). Rule 90 (01011010) starting with a single-site live cell until T  4. Historic and ahistoric models. Live cells:  last,  most frequent.

Table 1(c). Rule 233 (11101001) starting with a single-site live cell until T  5.

Historic and ahistoric models. Live cells:

 last,  most frequent.

monotonically, generating segments whose size increases two units with every time step. The dynamics are slower in the historic model (see Table 1(a)): the outer live cells are not featured as live cells until the number of times they “live” is equal to the number of times they were “dead.” Then the automaton fires two new outer live cells. As long as the local rules of both historic and ahistoric automata remain unaltered, and both the latest and most frequent states coincide after the first two iterations, the historic and ahistoric evolution patterns are the same until T  3. But after the third iteration, the last and the most frequent states often differ, and consequently the patterns for the historic and ahistoric automata typically diverge at T  4. This is the case in Table 1(a) and in Table 1(b) with rule 90 (01011010), in this case, after T  3 the most frequent state for all the cells is the dead one. Occasionally the last and the most frequent status may coincide after the third iteration. In this case the historic and ahistoric configurations will still be the same at T  4. This is the case for Rule 233 (see Table 1(c)). But the coincidence in the dynamics of the historic and ahistoric Rule 233 automaton ceases at T  5: the latest and the historic (most frequent) configurations do differ after T  4, as seen in the patterns generated from T  5.

Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

105

Table 2. Rules 90 and 150 are not additive in the historic formulation. The true evolution pattern starting with two adjacent live cells is shown on the left. The XOR superposed configuration of those evolved independently starting with a single seed is shown on the right. Evolution up to T  11.

In the standard (ahistoric) scenario, Rules 90 and 150 (together with the trivial 0 and 204) are the only additive legal rules; that is, any initial pattern can be decomposed into the superposition of patterns from a single-site seed. Each of these configurations can be evolved independently and the results superposed (modulo two) to obtain the final complete pattern. As illustrated in Table 2 starting with two adjacent live cells, additivity of Rules 90 and 150 is lost in the historic model. Every configuration in a standard (ahistoric) CA has a unique successor in time (it may however have several distinct predecessors). A (last) configuration in the historic scenario may have multiple successors: the transition rules operate on the most frequent state (MFS) configuration, not on the last, so the successor of a given configuration depends on the underlying MFS configuration. A primary example of this is given with Rule 254 (Table 1(a)): in the fully historic model, live configurations reproduce themselves for several iterations ( for example, twice) but at a critical time step (T  5 for the example), generate a new () one. We envisage that this will notably alter the appearance of the state transition diagrams in the historic scenario as compared to those of ahistoric CA [32]. The irreversibility (a feature necessary for the formation of attractors) of historic rules in Figure 1(a), often leads to the generation of oscillators that appear fairly soon. Thus, the number of different configurations generated by the evolution of a single-site seed is smaller in the historic model. This, generalized to any initial configuration, would lead to conjecture the existence of a higher number of “Garden-of-Eden” nodes (configurations for an automaton which could only exist initially) in the historic scenario. 4. Discounting

Historic memory can be weighted by applying a geometric discounting process in which the state a(T Τ obtained Τ time steps before the last i round is actualized to the value ΑΤ a(T Τ , with Α being the memory factor. i A given cell will be featured by the rounded weighted mean of all its Complex Systems, 14 (2003) 99–126

R. Alonso-Sanz and M. Mart´ın

106

past states. This well-known mechanism fully ponders the last round (Α0  1), and tends to forget the older rounds. After time step T, the weighted mean of the states of a given cell will be: Tt1 ΑT t a(ti (1 (2 (T m(T . i ai , ai , . . . , ai   Tt1 ΑT t Provided that states a are coded as 0 (dead) and 1 (live), the rounded weighted mean state f will be obtained by comparing its weighted mean m to 0.5, so that:  1 if m(T  i > 0.5   (T (T fi   if m(T ai  i  0.5   0 if m(T  i < 0.5. This is equivalent to studying the sign of: T

T

T

(1 (2 (T T t (t (T ai  ΑT t   (2a(ti 1)ΑT t i (ai , ai , . . . , ai )  2  Α t1

fi(T

 1    (T  ai    0 

t1

t1

if Ti > 0 if Ti  0 if Ti < 0.

(2 (3 2 (1 (2 (3 After T  3: (a(1 i , ai , ai )  Α (2ai 1) Α(2ai 1) 2ai 1. Of (1 (2 (3 course, if ai  ai  ai , history does not alter the series and it will be (2 (3 fi(3  a(3 i . Neither does history take effect until T  3 if ai  ai nor if (1 (3 (1 (2 ai  ai (see [42]). But the scenario may change if ai  ai a(3 i : (0, 0, 1)  (1, 1, 0)  Α2 Α 1   

1 5  0.61805 Α0 . 0Α 2 Thus, provided that Α > Α0 , cells with state history 001 or 110 will be featured after T  3 as 0 and 1 respectively instead of as 1 or 0 (last states). Rule 90 starting with a single live cell, as in Table 1(b), provides an excellent example of this: for Α > Α0 after T  3 all the cells, even the two alive at T  3 but dead at T  1 and T  2, are featured as dead, so that the annihilation of the live pattern happens as early as T  4. But for Α  Α0 , the two cells alive at T  3 are featured as live and will regenerate the pattern at T  4 as shown in Table 1(b) for the ahistoric model. After T  4: (0, 0, 0, 1)  (1, 1, 1, 0)  Α3 Α2 Α 1    0  Α  0.5437. (2 (T 1 In general, in the most unbalanced scenario, if a(1 i  ai    ai (T ai it holds that: T 1

(0, 0, . . . , 0, 1)  (1, 1, . . . , 1, 0)   Αi 1 i1



Α Α 1    0  ΑT 2Α 1  0. Α 1

Complex Systems, 14 (2003) 99–126

T

Elementary Cellular Automata with Memory

107

When T , it is: lim (0, 0, . . . , 0, 1)  lim (1, 1, . . . , 1, 0)

T

T



Α 1    0  Α  0.5. 1 Α

It is then concluded that memory does not affect the scenario if Α  0.5. Thus, the value 0.5 for the memory factor becomes a bifurcation point that marks the transition to the ahistoric scenario. Computationally, there is a saving if instead of calculating   2Ω  for every cell, we calculate T

Ω(ai , T)   ΑT t a(ti t1

all across the lattice and compare the Ω figures to the factor T

1 1 ΑT 1 1 (T)   ΑT t  . 2 2 t1 2 Α 1 This was done in the computer program written to perform the calculus and generate the figures. 5. Spatio-temporal patterns

Most of the legal rules are either unaffected (e.g., simple rules [28]) or minimally affected (e.g., Rules 250, 254, and 36) by memory, starting from a random initial configuration. In this scenario, only the nine legal rules which generate nonperiodic patterns in the ahistoric scenario [51, 52] are significantly affected by memory. Figure 1(a) shows the evolutive patterns of eight of these rules starting with a random initial configuration with uncorrelated values taken to be 0 or 1 with probability p(a  1)  0.5; the memory factor varied from 1.0 to 0.5 by 0.1 intervals. The lattice size is N  90 and evolution is shown up to 90 time steps. Rule 18 was one of the first rules carefully analyzed. History has a dramatic effect on Rule 18. Even at the low value of Α  0.6, the appearance of the spatio-temporal pattern fully changes: a number of isolated periodic structures are generated, which is far from the distinctive inverted triangles world of the ahistoric pattern. For Α  0.7, the live structures are fewer, advancing the extinction found in [0.8,0.9]. In the fully historic model, only a periodic pattern of live cells, appearing twice, survives. Rule 146 is affected by memory in much the same way as Rule 18. This is because although their rule numbers are relatively distant, their binary configurations differ only in their Β1 value. The spatio-temporal Complex Systems, 14 (2003) 99–126

Figure 1(a). Evolution of legal (Β1 Β2 Β3 Β4 Β2 Β6 Β4 0) rules significantly affected by

memory. The values of sites in the initial configuration are chosen at random to be 0 (blank) or 1 () with probability 1/2. The pictures show the evolution of CA with 97 sites for 90 time steps. Periodic boundary conditions are imposed on the edges. The evolution of the CA at successive time steps is shown on successive horizontal lines. Memory factor Α. Ahistoric model for Α  0.5, fully historic model for Α  1.0 with k  2, r  1.

Elementary Cellular Automata with Memory

109

patterns of Rules 182 and 146 resemble each other, though those of Rule 182 look like negative photograms (a  1 a) of those of Rule 146. The effect of memory on Rules 22 and 54 is similar. Their spatiotemporal patterns in Α  0.6 and Α  0.7 keep the essential feature of the ahistoric, although the inverted triangles become enlarged and tend to be more sophisticated in their basis. A notable discontinuity is found for both rules ascending in the value of the memory factor: in Α  0.8 only a periodic structure survives for both rules; in Α  0.9 two live structures seem to survive for Rule 22; extinction of live cells is found for Rule 54. But unexpectedly, the patterns of the fully historic scenario differ markedly from the others, showing a high degree of synchronization. The behavior of Rule 54 in the ahistoric model has been featured to some extent as transitional between very simple Wolfram’s Class I and II rules and chaotic Class III. Thus, Rule 54 appears among the two one-dimensional rules (with Rule 110) that seem to belong to Wolfram’s complex Class IV. The four remaining chaotic legal rules (90, 122, 126, and 150) show a much smoother evolution from the ahistoric to the historic scenario: no pattern evolves either to full extinction or to the preservation of only a few isolated persistent propagating structures (solitons). Rules 122 and 126 (close in terms of rule number), evolve in a similar form (particularly when comparing the ahistoric and fully historic patterns), showing a high degree of synchronization in the fully historic model. Some spatio-temporal patterns of rules in the ahistoric model are reminiscent of the patterns of pigmentation observed on the shells of certain mollusks [29, 53]. Wolfram [6] illustrates this idea by showing a natural cone shell with a pigmentation intended to be reminiscent of the pattern generated by Rule 90 in the ahistoric scenario. But the clearings in the shell would suggest Rule 22 for some value of Α between 0.6 and 0.7: with Α  0.6 the clearings seem to be scarce, with Α  0.7 the clearings appear to be excessive. Figure 1(b) shows the evolution of the average fraction of sites with value 1 at time T, density ΡT , up to 200 time steps, starting with an initial density Ρ0  0.5. The simulation is implemented for the same rules as in Figure 1(a), but with a notably wider lattice of N  401. A visual inspection of the plots in Figure 1(b), ratifies the general features observed in the patterns in Figure 1(a). Thus, starting with a disordered configuration of any nonzero density, the evolution of the density ΡT according to Rule 18 in the ahistoric model yields an asymptotic density Ρ  1/4 . Figure 1(b) illustrates how this value is reached soon, and how history induces a depletion of the asymptotic density, null for Α  0.8 and Α  0.9. In the fully historic model, a smooth period two density oscillator (0.105  0.115) is generated as early as T  8. Complex Systems, 14 (2003) 99–126

110

R. Alonso-Sanz and M. Mart´ın

Figure 1(b). Evolution of the average fraction of sites with value 1 at time step T, density ΡT , starting with an initial density Ρ0  0.5, up to 200 time steps. Results concerned with the same rules of Figure 1(a) are shown when operating in a lattice with size N  401. Note that in order to improve plot resolution, y-axes scopes have been tailored. Plots code: unjoined dots  fully historic model, dashed line  Α  0.8, continuous line  ahistoric model.

Rule 146 density plots resemble those of Rule 18, but with a sharper period two oscillator generated in the fully historic model (0.175  0.204). Rule 182 (not shown in Figure 1(b)) yields Ρ  3/4 in the ahistoric model, the shape of its evolution density curves resembling the complement to 1 of its equivalent Rule 146. Wolfram [28] proved that for Rule 90, Ρ  1/2, independent of the initial density Ρ0 (so long as Ρ0 0). When historic memory is taken into account, ΡT varies erratically around 0.5 as shown in Figure 1(b) Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

111

for Rules 90 and 150, without either periodic evolution or tendency to a fixed point. Not even in the fully historic model. Quoting Grassberger [51], Rule 22 is the only one of the elementary rules whose long-time behavior is not yet understood. Wolfram [28] resorts to simulations to report Ρ  0.35  0, 02 for evolution with Rule 22. Rule 54, very notably absent in Wolfram’s [28] considerations, again resembles Rule 22 with regard to Figure 1(b). For example, both rules present a very low density in the [0.8,0.9] interval, null for Α  0.9 in Rule 54. Wolfram [6, Table 6] reports Ρ  1/2 for Rules 122 and 126. In our simulation, ΡT oscillates around values slightly over 1/2 in the ahistoric model. Figure 1(b) makes apparent how the effect of memory differs in both groups of rules: memory sharply depletes the density curves of Rules 18 and 146, but leads them over the starting 0.5 value for Rules 122 and 126. In the fully historic model, these rules do not lead to low density values, but to synchronous behavior. The high degree of synchronization visually appreciated for Rules 22, 54, 122, and 126 in the fully historic model in Figure 1(a), stands out also in Figure 1(b) (unjoined dot plots). Synchronization11 in CA is not a trivial task since synchronous oscillation is a global property, whereas CA typically employ only local interactions; but the phenomenon of synchronous oscillations occurs in nature in fairly striking forms.12 6. Difference patterns

Figure 2 shows the difference patterns (DP) produced by evolution from the disordered initial configuration taken in Figure 1(a) resulting from changing the value of its initial center site value. The pictures show the damaged region as black squares corresponding to the site values that differed among the patterns generated with the two initial configurations. The perturbations in proper chaotic rules propagate to the right and left at a single (maximum) velocity at any time. This behavior illustrates the butterfly effect: a small perturbation grows, and finally rules the whole system. The velocity in the damage spreading is quantified by means of the left and right Lyapunov exponents (ΛL , ΛR ) which measure the rate at which perturbations spread to the left and right, and are given by the slopes of the left and right boundary of the growth of the DP. Thus, zero values indicate periodicity, whereas negative velocity indicates perturbation repair. The chaotic Class III rules in Figure 2 11 Given any initial configuration, the CA must reach a final configuration, within a finite number of time steps, that oscillates between all zeros and all ones at successive time steps. 12 The synchronization task has been investigated by Das, et al. [54] and Sipper [8], who conclude that Rules 21 and 22 are the best performing synchronizer rules.

Complex Systems, 14 (2003) 99–126

Figure 2. DP produced by evolution from the disordered initial configuration taken in Figure 1(a) and that resulting from reversing its center site value. The pictures show as black squares the site values that differed. Evolution up to 90 time steps. Memory factor Α. Ahistoric model for Α  0.5, fully historic model for Α  1.0 with k  2, r  1.

Elementary Cellular Automata with Memory

113

reach the maximum Λ attainable when r  1, Λ  1. The Lyapunov exponents of ahistoric elementary CA are tabulated in [6]. As a rule, the effect of memory on the DP mimics that of spatiotemporal patterns, so that the rule parallelisms found for the spatiotemporal patterns are again applicable to the DP. Rule 18 may serve as an example: a periodic structure (with only two elements) appears in the fully historic scenario; the differences die out when Α  [0.8, 0.9], the perturbation remains localized (again in the form of periodic structures) when Α  [0.6, 0.7]. Finally, in the ahistoric model, the perturbation grows at the speed of light: ΛL  ΛR  1. The pattern shown in Figure 2 conforms to this property, although the picture appears delimited in growth. This is because the lattice size N  97 is not large enough for free expansion during T  90 time steps (which would be feasible with N  181; recall that with r  1, after T time steps, an initial sole mutation may affect the values of at most 2T sites). This distortion operates on every ahistoric pattern in Figure 2 and is found, to a lesser degree in some rules and in Α > 0.5 values. Examples are Rule 90, 122, and 150 for Α  0.9. The DP of Rule 146 resembles that of Rule 18, and evolves in a similar way. In turn, the DP of Rule 182, fully resembles that of its equivalent Rule 146. The DP of Rules 22 and 54 are again similar:13 the DP is constrained when history begins to actuate [0.6,0.7], and ceases, or is localized near the central site, at higher Α values. The group of Rules 90, 122, 126, and 150 show a fairly gradual evolution from the ahistoric to the historic scenario, so that the DP appear more constrained as more historic memory is retained, with no extinction for any Α value. The patterns with inverted triangles dominate the scene in the ahistoric DP in Figure 2 (the exception is the peculiar DP of Rule 54), but history destroys this common appearance (and that of Rule 54), even at the lowest value of Α with memory effect in the figure: Α  0.6. Thus, there is a sort of discontinuity implied in the consideration of historic memory (perhaps with the exception of Rule 22) regarding the DP, which Rule 90 might exemplify: memory, at the low rate Α  0.6, destroys the structures characteristic of ahistoric DP. To avoid coined terms such as “chaotic” or “random,” the DP generated for Α  0.6 could be described as “helter-skelter.” Regarding the central site, for Rules 22, 54, 90, 122, 126, and 150, the disruption induced by its initial reversion, is in some manner more unpredictable in the historic model with Α  0.6 than in the ahistoric. Extreme examples are: Rule 90 (after initial reversion, the original evolution is restored in the ahistoric model) and Rule 150 (the initial reversion remains forever). 13 This coincidence in behavior agrees with their complexity: they are the only legal rules with no extreme Lyapunov exponents: ΛL  ΛR 0.75 for Rule 22 and ΛL  ΛR 0.55 for Rule 54 (see Table 6 in [6]).

Complex Systems, 14 (2003) 99–126

114

R. Alonso-Sanz and M. Mart´ın

The conclusions drawn from Figure 2 are supported by the N  401 simulations run for Figure 1(b). Again, the central site has been reversed and two types of plots are implemented to feature the DP: (i) the Hamming distance (, the number of nonzero site values of the DP), and (ii) the width of these patterns. From these plots (not included here to avoid graphical overloading) the overall conclusion drawn from Figure 2 remains valid: memory implies a depletion in the damaged region (i) and in the speed of propagation of perturbation (ii). In order to systematize the analysis of the DP, one can resort to equivalence classes. Memory is expected to affect all the rules of an equivalence class in a similar way. This was already observed in the legal class 146, 182. Figure 3 shows the DP of the remaining rules equivalent to the legal ones. A fairly consistent correspondence in the appearence of the DP is observed between equivalent rules: 18, 183, 22, 151, 54, 147, 90, 165, 122, 161, and 126, 129. The DP of Rule 150 resembles that of its complementary Rule 105. This is so despite the fact that Rules 150 and 105 do no belong to the same equivalence class (some classes are formed by a single rule, as in the case of Rule 150 [55]). We have also explored the effect of memory in asymmetric rules. Figure 4 shows the DP of some such rules affected by memory in the scenario stated in Figure 2(a). Rules in which almost all changes in initial configuration die out, and rules with ΛL,R  0 are not greatly affected by memory (e.g., Rules 156 and 100). A considerable number of equivalence classes of asymmetric rules have !ΛL !  !ΛR !  Λ with Λ  1 or Λ  1/2 as their minimal representative rule. These rules present a diagonal as DP in the ahistoric model, which is rectified (i.e., both Lyapunov exponents evolve to zero) and/or led to extinction by memory. But either extinction or rectification of the trajectory of the perturbation are not always achieved in a uniform way. Numerous examples of unexpected evolution have been found in this context. The important Rule 11014 and the others of its equivalence class, may serve as a paradigmatic example of the expected effect of memory: the damage induced by the reversal of the initial central site value becomes more constrained as the memory factor increases, with no discontinuities in the preserving effect. The same applies to all the rules of the equivalence classes of the three that have one irrational Lyapunov exponent:15 Rules 30,16 45, and 106. 14 This rule shows highly complex properties of information transmission, associated with particle-like structures [56]. 15 In Rules 22 and 54 (Figure 2(a)) both the Lyapunov exponents are irrational. 16 Wolfram uses Rule 30 for random sequence generation in [57].

Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory α 1.0

RULE

0.9

0.8

115 0.7

0.6

0.5

91(01011011) L LLL

L LLL

L LLL

LLL L L

L L LLL

L L LLL L

L

L L LLL L

LLL L L

L L LLL L

L

LLL L L L

L L LLL L

L

L L LLL

LLL L L

L LLL L

L L

L L LLL L L L LLL L

L

LLL L L L

LLL L L

LLL L L L

L

LL LL L LLLLL LL LL LLLLL LL LLLL LL LLLLL L L L LL L L LLL LLLL LLLL LLLLLLL LLLL L LLLLL LLL LLLLLLL LL LLLL LLLLLLL LLLL L LLLLLL LLL LLL LLLLL LL LLLL LL L LLLLL LLLL LL L LLLL LLL L L LLLLL L L LLLLL LL L LLLL LL LL L L LLLL LLLL LLLL LLLL LL LLLLL LLL LL L LLLLL LL LL LL L LLLL L L LLLL LLLL LLLL LLLL LLL LLLLL LL LLL LLLLL LL LL LL L LLLL LLL LLLL LL LLLL L LLLL LLL LLLLL LLLLLLL LLL LL L LLLL LL LL L LLLL L LL LLLL LLLL LL LLL LLL LLLL LLLLL LLL L LLLL LL LL L LLL LLLL LLLL LLLL LLLL LLL LLL LLLLL LLLL LLLLLL LL L LLL LLLL LLLL LLLL LLLL LLLL L LL LLL LLLL LLLLL LLL L LL L LL L LLLL LLLL LLLL LLLL L L LLLL

LL LL L L LLLLL LLLLLLL LL L LLL LL L LLL L L LL LL L L LL L LLL L LLL LLL LLLLLL LLL L L LL LLL LL LL LL LLLL LL L LLLLLL L LL LLL L LLL LL LLL L L LLL LL LL LL LLL LLLL LLL LLLL LLLL L LL LL LL L LLLLLLLL LL L LL L LL LLL LLL LL L LL L LL L L L L LL L L L LL L LL L LLLL L LLL LL L LL LLLL LL L L LLLLLL L L L LL L LL L LL L LL LLL LLL LL LL L LL L LL L LL LL LLL LLL LLL LLL LLLLL LLL LLL LLL LL L L L LLL L L L L L L LL LLL LLL LLLLLLL LLL LL LLL LLL LLL LL L L L L L LLL LL L L L L L L L LLL LLLLL LLL LLL LLL LLL LL LL LL LL L LL L LLL L L L L L L LL LL LL LL LLL LLLLLLLL LLL LL LLLL LLLL LLL LLL LLL LLL LL LL LL LL LL L L LL LL L L L L L L L LL LL LLL LLL LLL LLL LLL L LLLLLL LLL L LLL L LLL LLL LLL LLL LLL LL L L L L L L LLL LL L LL LL LL LL LL LL LL LL LL LL LL LL LL LLL L LL LL L L L L L L L L LL LL LL LL LL LL LL LL LLL L LLL LLL LLL LLL LLL LLL LLL LL LL LL LL LL LL LL LLLLLL LLL LL L L L L L L L LL LL LL LL LL LL LL LLLL LL LLLL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LLLL LL LL LL LL LL LL LL LL LL LL L L L L L L LLLL L L L L L L L LL LL LL L LL LL LL LL LL LL LLLL LLLL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LLLL LL LLLL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LLLL LL LL LL LL LL LL LL LLLL LL LLLL LL LL LL LL LL LL LL L LL LL LL LL LL LL LLLL LL LLLL LL LL LL LL LL LL LL LL L L L L L L LL LL L L L L L L L LL LL LL LL LL LL LL LLLL LL LLLL LL LL LL LL LL LL LL LLL LLL LLL LLL LLL LLL LLL LLLL LL LL LL LL LL LL LL LL LL LL L L L L L L L LLLL LL LLLL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LLLL LL LLLL LL LL LL LL LL LL LL L L L L L L L LLLL LL L L L L L L L

RULE 105(01101001) L LLL L LL LLLLL LLLL LLLLL LL LLL LL LL LL LLL LL LL LL LLLL LLL LL LL L L L LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLLL LLL LL LL L L L LL LL LLL LL LL LL LLLL LLL LL LL LLL LL LL LL LLL LL LL LL L L L LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLLL LLL LL LL LLL LL LL LL L L L LL LL LLL LL LL LL LLL LL LL LL LLLL LLL LL LL L L L LL LL LLL LL LL LL LLLL LLL LL LL LLL LL LL LL LLL LL LL LLL LL LL LL LL L L L LL LL LLL LL LL LL LLL LL LL LL LLL LL LL LL LLLL LLL LL LL L L L LL

L LLL L LL LLLLL LLLL LLLL LLL LLL LLLLL L LLLL LLL LLL L LLLL LLLL LLL L LL LL LLLLLL LL LLL LL LL L L L L L LLLL LLLLLLLL LLL L LL LLLLLLLL LLL L LLLLLL LLLLLLLL LLL LLLLL L LLL LL LLLL LLLLL L LL LL L LL L LLLL LLLL LLL L LLL L LLL LL L LLL LLLL L LL L L LLLL LL LLL LLLL L LL L L LLLL LL LLL LLLL L LL L L LLLL LL LLL LLLL L LL L L LLLL LL LLL LLLL L LL L L LLLL LLLL LLLLL L LLL L L LLLLL LLLL LLLLL L LLL L L LLLLL L L LLL LL L LLLL L L L LLL LLLLLLL LLL LL LLLLL L LLL LL LLL L LLL LL L LLL L LLL L LL LL LLL LL L LL L LL L L LL LLLLL LL L LL LLLL L L LLL LLL L LLL LLLLLLLLL L L LLLL LL L LL L LLLLLLLL L L LL L LL L LLLLLLLL LL L L LLLLLL LLL LL L L LLL LLL LLLLL LL LLLL LL LLL LLLLL LL LLL LL LL L L L L LLL L LL L LL L L LLL LLL LLL LLL LLLLL LL LLL LLLLLLLL LLLLLL LL LLL L L L LLLLL LLLLLLL LL LLL L L LLL LLLLLL LLL L L LL LLLLL LLL LL L L LLL LLL L LL LLL L L LLL LLL LLLL L L LLL LL L LL LLL L L LL LL L LL LLL LLL L L L L LLL L L LL L L

L LLL L LL LLLLL LLLL LLLLL LLLL LL LL LL LLLLL LL LLL L L LL LL LLL LL LLLLLL LLLLLL LL LLL LLLL L LL LLL LLLLLLLL LL L LL LLL LLLLLL LLLLLLLL LLLLLLLL L LLL LLL LL LLLL L LL LLLLLL LLL LLLLLL LL L LLL L L LLLLLLLLLL L L L LLL LLLLLLLL L LLLLL L L L LLL LLL LLLL LLLLLL L LLLLLL LLL LLLLLLL LL L L LLL L L LLLL L LLLL LL LLL L LL LL LL L LLL L LLLLL L L LLL LL LL LLLLLLL LL L LLL L L L LL LLL LLLLLLLLLLL L LLL LL LL LLLLL LL LL LL LL LLL LLLL LL LLL L L LLL L LLLLLLLLL LL LLL LL LLLL LLLLL LL L LLLLLLLL LL L LL LL LLLLL LLL LLL LLLLL LLLLLL LL LL L LL LL LLL LLL LLL LL LLLL LLLL LL LLLLLL LL LLLL L LL L L LLLLL L LL LL LLL L L L LL LLLLLLLLLL LLLLLLLLLLL LLL L LLL LL L L LLLLL L LLL L L LLLLLLL L LL L LLLL LLLL LLL L LLL LLLL LLLLL LL LL LLL L L LL LL LL LLLLLLLLL L L LL LLL LLL L L LLL L LLLL L L L LLLLLLLL LLL LL LL LLLLL LLLLLLL LL L LL L LLL L LLLL L LLL L LLLL LLLLLLLLLLLLLL LLL LL LL LLL L LL L LLLL LLLLLLL LL LLLLLL L L L LL LL LLLLL L LL LLLLLL LLLLL LLLL LL LLL LL L L LL LLL LLLLLL LL L LLLL LL LLL L L LLL L LLL LL L LLLLLL LL L LL LLLLLL LLL L LLL LL LLLL LLL L LL L L L LL LLL LL LLL LLLL L LL LL L L LL LL LLL LL L LLLL L LLLL L LLL LL LLLLLL L L L LLLL L LLL LL LLL LLL LL LLLLL LL LL LLL L LLL LLLL LLLL L L L L LL LLLL LLLL LL L LLLL LLLL LLL L LLLLL LL LLLLLLLL L L LL LLLLL LL LL LLL LL LL L LL L LLLLLL L LLL L LLL L LL LLL L L LLL L LLL L LLLLLL LLLLLLL LLLLL L LLLL LLL LLL LLLL L L L LL L LLLL LLLLLL LL LLL LL LL LLLL LLLL L L LL LL LL LLL L LLL LL L LL LL L LLLLLLLL LLLL L L

L LLL L LL LLLLL L LL L LLL L LLLLLL LL LLL LL L LLL LL L L L LLLL L L L LLLL LLLL L L LLL L LLLL LL LLL LLLLLLLLLL LLLL L L LLLLLLLL LLLL LL LL LL L LL L LLLL L L LL LLLLL L LLL LLLL LL L LL L LLL LL L L LLLLLLLLLLL LLLLL LLLLL LLLLLL LLLLLL LL LL L LL L LLL L LLL L L L L L LLL LLLLLLLL L L LLL L L LLL LLL L LLL L LL L L L LLL LL LL LLL LLL LLLL L LLL LL LL L LLLL LLLLL LLLL LLLLL L L L LLLLL L L LLLLL LLL L LLLLLL LLLL L LL LLLL L L LLLLLL L LL LLL LLLLLL LL LLLLLLL LLL L LL LL LL LLL L LLLL LL LL LL LLL LLLL LLL LLL LL L LL LLL L L LLL LL LLL L L LL LLL LLLLLLLLL L L LLL LLL L L LL LL L L LLLLLLLLL LLL LL L L L L LLLLLLL LLL L LLL LL L L L L LL L L LL LL L L LL L LLLLLL LL LL LL LL LLL LL LLL L LLL L LL L LLL LL L LLLLL LLL LLLLLLLLLLL L L L L L LLL LLL L LL LLLLLL LL L L L L LLLLLLLL LL LL LL LL LL LL LL L L LL LL LL LLLL L LLLLLL LLL LL LLLL LLLLLLLLL L LL LL LL LL L LL LL LL LLL L LLLLLL L LLLLL LLLLLL LL LL LL LL LL LL LLL L L LLLL L L L LLL LLLL L LLL LLL LLL LL L LL LL LL LL LL LL LL LLL LLL LLLLL LLL LL LL L L LLL LLL LL LLLLLLLLL L L L L L L L LLL LL L L L L LLLLL LLLL LLL LL LL LL LL LL LL LL LL LLL L LLLLLLL LLLL L L L LL LLLLLLLLLL LLL LLLLL LL LLLLL L L L L L L LLL LLL L LLLL L L LLLLLL L L LL LLLL LL LL LLL LLLL L LLLLLL L LLLLLL L LLL LLL LLL LLL LLL LL LLLLL L LLLLLLLLL LL LL LL L L LL LLLLLL LL LLL LL LLLL LL LLL L LL LL LL LL LL LL L LL L LL L L LL LLL L L L LL LLLL LL LL L L L LL LL LL LL LL LL LLLLLL LLL LL L LL LL LL LLLLLLL L LLLLLL LLLL LLL LLLL L L LLLLL L L L L L L L L LL LL LL LL LLLLL LLL LL LL LL LL LL LL LL LL LL L LL LL LL LLLL LL LLLL L LLLL L L L L LL LLL LLLL L L LL LL L L L L L L L L LLL L L LLLLLL LLLL L L L L L L L L L L L L L L L L L LL LLLLL LLLL LL LLLLLLLL L L LLLLLLLLLLLLLLL L LLLLLLLLLLL LLLLLL LL LL LL LL LL LL LL LL L LL L LLLLLLLLLLLLL LLLLLLLLLL LLL LLLLLLLLL LLLL LL LL LL L L LL L L LLLLLLLLL L L L L L L L LLL LLLL LL LL L LL LLL LLLLLL L LL L L L L L L LLLLLL LLL LLLLLLLLLLLLL LL L LLLLL L L L L L L L L LL LL LLL LL LL LLLLL LL LLL LLLL L LLL LL LL LL LL LLLL L LLL LLL LLL LLL LLL LLL LLL LLL LLL LLL L LLLLL LLL LL L L L LLLLL LLLL L L LL LLLLL LLL L L L L L L L L LLLLL LL LLL LL LLLLLLL LLLL L LL LLL LLLL L LL LL LL LL LL LL LL LL LLL LLLLL L LLLLL LLLL LL L LLLLL LL LL LL LL LL LL LL LL LL LLL LLL LLLL LLL L LL L LLLLLL LLLL LLLLLL LLL LLLLL LLL LL L L LLL L L L L L L L LLL LL L LL LLLLLL L LLL LL LL L LLLLL L LL LL LL LL LL LL LL LL LL LL L LLLL L LL LLLL LL L LL LLLLLLL LL LLLL L LL L LL L L LL LL LL LL LL LL LL LLL LLLLL LLLL LLLLL LLLLL LLLLLLLL LLLLL LLL LLL LLL LLLL LL LL LL LL LL LL LL LL LL LL LL LLLLL L LLL L L LLLLLLLL LL LLLLLL L LLL LL LL LLL LL LL LL LL LL LL LL L LLL LLL LL L LLL LLLLLLLL LLLLLLLLLL L LLL LLL LLLL LLLL LLL LL LLLLLL LL LLL LL LLLL L LLL LL L L L LLL LLLLL LL LLL LLL LLL LLL LLL LLL LLLL LLLL LL L LLLLLL LLL L LL L LLL LL LL LLL L L LLL L L LL L LL L L L L L L L L L

L LLLL LL LLL LL LL L LL L LLLLLLL LLLLL L L LLLL L LL LLLL LLL L LLLLLLL L LLLLLL L LL LLL LL LLL LLL LLLLLLL L LLLL LLL LLLLLL LLL LLLLLLLLLLL LLL L LLL L LL LL L LLLLLL LLL LL LL LLLLL L LLL LL LLL L LL LL LLL LL L LL L L LLLLL LLLLLLLLL LLLLLLLLLLLLLLLL L LL L LLLL LL LL LLL L L L LLLLLLL L L LLL LLLLL L LLLL LLLLLLLL LLLL LL LLL LL L LLLL L LLL LL LLLLL LLLL L LL LLL LLLLLLLLL LLL L L LL LLLLLLLL L LLLLLLLLL LL LL L LL LLLLL LL L L LL LL LLLL L LL LLLLLL LL LL LLL LLL LLL LLLLL LLLLLLLLL L LLLL L LL L LL LLLLLLL L LL LLL LL L L LLLLLLL LLLL L LLLL L LLL LLLLLLL LL LLLLL LLL LLLL LLLL L LLLLL LL L LLL LLLL LLL LLL LL LLL LL L LLLL L L LLL LLL LL L LL LL LL LL LL LL L LL LLL LLLLL L LL LL LLL L L L LL LLLLLLLL LL L L LLLLLLLL L L L LL LL L LLLLL L LLLLLLL LLLLLL LLL LLLL L LLL L LLL LLLLLLLL LLLLLL LLL L LLLLLL L LLLLLL LLLLLLL LL L L LLL LL LLLL LL LL LLLL LLLL LLLL LLLLLL L LLLLL LLLLLLLLLLL LLLLLL LLLL L LL LLL L LLL LLL LL LLL LLLL LLLL LL LL LL L LLL LLLLLLLLLLL LLLL LLLL L LLLL LL LLLLL L LL LL LLLL L LLL LLL LL L LL L L LL LLL LL LLL LLL L LLLLLL LLL LLLLL L LLLLLL LLLL LLLLL LL LL LLLLLL L L L L LLL LLLLLL LLLL L LL LLL LL LL LL L LL LL L L L L LL LLL LLLLLL L LLLLLLL L LLLL LLLLLL LLLL L LL LLLLL L L L LLLLL LLLL LLLL LLL LLLL LLLLL L L L L LL LLLL L LLLLLL L LLLL LLLLLLL LL LLLL LLLLLL LLL LLLLLLL LLL LL LLLL LL L LL LLLLLL L L LL LL LLLLLL LLLLL LLLL L LLLLLLLLLL L LL LLL LLLL LL L LL LLL LLLLL LLLLLL LLLL L L L LL LLLLLLLL LLLLLLLLLL LL L L LLL L LLLLL L L L LLLLL LL LL L LLLLLLLLLLLLLL LL LL LLLLL L LLL L LL L LLLL LLLLLLL LLLLLL LLLLLLL LLLLLLLLL LLLLLL LLL LLL LLL LL LLLL LLLLLLLL LL LLLL L L L LLLL L LLLLLLLLLL LL L LL LLL LLL LLL LL LLL LL LLLLLLL LL LLL L LLL LLL LLL LL LL LLLLLLLLLLL LL LL LLLL LLLL L L LL LL LLL L L L L LLL L L L LLL LL LL LLL LL LLLLLLLL LLL LLLLLL LLLLLL L LLLL L LL LL LL LLLLLLLL LLL LL LLLLLLLLL LL LL LLL L LLL LL LLLLLLL LLLLLLL LLLLLLLLL LL LL LLL LL L LLL L L L LLLLLL LLLLL L L LLL LL LLLLL LLLLLLL LL LL LLLLLL LL L LLLL L LLLLLL LLLL L L L L L L LL L L LLL L LL LL L LL LLLL L L L L LL LLL L LL LL LLLLLLL L LLLLLLLLLLLL LL LL LLL L LL LLLLLL L LLL LLL L LLL LL L LLLL LL LL L LLLL LL L L LL LL L LLLLLLLL L L LL LLL LLL L LL LL LL L LLLLLLLL LL LLLLLL LLLLL L LL LL L LLL LL LL L LLL L LLLLL L L LL LL LLLL LL LLLLLL LL L L LLLLL LLLL LLLL LL LLLLL LLL L LLLLLL LL L LL LL LL LLLLLL LLL L LLLLLL LL LL L L L LLLLL L L LLL L L LLLL LLL LL L LLLL L LLLLLLL LL LL L LLLL L LL LLL LL LL LL L LL L LLLLLLLLL LLLL LLL LLLL L LLL L LLLLL LL LLL LLLLLLLL L LL LL LLLLLLLLLLL L L L LLLL LL LL LL LL LLLLLL L LL LLLL LLLLL LL LL LLLL LLL L LLLLL LLLLL L LLLLL L LL LL LLLLLL LLL L LL L LLLLL LL LL LLL LLLL LL LLLLLL LLL LLLLL L LLL LLLLLLLLL L LL LLLL LLLLLL L LLLLL LLLL LLL L LL L L L L LL LL L L LLLL LLL L LLLL LLLLLLLLL LL LL LLLL L LLL L L LLLL LLL L L LL LL L LL L L LLL LLL LLLL LL LL L LLLLLLLL LLLL L LL LL LL L LLLLL LLLL LLLL L L LLLL L LLLLLLLLL LL LLLLLL LLLLLLLL LLLL LL L LLLLLL LL LL L LLLLL L L LLLL L LLLL LLL LLL LLLL LLL LLLLLLLL LLLLL LLLLLLLLL LLLLLL LLLLLLL LLLL L LLL LLLL LL LLLLLL L LLL LLLLLL LLL L LLLLLLLLLL LL L LL LLLLLLLLLLL LLLLLL L LLLLLL LLLL LL LL L LL LLLLL LL L L LLLL L LLL LLLL L LLL L LLLLL LLLLLLL LL L LLL L LLL L L LLLLL LL LL L LL LLL LL LLLLL L LL L LLLLLL LLLLL L LL LLLL L LLLLL LLL LLLL LLLL L L LL LLL LL LLL L LLL LLLL LLLL LLL LL LL LLLL LLL LL L LLLLL LLLL LL L LL LL LL LLL LLLL L LL LL LLL LLLLLLLL LLLLLL LL L LLLLLLLL L LL LL LL LL L L L L L LLLLL L L LLLLL L L L LLL LLL L L L LLLL L L L LL L L L L L LLL L L LL L L L LLLLL L LLLLL L L LLLL L L LLL LL

L LLL L LLL LL LL L LLL L LLL L LLLL LLL LLL L LL L LLL LL L L L LLL LLLL L L LLLLLL L LL L LL LLL LLL LL LLL LL LLLLLL L LLL LLLL LLL LL L LLL LL LLLL LL LLL LL LLLL LLL LL L LLL LL L L L LLL LLL LLL L LL LL L L LL LL L L LL LL L LLLLLL L LLL L LLLLLL L LLL L LLLLLL L LLL L LLL LLL LLL L L L LL L LLL LL L LLL LLL L LL L LLL LL L LLL LLL L LL L LLL LLL LL LLL L L L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLLLL LL LL LLLLL LL LL L LLLLLL LL LL L LLLLLL LL LL LLLLL LL LL L LLL LLL LLL LLL LLL LLL L LLL LL L LLL LL L LLL LL L LLL LL L LLL LLL L LL L LLL LL L LLL LL L LLL LL L LLL LL L LLL LL L L L LLL LLL L L L LL L L L L LL L LLLL LL LLL LL LL LL L L LLL L L L L L L L LLLL L LLL L LL L LLL LLLL L LLL L LLL L L LLLL L LLL L LLL L L LLL LL LLL L L LLL LL LLL L LL LLL LLL LL LLL L LL LLL LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LL LL LL LL LL LL LL LL LLL LL LL LL LL LL LL LL LL LLL LL LL LL LL LL LL LL LLL L L LLLLLL LLL LLL LLL LLL LLL LLL L LL L LL L LLLLLL L LLLLLL L LL L LL L LLLLLL L LL L LL L LLLLLL L LLLLLL L LL L LL L LLLLLL L LLL LL L LLLLLL L LLLLLL L LL L LL L LLL LL LL LL LL LL LL LL LL LL LL LL LL LL L L L LL LLL LLLL L LLLL L LL L LLLL L LL L LL L LLL LL LL LL LLLL LL L LLLLLL L LLLLLL L LLLLLL L LLLLL LLL LLL LLL LLL LLL LLL LLLLLL LLLLLLL L LLL LLL L LL L LLL L LL L LLL L LL L LLL L L LLL L L L LLLLL L LL L LL LL LL L L L L L L LLL LL LLL L LL LLLLL L LL LL LL LLL LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L L L L L L L LLL LLL LL LL LL LL LL LLL LL LL LLL LLL L LL LL LLLLLL LL LL LLLLL LL LL LLLLL LL LL L LLLLLL LL LL LLLLL LL LL L LLL L L LLLLLL LLLL LLL LLL LLL LLL LLL LLL LLLL LLLLLL LLLLLLLL L LLL LL L LLL LL L LLL LL L LLL LL L LLL LL LL L LLL LL L LLL LL L LLL LL L LLL LL L LLL LLL L LLLL L LL LLLLLL LL LLLL LL LL LLLLLL LL LLLL L L L L LLL LLL LLL L L L L L L L LL L LL LL L LL LL LL LLL L L LLL L L L LLL L L L LLL L LLLL L LLL L L L LLLL L LLL L L L LLLL L LLL L L L LLL LL LLL L LL LLL LLL LL LLL L LL LLL LLL LL LLL L LL LLL LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LLLL L L LL LL LL LL LL LL LL LLL LL LL LL LL LL LL LL LLL LL LL LL LL LL LL LL LLL LLLLLL LLLLLL LLLLLL LLL LLL LLL LLL LLL LLL LLL LLL LLL LLL L LLL LL L LLL LL L LL LLL LL L LLL LL L LLL LL L LLL LL L LLL LL L LL LLL LL L LLL LL L LLL LL L LLL LL L LLL LL L LL LLL LL LLLL LLL LLL L LLL LL LL LL L L L LLLL LLLL LL LLLL LLLL LL LLL LLLL L LL L LLLL L LL L LL LL LLLLLLL LLLL L LL L LLLLLLL LLLL L LL L LLLL LL LLLLLL L LLL L LLL L L LLL L LLL L L LLL L LLL L LLL LLLLLLLLLLLL LLL LLLLLLLLLLLL LLL LLLLLL L LLLLLLL L L LLL L L L LLLLL LLLLLLLL L LLL LLL L LL L LLL LLLLL LLLLLLLL L LLL LLL L LL L LLLLLLLL LL LL LL LL LLLL LL LL LL LL LLLL LLLL LL LLL L LL LLL L LL LLLLL L LL LL LL L L L L L L LLL LLL LLL L L LLL LLL LLL L

RULE 109(01101101) L LL

L

L

L LL L L

LLL LLLL LLL L L L L

L LL L

L

LL LLLLL L L L LLL L LLLL LLLL LLLL

L LL L L LLL L LLLL L L LLLLLLL L L LLLLLL L LL LLL LLLLLL L L L L LL L L LLLLLLL L LLL LLLLLL LL L

L LL L L LLLLLL LLLL L LL L L L LLLL LLLL L L L L L LLLLLLLLLLL L LLLLLL LL LLL L L LLLLLLL LLLL L L LLLL LLLLLLLLLL LLLL L LLLLLL L L L LLLL L LLL

L LL L L LLLLLL L L LL LLL L L LLLLLL L LL LLL LL L LLLLLLLLL L L LL LLLLLLLLL LLLLL L L LLL LL LLLL LLL LL LL LLLL LLLLLLLL LL LLL L L LLLL LLLLLL LL L L LL LLLL LL LL LL LL

RULE 129(10000001)

RULE 147(10010011)

RULE 151(10010111)

RULE 161(10100001)

RULE 165(10100101)

RULE 183(10110111)

F gure 3 DP o reflec on symme r c bu nonqu escen ru es Β1 Β2 Β3 Β4 Β2 Β6 Β4 1

a ec ed by memory n he scenar o o F gure 2 a

H story however has an unexpected effect on most of the ru es whose damage propagat on d rect on a ternates ΛL  ( 1 1) ΛR  (1 1) F gure 4 shows the case of fu y asymmetr c Ru es 43 57 and 184 Part cu ar y cur ous are the DP generated for Α > 0 6 n Ru es 43 113 The DP of Ru es 184 226 are rect fied when h story s taken nto account (that of Ru e 184 becomes ext nct for Α n [0 6 0 8]) Ru es 184 and 226 have proved part cu ar y effect ve n so v ng the dens ty prob em Comp ex Sy em 14 2003 99–126

R. Alonso-Sanz and M. Mart´ın

116

α 1.0

0.9

RULE 30(00011110) L LL LLLLLL LLLLLL L L LLL LLLLLLL LL LL LLL LL LLLL LLLLLL LLL LLLLLLLL LL LLLLLL LLL LLLLLLLL LLLLLL LLL LLLLLLLL LLLLLL LLL LLLLLLL L LLLLLLLL LLLLLLLLLLL LLLLLLLLL LLLLLLLLLLL LLLLLLLL L LLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLLL LLL LLLLLLLL LL LLLLLLL L LLLLLLLL LLLLLLLLL LLL LLLLLLLL LLLLLLL LLL LLLLLLLL LLLLLLLLL LLLLLLLLLLL LLLLLLLL LLLLLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLL LLLLLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLLL LLLLLLLLLLL LLLLLLL LLLLLLLLLLL LLLLLLLLL LLLLLLLLLLL LL LLLLLLL L LLLLLLLL L LLLLLLLLLL LLLLLLL LLL LLLLLLLL LLL LLLLLLLL LLLLLLLL LLLLLLLLLLL LLLLLLLLL L LLLLLLLL LLLLLLL LLLLLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLLL LLL LLLLLLLL LL LLLLLLL LLL LLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLL LLLLLLLLLLL LLLLLLLL L LLLLLLLL LLLLLLLLL LLL LLLLLLLL LL LLLLLLL LLL LLLLLLLL LLLLLLLL LLLLLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLL LLLLLLLLLLL LLLLLLLL LLL LLLLLLLL LLLLLLLL

0.8

0.7

≤ 0.5

0.6

[0.243,1]

LLL L LLLLL LL L L LLLL LLL LL LLLL LLLL L LLL LLL LL LLLLLL LL LLLLL LLL L LL LL LLL LLL LL LLLL LLL LL LL L L LLLLLL L LLLLLL L L LLLL LLLL LL L L L L LLL LLL LLL L LLLL LLL L LLL L LLL L LLLL LLL L LLLL LLL L LLLL LLL L L LLL L LLL L L LLL L LLL L L LLLL LLLL LLL L LLLL LLLL LL L L LLLL LLLL LLL L LLLL LLLL LL LLLLL LLLL LLL LLLLL LLLL LL LLLLL LLLL L L LL L LLL LLL L L LL LLL L L LL LLLL LL LL L L L LLLLL LL LLL LLLL L L LL LL L LLLLLLLLL LLLLL LLLLLL LLLLL L LLLLLL L LL LLLL L L LLLL LLL LLL L L LLLLLL LLLLL LLL L L LLL LL LLL LLLLLL LLL LL LLL

L LL L LLL LL LLLLLL L LLLL LL LL LL L LLLLLLL LLL L L LLLLL L LLLLL L L L L LLLLLLLL LL L LLLLLLL LL LL LLL LLL LL LLLL L LLLLLLL LLL LL LLLL L LLLLLLL LLL LLLL L LLLLLLL L LLLLLLLLLLL L L LL L LLLLL LL LLLLLLLL LLL LLL L LLLLL LLLLL L LLL LLL LLLL LL L LL L LLLLL LLL LLL L L LLLLL LL LLLL LLLLL LL LLLLLLLLLL LLLLLLLLLLL LL LLLLLLLLLLL LLLL LLLLLLLLLLLL LLLL LLLL L LLLLLLLL LLL L LLLL LLLLLLLL LLL LLLLL L L L LLLLLL LLL LLLLL L LLLL L LLLLLL LLLLLL LLLL L LLL LLLLL LLL L LLLLLLLLLLLL LLL LL LL LLL LLLLL LL LLLLLLL LLL LL LLLLLLLLLL LLL LL LLLLL L L LLL LLLLL L L LL LL LL LLL LLLLLL LLL LLLLLL L LLL L LL LLL L LL L LLLLLL LL LLLLL LLLL L LLLLLLLL LLLLL LLLL L L LLLLLL LLLLLL LLLL L L LLLLLL LLL LLLLL LL LL LLLLLLL LLLLLL LLLLL LLLLL LLLLLLL LL LLLLLL L L LLLLLLL LLL L LLLLL LL LLLLLLL LLL L

L LL L LLL LL LLLL L LLL L LLLLL LLL LLL LL L L LLLLL LLL L LLL LLLLLLLLLL LL LL LLL LL L LLL LLL LLLL LLLL LLLLL LL L LLLLLL L L LLLLL LL LL L LLL LL LLL LL L LLLL LLLL LL L L LLLL LLLLL LL L LLLLLL LLLL LLL LLL LLLLL LLL LLL LL L LLLLLLL LLLLLLL LL LL LLL LLLLLLLLL LLLLLLLL L L LLLLLL LL LL LLLLLLLLL LLLLLLL LLLLLL L L LLLLLLLL L LLLLLLLL L LLLLLL LLLL LLL LLL LLLLL LLL LLL LL L LLLLLLL LLLLLLL LL LL LLL L LLLLLLLL L LLL LLLLLLLLL LLLLLL L L L LLLLLLLL LLLLLLL LLLLLL L L LLLLLL L L LLLLLLLL L LLLLLL LL LLLL LLL LLLL LLLLL LLL LL L LLLLLLL LLLLLLL LL LL LLL L LLLLLLLL L LLL LLLLLLLLL LLLLLL L L L LLLLLLLL LLLLLLL LLLLLL LL LLLLLLLL LLLLLLLLL L LL LLLLLL LLLL LL LLLL LLLLL LLL LL LLLLLLLLLL LL LL LLL LLLLLL LLLLLLLL L L L L LLLLLLLLL LLLLLL L L L LLLLLLLL LLLLLLL LLLLLL LL LLLLLLLL LLLLLLLL

L LL LL LLL LL LLL L LL LL LL LL LL LLL LLL LLL L L LL

L LL LL LL LLL LLL LLL L L LL L LLL LLLL LLL LLLL L L LLLL L L LL LLLL L LLL LLL LLL L LL L L LL LL L LL LL LL LL L LLL LLL L LL L LLLL LL LLLL L L LLLLL LLL LL LLL LL LLL LL LL LL LL L LLL LLL LL LL L LLL LLLL LLLLLLL LLLLL LLL LLLL LLL LLL L LLL L L LLL LLLL L LL LL LLLLL LL LLLL LLLLLL LLLL LL LLLLL LL LLLL LL LL LLLL L LLLLLL LLL L L LLL LLLLLL LLLL L L LL LLL LLLLL L L L L LL L LLLLLLLLL LLL LL LL LL LL LLLL LLL LLL LLL L LLL LLLL LLLL LL LLLLLL LLL L LLL L LLLL LLLLLL LLLL

L LL LL LLL LL L LL LL L LLL LL LLLL LL LLL LLLLLL LLLLL L LLLLLLL L LLL LL L LLLLLL LLL LLLL LLLLLLL L L LL L L LL LLLLLLLLL L LLL LLL LLL L LLL LL LLLL LLLLL LLL LLLL LL L L LLL LLLLL LLL LLLL L LL LLL L LLLLLL LLLL L LL L LL LLL LL LLL L LLLL LL LLLL L L LLLL LL LLLLLLLLL LLLL LLLL LLLLLLLLL LLLL LLLL LL L LLLL LLLLLL LLLLL LLL LLL L LL LLL LLLL LLL LL LLL LL LL LLLLLLLLL LL LLL L LL L LL LLLLLLLL LL LLL LLLL LLLLLLLLL LLLLLL LLLLLL L LLL LLLL L LL LLLLL LLLLL LLLLLLLLL LL LL LLL LLLLL LL L LL LLLL LLL LLL LL LLLLL LL LLLL LL L LLL LLL LLL LLLLLLLL LLLL LLL LL LLLLL LL LLLLLL LLLLLLL LLL LL LL LLL LLLLLLLLL LLL LLL L L LL LL LLL L LL LL LLLLL LLL LLLL LL LL LL LLLLLL L LLLL L LLL LL LLL LL L LLLL LL LLL LLL LLLL L LL L L LL LL LLLL LLLLLLL LL LL LLL LLL LLL L LL L LL LL LLLLL LLL LLLLLLLLLL LLLLLLLLLL L LL L L LLLLL LLLLLL L L L LL L LLLLL L LLLL L LLL LL LLLLLL LL LL L LLLLL LL LLL LL LLLLL L LL L LLL LLL LLLL LL LLLLL L LLLLLLL LLL LL L LL L LL L L LLL L L LL LLL LLLL LLLLLL L L LLLLLL LLLLL LLLLL LL LLLL L LL L L LLL LLLLL LLLLLLLLLLLLLLL LLL LL LLLL LLLL LLLLLLLLLL LL L LL LL LL L LLLLL LLL LLL L LLLLL LLLLLL LL LLLLLLLLL LLLL LLLL LLLL LL LLLLL LL L L LLL L LLLL LLLLLLL LLLL LLL LL LLL LL LL LLLLL L LLLL LL LLLL LLLLL LL LLLL L LLLL LL LLLLLLLL LL L LLLL L L LLL LL L LLLL L L LLLLLLLLL L LL LL LLL LL LLL LLL LLL LL L L LLL LL LLL LL LL LLL L LLL LLL L LL LLLLLLL LL LLLLLLLLLLLL LLLL LL LLL L LL LLL L LL LLL LLLL LL LLL LLL LL L LLLLLLLL LLL L LLL LLLLL LL L LLL L LLLLL LLLL LLLL LLLLLLLLLLL L L LL L LL L LL LLL LLL L LLLLLLLL LLL LLLLLLL LLLL LL LL L L L LL L LLLLLLLLLLLL LLL LLLL LL LLLLLL LL L LL LL LLLLLL LLLLLLLLLLL L LL LL LL LL LLL LLLLLLLL LL L LLLL LL L LLLLL LLL LLL LLLLLLLL LLL LL LLLLLLLL LLLL LL LLL LLLLLLL LL LL L LL LL LLLL LLLL LLL LL LL L LLLL LLL LLLL L LLL LL LL LLL LL LL LL L L LL LLLLL LLLLL L LLLLL LLLLL LLLLLLLLL LL L L LLLLLL LL LLLLLLL LLLLL L LLLL LLL LLLL LLLLL L L LLLLL LL L LL L LLLLL LL LLLLL LL LLLL LLLL LLLLL LLLLLLLLLLL L LLL LLLL LLL LL LLLL L LL LLLL LLLLLL LLLL LLLL LLLL LL LL LLLL LLL LL LL L LLL LL LLL LLLLL LLLLLLL LLLL LLLLLLLLLLL L L LLL LL LLL LLLL LLLLL LL LL LLL L LLL LL LLL L LLLL LL LL LLL L LLLLLL L LLLLLLL LLL LL LLLLLLL LLLL L LLLL L L LLLLL LL LLLLLLLLL L LLLL LL LLLL LL LLL LL LL LLLL LLL LLLLLL L LLLL LL LLLLLLLLLL LL L LLL LLLLL L L LL LLLL LLL L L LLLLLL LLL LLL LL LLLLL L L LLL LL L LL LLLL L LLLLL LLLL LL LLL LLL LLL LL LLLLL LL LLL LLLLLLLLLLL LLL LLLLLLL LLLLLLLL L LLLLL LL LL LL LL L LLLL LLLLLL LLLL LLLLLL LLLLLLLLLLLLLL LL LLLL L LLLLL LL L LLLLLL L LL LLL LLL LL L LL L LL LLLL LLLL L L LL L L L L LL L LL LLL L LL LLL LLL L L LL L LLL LLL LL LL LLLL L LLLL LLL LLLLLL LL LLL LLLL L LLL LL LL L LL LLLLLLLL LLLLLLLL L LL LLL L LLLLL LL LL LLLLL LL L LLLLL LL LLLL LLL L LLLL LL LLLL LLL L LLLL LLL LLLLLLLLL LLL L L LLLL LL LLLLLL LLL LL L LLL LLL LLLLLL L LLLL LL LLL LLLLLLL L LL LLLLL L LLLL LLL

RULE 86(01010110) LLL LLLL LL LLLLL LLLLL L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L L

LLL LLLL LL LLLLL LLLLL L L L L LLLL L LL L L LLLL LLLLL LLLLL L LLLLLL LL LL L L LLL LL LLL LLLL L L LLLL LLLL L L LLLL LLLLL LLLLLLL LL L LL L L LLLLLLL LLLL LL LL L LLLL LL LLLLLLL LLLL LLL LLLLL L LLLL LLLLLLLLLL LLLL LLLL LLLLLLLLL L L LLLL LLL LLLLLLLLLL L L LL L L L LLLLLLLLLLL LLL LLL LL LLLLL LL LLLLLL LLL L LLL L LLL L LLLLL L L LLLL LLLLL LLL L L LLLLL L LL LLLL LLLLLLL LLLLLLLLL LLL LLLLLL LLLLL LLLLLLLLL LLLLL LL LLLLL L LLLL L LLLL LL LL LLLLL L LLLLLLL LLLL LL L LLL LLL LLLLLLL LLLL LLLLLLLLL LLL LLLLL L LL LL LLLLL LL L LLL LLL L LLL LL LLLLLLLLL LLL L LL LLLLL LLLLLLLL L LLL L L LL LL L LLLLLLLLLLL L L L LL LLLLLLLL L LLLLL L LLL LLLLL L LLLLL L L LLLLLL L LLLLLL L LLLLL LLLL LL LL LL L LLLLLL LLLLL LLLL LLL LL LL LLLLL LLLLLL LLLL LL L LL LLLLLL LLLL LLLLLL LLLL LL LLL LL LLLL LLLLLL LLLL L LL LL LLL LLLL LL

LLL LLL L LL LLL LL L L L LLL LL LLLLL LLL LL L L L L L L L LLL L L LL LLL LLLL L LLL LLLLLLL LL L L LL LLLLL LL LL L LLL LLLLL L LL LL L LL LLLL L L L L L LLLL L L LL L L L L L LLL LLLLLLLL LLLL LL LLLLLL LL L LL LLL LLL LL L LLL L LLLL L L LL L L LLL LL L LL LLL LL LLL L LL LL LL L L LLLLLL LLL LLLL LLLLL L LL LL LL L LL LLL LLL L LLL L LLLLLL L LLL LL L LLLL LL LLLLLL L LLLLLL LLLLLLL LLLL L LLLLLLL LLLL L LLLLLL LLLLLLL LLLL L LLLLLLL LLLL L LLLLLLL LLLL L LLLLL LLLL L LL LLLLLL

LLL LLL L LL LLLLL LL L LLLL LL LLLLL LL LLL LLLLL LL L LL L LLLLL LL LL L LLL LLL LLLLL LLLLLL L LLLL LLL L LLLLLLL LLLLLL LL L L LL L LLLL LL LL LLL LLL LLL LL L LL LLLLLL LLL LL LL LLLLLLL LL LLLL LLLLLLLLL LLL L LLLLLL LL LLLLL LL LLLL LL LLL L LLLLLLL LLL LL LLLLLL LLLL LLLLLL LL LL LLL LL LLL LL L LL L LLLL LLLLLLLLLL LL L L L LLL LL LLL L L LL LL LLLLL LL LLL LL LL LLL LLLL L LLL LLLLLL L LLLLLL L LLL LL LLLL L L L L L L L LLL LL L LLL LLLL L L LL LL LL LLLL L L L L L L L L LLLLLL L LLL L LLLLLL LLLL LLLLL LLLLLL L LLL LL L LL LLLLLL LLLLL LLLL LL LL L LL LLL LLL L LLLL LL L LLL LLL LLL LLL L LLLLLL L L LLL LL LLL LLL L LL L LL LL LL LLL LLL LLLL LL LLL L LLL LLLLLL LL LLLL LLLL L LL L LL LLLL LLLLL LLLLLL LLLLLL L LLL LLL LL LLLLL LLL

LLL LLLLL LL LLLLL L L LLLL LL LLLL L LLLL LLL LLLL LL LLL LL L LLL L L LLLLLLLLLL LLLLL L LLL LL LLLL LLLLLLLLLL LLLLL LL LLLLL LLLLLLLL LLLL LLLLL LL LL L LLLLL LLLLL L LLL L LL LL LLLL LLLLLLLLL LLL LLLLL L LLLL LL LL LLLLL LLLLLLL LL LL L LLLLL LL LLLLL LLL LL LLLL LLLLL LLLLLLLL LL LLLLL LLLL L L LLLL LL LL LLLL LLLLLL L L LLLLLLLL LLL LLL L LLLL L LLLL LLLL LLLLL LLLLL LLLL LL LL LLLL LLLL LLLLL L LLL L LL LLLLL L LLLLLLLLL LLLLL LL LLLLL LLLL LLLLLL LLLLL LLLL LL LL LLLLL LLL LLLL LLLL LLLLLLLLL LL LLL L LLLLLLL L LLL L LL LL L LLLL LLLL LLLLL LLLLL LLLL LLLLL LL LL L LLLL LLLL LLLL LLL L LLL L LLL L LLLLLLLLL LLLLL LL L LLLL LLLL LLLLLL LLLLL LLLL LLLL LL LL LLLL LLLL LLLLL L LLLLLLLL LLLLL L L L L L LLL LLLLL LL L LLLL LLLL LLLLLL LLLLL LLLL

LLL LLLLL LL LLL LL LLLLLL L LLLL LLL LLLL LLLLLLL LL LL L LLLL LLLLLL LLL LL LLLLLLL LLLL LLL LLLLL LL LLL LLL LLL L LL LL LLL LLL LLLLL L LLLLLLL LL L LL L LL LLL LLLL LLL LLL LLLLLLLL LL LLLL L LLLLLLLLLL LLLL LLL LLL L LLL LLLLLLLL LLLL LL LLL L LL LLLLLLL LL LL LLLLL L LLLLLLL LLLLLLLL LLLLLL L LLL L LL LLL L LLLL LLLLLL LLLL LLLL LL LLL L L LLLLL LL LL LLL L L LL L LL LLL LLLLL LLLLLL LL LLLLLLLLL LL LLLLLL L LLLLL L LLLLL L LLLLL LLLL LLL LL L LLLL LLL LLLL LLLLLLLLL LLL LLL LLLL LLLLLL LL LL L LLLL L LLLL LLLLLLLL LLLL LLLL LLL LL L L L LLL LL LLL LL LL LL LLLLLLL LLL LL LLLLL LLLL LLLLLLLLL LL LLL LLLLLL LL LLLLLLLLLL LL L L LLLLL LLLL L LL LLL LL L LL LLL LLLLLLL LL L LLL LLL LLL L LL LLLL LL L L LLL LLL LL LLL LL LLLLLLLL LLLL LLL LLLLLL LL LLLLLLLL L LLLLLLLL L LLLL L L LLL L LLL LL LLL L LL LLL L LLLLLLLLLLLL LLLL LLL LL LLLLL LL LLLLLL LLLL LL LLL LLLLLL L LLL LL LLLLLLL L LLLL LLLLLLLLL LLL L LLL LLL LL LL L LL LLL LLLLLLLLLLLL LL LL L L L LLL LLL LLL LLLLLLLL LL LLL LLLLLLLL LL LLLLLLLLLL LLLL LLLL LLL LLLL LLLLLLL LL LLLL LL LLLL LLL LLL LLLL L LL LLLLLL L LLL LL LLLL LLL LLL LL LLL LLLLL LLLLLLLL L LL L LL LL LL LLLL LLLL LL LLL LLLLL LL LL L L LLL LL LLLLLLL L L L LL LLLLLLL L LLLLL L LLLLLLLLL LLLL L LLL L LL LL L L LL L LLL L LLLLLLLL LL L LL LLLL LLLLL LL LL L LLLL LL LL LL LL LLL LLL L LL LLLLLLLL L LLLLL L LLL LLL L L LLLLLLL LLL L L LL LLL L LLL LLL LLL LLLL L LLLL L L L LLL LL LL LLLLL LLLLL LLLL LLL LL LLLL LLLL L LLLLLLLL LLL LL L L LLLLL L LL L L LL L L LL LL L L LL LLL L LLLL LLL LL LLLLLL LLL LLL L LLLLLLLLLLL LL LLLL L L LLL LLL L LLLL LLLL LLLLLLLLLLLLLLL L LL LL LLLL LL L LL LLL L LLLLL LL LLLL LLLL LL L LL LLL LLLL LL LLLLL LL L LLLLLLLLLLLLLLL L LL LL LLLLLL LL LLLLL LL L LL LLLLLL LL LL LLL LLLLL LL L LLLLLL LLLLLL L L LLLLLLLLL LLLLLLL LL LLLL LLLLLL L LL L L LL L LL LLLLL LLL LL L L LLLL LL LL LLL LL LLL LL LL LLL LLL LLLL LLL LLLLL LLLL L LL LLLLLL L LL LLLLLLL LL LLL LLLLL LLLL L LLL LL LLLL LL LL LLL LL LLLL LL LL LL LLLLL LLLLLLL LLLL LLLLL LL L LLLL LLLLL LLLL LL LLLLL LL LL LL LLLLLLLLLLL L L LL LLL LL L L LLLL L LL LLL L L LL L LLLLLL LL LLLL LL LLLLLL L LL LL L LLL LLLL LL LLL LLLLLLLL L L LLL L LLL LLL LL LLLLLLL L LLL L LL LL LL LL LL L LLL LL LLLLL LL LLLL L LL LLLL LLL LL LL LL LL LLLL L LL L LLLLLL LL L LL LLLLLLL L LL L L L LL LLLLL L LL LLLLLLLL LLLLL LLL LL LL LLL LL L LL LL LL LL LLLLL LLLL LLL LL LL LLLLL L LLLL LL LL LLLL LL LLLL L LLLL LLL L LL LLLLLLLLLL LLL LLLLLLLL LLLLL LL LLL L LLLL LL L L L LL L L LLL L L LL LL LLLL LL LLL LLLL L LLLLLL LL LL LL LLLL LLLL L LL L L LL LL LL L LLL L LLLL LLLLLL L L LL LLLLLLLL L LLLL LLL L LLL LLLLL LLL LLLLL LL LL LLL L L L L L LLL L LLLL L LLLLL LLL LLL LLLL LL LLL L LL LLLLLLL L LLLL LLL LLL LLLL LLL LL L LLLLL LLL LL L L LLLL LLL LLL LLLLLLL LLLL L LLL LLLLLL LLLLLL LLLLLLL LLLLL L LL L LLLLLLLLLLLL LLL LL LL LL LLLLLL LL L LLLLL L LLLLL LLLL L LLL LL L LL LLLL LLLLL LL L LLL LLLL L LLLL LLL L LLL LL L LLLLLL LL LLLL LLL LLLLLLL LL LLLL LLL LLL L LLL LL LL LL LL LLLLLLLLL LLLLLL L LLL LLL LLLL LLLLLLL LLLLLL LL LLL L LL LLLLL LL LLL LL L LLL LLL LLLLLL LLLL LLLLLLLLLLL LL L LLLLL LLLL LL LLL LLLLLLLLL LL L L L L L L LLL L LLL LLL L L LLLL LLL LLL LL LL LL L LLL L LL LL LLL LL LLLLLLL L L LL LL L L L LLL L LLL LLLLL L LLL L L LL LLL LLLLLL LLLL L LLL LL LLL LLLLL LL LL LLL L LL LL LLL LLLLL LL LLL LL LLL L LL L L LL L LLLLLLLLL LL LLLL LLLL LLL LLLL LL LL LLL LLL LLL LL LLLLL LLLLLLLL LL L LLL LLLLLLLLL LL LLLL LLLL L LLLLLL L L LLL LL LL LL L LL LLLLLL L LL LLL LL L L LL LLLL LL LLLLLLL LLL L LLLLLL LLL LLL LLLLLLLL LLLL LL LL LLLLLLLLLLL L LL LL LLLL LLL LLL LLLL L L L LLLLL LLLLL LL LLLLL LLL

RULE 135(10000111) LLLLL L LLLLLLLL LLLLL LL LLLLL LLLLLLL LLL L LLL L LLLLLL LLL L LLLLL LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L L LLL L LLLLL LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L L LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLL LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L L LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLL LLLLLL LLL L LLLLLL LLL L L LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLLL LLL L LLLLL LLLLLL LLL L

LLLLL L LLLLLL LL LLLLL LL L LLL LL LL L LLLL LLLL LL LLL LL LLLLLL LLLLLLLL LL L LLLLL LLL L L LL LLLL LLL L LLLLL LLL L LLL LLL LLLLL L LL LL LL L LLLLL LL L LL LL L LL LL L LLL LLL LLL LL L LLLL LL LLL LL LL L LL L L L LL LLLL L LLLLLLLL LLL LLL L LLLL L LL LLL LLL LLLLLL LL LLLLLL L LL L LL LL L LLL LLL LL LLLLL L LL LL LL LL LLLLLL LLLLL LL LL LLL LL LLL LLL LL LL LL LL LL LLLLLL LL LL LL LL LL LL LL L LL LL LL LL L LL LL LL LL L LL LL LL LL L LLLLLLL LL LL L LL LLLLLLL LL LL L LL LLLLLLL LL LL L LL LLLLLLL LL LL L LL LLLLLLL LL LL L LLL LLLLLLL

LL L LLL LLLLLL LL LLLLL LL L LLLLL L L LLLL L L LLL LLLL LLLLL LL L L LLLLLL LLL LLLLL LLL LLLLL LLLL LLL LL LL LL LL LLLLL L L LLL L L L LLL LLL L L LLL LLLLL LLLL L L L L LL L LLLLLL LL LLL LLLLLLLL LL L LL LL LL LL L L LL LL LL L L LL LLLLL LLLLLLL LLL LLLLLL LLLLL LL LL LLL LLLLLLL LL LLL LLLLL LL L LLL LL L LL L LLLL LLLLL LL L LL LLLLLLLLLLLL LLLLLLLLLLLL L LLLLLLLL LLLL LLLLLLLL LL LLLL LL L LLLLLL LLL LL LL L LLLLL LL LL L LLLLL LLLLLLLL LLLLLLLL L L LLL LLL LL LLL LLLLL LLLLL LLL LL LLLL LLLLLL LLLLL LLLLLL LLLLLL LLLLLLL LLLLLL L LLLLLL LLLLLL LLL L LL LLL L

LL L LLL LLLLLL LL LLLL L LL L LLL LL L LL LLLL LLL LL LLLLL LLL LL L LL LLLLL LLLLLL LLLLLL LL LLL LLL L LL L LL LLLL LLL L LLL LLLLLLLLL L LLLLLL L LLLL LL LLLL LL LL LLLLLLLL LLLLLL LLL LL LL LLLLL L L LLLL LLLLLL L L LLLL LLLLLLL L L L LLLL LL LLLLL LLL LLLLLLL LL LLLLLLLLLLLLLL LLL LLLLLLL LLL L LLLLLLLLLLLLLLLLL LLL LL LLLLLL LLL LLLLLLLLLLLL L LLLLL LLLLLLLL LL L LLL L LLLLLL L LLLLL LL LLLLLLL LL LLL LL LLLLLLLLLLLL LLLLL LL L LL L LLLL LLL LLL L LLLLLL L LLLLLLL LLL LLLL LLL LLLLLL L LLLLLLLLLL LL LL LLLLLLLLLLL LLL LLLLLLLLLLLL L LLLLL LL LLL LL L L LLLLLLLL LLLL LLLLL LL LLL LLLLL LL L LLLLL LL LL LLL LL LLLLLLLLLLL LLLL L LL LLL LLLLL LLLLL LLLLL LLLLLLL LLL LLL LLLL LL LLLL LLLLLLLL LLLLLL LLLLLLLLL LLL LLLLLL LLLLLL L LL LL LLL LL LL LLLLLL LLL LL LLLLL LLL L LLL LL LLLLLL LLLL LL LLL L LLL LL LL L L L LLLL LL LLLL L LL L L LLLL L L L LL LLL LLL L LLLL LLLLLLL LLLLL LLLLL LLLL LLL L LL LLLLL LLLLLL LLL LL L L LL L LLLLLL LLLL LL LLLL L L LLLL LLLLLL L LL LLL LLLL L LLLL L L LLLLLL LLL LL LLL LL LLLL L LLL LL LLLL LL LLLL LL LLLLLL LLLL L LLLLLLLLLLLL L LLL LLLL L LLLL LL LLLLLLLL LLLL LL L LLL LL LLLLL LLLLL L LL L LLL LL L L L LLL LLLLLL LLL L L LLL LLL L LL LLLL

LL L LLL LLL L LLLL L LL LLL LLL LLLLL L LLLLL LLLLL L LLLL LL L LL LLLLL LLLL LLL LL LLL LLL LLLLLLLL LLLL L LL LLLLLLLL L LLL LLL LLLLL LLLLLLLLL LLLL LLL LLLLL LLLL LL L LL L L L L LL LLLLLLLLL L LLLLLLL LLLLL LL LLLLL LLLL L L L LL LL LLL LL LL LLL L L LLL L L LL LLL LLL LL LLLLLLLLLLLLL LLL L LL LLL LL LLLL L LL LL LLLLLLLL L LL L L LLL LLLL LL LL LL L L LL LL LLLL LLLL L LL L LLLL LLL L LL L LLL L LLLL LLLL L LLL LL L LLL L L LLLL LLLLLL L L L LL L LLLL L L L LLL LL LL L L LLL LLL LL LLLL LLL LLL LL LLL LL L LL LL LLL LLLL LL LLL LLL LLLLL LL LLL L LL L L LL LL LL LL LL LLLLL LLLL L LLLLL LLL L LL LLLLLLLLLLL LL L LL L LLL LLLL L LLL LL LL LLLL LL LLLLLLL L LLL LL L L L L L LLL LL LLLL LLLLL LLLL LL LL LL LL LLL LLLLL LLLL LLLL LL LL L LLLLL LL L LL LL LLL L LLL LL LL L LLL LLL LLL LLL LLL LL L LLL LL L LL LL LLL LLL L LLL LL LLLL LLLL LLLL LL L LLLL L L L LLL L L LLL LL L LL LLL

LL L LLL LLL LLLLLL L LL LL L LLL LL LLL LLLL LLL L LLLLLLL LLL LL L LLLLLL LLL L LL LLLL LLL L LL LLLLL LLL LLLLLLL LLL LL LLL LLLL L LL LLLL LLLLLLLLL L LLL LLL L LL LLLLLLLL L LLL L LLLL LL LLL LLLLL LL LL LLLL LLLL LLLLL LLLLL LLLLL LLL LLL L LLL LLL LLL L LLLL L LLLL LLLL LLLLLLL LLL LLLL LLLLLL LLL LL LL LL LLL LL LLLLL LL LL LLLL LL LL L LL L LLLLLLLL LLL L LLLLLL L LLLLL LLLL LLL LL LLLL L L LLL L LLL L LL L LL LLLLL LLLLLLL L LL LL LLLLLL L LLLLLLL L LLLLLL LL LLLLL LLLLL LLLL L LL LLLLLL LLLL LLLLL LLLL LLLLLLLL LLLL LLL LL LL LLL LLLLL LLLLLLL LLL LL LLLLLL L LLL L LLL L LLL LL L LL LLLL L LL LL LLL LLLLLLLLLLLLL LLLLL LLLL LLLLL LLLLLL LLLLLLLL LL L LLLLLLLL L LLL LL L LLL L LL LLL LL LLLLLLL LL L LLLL L LLLLLLLL LL L LL LLL LL LL LLLLL L L L LLL LLLLLLL LLLL LLLLLLL LL LLL LLLLLLL LL L LLL LLLL LLLLLL LLLL L LL L LLLLLL LL L LLLL LLLLL LLLLL LL LL L LL L LLLLLL LL L L LLLL LLL LLLLL LL LLLLLL L L LLL LLLLLLLL LLLL LL LLL L LLL LL LL LLLLLLL LLLLL LLLL LL LL LL LLLL LLL L LLLLL LL LL L L L LL L L LLLL L LLLLLLL LL L LLL L LLL LLLL LLLLLL L LLLLL LLLLL L LL LL LLLL LL L LLLLL L LLLL LLLL L LL LL LLLL LLLL LL L LL LLLLL LLLL LLL LLLLL LLL LL LLL LL L LL LLL L L L L LLLLL L LL LL LL L LLL L LLLLLL LL LL L LLL L LLL LLLLL LLL LLL LLLLLLL LLL L L LLL L LL LLLL LLLL LL LL LLLLLL LLL LLLLL LL LLLLLL LLL L L LLLLLL L L LLL LLL LL LL LLLLL L LLL LL L LL L LLL LL LLLL L L L L L LL LLLL LLLL L LLLL L LLLLLL LLL L LLL LL L LLL LLLLLL LL L LL LL LL LLLLL L LL LLLL LL LL LLLL LLLLLLLLL LLLLLLLL LL LLL LLLL LL LL LLLLLLL LL LLL LLL LL LLL LLLL LLLL L L LL LL L L LL LL L L L LL LLLLL LLL L LL LL LLL L LLLLLL LLLLL L LLL LLLL LL LL LLLL LL L L LL LLL LLLLL LLL LLLLLL LLLLLLLLL LL LLLLLL LLLLLL LLL LL LLLL LL LLLL L LLL LLL LL LL LLL LL LLLLLLLLLL LLL LLL LLL LLLL LLLL L LL LL LL LL LLLL LL LLL L LLLLLLLLLL LLL LL LLL LL L LLL LLLLLLLLLLLL LL LL LL L LLL LLLLL LL LLLLLLLLLL LLL L LLL LLLLL LL LLLL LL LLLLLLL LL LLLLLLL LL LLLLL LLLLLLL L LL LLLLLLLL L LLLL LLL LLLLL L LLLLL LLL LLLLL LLLL L LLLLLLLLLLL LLLL LL LLL LL LLL LLLLLL LLLL L LL L LLLL LL LLLLLLLLLL L LLL LLLLLL LLLLL LLL LL L LL LL LL LLL L LL L L L LLL L LLL LL LLLL LLLL L L L LLLLL LL LLLLL LLLL LL L LLLL LL L LL LL LLLL L LL LLLL LLLLLLLLL LL LLL LLLLLLLL LLLL LL LLLLLLLLL LL LL LLLLLLLLL LLL LLL LL LLL L LLLLLL LLLL LLLLL LLLLLL L L LLL LL LL L L L L LLLLLL LLL LLLL LLLLL LL LL LLLL LLLL LL LLLLL LL L LLLLL LLL L LLL L LLLLLLLLL LLL LL L LLL LL LLLL LLL LL L LLLLL LLLL LLLLL LLLLLLL LL LL LLL LLLL LL LLL L LLLL LLL LLL LLLL LL LLLLLLLLL LLLLLLL L LLLLLLL LLLL LL LLLLL LLLLLLLLL LL LLLLL LLLLL LLLLLLLLLLLLL LLLLL LLLL LL LLL LL LLLLLLL LL LLL LLL LLLL L L LLLLL LL LL LLLL LLLLLLLLL LLLL L LLL LL LLLL L LLLL LLLL L LLL LLLL LL LL LLL L LL L LLLL LLL LLLLLLLLL L L L LLLLL LLLLLLL LLLL L LLLL L L L LL LLL L LLL LL LLLLLLLL L LLL LL L LLLLL LL LL LLLLL LL L LL LL L LLL L LLL LLLL LLLLL LLL LL LLL L LLLLL LLL LL LLLLL LLLLL L LLLLL LLLLL LL LLL LL LLL LL LLLLL LLLLL LL LLLL LLLLL LLL LL LLLLL L LL LLL LLLLLLL LL LLLLL LLLLLL L LLLLL LLLLLLL L LLL L LLLL LL LL LLLLL LLL LLLL LL LLLLLLLL LL LLLLLLL LLL LL LL LLLLLL LLLLLLL LL LL L LLLL LL LLL L LL LLLLLL LLLL L LLL L L L LL L LLL LL LL LLL LLL LLLLLL LLLLLLLL LLL LL L LL L LL LL LLLL LLLLLLLLLLLLLLLLL L LLLLL L L LLLLLL LL LLL LLL LL LLL LL LL LL LLL LLL LLL

RULE 149(10010101) LLL LLLL L LLL LLLL L LLLL LLL LLL LLLLLL LLL LL LLLLL LLLLLL LLL LL LLLLLL LLL L LLLLLL LLL LL L LLL LL LLLLLL LLL LLL LLLLLL LLL L LLLLLL LLL LL LLLLLL LLL LLL LLLLLL LLL L LLLLLL LLL LLL LLLLLL LLL L LLLLL LLLLLL LLL LLL L LLL L LLLLL LLLLLL LLL LL LLLLLL LLL LL LLLLLL LLL LLL LLLLLL LLL LL LLLLLL LLL LL LLLLLL LLL L LLLLLL LLL LL LLLLLL LLL LL LLLLLL LLL LL LLLLLL LLL LL L LLL LLL LLLLLL LLL L LLLLLL LLL LL LLLLLL LLL LL LLLLLL LLL LL LLLLLL LLL LLL LLLLLL LLL L LLLLLL LLL LLL LLLLL LLLLLL LLL L LLLLLL LLL LL LLLLLL LLL LL L LLL LLL LLLLLL LLL L LLLLLL LLL LL LLLLLL LLL LLL LLLLL L LLL LL LLLLLL LLL L

L LL LLLL L LLL LLLL L LLLLL LL LLL LLLL LL LLLL LL LL LLL LL LLL LLLL LL LL LL LLL L L L LLL LLL LLLLLLLL LLLLLL LLL LL LL LLLLLL LL LLL LLLLLL LL LLLLL LLL LLLLL LLL L LLL LL LL LLL LLLLL L LLLLL L LL LL LLLLLLL LLLLL L LL LL LL L LLLLLL LLLLLLL L LLL LL L L L LLL LL L LLLL L LLLL LL LLL LL L LLLLLL LLLLLL L L LL L L LL LL L LLLL LLLLLL LLLLLL LL LLL L L L L LLL LLL LLLLL LLLLLLL L L L L L L LLL L LLLLL LLLLLLL LL L LLL LLLLL LLLLL L LLL LLLL L L LLL L L LLLL LL LLL LLLLL LLLLLLLL L L LL LL L L LLLL L L LLLLLLLL L L L L LLL LLL LLLL L L LLLLLL LLL LLL LL L L L L LLLL LLLLLLLL LL LL L L L L LLLLL LLLL LLL LLL LLL L L LLL L LLLL LLL L L LLL LLLLL LLL LLL L LL LL LL LLLL LLL LL L LL L L LLLL LL LLL LLLL LL LLL LL LLLL LLL L L LLLL LL LLL L L LL LLLLLL LL L LLLLLLLLLLL LLLLLLLLLL LL L LL L LLLLL LLLLLLLL LLLLLLLLL L LLLLL LLLLLL LLL LLL LL LLLL LLLLLLLL LLLL LLLLLL LLL LLLL LLLLL LL LL LL LLLL L LLLL LL LLLL LLLL LL LL LLLLLL LL LLL LLLL LLLLL LL LL L LLLLL L L LLLL LLLLLLL LL LLL LLL LLLL L LL L LL LLL

RULE 43(00101011) LL LLLLL L LL LL L LLL LL LLLLLLL LLLLLLLL LLLL LLLLLLLL L LL LLLL LLLLLLLL LLLL LLLLL LLLLLLLLLLLLLLL LLLL LLLL LLLLLL LLLL L LL LLLL LLLLLL LLLL LLLL LLLLL LLLL LLLL LLLL LLLLL LLL LLLL LLLL LLL L LL LLLL LLLL LLLLLLLL LLL LLLL LLLLL LLLL LLLL L L LLLL LLLL L L LLLLL LLLL L L LL L L LLLL LLLL LLLLL L LL LLLL LLLL LLLLLLLLLLLLLL LLLLL L LLL LLLLL L L L L L LLLLLLLLLL LLLL LLLL LLLLLLLL LLLLL LLLLLL LLLLL LLLLLLLL LLLL L L LLLLLLLL LLLL LLLLLLLL LLLLLLL LLLLL LLLLLLL L L LLLLLLLL L L LLLL L LLLLLLLL LLLL LLLL LLLLLLLL LLLLLL LLLLL LLLL LLLLLLLL LLLL LLLL L LLLLLLLL LLLL L LL L LL LLLLLLLL LLLL LLL LLLL L LLLLL LLLL LLLL LLLL L L LLLL LLLL LLLL LLLL LLLLL LLLL LLLL LL LLLLLLLL L LLLL LL LLLL LLLL LLLL LLLL LLLLL LLLLL LLLL LLLL LL L LLLL LLLL LLLL L L L LL L LLLL LL L L LLLL LLLL L L LLLLLL L LLLL LLLL LL L LLLL LLLL LLLLLL LLL

L LL LLLL LL LLLL LLL LLLL LL LLL LLLLLLLLLL LL LLL LLLL L LL LLLLLL L L L L LLLL L LLLLL LL LLLLLLLLL L LLLL LLLLLL LL LLLL LL LLLLL LL L LL LLLLLLL LLL LLLLL LLL LLL LL LLLL LLLLLLL LLL LLL L LLLL LLLLLLL LL LLL LLLL LLLLLLL L LLLL L LLLLLLL LLLLL LLLL L LL LLL L L LLLL LL LLLL L LLLL LLLLL L L L LLLLLLLL L LL LLLL L L LLL L LL LLLL LLL LL LL LLLL L L LLLLLLLL LL LL LL LLLLLL LLL LL LL LL LLLL LL LL LLLL LL LLL LLL L L L LL LLLLL L LLLLLL LL LLL LLLLL LL LLL LL LLLL LLL LL LL L LL LLL LLL L L LLL L LL LL LL LL LLLLLL LL LL LLLLLL LLLLL LL LLLL LLL LLLL LLL LLLLL LLL L L L L LL L L LLLL L L L LLL L L L LL L LLLL LLLLL LL L L LL LLLL LLLLL LLLL L LLL LL LLLLL LLLL L LLLLLLLLL LLLL LLL LLLL LL LLLL LLLL LL LL LL LLLL LL LLLLL L LLL LLLLLL LLL L LLLLLL L LLLLLLL LLL LLLLLL LLLLLLL LLLLL LLLLLL LLL LL LLL LLLLL LLL LLLLL LLLL LL LLLLLL LLLL LLLLL L L LLLLLLL L LLL L L LL LL LLLLLL L LLL L L LL L L L LLL LLLL L L LL LLL LL LL LL L LLLLLL LLL LLL LL LLL L LLLLL L

L LL LLLL L LL L LLL LLL L LLL LL LLLL LL LLLLL LLLL LLL L LLLLLLL LLL L LLLLLLLLLLLL LLLL L LLL L LLLLL LLL LLLL LL L LLL LLL LLLL LLLLLLL L LL LLL LLLLL LLL LLLLL L LLL LLL LLLLLL L LLLLL LLL LLL L L LLL LL LLLLL L LLL LLLLL LL LLLLLLLLL LL LLL L LLLLLL LLLL LLL LLLLL L L L L LLLLLLL LLLLL LLL L LL LL LLLLLL LL LL LLL LL L LLL LLLL L L L LLLL L LL L LLL LLL L L LL LLLL LLL LLL LLLL LLLLLLL L LLL L L LLL LL LL LL L L LL L LL L LLL L LL LLL LL L L LLLLL LL L LLL LLL L LLLLLLL L LLLLL LLL L LLL LL LLL LLL L LLLLL LLLLLLL L LLLLL LL L LLL L LL LLLL LLLLLLLLLL LLL LLL L L LLLLL L LL LLL LL L L LLLLL LLLLL LLLLLLLL L L L LLL LL LLLL LLL LLLLL LL LL LL L LL L LLLLLL LL LL L LLLLLLL L LLL LLLLLL LLLLL L LLLLL LL LL LL LLLL LLLL LL LLLLLL LL LLLLL L L LLL LLLLL LLLLL LL LLLLL L LLLLLLLLLLLLL LLLLL LLLLLLLLL LL LLLLL L LLLL L LLLLLL LLL LLLLL LLLLLLLLLLL LLLL LL L LLLL LL LLLLL LL LLLLLL LLL LLL LLL LLLLL L LLLLLL LL LL LLL LLLLL LLLLLL LLLLL LLLLL L L LLLLLL LLLLL LL LLLL LLLLLLLL LL LL LLLLLLLLLLLL L LLLLL LLLLLLLLLL LL LL LLLLLLLL LLLL LLLLL LL LLLLLLLLLLL LLLLLL LLLLLLL LLL LLLLLLLL LLLLL LLLLLL LLLLLLLLLLLL L LLLLL LLLLLLL LLLLLLLLLLL LLLLLLLLLLLL LL LLLL LL LL LLLLLLLLLLL LLLLLLLLLLL

LLL LLLLL LLL LLL LLL L L L LLLLL LLLLLL LLLLL LLL LLLL LLLLL LL LL LLLLL LLLLLL LLLLLL LLLLLLL LLLLLL LLL L LLLLLLL LLL LL LLL LLL LL LL LLLLL LLLLLL LLLLLL LLLLLL LLLLL LLL LL LLLLLLLL LLL LL LLL LLL LL LL LLLLL LLLLLL LLLLL L L LLLLL LLLLL L LLLLLL LLL LL LL LLL LL L L LLL LL LL LLLLL LLLLLL L LLLLLL LLLLL LLLLL LLL LLL LLLLLLL LLL LL L LLLLL LL LL LLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLL LL L LLLLL L LLL LLL LLLLL LL LL LLLLL LLLLLL LLLLL L LLLLLL LLLLLL LLL LL LLLLLLL LLL LLL LLLLL LL LL LLLLL LLLLLL LLLLL L LLLLLL LLLLLL LLL LL LL LLLLL LLL LLL LLLLL LL LL LLLLL LLLLLL LLLLL LLLLLL LLLLLL LLLLLL LLL LL LL LLL LL

L LL LLLL LLL LLLLLL LLLLLL L LL LL LLL LL LLLLL L LLLL LL LLLLLLL L L LL LL LLLLL L LLL LL LL L LLLL LLLLLLLL LLL LLL LLL LL LL LLL LLL LLLL LLLLL LLLLL LL L LL LLLL LL LLLL LLLLLLL LL LLLLLLL LL L LL LLLLL LLL L LL LLLL LL LL LLLL LL LL LLL LLLLLL LL LLLLLLLL L LL LLLL L LL LLLL L LLLLL LL LLLLLL LLLL LLL LLLLLLLLL LLLL LLLLL LLLL L LLLLL LLLLLLL LLL LL LLL LL LLL LLLLLLLLL LLLL LLLLL LLLLL L LL L L L LL L LLLLLL LLLLL LL LLL LLLL LLLLL LLL LLL L LLLLLL LL L LLLL LLLL L L LL LLLLLLLLL LL L LLLLLL LL LLL L L L LLL LLLL L LL LL LLLL L LL LLL L L L L L L LLL L L LLLLLL L L L L LL LLLL L LL L LL LLL LL L LLL LLLL L L LLLL LLLL LLLL LL LL LLL LLLL LLLLL L LLLLL LL LLLLLLLL L L LLLLLLL LLLLLLL LLLLLLL LLL LLL LL LL LLLLLL LLLLL LLLLLL L LL LLLLLL LLLLLLLLL L LLLLL LLLLLLLLL LLLLLLLL L LLLLL LLLLLLLLLL LL LLLLLL LLLL LL LLLLLLLL LLLLL LLLL LLLLL LL LL LL LLLL LLLLLL LLL LLLLL LLLL LLL LLLL LLLL LLLLLLLLL LLLLLLLLL LLLLLLLL LLL L LL LL LLLL L L LL LL L LL L LL LLLL LL LLLL LLL LLLLL LLLLL L LLLLLL L LLL LL LLLLL LLL LL LLL LLLLLL LL LL LLLL L LL LL LLL LLLLL LLL LLLLLLLLL LLL L LL LL LLL LL LL LL LLLLL L LLLLL LLL LLLL L L LLLLLL LL LL LL L LLLL LLL L LL LLLL L LLL LL LL L LLLLLL LLL LLL LLL LL LLLLLLLL LL LL LLLLLL LLLL L LL LL L LLL L LLLLL LL LLLL LL L LLLL LLLLL LL LL LLL LLLLL LL L LL LLL LLL LLL LLLL LLLLL LLL LL LL LLL L L LLL L LLLL LLL LLLLLLL LLLL LLLLLL LLLLL LLLL LLLL LLLLLLLLLLL LL LLLLLLLLL L LLL LLLLLLL LLL LLLLL L LL L L L L L LLLLLL LL LL LL L LL LL L LLL L LLL LLLL LLLL L LLL LLL LLL LL LL LL LLLL L L LLL LLLL LLLL LLLL LL LL LL LLL LLL LLLL L LLLLLLLLLLL LLLLL L LLLLL LL LLLLLLL L LL LL LLL LLLLLL LL L LL LL LL L L L LL L LLLLLLL L L LL LL LLL LL LLL LL LL L LLL LL LLL LL L LL LLL LLLLLL L LLL LLL LL LL LLLLLL LLL LLLLLLL LLL L LLLLLLLLL LLLL LLLLL LLLLLL LLL LLL LLLLLL LLLL LLL LL LLLL LLL LLLL LLLLL LLLLLL LL L LL LLLLLLLL LLLLLLLLL LL LLLLLL LLLL LLLLLL LL LLLLL LLLLLL LLLL LLL LL LL LL L LL LLL LL LL LL LLLLLLL LLL LLLLLL L L LL LLLLL LL LLL LLL LL LLL LL LL LL LLLL L LLLLL L LL L LL LLLL LL LLLL LL LLL LLL LL LLLL LLLLL LLLLLLLLLLLL LLL L LLL L LLLLLL LL L LLLLLL LLL LLLLLLLLL LLLLL L LL LL LLLLL LLLL LLLLLL L LL LL LLLLL LLLLLL LL LL LL LLL LL L LLLL LLL LLLL L LLLLLLL LL LLLLLLL LL LLL LLLLLL LLL LLLLLLL L LLLLLL LLLLLLLLLLLL LLLL LL LLL LLLLL LLLLL LL LL LL LL LLL LLL LL LLLL L LLLLL LLLLL LLLL LLL LLLLLLLLL LLLL LL LLL LLL L LLLLL L LL LL LL L LL LL L LL LLL LLLLL LLLLLLLLLLLLL LLLL LLL LLLL L LL L LL LLL LLLLLL LLLLLL L L LLLLLLLLLL L LLLLLLL L LLLLL LLLLLLLLLLLLL LLL LLLLLLLLLLL LL LLLLL LLL LLLLLL LLLLLLLLLL LLLL LLLLLLLL LLLLLLLL LL LLL LLLLL LLLLLL L LLL LLLLLL LLL LLLL LLLLLLLLL L L L LL LL L LLL L LLLLL L LL L LLL L L LLLLLLLL LLL LLLLLLLLLL LLLLLL LLLLLLLLL LLLL L LL L LLL LLLL LLLL L LL LLL LLLLLLL LLLLLLLL LL LL LLLLLLL LL LL LLLL L LL LLL LL LL LLLL LLL LLLL L LLLLLLL L LLL LLLLLLLLLLLLLLL LL LLLLLLLLLLL LLLLLL LL LL LL LLL LL LLLLL LLLL LL LL LLLLL LLLLLLLLL L LL L L LLL LLL LLL L LLLLLLLLL L L LLLL LLLLLLL LLL LLLL LLLL L L L LL L LLLL LLLLLLLL LL LLLL LLLL LLLLLLLLL L LL LL LLLLL L LLLL LLL LL L LL LLL LLLLL LLL LL LLLLL LL LLLL LL L LLLLLL LLLLLL LLLLLL LLL LLLLLLLLLLL LLLLLLL L L LL LLLLLL L L LLLL L LLLL LL LLL L LLL LL LLL LL LLL L LL LL LLLL L LLLL LLLLL LLLL LLLLLLLLLL L LL LL L L LL LLL L LLLLL LLLL LLLL LLL L L LLLL L LLL LL LL LL LL LLLLLLLLLLLLLLLL LLLL LLL LLLL LLLLLL LLLLLL LLL L LLLLLLLLLLL L LLL LL LL L LLLL LL LLLL LLLLLL LLL LLL LLLLLL L

[(-1,1),(1,-1)]

LL LLLLL L LL L LLL LL LLLL LLLL LLL L L L LLLLL LL LLLL LLLLL LLLLL L LLL LLLLL L L L LLLLLL L LLLLLLLL LL L LLLLLL LL L L LLLLL L LLLLL L L LLLLLLLLLL LLLLLL LLL LLLLLLLLLL LLLLL LLLLLL LL LLLL LL LLLLL LL LL LL LLLLLL L LLL LLLL LLLLL LLLLLL LLLL L L LLLL LLLLLL LLL LLLL LLLLL L L LLLL LLLLLL LLLL LL LL LLL LLLL LLLLL L L LLLLLLLL LLLLL LLL LL L LLL LLLLL LLLL L L LLLL LLLLLLL LLLLL L L LLLLLL L LLLL LLLL LLLLLLL LLL L LL LLLLLLL LLLL LLLLLL LLL LLLLL LLLLLL LLLL L L LLLLLL LL LLLLLLLLL LL L L LLLLLL LLL LLLLL L L LLLLLL LLLLL LLL L LLLLLL LLL LL LLLLLLL LLLLLLLL LLLLLLLL LLL LLLLLLL LLLLLLLL L LLLL LLLLLLLLL LL LLLLLLLLLLLLL LLL LL LL LLL LL LLLLLLLLL LLL LLLLL LLLLLLLL LL LLLL LLL L LLL LLLLL LLLL LLLL

L LL LLLL LLL LLLL LLL L LL L LLLL L L LLLL LLLLLL LL L L L L LLLLL LL L LLLL LLL LL LLLL LLLLL L L L L LL LLLL LL LLLL L LL L LLLL L LLLLL LL LL L L LLLL LLL LLLL L L LLLLLL LLLL LLLL LLLLLLLLLL LLLLL LLLLL LLLLLLL LLLLLLLL LLLLLLL LL LLL LLLLLLLL LLLLLLL L L L LLLLLLLLL LLLLLLL LLLLLLL LL L LLLLLLLLL LLLLLLL LLLLL L LLLLLLLLL LLLLLLLL LLLLLLL LLLLL LLLLLLLLLLLLL LLLLLLLLLLLL LLLLLLLL LL LL LLLLLLLLLLLLLLL LLLLLLLL LLLLL LLLLL LLLLLLLLLLLL LLLLLLL LLLLL LLLLL LLLLL L LLLLLLLLLL LLLL LL LLL LLLL LLLLL LLLLLLLLLLLLL LLLLL LLLL LLLLLLL LLLL LLLLLLLLL LLLLL L L LLLLLLLL LLLLLLL LLLLLL LLLL LLLLLLLLL LL LLLLLLLLLL LLLLLLLLLLLLL LLLLLLLLLL LLL LLLL L L LLLLLLLLLLLLLLLL LLLLLLLLLLLLLLLL LLLLLLLLLLLLL LL LLLLLLLLLLLLLL LLLLLLLLLLLLL LLLL LLLLLLLLLLLLL LLLLLLLLLLLLL LLLLLLLLLLLLLL LLLLLLLLLLLLL LLLLL LLLLLLLLLLLLL LLLLLLLLLLLLLL LLLLLLLLLLLLL LLLLL LLLLLLLLLLLLL L LLLLLLLLLLLLLL LLLLLLLLLLLLL L L LLLLLLLLLLLLL LLLL

L LL LLLL LLL LL L LLLLLL LL L LL L LLL LL LL LLL L L LL LLLLLLLLLLLLLLLL LLLL L LLLLLLL LLLLLL LL LLLL LLLLLLLLLL LL L LLLLL L LLLLLLLL LLLLLLL L LLLLLL LLL LLLLLL LL LLLLLLL L LLLL LL LL LL L L LLLL LLLLL LLLLLL LLLLLL LLLLLL LLLLLL LL LLLLL LL LLLLLL LLL LLLLLLLL LLLLL LLLLLL LLL LLLL LLLLLL LLL LL LLLL LLLLL LLLLL LL LL LLL LLLLL LLL LLLLL LLLL LLL L LLLLLLLLL LLLLL LL LLLLL LLLL L LLLLLL LLL LLL LLLL LLLLLL LLLLL LLL LLLLLLLLLL LLLLLL LL LLLLL LLLLL LLLL LLL LLL LLL L LLLLL LLLLL LLLLL LLLL LL LL LLLLLLL LLLLLLLLLLLLLL LLLLLL LLLLLLLLL LLL LLL LLLLL LLLLLL LLLLL LLLLL LLLLL LLLLL LLLLLLL LLLLLL LLLLLLL LLLL LLLLL LLLLL LLLLL LLLLLL LLLLLL LLLL LLL LLL LLL LLLLLL LLLLLL LLLLLL LLLLLL LL LL LL LL LLLLLL LLLLLL LLL LLL LLL LLL LLLLLL LLLLLL LLLLL LLLLL LLLLL LLLLL LLLLLL LLLLLL LLLLLL LLLLL LLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLL LLL LLLLLL LLLLLL LLLLLL LLLLLL L L LLLLLL LLLLLL LLLLLL LLLLLL L L LLLLLL LLLLLL LLLLLL LLLLLL LLL LLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLL LLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL L LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLL LL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLL LLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLL LLLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LL LLLLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL L LLLLLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLL LLLL LLLLLL LLLLLL LLLLLL LLLLLL L LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLLL L LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LL L LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LL LLLLLL LLLLLL LLLLLL LLLLLL LLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLL LLLLLL LLLLLL LLLLLL L LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLL LLLLL LLLLLL LLL L LLLL LLLLLL L LL LLL LLLLLL L LLL

LL LL LL L LL LLLLL LL LL

LLLLL LL LLLL LL LLL LL

RULE 113(01110001) L LL LLLLLL LL LLL LLLLLLLLLL LLLLL LLLLLL L L LLLLL LLL LLLLL LLL L LLL LLLL LL LL LLLLL L LLLLLLLL LLLL L LLLLL LLLL LLLL LLLLL LL LLLL L L L LLLLLLL LLLLL L LLLLLLLLLLLL LLL LLLLLLLL LLLL LLLLLLLLLLLLLL LL LLLLLLL LLLLL LLLLLLL LLLLLLLLLLLLL LL LLLLL L L LLLL

L L LLLLL LLLLL LLL LLLL LLLLLL LLLL LL LLL L LLLLLL LLLLLL L LLLL LLLL LLLLLLLLLL LL LLL L LLL LLL L L L LLLLLLLL LLL L LLLL LLL LLLLLLL L LLLLLLLLLLLLLL LLL LLLLLLL LLLL LLLLLLLL L L LLLLLLL LLLLLLLLL LLL LLLLL L LL L LL LL

RULE 45(00101101)

L L LLLLLL LL L LL LL LLLL L LLLLLLLL LLL LLLL LLLLLLLLL LL LLLL LL LLLLLL LLLLL LLLL L LLLLL LL LLLLL L LLLLLLL L LLLLLLL L LLLLL LLL L LLLLL LLL LL LLLLLLL LLLL LL LLL L LL L LLLL L LLLLL L L

L LL LLLLLLL LL L LL LLL LLLLLLLL LLLL LLLLLLLLLL L L L LLLL LLLLL LL L LLLLLL LL LL LLLLLLL L LLLLLL LLLLLL LLLLLLLLLLLL LLLLLL LLLL L LLLLLLLLLL LLLLLL LL LLLLLL L LLLL LL LLLLLLLLLLLLLLLLL LLLLLL LLLLL LLLLLL L LL LL LLLLLLLLL LLL L LL L L LLLLLLLLLLLLL LLL L

L

L

[0.172,1]

RULE 75(01001011)

RULE 89(01011001)

RULE 101(01100101)

F gure 4 DP o some asymme r c ru es a ec ed by memory n he scenar o o F g ure 2 The ru es are grouped by equ va ence c asses The e and r gh Lyapunov exponen s o he owes ru e number o each c ass m n ma represen a ve n he ah s or c mode are g ven a er ru e codes

Comp ex Sy em 14 2003 99–126

Elementary Cellular Automata with Memory

RULE 57(00111001) LLL LL LL LL LLLL LL LLLL LL LLL LLL L LLL L LL LLL LLLL L LLLL LL LL L LL LLL LL LL

L LL LL LL LL LLLL LL LLLL LL LL LLLL LL LL LL LL L LLL LLLL L LLL LL LLLLLLL LLL L LL L L L LL LL LL LL LLL LL LL LL L L LLLLL LL LL LL LLLL LL L LL LL LLLL LL L L LLLL LL L L LL LL LL LLL LLL LLL LL LL LLLLLLL LL LLL LLLL LLLLLL LL LLLL LLLLLLL LLLL L L LL LL LLL LLL LLL LLLLLLLLLL LLLLL LLL LL LLL LLLL LL LLL LLLL L LLL LLLL LL LLL LL LL LL LLL LLL LL LLLLL LL LLLLLLL L L LL LLLLLLL LLL LLLL L LLLLL LLL L LLL L LLLLLL L LLLLL LL L LL LL LLLLLL LLLLLL LLLLLLLLL LLLLL L LLLL L LLLLLLLL LL LLLLLL L LLLLL LLLLL LL L LLLLLL L LLLLLLL LLLLL LL LLL L LLL LLL LLLLLL LLL LLL L L LLLLL L LL LLL LLLL LLLLLLLL LLLL LLL LLL LLLLL LL L LLL LLL L LLL LLL LLLLL LLL LLL LLL LL L LLL LLL LLLLL LL LLL LLL LLL LLL LL LLLLLLL LL LLLLLL LL LLLL LLL LLL LLL LLL LLLLL LLL LLL LLLLLL LLL L L L LLL LLL LLL L LLLLLLLL LL L L LL LLL LLL LLL LLL L LLL L L L LLL LLL LLLLLL LLL LLL LLLLLL LL L LLLLLLL LLLLLL LL LL LL LLLL LLL LLL LLL LLL LLLLL LLL LLL LLLL L L LLL L LLL LLL LLL LLL LLL L LL LL LLL LLL LLLLL LL LLL LLL LL LLLL LLL LLL LLL LL L LL LL LL L LL LL LL L LLLL LLLLL LLLLL LL LLL LL LL LLLLLLL LLLL LLL L LLL LL LL L LL LLLL

RULE 99(01100011) L LL LLLLL LLL LL LL L LLL L L LLL LLL LL LLLL LLL LLL LLL LLL L LL LL L LLL LLLL LL LLLL LLL LLL LLLLL LL LL LLLLL LLLL LLL LL LLLL LL L LL LLLLLLLL LL LL LLL LLLLLLLL LLL LLLLLLL LL LLL LLLLLLL LLLLLLLLL LLLLLLLLLL LL LL LLLLLLLLLL LLLLLLLL L L LL LLLLLLLL LL LLL LLLLLLLL LLL LLLLLLLLLL LLLLLLLLLL LLL LL LLLLLLLLLL LL LL LLLLLLLLL LLLLLLL L LLLLLLL LLL LLLLLLLL LL LLL LLLLLLLL LLL LLLLLLL LL L LLLLLLL LLL LL LLLLLLLLL LL LLLLLLLLLL L LLLLLLLLLL L LLLLLLLL LL LL LLLLLLLL LLL LLLLLLLL LLL LLLLLLLLLLL LLLLLLLLLL LLL LL LLLLLLLLLL LL LL LLLLLLLLL LLLLLLL L LLLLLLL LLL LLLLLLLL LL LLL LLLLLLLL LL LLLLLLL LLLLL LLLLLLL L LL LLLLLLLLL LLLLLLLLLL LLL LL LLLLLLLLLL LLLLLLLL LL LL LLLLLLLL LLL LLLLLLLL LLL LLLLLLLLLLL LLLLLLLLLL LLL LLL LLLLLLLLLL LL LLLLLLLLL L LLLLLLL LLLLLLL LL L LLL LLLLLLLL LLL LLLLLLLL LLLLLLL LL LLLL L LLLLLLL LL LLLLLLLLL LL LL LLLLLLLLLL LLLLLLLLLLL LL LLLLLLLL LL LLLLLLLL L LLL LLLLLLLL LLLLLLLLLLL LLLL LL LLLLLLLLLL LL LLLLLLLLLL LL LLLLLLLLL LL LLLLLLL LLLLLLL LL L LLL LLLLLLLL LLL LLLLLLLL LLLLLLL LL LLLLL LLLLLLL

L LL LLLLL LL LL LL LL LL LL LL LL L L LL L LLLLLLL LL LLLLLLLLL LLLLLL LLL LLL LL L L LLLL L LLLLLLLL LLL LLLL L LL LLL LL LL LLL L LLL LL LLLL LLLL L LL LLLLL LLLLL L L LL LL L LL L LLLLLLLLLLL LL LLLLL L LL LL L LLL LL LLLL LLL L LLLLLL LL LL LL LL LLL LL LLL LL LL LL LL LL L LLL LLL LL LLL L LLLL LLL LLLLL LL LLLLLL L LLLLLL LL LL LL LL L L LLLLL LL LLLL LLLLL LL LLL LL LLLL LLL LL L LLLLL LL LL LL LLL LL L LL LLL LL LL LL LLLL LLL LLLLLL L LLL LL LLL LLLL LL L LLLLLLLL LLLLLLLL L LLL LLLLL L LLLLL LLL LLLLLLLLLL LLLL L LL LLLLL LLL LL LL LLL LLLL LLLL LL L LL L LLL L LLL LLLLL LLLLLL LLLLLLL LLLL L L L LLLLL LLL LL LLLLLL LLLL LLLL LL LL LL LLLL LL L LLLL LLLL L LL LLLLLLL LLLL LLL LLL

117

[(-1,1),(1,-1)] L LL LL L L LL LLL LL L LL LL LL L L LLL LL L L LL LL LLL L L L LLLL L LLLL L L L LL L LLLL LLLLL LLL LL L L L LLLLLL LLLL LL LLL LLLL L L LL LLLLLLLL LL LLLLL LL LL L LL L LL LL L LL LL LLL LL L LLLLL LL LLLLL L LLLLLLL LLL L LL L LLLL LLLLL L L LL LLL LL LL LL LL LLL LLLL L LLLL LL LL LL LLL LL LL LL LL L LLLL LLLL L LL L LL L LLL LLL L LLL LLLLLL LLL L LL LL L LLLL LL L LLL LLL LLL L L L LL LLLLL L LLLL LL LLLLL LL LLLLLLLL LLL LLLLL LL LL LLLLLL LL L LLLLLLLLL LL LL LLLLLLLLLLLLL LLL LLLLLL L L LLL LLL LLLLLLL L LLL LLLL LLLL LL LL LLL LL LL L LL LLLLLL LLL LLLLL LLL LLLLL LLL LLLL LL LL LLLLLLLLLL L LLLLLLL LL LLLL LLL LL LL LL LL L L LLL L LL LLL LLL LLLL LLL LLLL LLL LL LL LL LL LLLLLL LLL LLLL LLL LL LLL LLL LL L LL LLL LL LLLL LLLL L LL L LLLLLL L LLL LLL LLLLL LL LLLLL L LL LLLLL LLLLLLL LL L L LLLLL LLL L LL LL L LLLLL L LL LL L LLL LL LLL L L LL LL L L L LL L L LLL L LLLL L LLL LLLLLL L L L L LL LLLL LL L LLLLLL LL LLL LLL LLL LLL LL LL LLLLLL LLL LLLL LL LL LLL LLL L LLLLL LLLLL LL LLLLLLL L LLL LLLLLLLLL LLLL LLL LL LLLLLLLLL LLL LL LL LL LLLLLL LL L LLLL L LLLLLL L L LLLLL LLLL L LLLLL LL LLLLL LLLL LLLLLL L LL L L LLLL L LL LLL LL LLL LL LL L L LLLLL LLLL L LL L LLLLL L L L LLL LLL LLL LL L LLL LL LLL LLL LLLL L L LLLLLL LLLL LLLL LLLLLLLL L LLL LLL L LL LL LL LL LLLLL LLL L LLLLL LL LL LLLL LL LLL LLLLLL LL LL LLLL LL LLLL L LL LLLLLLLL LL L L L L LL LL L LL L LLLL LL LLL L LL LL LL LL L LL L LL LLL LLLL LLLLL L LL LL LL LL LL LL LLLLLLL LL LLL LLL L LLLLL LLLL LL LLL LL LLL LL LL LLL LLL LLL LLLLL L LLL LLLL LLL L LL LL LL LLLLL LLLL L LL LL LL LLLL LL LL L L LLL LLLL LL LLLLL LLLLL LLL L LLL LLLL LLL L LLLLLL L LLLL LLLLL LLL LLL LLLLL LL LL LL LL LLL LLL LL LLLLLLLLLL L L LL LLLL LLLL LLLLL LLL LL LL LL LLL LLL LL LLL

L LL LL L L LL LLL LL L LLL LLLLL L LL LLLL LL LLLLLLL LL L LL LLLL LLLL LLLL LL LLL LL LL LL LL LL L LLLLLL LLLL LL L LLLL LLLL LLL LLLL L LL LLLL L LLL LL LLL LLLL LLLLL L LL L L L LLLLLLL L LL LLLLL LLL LLLL LLL L LLL L LLLLLLLLL LLLLLLLL LLLLL L LL LLLL LLL LLLL LLLLL LLLLLLLLLL LL LL LL LL L LLL LL LLL LL LL L LL LLL LLLLLLL LLLL LLLLLLLLL LL LLL L LLLLLL LLLLL LL L LL LL LLLLL LLLLLLLL LLLLLLLL LL LL LLLL L LLLLLLLL LLLLL LLLL LL LLLLLLLL LL LL LLLL LLLL LLL L LLLLL LLL L LL LL LLLL LLLLLLLLLL LLLLLLLLL L LLLLLLLL L L LL LL LLLL LL LLLLLLLL LLLLLLL L LLLLLLLLLLLLLL LLLLLLLL LLL LLLLL LL L LLL LLLLLLL LL LL LLLLLLLL L LLLL LL LL LLL LLL L LLL LLLLL L LL L LLLLLL LL LLLL LLLLLLLLL LL LLLLL LLLLLLLL LL LLLLL LL LLLL LL L LLL L LLL LLLLL LL LLL LLLLL LL LLLL LL LLLLL L LLLL LL LLLLL LL L L LLL LLL L LL LL LLLLLL LLLL L LL LLLLLLL LLLL LLL L LL LLLLLL LL LLL L L L LLLLL L L LLL LLLLLLLLLLL LLLLL LLLLLLL LLLL LLLL LLLLLL LLL LL LL L L LLLLLLL L LLLLLLL LLL LL L LL LLL LLLLLLLLL LL LLLL LLLL L LLLL LLLLLLLLL L LLL L LLLLL LL L LLLLL LLL LLLL LLL LL LLL LLLLL LLLLLLLL LLLLL LL LLL LLL LLLL L LLLLLLLLL L LLLL LL LLLL LL LLL LLLLL LL LLL L LLL L LL L L LLLLLL L L LLLL L LL LL LLLL LLLL L LLLLL LL LLLLL L L LL LLLL LLLLLLL LL LL LL LLLL LLLL LLLL LLLLLL LLLLLLLL L L LLLLLLLL LLLL LLLLLL LLLLLLLL LL LLLL L LL LL LLLL LLLLLL LLLLLL LLLL LLLL LL LL L LL LL LLLL LLLL LLLL LLLLLLL LLLL LLLLLLLLLL LL LLLLLLLL L LL L LLLL LLL LLLL LLLL LLL LL L L LL LLL LL LLLLL LL LLLLLLLL LL LLLL LL L L LLL L LLLLL L L LLLLLLLLL LLL LLL LLLLL LLLLL LLLLLLLL LLLLLLLL LLLLLLLLL L LL LLL LLLLL L LLLLLL LL LL LLL LLLLLL L LLLLLLLL LLLL LL LL LL LLLLLLLL LLL LL LL L L LLLL LL LLLLL L L LLLLL LLL LLLLLLL L L LL LL LL LLL L LL LL LL L LLLL L LLLLLLLLL LL L LLLL LL LLLL LLLL LLLL LLL LLLLL LL LLL L L LLL LLLLLLL LLLL LL LL LL LL L LL LL LLLLL LLL LL LLLLLL LLL L LLL LLLL LLLLLLLLL LLL LLLLLLLLLLLL L LLLLLLL LLLLLLLLLL LL LLL L LL LL LL LLLLLLL L LLLLLL L LLLLLL LLLLLL LLLL L L LL LLL LL LLLL LLL LLL LLLL LLLLL L LLL LL LLL LLL LL L L LL LLLL LLL LLLLLL LLLL L L L LL L LLL LLL LL LLLLL L LLLLLLL LLL LLLLLL LLLL LL LL LLLL L LLL LLLLLLLLL LLLLLLLL LL LLLLLL LLLLLL LL LL LLLL LL L LLLL LLL LLLL LLL LLLL LL L L LLL LLLLLLLLLLL LLLL L LL L LLL L LLLL LLLL LLLLLLLLLL L LL LLLLL LLLLLLLLL LLLL LLL LLL LL L LLLLLLLL LLLLLLL L LLL LLL L L LLLLLLLLL LLLL LLLL LLLL LL LLLLL LL LLLLLLL LL LLLL LLL LLL LL L LLL L LL LL L L LLLLLLLLL LLLL L LLLLLL LLL L LL LL LLL L LLL LLLLL LL LLLLL L LL LL LLLL L LLL LL LL LL LL LL LLLL L LL L LL LL LLLL LLL LLLL LLLLLLLL LLLLLL L LL LL LLLL LLL LL LL L LLLLLL LLL LLLLLLLL L LLLL LL LL L L LLL LL L LL LL LLL LLLLLL L LLL LLL L L LLL LLLL LLLLLLLLL L LLLLLL LLL L LL LL LLL LLLLL LL L L LLLL L LL L LL L L L LLL LLLL LLLL LLLL L LLL LLLL LL LLL LLL LLL LLLLL L L LL LL LLLLL LL L LLL LLLL LLLL LLLLL LL LL LLL L LLLLLLLLLLL LL LLL LL LL LLLL LL LL L LLL L LLLL LLLLL LLL LL LLLLL LL LLLLLLLLLL L L L L LL L LL LL LLLLL LL LL LLLLLLLL LL LLL LLL L LLLLL LL LLLLLL LL LLLLLL LLLL LL L LLLLLLL LLL LLLLL L L L L LLLLLL LLLLL L LL LL LLLL LLLL LLL LL LL LL LLLL L LLLLLL L L LL LLLLLL LLLL LLLL LLLL LLLLLL L LLLL LLLLLL L LL LLLL LLLL L LL LLLLLLLL L L L L LL

L LL LLLL L L L L L L LLL LL LLLLLLLL LL LLLLLLLL L LL L LL LL LLLL L L L L L L L LL LLLLL LLL LLLL LLL LLLL LLL L LLLL L L L L L L LLLLLLL L LLL L L LLL L LLLLLL L L L LLLLLL LL LL LLL LL LLLLL LLLLLLLL LLL LLLLLL L LLLLL LLL L LL LLLLLL L L L L L LLL LLL LL LLLL L L LL L LL LLLLL L LLLLL LL L L L LLL LLL LLLL L LL LL LL LL LLLLLLLLL L LLLLLLL LL LL LLL L L L LL LL LL L LLLL LLLLLLLLLLLLLLLLL LLLLL LLL LL LLL LLLLL LLL LL LLLLL LLLL LLLLLL LLLLL LL LLL LL LLLLL LLLLLLL L L LL LL L LLLL LL L LLL L L LLLLL L L L LL L L LLLLL LLLLLL LL L LLLL LL L LL LLLL L LLL LLLLL L LLLLL LL L L LLLLLLL LLLL LL L L L LL LLLLL LL LL L LLL L LLL L LL L LLLL LLL LLLLLLLL LLLL LLLL LL LLLLLLL LL LL L LLLLLLL L LL L LLLLLLL LL L LLL L LLLL LLLLLL L LLL LLLL LLLLL LLLLL L LLLLLLL L L LL LLLLLL LL LLL LL LL LL LLLL L L L L L LL LLLLLL L L LLL LLLLLL LLL L LL L LL LL LLLL L LLLLLL L LLLLLLLLL LLLLLLL LLLL L LLL L LL LLL LLL LLLL LLL LLLLLLLLL L LL LL L LL LL LLLLL L LL LL LL LL LL LLLLL LLLLLLLLL LL LL LL LL LL LLL LLLLL LLLLL LL LL L L LLLLLL LL LL LL LL LL L LL L LLLLLLLL LL LL L LLLLLLLL L LL LL LLLLLLLL LLL L LL LLL LLLLLLL LLLLLLLL LLLLLL L LLLLLL LLL LL LLLLLLL LLLLL L LL LL

L LL LLLL L L LL LL LLLLL LL LL LL LL LL LLL LL LLLL LL LL LL LL L LLL L LL LLLLL LL L LL LL L LLL LLLLL L L LLLLL LLLL LL LLLLLL LLLL L LLLLLL LL L LLLLLL LL LLLL LL LLLLL LL L LLLL L L LLLL L LLL LL LLLL LLLL LLL L LL LL LLLLL LLLLLLLLLLLLLLLL LLLLL LLL LL L L LL LLLL L LLLL LLL LL LLL LL LL LLLLLL L LLL LLL LLLL LLLLL LLL L LL LL L LLLLLLL LL LLLLLLL L LLLL LLLLL LLLLLLLL LLL LL LL LLLL L LLLL LLL LLLLL L LLLL LL L L LLLLLLLL LLLL LL L LL LLLLLLLL LL L LL LLLLL L L LLLLLLLLLL L LLLLLL LLLL L LLL LL LLLLLLLL L L L L LLL L LLLLLL LLLLLLL L LLL LLLLL L LLLL L L LLLLLL LLLLL LL LL LLLL LLLLLL LLLL LL LLL LLLL LLLLLL LLL LL LLL L LLLLL LL LLLLLL L LLL LL LLLLL LLLL LLL LL LLLLL L L LL LL LLLL LLLL LLLL LLLLLL L LLL LLL LL LL LL LLL L LLLLLLL LLLLLLLLLL L LLLL LLLLL LL LL LLLLLLLL LLLLL L LL LLL LL LLL LLLLLL LLLLLLLLLLL LLLLLL LL LL LLLL L LL LL LLLL LL LL L LL LLLLLL L LL LLL LLL L L L L LL L L LLLLLLL LLLL L LLLLLL L LLL L LL L LLLLLLL LL LL LL LLL LL LLLLLLLL L LLL L LLLLL LL LLLL L LL LL LLLL L L LL L LLL L LLLLL L L LL LL LL L LLL LL LLLLL LLLLLLL L LLL L LLLL LLL LLLL L L LL LLL LLLL L LLLLL L LL LLLL LLLLLL LLLL LL L LLLLLL LL LL LLL LLLL LLLLLLLL LL LLL LLLL L LLLLLLL L L L LLL LLL LL LLLLLLL LLLL LLL LLLLL LLL LLL L L LLLLLLLL L LL LL LLLL LLLLLLL L LLLLL LL LLLL LL LLLLL L LL LL L LLLLL LL LLL LLL LLL LL LLLLLLL LLLL LLLLL LLLLLLL L LL LLL LL L LLL LL LLLLL LL L LLL LLLLLLL LLLLLLLLLL LL LLLL L LLLLL L LL L LLLLLL L LLL L LLLLLLLL LLL L LLLLLLLL LL L L LLL LLL LL LL LLL LLLLLLLL LL LL L LL L LLLL LL LLL LLLL LL L LLLLLLL LLLLLL LL LL L LLL L LLLL LLLLLL LLLLLL LLL L LLL LL LL LLL LL LL LL LLL LLLLLL LLL L LLLLLLLLLLLL L LLL LL LLL LLL L L L LLL LLLLLL LL LLLLLL LLLL LL LL L LLLL L LL LLLLLL LL LL LLLLL LL LL LLLLLLL LLLLLL LLLL LLLL LLLL LLL LLLL LLLLLLL LLLLLLLL L LL LLLLL LLL LLL LL LL L LL LLLLLLL LLL LLLL L LLL LLL LLL L LLLLLL LL LLLLLL L LLLL L LLLLLLLLL LLL L LLL LLLLLLL LLLLL LLLL LLLLL LLLLLL LLL L LL L LLLLLL L LL LLLL LL LLL LLLLLLLL LL LLLL LLLLL LL LLLLLLL LL L L LLLL L LLLLLL LLLLL LL LLLL L LLL L L LL LLLLLLLLL LLLLLLL LL LLL LLLL LLLLL LLLLL L LL LL LLL LL LLLLLL LL LLLLL LLLLLLLLL LLLLLLLL L LL LLLLLLLLLLL L LL LLLL LLLLLLLL LL LLL LLLLLL L L LL LL LLLL LLLLL LLLLLLLLL LLLLLL L L LLL LL LLL L LLLLLLL L LLLLL LLL LLLL LLL LLL LLLL LLLLLL L

L LL LLLLL LL LL

L LL LLLLL LL LL

L LL LLL L L LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLL LL

L LL LLL L L LLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLLLL LLLL LL

RULE 106(01101010) [1,-0.134] L LLLL LLLLL LL LL L LL LLLLLL LL LLL L LLLLLL LLL LLLL LL LL L LL LLLLLLL LL L LLLL LL L LLLL L L LL L LL LLLLLL L LL LL L L LLL L L LLLLLL LL L L LLLL LLLL L LL LLLLLL L LL L L LLLL L L LLLL L L LLLLL L LLL L L LL L L L LLLLL L L LLLLLL LL L LL LLLLLLL L L LLLLL LLL LL L L L L LL L L L LL LL L L L LL LL LL L LL L LL LL L L LL LLL L L LLLLLL L L L LL LL L LL LLL LL LL L LLL LLLL LL LL L LL LLL LL L L LL LL LL L LL LLLLL L LL LL L LL LLL LL L LL LLL LL L LLL LLL LL L L LLL L LLL LLLL L LLL L LL LLLLL L L LL L LL LLL LL LL L LL LLL LL L L LL

L LLL L LLLLL L LL LLL LLLLL L LL L LLLLLL L LLL LL L LLL LL LLLLLLL L LL L LL LLLLLLL LL L LLL LL LL LL LLLLLLL LLLLL LL LLLLLL LLL LL LLL LLLL L LLLL LLLLL LL LLLLL LLL LL L LL LL L LL LLL LL L LL LL LL L LLL LLLLLLLL LLLLLL LL LL L LLL LL LL LLLLLL LL LL L LLLLLL LL L LLL LLL LL L LLL L LLLLL LLL LL L LLLL L LL LLLL LLLLLL LLL LL LLL LLLL L L L L LLLLLLLLLLL LLLLLLL L LLLLL LLLL LL LLL LL LL LLLLL LLL L LLL L LLLLLLL LLL LLLL LL LL LL LLLL LLL LL LL LLLL L LL L LL LL LL LLLLL LLLLLLL LLLL LL LLLLLLL LLL L LLLLLLLLL L L LL L L L L LL LL LLLLLLLL LLLLL L L LLL L LLLLL LL L LLLLL LLLLLLL LLLLLLLLL LLLL LLL LL LL L LLLL LL LL LL L LLLLLLL LL L LLL L LL LLLL LL LLLLL LL LL LL LLL LLLLL LL L LLLLL L LLLLL L L LLLLL L L LLLLLL LLLL L L L LL LL LL LL LLL LLLLLLLL LL LL LLLL LL LLL LL L LLL L LL LL LL LLLLL L LL LLLL LL LLLLL L LL LL LLLL LLLLLL L LLL L LLLLLLL LL L LL L L L LLL LL LLLLLLLLLL LLL LL LLL LL L LLLLL L LLLL LLLLLL LLLLLL L L L LLL LLL LL LLLL LLLL L LLL LLL LL LL L LLLL L L L L LL LL L LL LLLL L L LL LLLLLLLLLL LLLL LLLL LLLLLL LLLLLLL LL LL LL L L L L LL L LLL LLL L LL L L LLL L LL LL L LLLLLL L LL LLL LL LLL L LLL

L LLL L L LL L L LLLL LLLL L L L LL LL L LLL LL LLLL LL LLL LLL L L LLL LLLLLLL LLLLL L LLLLL LL LLLLL LL LLLLL LLL LLLLL LLLL L LL LL L L LLLL L L LLL LLL LLL L L L L L LLL LL L L L LL L LLL L LL L LL L LLL LL LLLLL LLLL L LLL LLLLL LLLL L LLLLLL L LLL L LLLLL LLLLL L LLLLL LLL LLLLLL LLLLLLL LLLLLL LL LLL LLL LLLLL LL LL L LLLL LL LLLL L LL LLLLL LLL LLLLL L LL L LLLL LL L LL LLL LL LLLLLLLLLL LL LLLLLLL LL LLL LLLLLL LL L LLLL LL LLLLLL LLLLL LLLLL L LLLLLLL LL LLLL LL L L LL LL LL LLLL LLLL L L LLL LL LLL LL LL LL LL L L LLLL L L L LLL LLL L LL L L L L L LLL L LLL LL L LLLLL L LL LL L LL LLL L LLLL L LLL L L L L L LLLLL LL LL LLLL LLLL LLL L LL LLLL LLL LL L L LLLLL LLLL LLL L L LLLL LL LL L L L LLL L LLL LLLL LLL L L LL L L LLLLLLL L L LLLL LLLLL LLL LLLL L LLLLLL LLL LL LLLLL L LLLLLL LLLL LL L LLL L L LLLLL LLLL L LLLL LLL LLL LLL L LL LLL LL LLLLLLLLLL L LL LLLLLLLLL LL LLLLLLL L LL LLL L LLLLLLL L LLLL L LLL LLL L LL L LL LLL LLL LL LLLL LL L L L LL L L L LLLLLLL L LLLL LLL L LL LLL LL LL LL L LLLLLLLLLLLL L LL L LL L LL L LLLL L LLLLLL LL LL LLL LLL LLLLLLLL LLLLLLLL LLLL LLLLLLLLLLLL LLLL L LLLLLL LLL LLLLLLLL LL LL LLLLLL L LLL L LLLLLL LLLL LLL L LLLLL LLL LL LL L L LLLLLL L L LL LLLL LL L LL L LLLLLLL LLL L LLLL L L LL L LLLL LLLLL L L LL LL LLLLLLLLLLLLLLL LLL LLL L LL LLLLL L LLL LL LLL L LL LLLLL LL LLL L LL LL LLL LLLLLL L L L LL LLL L L LLLLL L LLL LLLL LLLLL L LLL LL LLLLL L LL L L LL LL L LLL LLL L LLL L LLLLL L L L L LLL LLL LLLLLLL LLL LLLLLL LL LL LL LLL LL LLLLLLLL LL L LL LL LLL LL LL LLL L L LLL LLLLL LLL L LL L LLL L L L LLLLLL LL L LLLL LL L

L LLL L L LL L L LL LL LL L LLL LLL L L LLLLLL L LLLL LL LL LLL LLLL LLLLLL L LL LLLL LLLL L LLLLLLLLL LL L LLLL LLLLLL LLLL L LLLLLL LL LLLL L LL LL L LLLLLL LL L LLLLLLLLLLL LLLL LL LLLL LLL L L LL L LLLL L LL LL LLLLLLLL LLLLLL LL LLLL LL LLLLLLL LLLLLL LLLLLLL LLLL LL LLL L LLL LL LLLLLLL L LL LLLL LL LL LL L LL LL LLLL LLL LL LLLLLL LLL LLL LLLL L LL LL L LL L LL LLLLLL L LLLLLL LLLL LLL L LL LLLL LLL LL L LL L L LLLLLL L LLL LLLL LLL LL LL LL LL L LLLLL L L L LLL L LLL LLLLL LLLL LLLL LL LL LL LLL L LL LL LL LLL LLL LL LLLLLL LLL LLL LLLLL LLLLLLLLL LLLLLL LL LLLL LL LL LLL L LL LLLL LLLLLL L LL L L L LLL LLL LL LLLL LLLLLLLLL L LLL L L LLLLL L L LLLLLLL LLLL LL LL LLLLL LLLL L LLLLLLL LL LLL LL LLL LLLL LL LL LL LLLLLLL LL LLL LL L LLL LL LLLL L LL LLL LL LLL LLLLLLLLLLLLL LL L LL L LLLLLL LLLL LL LLLL LLL LLLLLLLL LL LL L LLLL LLL LL LLL LLL LLLLLLLLLLL LLL LL L LLLLLLLLL L L LLL LLL LLLL LLLLLLL LLLLLL LLLL LL LL LLLL L LLL L L LL L LLLLLLLLL LLLLLL LLLLLL LL LL LLLL LLLL L LLLL LL LL LL LL LL LLLLL LLLLLLL L L LLL LL LLLL L LL LL L LL L LL LLL LLLLL LLL L L LLLLL LLL LLLLLLL LL LL L L LLLLLL LLLL LLLL LL LL LLLLLLLLL L LLLLL LL LLLL LLLL LL LL L LL LLLL LLLLLL L LLLL L L L LLLL LLLL L LLLLL LL LL LL LLLLLLLL LL LLL L L LLLL LLLL LLLL L LL LLL LLL LL LLLLLLL LLL LL LLLL LL LLLLLL L L L LLLLL L LLLLLLLLLLLL LLL LL LLLLL LLL L LLL LLLLL L LL LL LLLLL L L LL LLL LLL LL LLLL LL L LL LL LLL L L LLLLLLL LLLL L LLLLL LL LLLL LLLL LLL LLLL L L LLLL L LL LL L L L L LLL LLL LL L L LLLL LL LLLLL L L LLLL L LLLL L LLLL LL LLL LLLLLLL LLL L L L LL L L LL L LLLLL L L LLLLL L LLL LLL LLLL LL L L LLLLLL LLLLLL LLL L L LL LLLL LLLL LL L LL L LL LL LLLLLLL LL L L LLL L LLL LLL L L LLLLLL L LLL LLL LL L LL LLLL L LL LLL L LLL LLLLLL LL LLLLL LL LL L LLLL LLLLLLLL L LL LL LLLLL LLLLLLLL L LL LLL LLLLLLLL LL L LL L LLLLLLLL L LL LL LL LL LL LL LLLLL LLLL LLLLL LLLLL LLL L LLLL L LLL L LL LLLLLL L LL LLL LL LLL L LL L LLL LLLLL LLLL LLL LLLLLLL LL LLLL LLL LLL L LLLLLL LLL LL LL LL LLL LLLLLLL LL LLL LLLL L L LL L LLLL LL LL L L LLLLL LLL LL LLLLLLLLL L LLLLL LLL LL L L L L LLL L LLL LLL LL LLLLLLLLLLLLL LLLLL L LLLLL LL LL LLL LL LL LLLLL L LL LLL LLL LLL L LLL LLLL LL L LL LL L LLLLLLLL LLLLLL L L L LLLLLLLLLLL LLL LL L L LLL LL LLL LLLL LLL LL LLLLLLL L LLLL LLL L LLL L LLLLLL LLL LLLLL LL LL L LLLLLLLL LL L L LLLLLLLL L LLLLLL L LLLLL L LL LL LLLLL L LL LL LLLLLLLLL LLL LLLLLLLLLLLL L LLLL L LLL LL L LL LL LLLLL LL LLLL LL LLLL L LL L L L L LL LL LLLLLL LL L LLL L L LL L LL L

L LLLL LL

L LLLL LLL L LL LL LLL LLLLL LLL LL LL LL LLLLLL LL L LL L LL L LL LL LL L LL LL L LLLL L LLLL L L LL LLLLL LL LLL LL L L LLLLL L LLL LLLL LL LLLLLLL LLL LLLLLL LL L LLLLLL LLL LLL LLLL LLL LL LL L L LLLL L LLLLL L LLLLL LLLLLLL L LLLL LLL LL LLL L LL LLLLL LLL LLLL LL LLL LLLL LL LLLL LLL LLLL LL L LLL LLLL LL LLL LL LL L LLLL LL LLLLLLLLL L LLLL L LLLLLL LLLLL LLL LL LL LL LLLL L LL L L LLL LLLLLL LL L LLL LL L LL LLLL LLLLLLL LL LL LLLLLL LL LLLL LL L LL LLLLLLL LLLLLL LL LLL LL LL LLLLLLLLLLL LLLL LLL LLLLL LL L LLL LLLLL LLL L L L LL LLL LLLL LLLL LLLLLL LLLLL LLL L LLL LLL LL LL LLLL LL LLLL LLL L LLLLL LLLLLLL LL L LLLL LLL LLL LL LL LL LLL LLL L LL LLL L LL LL L LLL LLLLL LL L LL LL LLLL LL LL LLLLLLLL LLL LLLL L LLLL LL L LL L LL LL LL L L LL LLL LL LL LLLLLL LLL LLLL LLLLL LLL L L LL LL LL L LLLL LLL LLL L L LL LLLL L LLLL LL LL LLLLLL LLLL LL LL LLLL LLL LL LLL LLLLLL LL LL L LL LL LL LL LLLLL L LL LL LLL LLL L LLL L LLLLL L LLLLLL LLLL LL LLL LL LLL L LL LL LLL L LLLL LLLLL LLLLLL LLLLL LLL LL LLLLLL LLLLL LL LL LLL LLL LLL LL LLL LLLL LLL LL L LL LLLL LLL LLLL LLLL LLLLL LL LL LLLLL LL LLL LLLLL LL L LL LLLLLLLLL LLLLL LL LLLLL LLLL LL LL LLLLLLLL LL L LL LLLL L LL L L LLLLL L LL LLLL LL LLL L LLL LLL LL LL LL L LL L LL LL L L L L L

L L LL LL LLL LLL

L LLL L LLLL L LLLLLLLL LLLL LLLLL LLL LL L L LLLL L LLLLL LLL LLL L L LLLLLLLL LL LL LL LLL LLLL LLLL L LL LLLL LLLLLLLLLLLL LL LL LLLLLLLLLLLL LL L L L LLLL L LLL L L LL LL LL LLLLLLL L LLLL LLL LLL LL LLLL LL L LLL LLL LLL LLLLLLLLL LL LLLLLLL LL LLLL LL L LL LL LLLLL L LL LLL LLL LLLL L LLLL LL L LL LL LL L LLL LLLL L LL LLL LLL LLLLLL LLLL L LL LL LLLLL L LLL LLLLLL LL LLLL LL LL L L LLL LLL LLLLL LLLLLLLLL LL L LL LLL LL LL LLL LLL LL LL LLLLLL LLL LL LL LLLLL LL L L LLLLL LL L LLLLL L LLL LLLL LL LLL LL LL LL LL LLL L L LL LL LLLLL LL L LL L LLLL LL LL LLL LLLLLLL L LL LLL LL LL L LLLL L LL LLLL LL LLL LL LLLL LLLLLLLLLLL L LL LL LL LLLL L LLL LLLL LLLL LL LLL L LLL LLLLL LL LL LLLLLLLLLLLLLL L LL LL LLLLLL LLLL LLL LLLL LLLL LLL LL LL LLLLLLLL LL LL LLLLL L LL LL LLLLL LLLL LLLLL LL LLLL LL LLL LL LLLLLLLL LL LLLLLLL LL L LL LLLL LLLLLLLLLLLLL L LL LLLL LLL L LL LLLL LLLL LL LLL LLLLLLLLL LL LLL LL LLL LL L LLL LL LLL LLLL L LL LLLL LL LLL L LLL L LL LL LLLLLL LL LL LLLLL LLLLLL LLL LLLLL LLLLLL LLL LLLL LLLLL LL L L LLL LLLLLLL LLLLLLL LLLLL LLLL LLLL LL LL LLLLLLL L LL LLL LLL LLLL LLLLL LL L LL L L LLL LLL LL L LL L LLLL LL LLLLLL LL LL L L LL LL LLLLLLL L L LL LLL L LLLLLL LLLLLLL L L L LLLLL L LL LLLLL L LL L LL LL LLLLLLLLLL LLLL LL L LLLL LL LLLL L L LL LL L LLLLLL LLLL LL LLLL LL LLLLLLL LL L LL LLL LLLL LL LLLLL LL LLL LLLL LL LL LL LL LLLL L LL L LLL L LL L LLL LLLLL L LL LL LL LLLL LLLLL L LL LL LLLLLL LL LLL L L LL LL LL LL LLLL LL LLL LL LLLLL LL LLL LL LL LL LLLLL LL LLLL LL LL L LLLL LL LL LL LL L LLLLLL LL LLLLLLLLLLLL LLLL LL L LL LL LL L LLLL LL LLLL LLLLLLL LL LLL LL LL LLLLLL LLL LL L LL LLLLL LLLL LLLL LLLLL LL LLLLL LL LLLLL LLLLLL LLLLLLL LL LL LL L LLLLLLLLL LLL LLL LL LLLLLL L LL LLLLLL LLL LLLLLL LL LLLL L LL LLLL LL LLLL LLL L LLLL L LLLLLL LL LLL L L LLLL L L LL L L L LLLL LL LLLLLLLL LL LLL LL LLLLL LL LL LLL LLLL LLL L L L L L L LL LL LL L L LL L L LL L LLLLLL L L LLL LL LLL L LLLLLLL LLLL LLLLL LL LL LLLL LLLLL LL LLLLLLLLLL LLL LLLL L LL LLL LLL LL LLLLL LLLL L LL LLLL LL LLL LLLLL LL LL LLLL L L LL LLL LLLL L LL LL LL LLL L LLL LLLLLLLL L LLL LL LLL L L LLL L L LL L LL LL LL LLLLLLLL LL L LLLLL LLL L LL LL LL LLL LLLLLLLL LLL L L L LLLL L LLLLLLLLLLL L LLL LL LLLLLL LLLL L LLLL LLLLL LL LLLLLLLL LLLLL LLLL LLL LL LL LL LLL LL L LLLLL LLLLLLLL LLLL LLL L LL LLLL LLL LLL LL LL LLLL L L L LL L LLLLL L LL LL LLLL LLL LLLLLLLL LL LLLLLL LLL LL LLLLL LLLLLL LLL LL LL LLLLL LLLL LL LLLL LLLLL L LLLL LL L LLLLLL L L LLLLL LLL LLLLLLL LL LLLLLL L L L L LLL LL LLLLLLLL LL L LLLLL LL LL LLL LL LLL LL LL LLL L LLLLL LLLLL L L LLLL LLL L LLLL L LLLLL L L LLL LLLLLL L LL LLLL L LL LL L LLLL LLLLLL LLL LLLLLLL LL LL LLL LL LL LLLLL L LL LLL L L L L LLLLL L LLLLLLLLLL LLLLL LL LLLLLL L LL LLLLLL L LL L LL LL LLL LLLL L L LL L L LLLL LLLLL L L LLLL LL L LL LL L LL L LL L LL L LL L LL LLL L L LL LLL LLLL L L LLLLL L L L LLLLL L LLLLL L L LL LL L LL LLLLLLL LL L LLLL LLLL LL LL LLLLLLL L LL LLLL L L LLL L LL LL L L L LLLLLLL LLLLL LLL LLLL LLLLLLLL LL L LLL LLL L LLLLLLLL LL LL LLLL LL LLL LL LLL LLL L LL L LLLL LLLLLLLLL L LLLLLLLL L LLL L LLLLLLLLLLLLLLL LLLLL LLL LLL LL LLLLLL LL LLLLLLLLLLLLLLLL LL LL LLL L LL LLLLL L LLLLLL L LL LLL LLLLLL LL LL LL LL LLL LL L LLLLLLLL LL LL LLL LLLLL LL L LLL LL LLL L L LLLLLL L L L LL L LL L L L LL L L L L LL L LLL L LL LLL LLLL

LLL LL LL LLLLL L LLL LLL LL LLLL LLL LLL L LLLLLLL LLLLL LL L LLLL LLL L LLL LLL LLLLL L LLL LLLL L L LL LL LL L L L LL LL LLL LLLL L LLL LLL L LLLLL L LLLLL L L LLLLLLLL L L LLLLLLL L LL LL LL LLLLLL LLL L L LLLLLLL LLLL LLLL L LL L LLLLL L L LLLLLL LL L LLLL LLLLL LL LLLLL LL LLL L LL LL LLL LL L LLLLLLL L LLLL LLLLLLLLLLL LLLL LL L L LL LLL LLL L L L LL LLLLL L LLL LLL LLL L LL L LLLLL LLLLL LLL L LLLLL LL LL LLLLL LL LL L LL LL LL LLLLLLLL L LLLL L LLLLLL LLLLL L LLLLLLLL LL L LLLL L LLLLLL L LLLLL LLL L L LL L LL LLL LLL L LL LLLL LLLL LLL L LL LLL L L LLLL LLL L L LLL LLLL LL LLL LLLL L LL LLL LL L LLL LL LLL L LL LLLL L LL LLLL L LLL L LL LLLLLL LLL LL LLL LL LLLLL L LLL LLL L L LLL LLLLLL L LL L L LLLLLL LLLLL L LLLLLL LLLL LLLLLL L L LLLLL L LL LLL LLLLL L L LLLL LL L LLLLLL L LLL LLL L LLLLLLLLLLL LL L L LLL L LL LLLL L LL L LLL L LLL L LL LLLLLLL L LLLLLLLLLLL LLL L LL LLL LLLLLLLL L LLLL LLLL L LL LLLL L LL L L LL L L L LLLLLL LLLL LL L LLL L LLL LLLL LLLLLLLL LLLLL LLLL L LL LL LLLL LLLLL LLLLLL L LL L L LL LLLLLL LL L LLLL L L LLLLLLLLL L LL LLL LLL L L LL LLLL LL LLL LL LLLL LLLLLL LL LLLLL L L LLLLLLLLLLLL LLLL LLL LLL LLL LLLL LL LLL LLLL LLL LL L LL LL LLLLLLLLLLL L LL LL LL LL L LL LLLL L LLL LL L LLLLLL L LLLLLLLL LLLLLLL LLLLL LLLLLLLLLLL L LLL LL LLLLL LL LLLLL L L LL L L LLL LLLL LL L LLL LLLLL LLLLL L LLLLL L LL LLLLL L LLLLLLLLLL LLLLLLLLLLL LLLLLL LLLLLLL L LLL LLLL L LLLL LLLLLLLLLL LLLL LLLL L LLLLLLLL LLLLL LLLLL LLLLLLLL LLLL LLLLLLL LL L L L L L L LLLLLLLLLL LL L L LL LLL LLLLLLLLL LL LLLLLLL L LLLLL LLLLLL LLL LLLLLLLLLLLL LL LL L LLL LLL LL LLLLLLL L LLLL LLLL LLL LLLL LLL L L LLLLL LLL L LLLLLL LLL LLL L L LLL L L L LL L LLLLL LLLLLL LL LL L LLLLLL LLL LLLLL L LL LLL LLL LL LL LLLLLLL L L LLL LLLLL LLL LLL LLLL LL LLL LLLLL LL L LLL LLL L LLL LLL LLLLLLL L LL LLLLLLLL L LL LLLL LLLLLL LL LLL LLLLLL LL LLL LLLL L LLLLLLL L LL L LL LLL LL LLLLLLLLLLLL LL LL LLLLLL LL LLLL LLL LLLLLLLL LLLLLLLLLL L LL LLL L L LLL LLLL LLLLLLLLLL LLLL LLLLLL L LLLL LL L LL LLL L LLL LL LLL LL L L LLL LL LLLL L LLLLLL L LL L LL L LLLLLLL L L LLLLLL LL LL LL LL LLL LL LL LL LLL LLLL LLLLLLL LL LLL LL L LLLLLL LLLLLLL L L L LLLLL LL LL L L L LLL LL LLLL LLLLLL LL LL LLL L LLLL LL L LLL L LL LLL LLLL L LLLLLLL LLLLL LLLL LL LLLL LL LL LLLL LL L LL LL L LL LLLLLLL L LLLL LLLLLL LL LL LLLL L LLLLLL L LLLL LL LLLLLLLL LLLLL LLLLL L LL LLLL LL L LL LLLLL L LLLLLL LL LLLL L LLL LLLLL LLL LLLLLLLL LLLL LLL LL LL LLLLLLL LLLLL LLLLLL LLLLLL LL LLLLLLL LL L LLLLLL L LLL LL L LLL L LLL LLLL LL LLLL L L LL LLLL L L LLL LLLLLLLLL LLL LLLL L LLLLLL LL LLLL LL LLLLLL LL LLLL L LL LL LL LL L L LLL LLLLL LL L LLLL LLL L LL LLLL L LLLL LLL LL LLL LLLLL LLLLL LL LLL L LLLL LL LLL L LLL L L LL LLLLLLLL L LL L LL LL LLLLL LLLL LL LLLL L L L L LLL LL LL L LL LLLL LL LLLL L L L LLLLLLLLL L LLLLLLL LLL LL LLL LL L LLLL L LLL LL LL L L L LLL LLL L LLLLLLL L L L LLL LLL LLL LLLL L LLLLLLL LL LLLLLLLL LLLLLLLLL LL LLLLL LLLL L LL LLLL L L LL LLL LL LLLLL LLL LL LLL LLL LLLLLL L LL L LL L LL LLLLL LLLL LL LLLL L L LL LLLL LLLLL LL LLL LLL L LLLLLLLLLLLLLLL LL LLL LLL LLL LLL LL L LLLL LL L LL LLLL LL L LL LLL LL LLLLLL LLLLL LL L LLLL LLLLL LLLLLLLL LLL LL LLLL LLLL LL LLLLL L L LL LLL LLLLL LL LLL LL LL LLLLL LLLLLLLLLL L LLL LL LLLL LL LL LLLLLLL LL L LLL LLLLLLLLLL LL LLLLLLLLLL LL LL LLLLLL L L LLLL LLL LLL LLL L LLL LLL L L L LLL LL LL LL L LLLLL LLLLLL L L LLLL L LL LL LLLL

LLL LL LL LLLLL L L L LLLLLLLLLLLLLL LL LLLLLL LLL L LLLLLLLL LLLL L L LLLLLLL LL L LLL LLLLLLLL L LLLLLLL L LLL L LL L LL LL L LLLLL LLL LLL L L LL LLLLLLLL LLL LLLLLLLLL L L L LLL LLL LLL LLLLL LLLL L LLLL LLLLL LL LLL LL L LLL L LL LLLL LL LL L LLLLLL LLLLLL L L L LL LLLLLL LL LLL LLLL LLL LL LLL LL LLL L LL LL LLLL LL LL LLLLLL LL LL L L LLLLLL LLLLL LL LLL LLLLLLLLLLLL LL LLLLL LL L L LL LLLL LL LLLL LL L LL LLL LL LLLLLL LLL LLL LLLL LL LLLLLLL LLL L LL LL LL L LL L L LL LLLL LLL LL LLLL LLL LLL LL LL LLLLLL LLL LL LL LLLLLLLLLLLLLLLLLLLLL L L LLL LL LL LLLLLL LLL LL LL LL LLLLLLLLL LLL L LL LLLLLL LL LL LLLLL LL LL L LL L LLL LLLLLLLLLLL L L LLLLLLLLLL LL L L LLLL LLLL LLLLL LLL L LLL LLLL LLLL LL LLLLLLLLLLL LLL LLLL LL LLL LL L LL LLLL LLL LL LLLLLLLLLLL LLLLLLL L LLLLL L LL LL LLL LLL LL L LLLL LLLLL L LLL LLLL L L LLLL LL LLLL LLLLL LL LL LL L L L LL LL LL LL L LL L LLLLLLL L L LL LLL LLLLLL LL LLLLLL LL LLL LLL L L L LL LL LLLLLL LLLLLLL LLLL LLL LLLL LL LLL LL LLLL LLLLLL

RULE 120(01111000) LLL LLLLLLLLLL LL LLLLLLL LL LL LL LLL L L LLLL LLLL LLL LLLLL L LLLLL LLL LLLLL L L LLL LL LL LL L L LL LL LL LLL LLLLLL LL LLLL LL LL LL LLLLLLLLLLL L L LL LLLLLLL LLL LL LLL LLLLLL L L LLL L LL LL LLL LLL LL LLLLL LLL LL LL LL LLLLLLLL LL L L L LLL LLLL L LLL LL LLLL LLLL LL LL LL LLL L LLLL LL LL LL L LL LLLL LL LL LL L LLLL LL LLL LLL LLLLLLLLLLLL LL LL LL LL LLLLLLLL LL LL LL L LLLLL LL LL L LLLLL LL LL LL LLL LL LL LLL L LL LLL LLL LLLL LL LLL LLLL LL LL LL LLLL LL LL LL LLL LL LL LL LLL LL LL LL LL LL LLL LL L LLLLLLL LLL L LLLLLLLL LLLLLL LL LL LL LLL LLL LL LLLLL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL LL L LLL L LL L L LLL LL LLL LLL LLLL L LLLL LL LL LL LL LL LL LL LL LL LL

LLL LLLLLLLLLL LL LLLLLLLL LLLLLLLLL LLLL LLLLLL LLLLL LL LLLL L LL LL L LLL L LLLLLLL LLLLL LLLLLLL LLL LL LL LL L LL LL L L LLLLLLL LLLLL LL LLLLLLL LLLLL LL LLLLLL LL LLLLL L L LL LL LLLLLLL LL LL LLL LLLLL L LLLL L LL L LLLLLLL LL LLL LL L L LLLLLL LLLLL L LL L LLLL L LLLLL LLL LL L LLLLLL LLLLLL LLL L LL LL LLL LLL L LLLLLLLLLLL LL LLLL LLLL LL LLLL L LL L L LLL LLL LLLLLLL L LLL LLL LLLL L LLL LLL LL LLL LL LLL LLLL L L LLLLL LLL LLL LL LL L LLLLLL L L L LLLL L LLLLLLLL LL L LL LL LL L LLLLLL L LLL LL L LLLLLL L L LL LL L LLLLL L LL L LL L LLL LL LL LL LLLLL LLLLLLL LLL LLLLLLL LLL LL L LLLLL L LL LLLL L LLL LL LLLLL LL LL L LLLL LLLLLLLL L L LLL LLLLL LL LLLLLL LLL LL L LLLLL L LL LL LLLLLL LLL LLLL LL LLL LLLLL L

LLL LLLLL L LL LL LL LLLLLLL LLLLLLLL L L LLLL LL LL LL LL LLLLLLL LLLL LLLL L LL LLLLL L L LLL LLLLL LLL L LLLL L LLLLLLL L LLLLLLLL LL L LLL L LL L L LLL LLL L L L LLL L LLLLL LL LLL LL LL LLLLLLLL L L LLLLLL LL LLLLL L LL L L LLLLLLL LL LL LLL L LLLLLLLL L LLLLL LLLLLL LL LL LLLLL L LLL L LLLL L LL LLL L LL LL L LLLLL LLL LL L L LLLL LLLL LLLL L L LLLL LLLL L LL LLLL LL LL L LL LLLLLLL LL L LL L LL LLLLLLL L LLL LLLLLLL LLLL LLL LLLLL LL LL L LLLLLL LL L L LLL L LLLLL L LLL L LLL LL L L LLL LL LLLLL LLLLLL LL LLL L LL LL LLLLLL LL LLL LLLLLL LL LLL LLLLLLLLL L LLL LL LL L LL L L LLL LLLLLLLLLL L L LLLL LLLL LL L LL LLLLLL LLLL L LL L LL L LL L LLLLL LLL LLL L LLLL LL LLLLLLLL L L LL L LLLL LL LLLLLLLLL LLLLLL L L LLL LL L L LLLL L L LLL LLLLL L L LL L LLLL L LL L L L LLLLL LLLLL LLLLLL LLL L LLL L LLLLLL LLL L L LL LLL LL LL LLLLL L L L LLL L L L L LLLL L LL L LLL L LLLL LL L LL LLL LLLL LLLL LLLLL L L LL LLLL LLL LLL L L L LL LLL LLLLL L L LLL LL LL LLLLL LL LLLLLLL LL L LL L LL LL LL LLLLLLLL L L LL LL LLL LLLL LLLL L L LLLL LLLL L LL LLLL L LLLLLLLL L L LLL LLL LL LL LLLL LLL LL LL LL LLL LL L L LL LL LLLLLLL LLLLLL LLLLLLL LLL LL L LLLL LL LL LL L LLL LL LLL LL L LLL LL LLL LLLLLLLLL LL LLLLLL LL LL LLL LL LLL L LLLL L L LL LL LL L LLLL LLLL L LL LLL L LLL LL L LL LLLLLL L L LLLL LLL L LLLL L LLLLLLL LL LLL LL L LLLL L L LL LLLLLLL LL LL LL L LLLL LL LLLLLL LLLLLLLLL LL L L LL LLLLLL LLL L LLLLLLLL L LLL LLLL L LLLLL LL LL LL LLL LLL LL L LLLLL LL LL LLLL LL LL LLLLL L LLLL L L LLLLLLLLLLL LL L LLL LL L LL L LL LLLLLL LLLLLLL LL LLL L L LL LLLLL L LLLLLLLLLLL L LL LL LLL L LL LL L LLLLL L LLL L L L LLL LL LLL L L

LL LLL LL LL LLLL

LLL LL L LL LL LLL LL L LLLLL LLLLL LLLLL L L L LL LL LLL LLLL LLL L L LL L LLLLLL L L LLL LL L LLLLLL LLLLL LLL L LLLLLL L LLL L LLLL L L LLL LL LLL L LLLL LLLL LLLLL LLLLL LL LLLLLLLL L L LLL L L LLL LL LLLLLL LL LL LL LL LL LLLLLLLL LLL LLL LL LL LLLL L LL LLL L L LLL LLLL LLLLLL L LL LLL LLLL LLLLL LLL LLL LL LLLL L LL LL L LLLL LLLLLL LLL LL LL LL LL L LLLL L LLL L LLLLLLLLL LL LLLLLL L L LLLL LL LLLLLL LL LL LL LLLL LLLL LLLLLLL LLLLLLL L L L LLL LL L L LL L LLLL LLLLLLL L LL LL LLL LLL LL LL LL LLLL L LL L LL L LLL LLLLL LL L LLLLL LL LLLLLL LLLLL L L LLLLLLL LLLL L L LLL L LL LL L LLLLLLL LLLL LLLLLLL L L LLLLLLLLL LLLL LLLLLL LLL LLLL LLL L L LLL L LL L LLLLL LL LL LLLL LLL L L LLLLLLL LLLLLL LL LLLL LLL LLLLL L LLLLL LLLLLL LLLLLLLLLL L LL LL LLLL L LL L L LL LLL LLL L L L L L LL LLL L L L LL LL L LL L LLL L LL LLL L LLLLLL L LL LLLL LLLL LLLL LLLLL LLL LLL LL L L LLLLL LLLL LLLLLLLLLL LL L L LLLLL LL LLLL LL LLLL L L LLLL LLLLLL LL L LL LL LL LLLL L LLLL L LLLLLLLLLLLLLLLL LL LLLLL L LL L L LL LLL LLLLL L LLLLL LLLL LL LL L L LL L LL LLL LL LLLLL L LLL LL L LLLLLL LL LLLL L LL LL L LLLL L LLLLL LL LLLL LL LLLL LLLLL LLLLLLLL L LL LLLLLLLLLL LL LLLLL L LLL LLLLLLLLLLL LLL LLL L LLLL L L LLL LLL L L L LLL L LLLL L LLLL LLL LLLL LL LLLLLLLL LL LL LLLL LLL LLLLLLLLLL LL L L LL L LLLLL L LLLLL L LL LLLLL LL LL LL LLLL L LL L L L LLL L LL L LLLLLL L LL LLL LLLL L LLL LL LL L L LLLL L LLL LL LLLLLLLL LLLLLLL LLLL L LLLLL L LL LLLLLL LL LLLLLL L LL LLLLLL LLLL LLLLL LLL LL L L LLL LLL L L LLL LL LLL L LLL LL L L L LLLLLL L L L LLLLLL L L L LL LL L LLL L L L LLLLL L L LLL LLL LLL LLLL LL L L LLLLL LLLLL L LLL LL LL LLLLLL LLL LL LLL L LLL L LLLL LLLLL LLLL L L LLLLLLL LLL LLLL LLLLLL L LLLLLLLL L LLLL L LL LLL LLL L LLLLLLLLLL LLLLLL LLLLL LLLLLLLLLLLLLLL LLLLL LL LLLLLL LLL LL L LL LL LLL LLLL L LL LL LLL LL LLLL LL L LLLLLLL L LL LLLL LLLLLLLLL L L L LLLL L LLLLL L L LL LL LLL LL LL L LL L LLL LLL LLL LL LLLL LLLL LLLLL LLLL LL LL LLLLL LLL L LLLLL L LLLL L LLLLL L LLLL LL LLLLL LLL L L L L L LLL L LLLLL LLLL LLL LLLL L L LL L LL LLL LLL L L LL L LLL LLLLLLLL LL LLLLLLL L LLLLL LLL LLLL LLLL LL LL LLLLLLLLL L LLLLL L LLLLL L L LLLLLL LLLL LLL L LLLLL LL LLLLL LLLL LLL L L LLLL LL LLLL LLL LL LL LLL L LLLLL LL LLL LLL LLLLLL L LLLLLL LL LLL LLLLLLL L LLLLLLLL LLLL LLLLLLLL LL LLL LLLLLLL LLLLLL LL LL LLLL LL LL LLL LL L L LL LL LLLL LLL LL LL LLLLLLL LL LLL L LL L LLLL LLL LL LLL LLLLLL L LLLLLL LL LL LLL L L LL L LLLLLLLLLLL LLLL LL L L L LL L LLL L LL L LLLL L L LLLLLLLL LL L LLLLLLL LLLL L LLL LLL LLLLLLL L L LLL LLL LLL LLLL L L LLLLLLLLLLL

LLL LLLLL L LLLLL L LLL L LLLLLLLLLL L LL L LLL LL LLLLLL LLLL LL LLLLLLLLLL LLL LL LL LLLLLLLLLLL LLL L LL L LL LLLLLL LLL L LL L LL LL LLLLLLLL LLLLLLLLLLLLL L LLL LL LLL LLLL LL LL LL LLL L LLLL LLLL LL LL LL LL LLLLLLLL LLLLL LLLL LLLL LLLLL LLL LLLLLLLLL L LL LL LLLLL LLLL L L LLLLL LLLL LL L LL LLL L LLL L LLL LLLLLL LLLL LL LL LL LL LL LL LLLLLLL LLLLL LLLL LL LL LLL LLLLL LLLLLL L LLL LLL L LL L LLL L LLLL LLL LLLLLL LL L LLLLL LLLLLLLLLLLLLLL LLLL LL LLL LLLLLLLL LL LL LLL L L L L L LLLLLLLLL L L L LL LL LLLLLLL LLLLLLLLLLLL L LLL LLLLLL LL L LLLLLLL L L L L L LLLLLLLL L

RULE 169(10101001) L LL LL LLLLL LLLL LLLLL LLL LLLL LLLL L L L LLLLLL LLLL L LLL LLLL L LLLLLLL LL LLLLLLL L LLLLLLLL L LL LLLLLLLL L L L L LLL L LL LLLL L LLLLL L L LLLLLLLL L LLL L LLL LLLLLL L LL L LLLLLLL L

L LL LL LLLLL LLLL LLLLL LLL LLLL LLLL L L LLLLL LL LLL L LLLL LLLLLL LLL L LLLL LLLLL LLLLLLL LL LLLL LLL LL LL LLL LL L LLLL L LL LLLLLL LL L LL LLLLLL LLL LLL L L L LLL LLLLLL LLLL LLLLLLLLLL L L LL LLLLLL L L LLLL

L LL L LL LLL LL LLL LLLLL LLL LLLL LLLL LLLLL L L LLLL LLL L LLL LLL LLLL LL LL L LLLLLL LLL LL LLLLLLLLLLLLL LLL LL L LLLLL LL LL LL LL LLLLL L L LLLL LLLLLLLL LL LLLL LLL LL L LLLL L LLLL LLLL LLLL LL LL LL LL LL LLL L LL LLLLLL LL

L LL L LL LLL LL LLL LL LLLLLLL LL LLLLLL LL LLLL LLL LL LL LLL L LLLLL LLLLLL LLL LLLLL L L LLLLL LL LL LL L LLLL LLLL LLLLL LL LL LL LL L L L LL LLLLL LLL LL LLLL LLLL LLLLLLLLL LLL LL L LLL LL LL LL LLLL LLL LLLLLL LL LL LL LL LLLLLLLL LL L LLLL LL LL LLL L

L LL L L LL LLL L LL LL LL L LLLL L L LLLL LLL LLLL LL LL LL LLLLLLL L LLL LL LLLLLL LLL LLL L LLLLLL LL LLL LL LL LLLLL LL LLLLL LL LL LLLLL L LL LL LLL LLL LLL LLLLLLL LLLL LLLLLLL LL LLLLLL LLL LL LL LLLL L LLLLLLL L LL LL LLLLLLLLL LLLLL LLL LL LLLLLLL LLLLL LLLL L LL

LLLL LLLLL L LL L LLL LL LLLLLLLLLL L LLLL LL LL LL LLL LLLLLL LLLL L LL LLLLLLLL LLLLLLL LL LL LLL LLL LLL LLLL LLLLL LL LLLLLLL LLLLLLLLL LLL L LL L LLLL LLLLLLLLLL LL LLLLLLL LLLL LLLL LLLLL L L LL LLL L LLLLL LLLLLL LL LLL LLLLL L L LLLLL LLL L LLLLLL LL LLLLLLLL LLLLLLL L LL L

RULE 225(11100001)

RULE 110(01101110)

[0.26-5,-0.27-0.0]

RULE 124(01111100)

RULE 137(10001001)

RULE 193(11000001)

RULE 184(10111000) [(-1,1),(1,-1)]

RULE 226(11100010)

F gure 4 con nued

Comp ex Sy em 14 2003 99–126

R. Alonso-Sanz and M. Mart´ın

118

[58]: to decide whether an arbitrary initial configuration contains a density of ones above or below Ρc . As in synchronization, the density task comprises a nontrivial computation for small-radius CA: again, density is a global property of a configuration, where small-radius CA rely solely on local interactions. 7. Other memories

This work deals only with the memory mechanism described in section 4. Nevertheless, some alternative memory mechanisms are reported in this section. We expect to feature them in a subsequent work. 7.1 Exponential weighting

The decay in the weighting factor can be implemented in an exponential way instead of in the geometric one: (t

(1 (2 (T m(T i ai , ai , . . . , ai 



Tt1 exp  Β(T t)ai Tt1 exp  Β(T t)

.

This is equivalent to a geometric discount model with Α  e Β , so Β 

ln Α. Thus, the fully historic model (Α  1) corresponds to Β  0. The parameter Β ranges in , but as has been seen in section 4, below Α  1/2 memory has no effect, therefore history has no effect for Β beyond ln 2  0.6931. 7.2 Inverse memory

The weighting mechanism may be designed in an inverse way, so that the older rounds are remembered more than the more recent ones. The idea is implemented in the following way: Tt1 Αt 1 a(ti Tt1 Αt 1  1 if    (T  1

a if  i  0 if 

(1 (2 (T m(T i (ai , ai , . . . , ai ) 

fi(T

m(T i > 0.5 m(T i  0.5 m(T i < 0.5.

Recall that in the case of an equality, the opposite to the most recent state is registered, so the configuration at T  3 will be the same as at T  2 regardless of the rule Φ. This will alter the evolution patterns in the inverse memory mechanism as compared to the standard (direct) one as from T  3, even in the fully historic model. For example, starting with a single-site live cell, the pattern at T  3 for any quiescent, Β1  1 rule will be again, as at T  2, a 3 " 3 square. After T  3 these cells are featured as live: Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

119

            7.3 Accumulative memory

The weighting memory mechanism described in section 4 is holistic in its information demands: for evaluating the weighted mean of a cell, it is necessary to know the entire series of states in time. In order to avoid this demand, previous states can be pondered in the following way: (1

(2 (T m(T i ai , ai , . . . , ai  

Tt1 tk a(ti . Tt1 tk

The rounded weighted mean state f will be obtained, as in section 4, by comparing (t Ωai , t

T

 T   tk a(ti t1

to T

(T) 

1  tk , 2 t1

or, in order to work only with integers, by comparing 2Ω to 2. This mechanism is not holistic but accumulative in its demand of knowledge of past history: for evaluating Ω(ai , T) it is not necessary to know the whole a(ti  series because it is determined by the accumulation of the contribution of the last state (T k a(T i ) to the already accumulated Ω(ai , T 1) # Ω(a(ti , t  T)  Ω(a(ti , t  T 1) T k a(T i . In fact we have already considered two such accumulative mechanisms, (i) inverse memory: Ω(a(ti , t  T)  Ω(a(ti , t  T 1) ΑT 1 a(T i and (ii) the full memory scenario described in section 3: Ω(a(ti , t  T)  Ω(a(ti , t  T

1) a(T i . For k  0, we have the full historic model of section 3; for k  1 it is T

2(T)   t  t1

T(T 1) . 2

The larger the value of k is, the more heavily the recent past is taken into account and consequently closer to the ahistoric scenario. Choosing integer k values allows working only with integers (a´ la CA). There is a clear computational advantage of this model over that of section 4. But the accumulative memory mechanism just described Complex Systems, 14 (2003) 99–126

R. Alonso-Sanz and M. Mart´ın

120

has a serious drawback: tk “explodes,” even for k  2, when t grows (see [59]). 7.4 Parity memory

Cells can be featured by the parity with the sum of previous states: T (t p(T i    ai  mod 2. t1

7.5 Interest memory

In the interest memory scenario, a cell will be featured as dead if all its previous states are equal (zero interest means boring), and as alive if any of them are different. Toffoli and Margolus [1987] use the interest idea when considering the time-tunnel, a reversible mechanism based on the rule: Φ((a(T i ))  0 if all the states in  are equal, 1 in the contrary case. 7.6 Continuous valued memory

Historic memory can be embedded in Coupled Map Lattices, in which the state variable ranges in , by considering m instead of f in the application of the updating rule: a(T 1  Φ((m(T i i )). 7.7 Fuzzy rules

Fuzzy CA, with states ranging in the real [0,1] interval, may be obtained by fuzzification of the disjunctive normal form of boolean CA rules by replacing: a  b by min(1, a b), a  b by ab, and $a by 1 a [60]. Featuring cells by the unrounded weighted mean of their past states; that is, m, will provide fuzzy historic CA. An illustration of the effect of memory in fuzzy CA, starting with a single crisp live cell, is given in [42]. The illustration operates on Rule 90: (T

(T

(T

(T

a(T 1  ai 1  $ai 1   ($ai 1  ai 1 , i after fuzzification: (T

(T

(T

(T

a(T 1  ai 1 ai 1 2ai 1 ai 1 , i which gives the historic formulation: (T

(T

(T

(T

a(T 1  mi 1 mi 1 2mi 1 mi 1 . i 7.8 Imperfect memory transmission

The standard CA and the CA with memory models can be combined by considering two types of cell characterization of the neighborhood . Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

121

Thus for a subset of , let it be , the cells can be featured by their last state and the remaining   by their weighted mean states. For example, history can feature the cells of the strict neighborhood but not (T

(T

the cell to be updated: a(T 1  Φ(f i 1 , a(T i i , fi 1 ), or, the contrary, the cell (T (T to be updated but not the outer cells: ai(T 1  Φ(a(T i 1 , fi , ai 1 ). i The different memories mentioned can be considered to implement new reversible mechanisms. For example, by using partial memory (T 1 (T (T transmission, a(T 1  Φ(a(T . i 1 , fi , pi 1 )  ai i 8. Conclusion

The consideration of historic memory of past states has been found to produce an inertial (or conservation) effect. According to this principle, historic memory: 1. Leads chaotic Class III rules in the ahistoric model to turn to Class IIlike behavior (separate simple stable or periodic structures) in the fully historic model. 2. Induces a preserving effect regarding the damage caused by the reversal of a single site value. Thus, memory tends to confine the disruption measured by the difference patterns (DP). Discounting memory, as implying less historic information retained, implies an approach to the ahistoric model. Considering discount, 3. On increasing the value of the memory factor from Α  0.5, for which the evolution corresponds to the standard (ahistoric) model, to Α  1.0 (fully historic), there is usually a gradual variation of the effect of memory. Nevertheless, 4. An appreciable number of exceptions (and discontinuities) in such a gradual effect have been found.

These conclusions agree qualitatively with those found when starting with a single-site live cell [42, 43], which could be considered as a special case of reversion of a single site. In general, it is possible to distinguish two main approaches to cellular automata (CA): forward and backward. The forward (theoretical) approach implies the study of transition rules of a given cellular space in order to establish its intrinsic properties (dynamical behavior, pattern growth, and so on). The backward (practical) approach involves the design of sets of transition rules to match the “correct” behavior of the CA system of a given complex system (physical, biological, social, and so on). This paper adopts the forward approach: it introduces a kind of CA with memory (), and surveys its properties in the simplest (elementary) scenario in a fairly qualitative (pictorial) form. Borrowing Complex Systems, 14 (2003) 99–126

122

R. Alonso-Sanz and M. Mart´ın

an expression from Vichniac [1990], this paper deals with the  “zoology,” that is, the study of  for its own sake. Following this metaphor,  increases the CA-(bio)diversity. A more complete analysis of the class  is left for future work. For example, the dynamics of  in the state (phase) space is to come under scrutiny. Another interesting open question is the analysis of the effect of memory on solving computational tasks such as synchronization, density, and ordering. The study of r  2  will inform on the effect of memory in complex (Wolfram’s Class IV) rules. The study of two-dimensional  will follow. In this future work, the pioneer papers by Wolfram [62, 63] will again be an important reference. The forward and backward (or inverse) approaches are obviously interrelated. A major impediment in the backward approach stems from the difficulty of utilizing complex CA behavior to exhibit a particular behavior or perform a particular task. This has severely limited their applications and hindered computing and modeling with CA [64].  structurally exhibit a growth inhibition feature found exceptionally in standard CA [28]. The extent of the growth inhibition can be modulated by varying the memory factor Α. This could mean a potential advantage of  over standard CA as a new tool for modeling slow diffusive growth from small regions, a common phenomenon in nature. The question of how errors spread and propagate in cooperative systems has been studied in a variety of fields. Given the difficulty of creating analytical models for any but the simplest systems, most investigations have been conducted by computer simulations, especially in the area of statistical physics of many-body systems [8, 65]. Errors reverse the state in elementary CA. In , the damage induced by a single error is generally confined to the proximity of the site where it occurred. From this point of view,  can be featured as resilient in the face of errors. Fault tolerance is an important issue when considering systems with a large number of components, in which faults will be highly probable. The robust  could play a role in this (nanotechnological) scenario. Acknowledgments

This work was supported by Comunidad de Madrid (08.8./0006/2001.1) ˜ ´ and INIA(MCYT) RTA02-010 . Dr. M. Iba´ nez-Ruiz (ETSI Agronomos, Estad´ıstica) contributed to the computer implementation of this work. References [1] T. Toffoli, Encyclopedia of Physics, edited by R. G. Lerner and G. L. Trigg (VCH Publishers, NY, 1990). Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

123

[2] T. Toffoli, “Cellular Automata as an Alternative to (Rather than an Approximation of) Differential Equations in Modeling Physics,” Physica D, 10 (1984) 117–127. [3] T. Toffoli, “Occam, Turing, von Neumann, Jaynes: How Much Can You Get for How Little?” in ACRI-94, edited by S. Di Gregorio and G. Spezzano (ENEA, Italy, 1994). [4] M. Gardner, “The Fantastic Combination of John Conway’s New Solitaire Game ‘Life,’ ” Scientific American, 223(4) (1970) 120–123. [5] W. Poundstone, The Recursive Universe (William Morrow, New York, 1985). [6] S. Wolfram, Cellular Automata and Complexity (Addison-Wesley, Reading, MA, 1994). [7] G. Y. Vichniac, P. Tamayo, and H. Hartman, “Annealed and Quenched Inhomogeneous Cellular Automata,” Journal of Statistical Physics, 45 (1986) 855–873. [8] M. Sipper, Evolution of Parallel Cellular Machines (Springer, 1997). [9] S. A. Kauffman, The Origins of Order (Oxford University Press, 1993). [10] L. O. Chua, CNN: A Paradigm for Complexity (World Scientific, Singapore, 1998). [11] T. E. Ingerson and R. L. Buvel, “Structure of Asynchronous Cellular Automata,” Physica D, 10 (1984) 59–68. [12] K. Kaneko, “Phenomenology and Characterization of Coupled Map Lattices,” in Dynamical Systems and Sigular Phenomena (World Scientific, 1986). [13] D. A. Wolf-Gladrow, Lattice-Gas Cellular Automata and Lattice Boltzmann Models (Springer, 2000). [14] J. R. Weimar, “Coupling Microscopic and Macroscopic Cellular Automata,” Parallel Computing, 27 (2001) 601–611. [15] C. Wissel, “Grid-based Models as Tools for Ecological Research,” in The Geometry of Ecological Interactions, edited by Dieckmann, et al. (2000). [16] S. Di Gregorio and G. Spezzano, Proceedings of the First Conference on Cellular Automata, ACRI-94 (ENEA, Italy, 1994). [17] S. Bandini and G. Mauri, Proceedings of the Second Conference on Cellular Automata, ACRI-96 (Springer, 1996). [18] S. Bandini, R. Serra, and F. Suggi (editors), Proceedings of the Third Conference on Cellular Automata, ACRI-98 (Springer, 1998).

Complex Systems, 14 (2003) 99–126

124

R. Alonso-Sanz and M. Mart´ın

[19] S. Bandini and T. Worsch, editors, Proceedings of the Fourth Conference on Cellular Automata, ACRI-2000 (Springer, 2000). [20] A. Ilachinski, Cellular Automata: A Discrete Universe (World Scientific, 2000). [21] A. Wuensche, “DDLab: Discrete Dinamics Lab,” www.santafe.edu/%wuensche/ddlab. [22] B. Chopard and M. Droz, Cellular Automata Modeling of Physical Modeling (Cambridge University Press, 1998). [23] R. E. Wilde and S. Singh, Statistical Mechanics (Wiley, NY, 1998). [24] P. Rujan, “Cellular Automata and Statistical Mechanical Models,” Journal of Statistical Physics, 49(1,2) (1987) 139–222. [25] J. L. Lebowitz, C. Maes, and E. Speer, “Statistical Methods of Probabilistic Cellular Automata,” Journal of Statistical Physics, 59 (1990) 117–170. [26] F. Alexander, I. Edrei, P. Garrido, and J. L. Lebowitz, “Phase Transitions in a Probabilistic Cellular Automaton: Growth Kinetics and Critical Properties,” Journal of Statistical Physics, 68 (1992) 497–514. [27] U. Dieckman, R. Law, and J. A. J. Metz (editors), The Geometry of Ecological Interactions: Symplifying Spatial Complexity (Cambridge University Press, 2000). [28] S. Wolfram, “Statistical Mechanics of Cellular Automata,” Reviews of Modern Physics, 55 (1983) 601–644. [29] S. Wolfram, A New Kind of Science (Wolfram Media, Champaign, IL, 2002). [30] J. L. Casti, “Science Is a Computer Program,” Nature, 417 (2002) 381– 382. [31] J. Giles, “What Kind of Science Is This?” Nature, 417 (2002) 216–218. [32] A. Wuensche and M. Lesser, The Global Dynamics of Cellular Automata (Santa Fe Institute Studies in the Sciences of Complexity, Volume 1, Addison-Wesley, 1992). [33] R. Feymann, Feymann Lectures on Computation, edited by T. Hey and R. W. Allen (Perseus Publishing, Cambridge, MA, 1996). [34] T. Toffoli, “Computation and Construction Universality of Reversible Cellular Automata,” J. Comp. System. Sci., 15 (1977) 213–231. [35] E. Fredkin, “Digital Mechanics: An Informal Process Based on Reversible Universal Cellular Automata,” Physica D, 45 (1990) 254–270. [36] N. Margolus, “Physics-like Models of Computation,” Physica D, 10 (1984) 81–95. Complex Systems, 14 (2003) 99–126

Elementary Cellular Automata with Memory

125

[37] T. Toffoli and N. Margolus, Cellular Automata Machines (MIT Press, Cambridge, MA, 1987). [38] G. Y. Vichniac, “Simulating Physics with Cellular Automata,” Physica D, 10 (1984) 96–115. [39] G. ’t Hooft, “Equivalence Relations Between Deterministic and Quantum Mechanical Systems,” Journal of Statistical Physics, 53(1,2) (1988) 323. [40] K. Svozil, “Are Quantum Fields Cellular Automata?” Physics Letters A, 119 (1987) 153–156. [41] A. Adamatzky, Identification of Cellular Automata (Taylor and Francis, London, 1994). [42] R. Alonso-Sanz and M. Martin, “One-dimensional Cellular Automata with Memory: Patterns Starting with a Single Site Seed,” International Journal of Bifurcation and Chaos, 12 (2002) 205–226. [43] R. Alonso-Sanz and M. Martin, “Two-dimensional Cellular Automata with Memory: Patterns Starting with a Single Site Seed,” International Journal of Modern Physics C, 13 (2002) 49–65. [44] R. Alonso-Sanz “Reversible Cellular Automata with Memory,” Physica D, 157 (2003) 1–30. [45] R. Alonso-Sanz, M. C. Martin, and M. Martin, “Historic Life,” International Journal of Bifurcation and Chaos, 11 (2001) 1665–1682. [46] R. Alonso-Sanz, “The Historic Prisoner’s Dilemma,” International Journal of Bifurcation and Chaos, 9 (1999) 1197–1210. [47] R. Alonso-Sanz, M. C. Martin, and M. Martin, “Discounting in the Historic Prisoner’s Dilemma,” International Journal of Bifurcation and Chaos, 10 (2000) 87–102. [48] R. Alonso-Sanz, M. C. Martin, and M. Martin, “The Historic Strategist,” International Journal of Bifurcation and Chaos, 11 (2001) 943–966. [49] R. Alonso-Sanz, M. C. Martin, and M. Martin, “The Historic-Stochastic Strategist,” International Journal of Bifurcation and Chaos, 11 (2001) 2037–2050. [50] R. Alonso-Sanz, M. C. Martin, and M. Martin, “The Effect of Memory in the Continuous-valued Prisoner’s Dilemma,” International Journal of Bifurcation and Chaos, 11 (2001) 2061–2083. [51] P. Grassberger, “Chaos and Diffusion in Deterministic Cellular Automata,” Physica D, 10 (1984) 52–58. [52] E. Jen, “Aperiodicity in One-dimensional Cellular Automata,” Physica D, 45 (1991) 3–18. Complex Systems, 14 (2003) 99–126

126

R. Alonso-Sanz and M. Mart´ın

[53] I. Kusch and M. Markus, “Mollusc Shell Pigmentation: Cellular Automaton Simulations an Evidence for Undecidability,” Journal of Theoretical Biology, 178 (1996) 333–340. [54] R. Das, J. P. Crutchfield, M. Mitchell, and J. E. Hanson, “Evolving Globally Synchronized Cellular Automata,” in Proceedings of the Sixth International Conference On Genetic Algorithms, edited by L. J. Eshelman (Morgan Kaufmann, San Francisco, CA, 1995). [55] N. Pitsianis, Ph. Tsalides, G. L. Bleris, A. Thanailakis, and H. C. Card, “Deterministic One-dimensional Cellular Automata,” Journal of Statistical Physics, 56 (1989) 99–112. [56] W. Li and M. Nordhal, “Transient Behavior of Cellular Automata Rule 110,” Physics Letters A, 166(5/6) (1992) 335–339. [57] S. Wolfram, “Random Sequence Generation by Cellular Automata,” Advances in Applied Mathematics, 7 (1986) 123–169. [58] M. S. Capcarrere, M. Sipper, and M. Tomassini, “Two-state, r  1 Cellular Automata that Classifies Density,” Physical Reviews Letters, 77 (1996) 4969–4971. [59] R. Alonso-Sanz and M. Martin, “Cellular Automata with Accumulative Memory: Legal Rules Starting from a Single Site Seed,” International Journal of Modern Physics C, (in press). [60] G. Cattaneo, P. Floccini, G. Mauri, et al., “Cellular Automata in Fuzzy Backgrounds,” Physica D, 105 (1997) 105–120. [61] G. Y. Vichniac, “Boolean Derivatives of Cellular Automata,” Physica D, 45 (1990) 63–74. [62] S. Wolfram, “Universality and Complexity in Cellular Automata,” Physica D, 10 (1984) 1–35. [63] N. H. Packard and S. Wolfram, “Two-Dimensional Cellular Automata,” Journal of Statistical Physics, 38 (1985) 901–946. [64] A. Cowan, D. Pines, and D. Meltzer, Complexity: Metaphors, Models, and Reality (Perseus Books, Cambridge, MA, 1999). [65] N. Boccara, E. Goles, S. Martinez, and P. Picco, (editors), Cellular Automata and Cooperative Systems (Kluwer Academic Publishers, Dordrecht, The Netherlands / NATO ASI Series C, 1993).

Complex Systems, 14 (2003) 99–126