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Elementary Cellular Automata
with Memory
Ram ´on Alonso-Sanz ETSI Agr ´onomos (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 dynami- cal systems. A CA system is composed of adjacent cells^1 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 digi- tal computers, enabling exact computation. The synchronicity of the updating mechanism, the regular topologies, and the locality of interac- tions^2 make the CA paradigm ideally suited for parallel computers.
Electronic mail address: [email protected]. (^1) The 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.
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 equa- tions [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. Of- ten, 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 pro- gramming language, as postulated by the Artificial Intelligence com- munity (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 Neu- mann. 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 Gard- ner 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 dur- ing 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 lightning- quick 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 out- come 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 ap- proach to real problems (connecting different levels of detail) enlarged the basic paradigm, leading to systems related to CA (mere extensions in
102 R. Alonso-Sanz and M. Mart´ın
triplets: ( a
( T i 1 ,^ a ( iT ,^ a
( T i 1 )
111 110 101 100 011 010 001 000
Β 1 ,^ Β 2 ,^ Β 3 ,^ Β 4 ,^ Β 5 ,^ Β 6 ,^ Β 7 ,^ Β 8 binary
8 i 1
Β i 28 i decimal_._ The rules are conveniently specified by a decimal integer, to be re- ferred 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 re- flection 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: Β 1 Β 2 Β 3 Β 4 Β 2 Β 6 Β 4 0. Another set of 32 categorized rules are those re- flection symmetric but nonquiescent , with binary codes of the form: Β 1 Β 2 Β 3 Β 4 Β 2 Β 5 Β 4 1 (the binary specification of any nonquiescent rule ends in a 1). There are 128 semi-asymmetric (either Β 2 Β 5 or Β 4 Β 7 , but not both) rules and 64 fully asymmetric (Β 2 Β 5 and Β 4 Β 7 ) rules. There are 88 equivalence classes, that is, 88 fundamentally inequiva- lent 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 ( i T is taken to denote the value of cell i at time step T , the site values evolve by iteration of the mapping: a ( iT^1 Φ(( a ( iT )), 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 ( iT^1 Φ(( f (^) i ( T )), with f (^) i ( T being the mean state of cell i after 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
Elementary Cellular Automata with Memory 103
are the second order in time (memory of capacity two) reversible 10 rules due to Fredkin. These rules (referred to by Adamatzky in [41] as the class ), are of the form:
a ( iT^1 Φ( a ( iT ) a ( T 1 i ,
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]:
a ( iT^1 Φ( f (^) i ( T ) f ( T 1 i.
But to preserve the reversibility inherent in the XOR operation, the re- versible mechanism incorporating memory must be of the form (see [44]):
a ( iT^1 Φ( f (^) i ( T ) a ( T 1 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:
a ( iT^1 Φ( a
( T i 1 ,^ a
( T i 1 ). Rule 90:^ a ( iT^1 ^ a
( T i 1 a
( T i 1 mod 2, would be an 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 two- dimensional 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.
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 ini- tial pattern can be decomposed into the superposition of patterns from a single-site seed. Each of these configurations can be evolved inde- pendently 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 succes- sor 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 configura- tions 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 num- ber of different configurations generated by the evolution of a single-site seed is smaller in the historic model. This, generalized to any initial con- figuration, 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 ( iT Τobtained Τ time steps before the last round is actualized to the value ΑΤ a ( iT Τ, with Α being the memory factor. A given cell will be featured by the rounded weighted mean of all its
106 R. Alonso-Sanz and M. Mart´ın
past states. This well-known mechanism fully ponders the last round (Α^0 1), and tends to forget the older rounds. After time step T , the weighted mean of the states of a given cell will be:
m ( iT a ( i ,^ a
( i ,^...^ ,^ a
( T i ^ ^
Tt 1 Α T^ t^ a ( it Tt 1 Α T^ 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:
f (^) i ( T
1 if m ( i T> 0_._ 5 a ( i T if m ( i T 0_._ 5 0 if m ( i T< 0_._ 5_._ This is equivalent to studying the sign of:
( iT ( a (1 i , a (2 i ,... , a ( iT ) 2
T t 1
Α T^ t^ a ( it
T t 1
Α T^ t^
T t 1
(2 a ( i t 1)Α T^ t
f (^) i ( T
1 if Ti > 0 a ( i T if Ti 0 0 if Ti < 0_._ After T 3: ( a (1 i , a (2 i , a (3 i ) Α^2 (2 a (1 i 1) Α(2 a (2 i 1) 2 a (3 i 1. Of course, if a (1 i a (2 i a (3 i , history does not alter the series and it will be f (^) i (3 a (3 i. Neither does history take effect until T 3 if a (2 i a (3 i nor if a (1 i a (3 i (see [42]). But the scenario may change if a (1 i a (2 i a (3 i :
(0, 0, 1) (1, 1, 0) Α^2 Α 1
0 Α
Thus, provided that Α > Α 0 , 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 > Α 0 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 Α 0 , 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) Α^3 Α^2 Α 1 0 Α 0_._ 5437. In general, in the most unbalanced scenario, if a (1 i a (2 i a ( iT^^1 a ( iT it holds that:
(0, 0,... , 0, 1) (1, 1,... , 1, 0)
T 1 i 1
Α i^ 1
Α T^ Α
1 0 Α T^ 2 Α 1 0.
Figure 1(a). Evolution of legal (Β 1 Β 2 Β 3 Β 4 Β 2 Β 6 Β 4 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 spatio- temporal 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 Ρ T , up to 200 time steps, starting with an initial density Ρ 0 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 Ρ T 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.
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, Ρ T 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). Synchronization 11 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 configura- tions. 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 (Λ L , Λ R ) 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.
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.
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 re- versed 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 perturba- tion ( 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 Λ L , R 0 are not greatly affected by memory (e.g., Rules 156 and 100). A considerable number of equivalence classes of asymmetric rules have !Λ L! !Λ R! 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 110^14 and the others of its equivalence class, may serve as a paradigmatic example of the expected effect of mem- ory: the damage induced by the reversal of the initial central site value becomes more constrained as the memory factor increases, with no dis- continuities 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].
Elementary Cellular Automata with Memory 115 α 1.0 0.9 0.8 0.7 0.6 0. RULE (^) v v v v 91(01011011) vvv v vv v v vv v v (^) v vvvvvv (^) v vv vvvv v (^) vv vv (^) v vvvvvv v (^) v v^ v^ v (^) v v v v v^ v^ v v (^) v (^) v v v^ vv^ vv (^) vv (^) vv^ v^ vvvvv^ vvvv^ v^ v^ vv (^) vvvvvvvvv vvvv vvvvvv (^) vvvvvvvv vvvv^ v^ v vv (^) vvv vv v^ v (^) vvvvvvvv vvvvvvv vvv vv^ v^ vvvvvv^ v v^ vv^ v^ vv^ v^ v^ vvv vvvv^ vv v v v^ v^ v^ v^ v^ v^ v^ vv^ v^ v^ v^ v^ v^ v^ vvv^ v^ v v v v v^ v^ v^ vvv^ v^ v v vvvvvvvvv^ vvvv^ vvvvvv^ vvv^ vvvvv^ vvvvvvvvv^ vvvv^ vvvvvv^ vvvvvvvvv vvvv^ vvv^ vv^ vvv^ vv vv v vv^ v vv^ v^ v^ vv vv^ v^ v^ vv^ v v vv^ vvv^ vvvv^ v^ v^ vv^ v^ v^ vv v v v^ vvvvvvv^ vvvvvvv^ vvvvvvv^ v^ vvvvvvvvvvv^ v^ vvvvvvvvvvv v vvvv^ v^ vvv^ vv^ v^ vv^ vvvv^ vv^ v^ vv^ vvvv vv v^ v^ v^ v vvv^ v^ v^ v^ v^ v v vvv^ v^ vv vvvv^ v v vvvv vv v v v v v vv (^) v v (^) v v vv v v v (^) v (^) v (^) v v v v vvv vv v v v v v (^) vv vv v vv (^) v v vv v (^) v v (^) vv v v (^) v vvvvvvvv vvvvvv vvvvvvvvv vvvv vvvvvvvv vvvvvv vvvvvvvvv vvvv (^) vvv (^) vv (^) vvv 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v v v vv^ vv^ vv^ vv^ vv (^) vv vv vv v^ v^ v^ v^ v^ v^ v v v v v v (^) v v v^ vvvvvvv^ vvvvvv vvvv vvvv vvvv vvvvv^ vvvvvvvvvvv^ v^ vvvvvvvvv vv (^) vv vv^ v^ vv^ vvvv^ vv^ v^ vv vvvv vvvv vvvv vv^ v^ v^ v^ v^ v^ v^ v^ v^ v v (^) v v v (^) v v v v^ v^ vv^ vv^ vv (^) vv vv vv v^ v^ vv^ v^ v^ v^ v^ v v v v v v (^) v v vv^ v^ v^ v^ v^ v^ v^ v^ v^ v v (^) v v v v v v v^ v^ v^ v v v v vv^ v^ v^ v^ v^ v^ v^ v^ v^ v v v v v (^) v v v v^ v^ v^ v^ v^ v (^) v v (^) v vvvvvvvvvvvvvvvvv v v v v v vv v v v (^) v vvv^ v^ v^ v v v (^) v v vv^ v^ v^ v^ v^ v^ v^ v^ vv vv (^) v v v v v vv^ v^ v^ v^ vv v v (^) v vvvvv^ vvvvvv^ vvvvvvvvv^ vvvv^ vvvvvvvv vvvvvv (^) vvvvvvvvv (^) vvvv vvvv^ vvv^ vv^ vvv vv v^ vv^ vv^ vv^ vv vv (^) vv vv v^ v^ v^ v^ v v v v vv^ v^ v^ v^ v^ v^ v^ v^ v v v v v v (^) v v v^ v^ v^ v^ v (^) v v (^) v v^ v^ v^ v^ v^ v^ v^ v^ v v v (^) v v v (^) v v v^ v^ v^ v^ v^ v^ v^ v^ v v v (^) v v v (^) v v vv^ vv^ vv^ vv^ vv vv vv (^) vv v^ v^ v^ v^ v^ v^ v^ v v v v (^) v v v v v^ vvvv^ 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v (^) v v v (^) v v (^) v v v v v (^) v v v (^) v v v (^) v v v (^) v v (^) v v v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvv vv (^) vv (^) vv vvvvv (^) vv (^) vv (^) vv vv v v (^) v vvv (^) vv vvv vvv (^) vvv (^) vvv vvvv (^) vv vvvvvvv v vvvvv v v v vv vvv (^) vvv vvvv vv vvvv (^) vvvv vvvvv vvvvvv vv vv (^) vv (^) vv vv v (^) v vv v (^) vv v vv v vvv (^) vv v (^) vv v vvv (^) v v (^) v (^) v v (^) v (^) v vvvvvvvv vv (^) vv (^) vvvvvvvvvvvv (^) v v (^) v v vv v v (^) vv v vv (^) vvvv vvvvv vvvv vvv v vvvvvv vvvvv v vv vvvvv vvvvvvvvv v vv (^) vvvvvvvvv (^) vv vv (^) v v (^) v vv v vvvvvvvvvvvvvv (^) v vvv vv v v vvv vvvv vvvvvv v (^) vvvv v (^) vvv v vv vv (^) vvv vvvvvvvvvvv (^) vv vvv (^) vv vvv vv (^) vv (^) vvvv vvv vvv vv vv v vv vv (^) v vvv v v (^) v vvv vv (^) vv v vv (^) vvvvvvvv vvvv vvv (^) vv vvvv vvv v (^) v vvvvv v v (^) v v (^) v (^) v v (^) v (^) v vv vvvvvvvv vvvvv vv (^) vvvvvv vvvvv vvvv vvvvvvvvvvvvv (^) vvv vvvvv (^) vv (^) vv vvvvvvvvvvv vvvvvvvvvvvv v (^) 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vv (^) vvv (^) vvv v (^) v v v (^) v vv vv v (^) v (^) v v (^) vv (^) v v v vv v (^) vvv vvvv vvvv vvvv (^) vvv v vvvv vvvv v (^) v (^) v (^) v v (^) v (^) v vvvvvvvvvvvv v vv vvvvvvvv (^) vvvvvv (^) v vvvvvvvvv vv v v v (^) v vvv vv (^) vv (^) vvvvvv vvv (^) vvvv vv v (^) vvv v (^) vv (^) v v (^) vvv vvvvvvv (^) vvvv vv (^) vvvvv (^) vv v vv v vvvvvvv v (^) vvvvv v v (^) v v v (^) v v (^) vvv v v (^) v v vvvvvvv vvvvv vv vvv vvvvvvv vvv vv vvvvvvv (^) vvv vvv vvv (^) vvvv v vv (^) v v v (^) v v (^) vv vv vv vv (^) vvv v (^) vvvv (^) v v (^) v v (^) v vvv vvv (^) v vvvvvv vv vv v vv (^) vvv vv vvvvvvvvvv vvvv (^) v (^) v vvvvvvvvvvvv vvvv vvvvv vvvvv vv (^) vvvvv vvv vvvv v vvvv vvvvvvv (^) vv (^) vvv vvvvvvvvv (^) vvvvvvv v v vv (^) vv vvv vv (^) vvvv vvv vvvvvvv v vv vvvvvvv (^) v vv (^) vvvv vvvvv vvv vvv v vv vvvv (^) v (^) v (^) vvv vvvv v (^) vvvvv vvv vvvv vvvv v vvvvvvvvv vvvv vvv v vv vvvv (^) vv vvvvvvvvvv v (^) vvvv vvvvvvv (^) v v v (^) vvvvvvvvv vvv v vvv vvv v vv (^) vv vv vvvvvv (^) v vvvvvv (^) vv vv (^) v (^) v (^) v (^) v vv vvvv vv (^) v vvvvvv vvvv vvvv vv (^) v vvvvvvv vvvvv (^) vvvv v vvvvvvvv v (^) vvv v vvvv v v (^) v v v v v v (^) v v v (^) v v (^) v v v (^) vv vv (^) vv v v v (^) v v v (^) v v (^) v v v (^) v v v (^) v v v vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv v v v v (^) v v v v v v v v v v v v vv v v (^) v v v (^) v v (^) v v v (^) v v v (^) v v v v vv vv vv vv vv vvvv vvv v vvv (^) v vv vv vv vv vv vvvvv vv v (^) v vvvvv vv (^) v v (^) v vvvv (^) vv (^) vvv v v (^) vvvv v (^) vvvv (^) v v (^) v v vvvvvvvvvvvvvvvvvvvv vvvv vvvvvvvvv (^) vvvv (^) vv v (^) vvvv (^) v vv vvvv (^) v v vv (^) vvvv v vv vv vvv (^) v v v v vvvv v (^) vvvvvv v vvv vv vvv (^) v v vvvvvvvv v (^) v vvv (^) v (^) vvvvvv (^) v vvv (^) vvvv v vv vvvvvv (^) vv (^) v vvvv vvv vvvvvvvvvvv vv (^) v vv (^) vvvvvvv (^) vv vvv v vvvv (^) v (^) vv vv vvv vvvv vvvvvv v vvv (^) v (^) vv (^) vvvvvvvv vvv v (^) v vv v (^) v vv (^) v vvvvvv v (^) vvv (^) vvv (^) v v (^) vvvvv vv vvv v vvvvv vvv vvv vvvv vvvv v v v v vv v v (^) v vvv v (^) vvvvvvvvv vvvvvv (^) vvv vvvvvv (^) vvv (^) vvv vvvvvv vv vv v (^) vvvv (^) v vvv vvvvvvvv (^) v (^) vv vv vvvvvvvvvv vvvvv v vvv vvvvvv vv (^) vvvvv (^) v vvvvvv vvvvv v v (^) vvvvvv vvvv vv v (^) vv (^) vvv v vvvvvvvvvv vv v vv (^) vvvvv (^) vvv vvvv vv vvv v vvv (^) vvv vvvv v vv vv vvv (^) v vvv v v (^) v vv vvvv vvv (^) vvv (^) v v (^) v v v vv vvvv (^) vv v vv (^) vv vv vvvvvvv vvvv v (^) vvvvvvv vvvvvvvvvv (^) v vvvv v (^) v v v (^) vv vvvvvv vvvvv vv (^) vvvv vv v vvvvv vvvvv (^) v vv vvv vvv v v vv v (^) vvv v vvvv vvvv v (^) v vvvv v (^) v vv vvv v v vvvv vvv (^) vvvvv (^) v vv v vv v vv v vv (^) vvvvvv vvvvvvvvvvv v (^) vvvvvvvv vv (^) vvvvvvvvvvvv (^) vv vvvvv (^) vvvvvvvvvvvvv (^) v vvv vv v v (^) vv (^) v v (^) v vvvv vvvvvv vvvvvvvvvvv v vvvv vvvvvv vvv vvvv v v (^) vvvvvvv (^) vvv (^) vvvvv v v (^) v vv v v (^) vvv (^) vv v (^) vv (^) v vvvv vvv vv v vv v vvv v vv (^) vv vvvvvvvvvvvvvvv v vv v (^) v vvvv vvvvvvv (^) vvvv vvv v v v (^) vvvv (^) vv vv (^) v vv (^) v (^) vv (^) vvvvv v (^) vvvvvvvvv vvvv vvv v v v (^) vvv v (^) v v vvv v v v (^) vvvv vv (^) v vv v (^) vv (^) v vv vvv vvv vvv vv v (^) v v v vvv v v v (^) vvv v (^) vv v vv vvv vvvvvvv v (^) v v vvvvvvvvv vv (^) vv vvv (^) vvvvvv vvvvv v (^) vv vvv (^) vvv v (^) vv v vv (^) vvvvvvvvv (^) vvv vv vvv (^) v vv vvv (^) v v vvv vv (^) vvvv v vvvvvvv vv (^) vvvv (^) vv vvvv vv (^) vvv v (^) v vv vvv (^) v vv vvv (^) v vv vvv (^) v vv (^) vvvvv vvvvvv vv (^) vvvv (^) vv (^) v v (^) v v v (^) v (^) vv (^) v vv v vvvv vvvvv vvvv (^) v (^) vvvvvvvvvv vvvv (^) vv vv (^) vv vvvvvv (^) vv v (^) v vvvv vvvvvvv (^) v (^) vv (^) vvvvvv v vv vvvv vv vv (^) v vv (^) v (^) vvvvvv vv vv vvvvv (^) v vvvvvvvv v vvvvvv v vvvvvv v vv vvvv (^) v vv (^) v vv (^) v vv vv (^) vv (^) vv v (^) vvvv vvvvvvv v vvvvvv v vv v v vv (^) vvvvvvvvvv (^) vv (^) vvvv vv (^) vv vvvvvvvvv v v v v v v v v vvvvvv v vvv v vv v vvv v v vvvv v v vvvv vv vv vv v vv vv vv vv v v vvvv vvvvv vvv vvvv vv vv v v vv v vvvvv v v v vvvvvvv vvvv vvv vvv vv vvvvvv v v vv vvvv vvvvvvv vv v v v vv v vvvv v vvvvv vv v v v v v v v vv vv v v v v v v v v vvvv vvv vv vvvv v vvvvv v v v v vvv v v vv vvvv vvvvvv vv v v vvv vvv v vvvvvv vv v vvv v vvv v v v v vvvv vv vv v vvvvvv vvvv v v v vvvv vv v vvvv vv vvvvv vvvvv vvv v v v vv vvvvvvv v vvv v v v vvvvvv vv v vvv v v vvv v vv vvvvvvv vv vvvv vvv vv vv vvvvv vv vv vvv vv v vv vvvvv vv vv vvvvvv v vv vv vvvvv vv vvv vv vv v vv v v vvvvv vv vvvvvvv vv vvvv vvv vv vvvv RULE 109(01101101) v vv v v vvvv v v (^) vvv v (^) vv v v (^) vvv (^) v v v vvvvvv v vvvvvv (^) vv (^) vv v v (^) vvv vv vvvvvvvvvv v vv vvvvv v vv v vvvv v vv vvvv v vv v vv v (^) vv v vv (^) v v v (^) v (^) vv^ vv^ v^ vvvv^ vv v vvv (^) vvv vvv v vvvv (^) v vv v^ v^ v v (^) v (^) v vvvv vvvv^ v^ v vvvvvvvv v vvvvvvvvvvvvvvv^ vvvvvvv vv vv (^) vv (^) vv vvvv^ v^ v vvvv (^) vvvvv vvvv vvv vvv v^ v^ v (^) v (^) vvv v v vv v v v vvvv v vvvvvvvvvvvvvvvv^ vvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvvv v^ vvv^ v (^) vvvvvv (^) vvv v v^ v^ v^ v vvvvvvv v vv (^) v vvvv v^ vv^ vv vv v^ v^ v (^) v vv v (^) vvvvvvvvv (^) v (^) v v vvvv v vvvvvvvvvv^ vvvv vv vvvvv^ vvvvvvvvvvvv vv vv (^) vvvvvvv^ vvvv^ v^ vv^ v vvv v v vvvvv vv (^) v vvv (^) vvvvv^ v v vvv (^) v^ vv^ v^ vv vv v (^) v vvvv vv v v vvv v vvv^ v^ v vv (^) v v vvvvv vvvv vvvvvv v vvvvv (^) v vv vv (^) vvvvvvv v vv v vv v vv v (^) vv v vv vvvvv vvv v (^) vv v vv v (^) v v v (^) v v (^) v (^) v v (^) v vvv v vvvvv (^) vvvv vvvvv vvv (^) v vv (^) vvv vv vvv (^) vvv v vvvv v (^) v v (^) vvvvvvvv (^) v (^) vvvvv v (^) v v (^) v v (^) v (^) v vvv v (^) vv v vv (^) vvv (^) vv v (^) vv vvvvvvv vvvvvvvvvvvvv vvvvvvvvvvvvvvvvvv vv vv v vv (^) vv vv vv vvvvvvvvvv vvvv (^) v v vv vvv vv v v vvvv (^) vvvv vvvvvvvv (^) vvvv (^) vvv v vvv (^) vvv (^) v (^) v vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv (^) vvvvvv v (^) vvv v vvvvv (^) vvvvvvvvv vvvvvvv vv (^) vvvvv (^) vvvvv v v v v (^) v v (^) v v v v vvvvvvvvv v (^) vv v v vvv (^) vv vv vvvv v (^) v v v (^) v vv (^) v vv v (^) v (^) vvvv (^) v vvvvvvvvvvvvvvvvvvvvvvvv vv (^) vv (^) vv (^) vv vv v vvvvvv vvvv vvvvvv vvvv vvvv v vvvv vvvv v vvvv vvvvv vvv v vvvvv vvv vv vvvv (^) vv (^) vv (^) v (^) v (^) vv vvvvvvv (^) v vv v v v (^) vvvv vv vv vvvvvvvv v (^) vvv vvv vv vvv vvvv v vvvv v (^) v vv (^) v vvvvvv v vv vvvv vvvvvvv v vvv vv vvvv vvvv v v v (^) vv vv v vvv vvvvv vvvvv v vv v (^) vv v (^) vv v vv v vv vvvvv v vv v vv v v v v v (^) v vv (^) v v (^) v v vvvvv vvvv vvvvv (^) vvv vvv vv (^) vvv vvv v vv (^) vvv vv vvv vvvvvvv v vvvvv v (^) v vvvvvvvv v (^) vvvvv v v (^) v v (^) v (^) v v v vvvvvvvvv v vvvvvvvvvvv vvvv vvvv (^) vvvvvvvv (^) vvvv (^) vvvv vv (^) vv vv vvvvvv v vvvvv vvvv vv v (^) v (^) vvvvvvv vv vvvvvv vvv (^) vv vvv v (^) v v vv (^) v (^) v (^) vv v v (^) vvvv (^) vvvv vvvv vvvvvv (^) v vvv (^) v (^) vv v (^) vv v vv (^) v v vvvvvvvvvvvvvvvv vvvvvvvvvvvvv vvvvvvvvv vv vvvv v (^) vv vvvv v (^) vv vvvvvvvvv (^) v (^) vv v (^) vv v v v vvv (^) vv v (^) vvvv v (^) v vv v (^) vvvv v (^) v vv v v vv (^) v v (^) vv (^) v v (^) vv vvvvvvvvvvvv (^) v (^) v (^) v vvvvvvvvvvvvvv vvv (^) vvvv v vvvvv vvv vvvvvvvvvvv vv vv vvvvvv vvvvvvvv v (^) v v (^) vvv vv (^) vvvvvv (^) vvvv vvvvvv vv (^) vv vv vv v (^) v v v (^) v v (^) vvv v (^) v v v (^) vv vvvvv v v (^) v vvvvvvv (^) v vv (^) v (^) v vv vv vv v v (^) v vvvv v vv (^) vv v v vvvv vv (^) v v v (^) v v (^) v vv vvv v vvvvv vv vv vv vv vvvvvv v vvvvv v vvvvvv vvvv vv vv v vvv vvvv v vv v vv vv v v v v v vv v vv v (^) vv v vv v vv v (^) vv v v (^) v vvv vvvv v vv v v v (^) v v vv (^) v v (^) v v vvvvv vvvv (^) vvvvvvv (^) v vv (^) vvv vv (^) vvv vvv v vv vv v (^) vvvvvvvv (^) v vvvvv v (^) v vvvvvvvv v (^) v v (^) v (^) v v v (^) vv v vv (^) vvvvvvvvvvvvv v vvvvv vv v vvvvvvvvv vvvv (^) vv vvvvvv vvvvvv (^) vvv (^) vvvvvvvvvvvvv (^) v v v vvvv (^) v v vv v (^) v vv vvv v (^) v (^) v (^) v v (^) vv vvvvv vv v vvvvvvvvvvvvvvvvvvvv vvvvv v (^) vvvvvvvvvvvvvvvvvvvvvv (^) vv v (^) v (^) v vvvvvv v v (^) vvvvvv vv v (^) vvvv (^) vv v vv vvvv vvvvvvvvvvvvv vvvvvvvvvvvvv vvvvv v vv (^) vvvv v vv (^) vvvv (^) v vv (^) vv v (^) vv v (^) vv vvvv (^) v v vv (^) v vvvv (^) v v vv (^) v vvvv v v v (^) v vv (^) v (^) v vv (^) v v vv vvvvvvvvvvvv (^) v (^) v vvvvvvvv vvvv v vv vv (^) vvvvvvvvv (^) vvvvvvvvvvvv vvvv vvvvv (^) v vvv vvvv v vvvv (^) vvvv v (^) v v (^) v (^) vv (^) vv vv (^) v v (^) vvvv v vv vv v (^) vvvvvv v vvvv v v vvvvvvv v v vvvv v (^) v v vv (^) v v (^) v v vv v (^) vv vvvvvv (^) v v (^) vvvvv (^) v v (^) v vv vv vvv (^) vv vvvvvv vvvvvvvvvv (^) v vvv v vvvvvvvvv vv v vv v vv v (^) vv v (^) vv v vv v v v vvv vvvv v vv v (^) v v v v (^) vv (^) v v v vvvv vvvvv (^) vvvv v vv vvv vvv (^) v vv (^) vvv vv vvv (^) vvv v vv vvvvv (^) v v (^) vvvvvvvv v (^) vvvvv v (^) v vv (^) v v (^) v (^) v (^) v vvv (^) vv v vv (^) vvvvvvvv v vvvvvv vvvvvvvvvv vvv vvv (^) vvvv vvvvvv vvvvvv v vv v v v vv (^) v v v vv (^) v v vv vv vv (^) v (^) vv vv v (^) v vv v v vv v vvvvv vvvv vvv v vvvvvvvvvv v (^) vvv vvvv v vvv v (^) v (^) v v vv v (^) v vvvv v (^) vv vv v v v vvvvvvvvvvvv (^) v vvvvvvvvvv (^) vv vvvvvvvvvvv v v vv (^) v v v (^) v v (^) v vv (^) v vv (^) v vv vv v vv v v vv (^) v vv vvv (^) vv v vv (^) vv vvvv vvv (^) v vvvvvvvv vvvvvvvvvvvvv vvvvvvvvvvvvv vvvv vvvv (^) v vv vvvv v (^) vv vvvv v v (^) vv (^) v (^) vv v (^) vv vv v vvvv v v (^) vv v (^) vvvv v (^) v vv v (^) vvvv v v v vv (^) v v (^) vv (^) v v vvvvvvvvvvv (^) v (^) v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvv v v vvvv (^) vvvvv (^) vvv v (^) vvvvv vvv (^) vv vv v (^) v vv v (^) v (^) vv vv vv vv (^) v vv (^) vvvv (^) v v v (^) v v v (^) vv v v (^) vv (^) v (^) v v v (^) v v v (^) v vvv vv vv vv vv vv vv vvv (^) v vv vv vvvvvv vvv v v vvvv vv (^) vv (^) v v (^) v (^) v (^) v (^) vvvvv v v v vvvvv v vvvvv vvvvvvvv v vvv vv v vvv (^) v (^) v v vvv vvvvv RULE 129(10000001) v v (^) v v vv vvvv vv vv (^) v v vvvvvvvv v (^) vv (^) vvvv v vvvv v (^) vv (^) vvvvv v vvvv vvvvvvvvvvvvvv (^) vv v vv (^) vv vvvvvvvvvv vvv v vvvv vvvv vv vvvv vv vvvv vv vvvv vv vvvv vv vvvv vv vvvv vv vvvv vv vvvv vvvvvvvvvvvvvvvvvvvvvvvvvvv (^) vvvv vvvv vvvv v (^) v (^) v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v v (^) vvvvvvv vvv (^) v (^) v (^) v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v vvv (^) vvvv vvvvvvvvvvvvvvv v vv vv (^) v vv v (^) vvvvv vvvvvv (^) vvvvv vv v (^) vvv vv vv vv vvv v vvvvvvv vvvvvv (^) v v vv vv v (^) vvv (^) vvvvvvvvvvv (^) vv vv (^) vv (^) vv (^) vv vvvvvv vvvv (^) v vvvvvvvvvvvvvvvvvvvvv v (^) vvvvv vvvvvvvv (^) vv vvvvvv (^) vvvvvvv (^) vvvvvvvvvv vvvvvvvv (^) v v v (^) v vvvv vvvvv vvvv vvv (^) vv v vvv vv (^) vvvv v vvvvvvvvvvvvv vvv v v vvvvvvvvv v (^) vv vvvvv vvvv vvvvv v v vvv v vv vv (^) vv (^) vv (^) v (^) v (^) vvv vv (^) v v v v (^) vv vvvvvv vvv v v (^) vvvv (^) vv v vvv v (^) v v vvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvv vvvv vvvvvvvv vvvvvvvv (^) vvvvvvvvvvvvvvvvvvvv v (^) vvv (^) vv (^) vvvvvvvv (^) 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(^) v v v (^) v vv v v vvvvvvvv (^) v v (^) v vvv v vvvvvvv v v v vv (^) v v v vvvvvvv v v v vv (^) v v v vvv v v v v v (^) v v v vvv v v v vv v vv vv v vv vvv vv vvv vv vv vv^ vvv^ vv^ vvv^ vv^ vvv^ vv^ vvv vv (^) v v^ v^ v^ v^ vv v vv v vv (^) v vv v v^ vv^ vv^ vv^ vvvvv vvvvv (^) vvvvv vvv vvv v^ v^ v^ v^ v^ v^ v^ vvv vvvvv vvvvv^ vvvvvvvvvvvvvvvvvvvvvvvvvvvvv vv vv vv vvvvvvvvvv^ vvv^ v^ vv^ vvvv (^) v v vvv v^ v^ v^ vv^ vvv (^) v (^) vvv^ vvv^ vvv^ vvvvvv^ vvvvv vv vv vv vv vv v v v^ v^ vv vv (^) v vv (^) vvv vvv vvv v (^) v v vvvvvvvvvvvvvvvvvvv vvvvvv (^) vv v (^) vvvvvvvv v vvvvv^ v^ vv^ v vvvvvvvvvvvvvvvvvvv vvvvv^ v^ vv^ vv^ vvvvvvvvv (^) vv vvv vvv^ v^ vvv^ vvvvv^ v^ vv^ vvvv v v^ v^ vvvvvv (^) v vvvv vvv^ v^ v^ v^ v^ vv vvvvvvvvvvv v vv (^) vv v v vvvvvvvvvv v vv vv vvvvvvvvvvvvvvvvvvvvv (^) vv (^) v v vvvvvvvvv^ vv^ vvvvvvvvvvvvv vvvvv^ v^ vv^ vv^ vv^ v^ vvvvv v (^) v v v v v^ vv^ vv v v vvvvv vvvvv^ vvvvvv^ v^ v^ v^ v^ v^ vv vvvvvvvvvvvvvv v v v^ v^ vvvvv^ v^ vvv vv v (^) 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vvv vvvvvvvv (^) vv vv vvvvvv vv vv vvv vvv v v vv vvvvvvvvvvvv (^) vv vv vvvvvvvvvvvvvvvvv v vvvv vvvvvvvvv (^) v v v vvvvv v vv vvvvvv (^) v vvvvvv v v vv v (^) vvvvvvvvv (^) vv (^) v vvv (^) vvv vvv vvvvv v v vvvvvvv v (^) v (^) vvvvv vvvvv (^) v vvv v vvv vvvvv vvvvvvvvv v vvvv v vvvvvvvv (^) v vvvvvv v (^) v v (^) vvvvvvvvvv vvvv vvv v v vv v vv vvvvv vv (^) v vvvvv v v (^) v vvvvv v (^) vv vv vvv vvvv vvvvvv (^) vvvvv vvvvvvv v vvv (^) vvv (^) vvvv (^) vvv (^) v vv vvv vvv (^) v v (^) vvvvvvv vvvvv vvvvvvvvv vv vvvvvvvvvvvvvvvvv (^) vvvv (^) vvvvvvvvvvv (^) v vvv v vvvvvvvvv vvvvvvvvvvv (^) vvvv (^) vvvvvv (^) v vvvvv vv v v v (^) vv vvvvvvv v (^) vv vvv (^) v vvvvvvvvv vvv v v vvvvv vv vvvvvv vvvvvvvvvvvvv vv (^) vvvvvv v vv (^) vv vvv (^) v vvvvvv vvv v vvv v (^) vvv (^) vv vvv vvv v v v v v vvvv vv vv vvv (^) vvvv (^) vv vv v v (^) vvv v (^) vvvvvvv (^) v vvvvvvvvvvvvvvvvvv vvv v vvvvvvvv (^) vv (^) v v (^) vv vvv v (^) vvv vvvvvv vvvvvvvvvv (^) vv vvvvvv v v (^) vvvvvvvv vv 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v v vv vv vvvvvv vvvv vv vvvvv v vv vvvvvv vv vvvv v vv vv vv v vvvvvv v v vvvv vvvv v v vvvv vvvv vvv vvv vvvvv vv vvv vvvvvvvvv v vv vv vv vvvv v v vv vvvv vvvvvv vvvvv v vv vvvvv v vv vv vv vv vvvv vv v vv vvv vvvvvv v vv v vvvv v v vvvv vvv vvv vv vvv vvvvvv vv vv v vvvvvvvv v vv vv v vvv vvv vvvvvv vv v vv vvvvvvvv v vvvvvvv v vvv vvvv vvv vvv v vvvv v vvvvvvv vvv v v vvvvvv v vvvvvv v v v vv v vv v vv v vvvvvv RULE 183(10110111) v v v v v vv v vvvv v v v vv vv (^) v vvv v vvvv vv v v v (^) v vv (^) v vvv v vv vv (^) vvv (^) v v (^) v vv vv (^) vvvv vv (^) vvv (^) v v v (^) v vv vvvv v vv vvvv v (^) v (^) v v v (^) v vvv vvv v vv vvv vvvv v vv vvv v v v v v v v (^) v v (^) v v v v v v (^) v v v v v v (^) v v v (^) v v (^) v v v (^) v v v (^) v v v vv v vv (^) v vv v v v v (^) v (^) v v v (^) vv vv (^) vvv (^) vv v v (^) v v v v (^) v v v (^) vv v vv v v (^) vv v (^) v (^) v v (^) v v v v v (^) vvv vv v (^) vvv vv v v v v v v v (^) v v (^) v v v v v v v (^) v v v v (^) v (^) vv vvvv vvvv (^) v v (^) v v vvvv (^) v vv vvvvvvvvvvvvvvvv v vv (^) v vvv v (^) v vvvvvvvvvvv v v vvv vv v (^) v (^) vvvv v (^) v v (^) vvvvvvv vvvvvvv v vvvvvvvvvv vvvvvvvv vvvvv v vvv v (^) vv (^) v vv vvvvvv v vv v vv vvv (^) v (^) v v (^) v v (^) vv v (^) v vvvvvv (^) v (^) vv v v vv v (^) vv (^) vvvvvvv (^) vvvvv v v v vvvvvvvv (^) v (^) vv (^) vv vvv vv v (^) vv v v vv v vv v (^) v v v vvvv (^) v vvv vv v v v v v (^) v v v v v v (^) v v (^) v v vv v (^) v v v (^) v v v (^) v v v (^) v v (^) v v v v v (^) vvv (^) vvv v (^) v v v (^) v (^) v vv v vv v (^) vv v v vv v v (^) v v v v v (^) vvv vv (^) v vvv vv v v v v v v (^) v v v (^) v v v v (^) v v (^) v v v v (^) v (^) vv vv (^) v v v vvv (^) v v vv (^) v v v v vv v v vv (^) v v v vv v (^) v vvv v v (^) v v (^) vv (^) vv v (^) v v vvvv v vvvvvv vv v v v v vvvvvvvvv v vv (^) v vv v v (^) v (^) v (^) vv vvv vvvvvvv vvvvvvvvvvvv vv vvvvvvv v vvv vvvvvvv v vvv v (^) v (^) vvvvvvv vv vvv v v v v vvvvvvvv vv vv (^) v (^) vv v vvv v (^) vvvvvvvvvv (^) vv vvv vvvv v v (^) vvvv (^) v vv v (^) v v vv vvvvvv (^) vvvvv vv (^) v (^) v v v v v (^) vvvvvvvvvvvvv (^) v v (^) vv v (^) v vv vv vvv vvvvvvvvv vv vvvv (^) v (^) v (^) vvv (^) v vv (^) vvvvvvv v v v (^) vv vvvvv vv (^) vvvv vvvv v v v (^) vv v v (^) v v v vvv v v vvv (^) v (^) v v v (^) vv v v v v (^) v v v vv v vv v v (^) v v (^) v (^) v v vv vvvvv vvv (^) v vv (^) vvv vvv v vvv v v (^) v v (^) v v vvvv (^) v vvvv v v v v vv v v v (^) v v v v v v v v (^) v v v v v v v v v (^) v v v (^) v v (^) v v v (^) v v v (^) v v v vvvvv (^) vv vv v vv v vv (^) v v v v v v v v v (^) vv v (^) v v (^) v (^) v v v (^) v v vvv (^) vv (^) v vvv (^) vv v v v v v (^) v v v (^) v v (^) v v v v (^) v v v (^) v v v v (^) v vvvv v (^) v (^) vvv vv vv (^) v vv vv (^) v vvv v v v vv v (^) v v vv v (^) v (^) v v (^) v v v (^) v v v v vv vv (^) v (^) vvv vv (^) vv v (^) v v v v (^) v (^) v v v v v (^) vv vvv (^) vv vv v vv v v v (^) vv (^) v v v (^) v v v (^) v v v v (^) v v v v v v vv (^) v v v (^) v v v v v v v (^) vv (^) vvv v v v v (^) v v (^) v v (^) v v v (^) v vvvvvvvvvvv (^) vvv v vvvvvvvv (^) vvv vv (^) v (^) vvvvvvvvvvvv v (^) vvvv v v v (^) vv vv (^) vv vv (^) vv vvvvvvvvvvvvv vvvv vvvvv (^) vvvvvvvvvvv (^) v vvvvvv vvv v (^) vv (^) vvvvvv (^) vvv vv vvvvv v v (^) vvvvvv (^) vvv (^) vv v vvv v (^) vvvvv vv vvvv v vvvvvv vv vvv v (^) v v v vvvvvvvvvv vvv vvv v vv vvvv vv vvvvvvvvvvvvv (^) v (^) v (^) vv vv vvv v vvvvv vv vvvv v v (^) v v (^) v v v (^) vvvv (^) v vv (^) vv v v vvvv v v (^) v v v vvvvvvvv v v vvvv v (^) v v vvv v (^) v v (^) vvvvvvvv v v (^) vv (^) v v v (^) v v v v v vv v (^) vv v (^) v v v (^) v v vvvvvv v (^) v v vvv vvv (^) vvvvv vv v (^) v vvvv v (^) vv (^) v v (^) v v v vvv (^) vvv (^) vvvvv v v v (^) v v v v v v v v vvv vv v (^) v v (^) v v (^) vv v v v v (^) v vvv vv v v v vvv (^) vv v vv v (^) v v v v v v (^) v v v v v v vvv v (^) v v (^) v v (^) v v (^) v v vv vvv vv v v vv vv v (^) v vv v vvv vvv v v (^) v v v v v v (^) v v v v v v v v v v (^) v v v (^) v v v (^) v v v (^) v v (^) v v v vvvv (^) vv (^) vv v vv vv vv (^) vv vv v (^) vv (^) v v v (^) v v v v (^) v v (^) v vvv (^) vv v (^) vvv (^) vv v v v v (^) v v v (^) v v v (^) v v v v v v (^) v v v v (^) v vvvvvvvvvvvv vvv v (^) v v v (^) vvv v v vv vvvvv (^) v (^) v (^) v v v v (^) v vvv (^) v v v vv v v v vv (^) v v v vv v (^) vv (^) v v v vv v v v (^) v v v v v (^) v v vv v v v (^) v (^) v v v vvv vv v vv v v (^) v v vv v v (^) v vv v (^) v v (^) v v v (^) v v (^) v v (^) vv vv vv vv v vvvvvvvv (^) vvvv v (^) v vvvvvv v vv vvvv vvvv (^) vv v vv vvvvvv v v (^) vv (^) v vvvvvvv v (^) vvv vvvv (^) v v vvvvvvv vv (^) vv vv (^) vvvvv v vv vv vvvvvv (^) vvvvvv (^) v vv (^) v v vvv vvvvvvvvv vv (^) vvvv v vvvvv v vvvvvvv (^) v (^) vvvv v vv v vvv v vv (^) vv vv vv (^) vv vv (^) vvvvvvvv v (^) vvvvvvvvvv (^) vvvvvvv (^) vv vv vvvv v v (^) vvvv vvv vvvvvvv vvvvvv (^) vv vv (^) vvvvvvv vv vvvvvv v v (^) v vvvvvv (^) vvv v vvv vvvvvvvv vvvvv v vv vv (^) v (^) vvvvv vv v v (^) v vvvvv vvvvvvvv vvvvvvvv v (^) v vv v (^) vv (^) vv vvvvv vv (^) v v (^) v vv (^) vvvvvvvvvv v (^) vvvvv (^) vvvvvvvvvvv (^) v v vv (^) v vv (^) v v v (^) v (^) v vv v (^) v v vv v v v (^) v vv (^) v vvv v (^) v v v (^) v v (^) vv (^) v v (^) v vv vvv vv v v (^) v vvv (^) v v v (^) v v v (^) v v vv v vv v v v v (^) v v v vv v (^) v v v v (^) v vv v vvvv (^) v v v (^) v v (^) vvv vvvvvv v v v v v v v (^) v v (^) v v v v v v (^) v v v v v v (^) v v v (^) v v (^) v v v (^) v v v (^) v v v vvvv vv vv (^) v vv v vv v vv (^) v v v v v v v v v (^) v v v v v v (^) v (^) v vvv vv v (^) vvv vv v v v v v v v v (^) v v (^) v v v v v v v (^) v v v v (^) v (^) v vvvvvvvvvvvvvvvvv vv vvvvv v vv vvvvvvv v v vvv v (^) v (^) vvvv (^) v vvvvvvv v (^) v vv (^) vv (^) vvvvvv vvv vvv v (^) v v (^) v v (^) v v v vv vv vv (^) v vv v v (^) v v (^) v vvvv (^) v (^) v (^) vv vvv v v (^) v vv v v v (^) vv v v v v v v vv (^) v v (^) v v v v v v (^) vvv vv vvv v vv v (^) vv vvvv (^) v v v (^) v v (^) vvvvvv v v v v (^) vvvvv vvv vvv vvvvvvvvv v (^) v vvvvvvvvv (^) v v (^) vv v (^) vvvv vvvvvv vv vvvv v (^) v v vvv vvv vv v v (^) v vvvv v (^) v vv vvvvvvv v v (^) v v vvvvvvv vvvv v vvvvvv (^) v vvvvvvvvvv (^) vvvv (^) vvvvvvv vvvvv vv v (^) vv vv vvvvv v (^) vv vvvvvv vv vvv v v v v (^) v (^) vvvvvv (^) v vvvv v (^) vvvvvvvvvv v (^) v (^) vvvvvvvv v vvv v v (^) v vvvv v v (^) vv vv vvvvvvvvv vv v (^) v vvv v v vvvv v v v vvvvv vv (^) v (^) v vvv vvv (^) vvv (^) vv vv vvvvv (^) vv vv vv (^) vv vv (^) v vv vvvv v vv vvv v vvvvvvvvvvvvvvvvvvvvvvvvv v (^) v (^) vvvvv vv (^) vv vvvv v (^) vvvv v v v vvv (^) v (^) v v v v v (^) vv v v v (^) v v v v (^) v v (^) v v v v v v (^) v vvv v (^) v v v (^) v vv vvv (^) v (^) v (^) v (^) v vv v v (^) v v v (^) v v (^) v v vvv v vv v v (^) vvv (^) v (^) v (^) v v v v (^) v vv v (^) v vvv (^) v v v (^) v vv v v (^) v vvv v v v v v v v (^) v (^) v vv v vvv (^) v vvv vvv vv v vvvvvv v v Figure 3. DP of reflection symmetric but nonquiescent rules (Β 1 Β 2 Β 3 Β 4 Β 2 Β 6 Β 4 1) affected by memory in the scenario of Figure 2(a).
History however has an unexpected effect on most of the rules whose damage propagation direction alternates: Λ L ( 1, 1), Λ R (1, 1). Figure 4 shows the case of fully asymmetric Rules 43, 57, and 184. Particularly curious are the DP generated for Α > 0_._ 6 in Rules 43, 113. The DP of Rules 184, 226 are rectified when history is taken into account (that of Rule 184 becomes extinct for Α in [0.6,0.8]). Rules 184 and 226 have proved particularly effective in solving the density problem
Elementary Cellular Automata with Memory 117
RULE 57(00111001) vvv v vvv vv vv vv vv vv (^) v vv vvv vvv v (^) v v (^) vvvv (^) v vvvvvvvvv vvv (^) v vvvvv v v v (^) vv vv [(-1,1),(1,-1)] vvvvvvvv v vv (^) v v vvv vvvv (^) vv v v vvvvvvvvvvv v vv (^) v v vvv vvvvv (^) vvvvv vvvvvv (^) vvv vv (^) vv vvvvvvvvv (^) vvv vvvvvvvvv vvv v v v v vv vv vvvv vvvv vv v vvvvvvv vv v v v v v v v v v v vv (^) vv v (^) v v v vv (^) v v v (^) v v v (^) v v vvvvv (^) v vvvvv vv (^) vvvvvvvvvvvvv (^) vv vvv vv vvvvvv v (^) v vv (^) vvv vv v vvvvvvvvv vvvvvvvvvv vv vvvvvv v vv v (^) v v v (^) v v v (^) v vv (^) vv vvvv v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vv v vv v (^) v v vvvvvv (^) vvvvvvvvvvvvvv vvvvvvvv vvvv vv vvvv v (^) vvvvvvvvvv vvvvv vvvvvvvv vvvvvv vvv vvvvvv v (^) vvv vvv (^) vvvvvvv vvvvvvv vvvvvvvv (^) vvv vvvv (^) v (^) v (^) vvvvvvvv vv (^) vvvv vvvvvvvvvvvvvv v vvv (^) vvvvvvv v (^) vvvv v vv vvvvv (^) vvvv vvvvvvv (^) v (^) vvv v vvvv (^) v vvvvv vvvvv vvvv v v vv^ vvvv^ vvv^ vvv^ vvvvv^ vvvvvv^ v^ v^ v^ v^ v^ vv^ vvv^ v^ vvv^ vv^ vv^ v^ v^ v^ v^ v^ v^ v^ v^ v^ v^ vv^ vv^ v^ v^ v v vvvvvvvvvvvvvvvvvvvv^ vvv^ vvvvvv vvv^ v^ vvvvv^ vv^ v^ v vv vvvvv^ vvv^ vvvvvvvv^ vvvvvvvv vv^ v^ vv^ vvvv^ vv^ v^ vv vv^ v^ vvvvvvvvvvvvvvvvvvvv^ vvv^ vvvvvvv^ v^ vv^ vvvvvvv^ vvv^ vvvvv^ vvvvvvvv vvvvvv^ vv^ v^ vv^ v^ vv v^ v^ vvvvvvvvvvv^ v^ v^ vvv^ vvvvv^ vv vvvvvvvv^ vv^ v^ vvvvvvvvv^ vv vvvvvvvvvvvvv v^ v^ v^ v^ vvvv^ vvvv^ vvv^ vvv^ vvvvvvv^ v vv^ v^ vv^ v^ v^ vvvv^ v^ vv vvvvvv^ vvvvvv^ v^ vvv^ vvvvvvvvvv^ v^ vvvv^ v^ v^ vvv^ vv^ vvv^ vv^ vvvvvvvv^ v^ v^ v^ vvvv^ vv v vvvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvv^ v vvvvvv^ vvvv^ v^ vvvvv vvvv^ vvvvvvvv^ vv^ v vvvvvvvvvv^ vvv^ vvvvvvvvvvv^ vvvvvv^ vvvvvvvv^ vvvv vv^ vvvvvvvvvvvvvvvvvvvv vvvv^ vvvv^ vvv^ vvv^ vvvv v^ vv^ vvv^ vv^ vvvvv^ vvvvvv^ vvvv^ v^ v vvvvvvv^ v^ v^ vvvvvvv^ v^ v^ v^ v^ v^ v^ vv^ v v vv^ vvvvvv^ vvvvv^ vvv^ v^ v^ vvvvvvvvvvv vvvvvvvvvvvvvvv^ vv^ v^ vvvv^ v vv vvvvvvvvvvv^ vvvvvvvvvvvvvvvv^ v^ vvvvvvvvvvv^ vvv^ v^ v^ vvvvv v vvvvvvvvvvv^ v^ vvvvvv^ vvvvvvvv v^ v^ v^ v vvvvv^ vvvvvvvvvvvvvvvvvvv v vvvvvv^ vv^ vv vv vvvv^ vvvvvvvvvv vvvvv vvvvvvv vvvvvv v vvv vvv vvvvv vv vvv vvvv vvvvv vvvvv vvvvv vvv vvv vvv v (^) v v (^) vv (^) v vvv v v (^) vvv vvvvv v v (^) v vv vvvvvvvv vv vv vvvvvvv vvvvvvvvvvvvv (^) v (^) vv v (^) vvvvvvvvvvv vvvvvvvvvvvvvv v vvvvvvvvvv vvvvvv vvvv vv (^) vv vv v (^) vv v v (^) v (^) vv vvvvvvvvvvvvvvv vvvvvvvv vvv (^) vvvvv vvvvvvv vvvv vvv v (^) vv (^) vv v vv vvvvv v v (^) vv vvvv vvvvvv (^) vv vv vvv vvvv vvvvvv vv vv (^) vvvvvv (^) v v vvvv vv vvv vvv v (^) v v v (^) v v v (^) vv vvv vv vvvv v v (^) vvvvvvvvvvvv v vvvvvv vvv v v vvvvvvv (^) v (^) vv vv vvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvv vv vvvvv (^) v (^) v v vvvv (^) vvvvvvvvvvvvvv v v (^) v (^) v v v v (^) vv vvvvvvv vv (^) vvvvvvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvvv v vv (^) vvv (^) vv v v v (^) vvvvvvvvvvvvvv vvvvvv vvvvvv (^) vvvv v (^) vv v vvvvvvv vvvvvvvvvv (^) v vv (^) vvvvvvvv (^) vvvvv vvvvvv (^) vvvvvvvv vv vvv (^) v (^) vvvvvv v vvv (^) v (^) vvvvvvvvvvvvvvv vvvv vvvvvvvvvvvv vvvvvvvvvv vv vvvvv v (^) vvvvvvvvvv (^) vv vv v v (^) v (^) v vv vvvvv (^) vvv v vv vv vvvvv v (^) v v vv (^) vvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvv (^) vvvvvvvvvv vvvv vv v (^) vvvv vv (^) v vvvv v (^) vvv (^) vvv vvvv (^) vvv vvvv v vvv (^) v v v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) vvvvvv (^) vvvvvvv (^) vvvvvvvvvvvvvv vvv vvvvvvvvvv vv vvvvvvvvvvvvvvvv v (^) vvv (^) v vvvvvv v v vvv vvvvvvvvv v v vvvvvvv v vvvvv vvvvvvv (^) vvvvv vvvvvvv v vvvvvv vvvvvvv v vvvvvv vvvv vvv v vvv vvv vvv vvv vvv vvv vvv vvv vvv vvv vvv vvv vvv vvv (^) v vvvvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvv v v vv (^) v vvvvvvvvvvvvvvvvvvvvv vvvvvvvvv vvvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvv v (^) v v (^) v vvvvvv (^) v v (^) v v (^) v (^) vvvvvvv vvv vv v vvvvvvvvvvvvvvv (^) vvvvv vvvvvvvvvvvvvvvvvvv (^) vv (^) v (^) v vv vvvvv (^) v v v (^) v vv (^) vv (^) v vvv vv v vv vvvvvvv vvvvvvvv vv (^) vv vvvvvv vvvvvvvvvvvvvv vvvvv (^) v v v v v v (^) vvvv vv v v (^) v vvvv v v v (^) vv vvvv vvv (^) vvvv vv vvvvvvvvvvvv v v (^) v (^) vvv vvvvv v v v (^) v v vv (^) v (^) v v (^) v v vvv vv vv v vvvvvvvvv (^) v vvvv v vv v vvvvvvvvvv (^) vvvvvvvvvvvvvvvv (^) v v vv v v vv vvv (^) vvvvv vvv v (^) v v v (^) vvv v v v vvvvv v v v vvvvvvv v (^) vvvvvv vv vvv v (^) v v (^) vvvvvv v v v (^) v v (^) v (^) v (^) v (^) v vvvvv (^) v (^) vvvvvvvvvvvvvvvvvvvvvv (^) vvvvv vvvvvvvvv vvvvvvvvvvvvvvvvv (^) v (^) v vv (^) v vvvvvvvv v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvv v v (^) vvvvvv (^) vvv v vv (^) vvvvvvvvv v v v v (^) vvv vv vvvv vvvv vvvvvvv vvvvv (^) v v vvvvvv (^) v vvvvvvv v v (^) v v v vvvvvvvvvvv v (^) vvv vvvv (^) vvvv v vvvvvvvvvvvvvvvvvvvvvv vvvvvv vvv (^) vvvvvv vvvvvvv (^) v (^) vv vvvvvv (^) v v (^) vvvvvvvvvvvvvvvv vvvvvvvv vvvvvvvv (^) vvv vvvvvv (^) vvvvvvv vv (^) vvvvv v (^) vvvv (^) vvvvvv vvvvvvvv (^) v (^) v (^) vvvvvvvvv vv vvvv vvvvvvvvvvvv vv (^) vvvvv v (^) v v vvvvvvvvvv vvv (^) v (^) vvvvvvvv vvvv v vvvvvvvvv vvv v v (^) v v vvv (^) v vv vv vvvv v (^) vvvv v (^) vvvvvv (^) vvvvvv vvvvvvvvv (^) v (^) v v (^) v v vvvvvvvvvv (^) vvvvv v vvvvvvv vvvvvv vvv v vvvvvvv v vvv (^) vvvv v vv vv (^) vv vvvvvv vvvvvvvv v vvv v vvvvv v (^) v vvvvv vvvvvvvvvvvv vvvvvvvv v vvv vvv vv vvvvvvvvv (^) vvvvvvvvvvvvvvv vvvvvvvv vvvvvv vvv (^) v vvvv vv vvv v v (^) vv (^) vvvvv vvvvvvvvvvvvvvvvvvvvv (^) vv vvvvvvvvvvvvvvvv v vvvv (^) vvvv (^) v vvvvvvv vv vvvvvvv vvvvv (^) vvvvvvvvvvvvv vvv v vvv (^) vvv vvv vvv vvv vvv vvv vvv vvv vvv vvv (^) v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvvv vvvvvv vvvvvvvvvvvvvvvvv (^) vvvvv vv vv v v (^) vvvvvvvvvvvvvvvvvvvvvvv (^) vv v (^) v v vv v v v v (^) vvvv vvvvvvvvv vv vv (^) vvv v (^) vvvvv vv vvv vv vv (^) vv v vvvv vvvv (^) vvvvvvv vv vv v vv vvvvvv vv v vvvvvv vv vvvvvvvvvvvvv (^) vv v v vv (^) vvvv (^) v vvvvv vvvv (^) vv vv vvvvvvvvvvvvvvvvvvvv (^) v vvvvvvvv vvv vvv vvvvvvvv (^) vv (^) vv vvvvvvv vvv (^) vvvvvvvvvvvvvvv v (^) vvvv (^) vvvv vv vv (^) vvv vvvv vv v v vvvv (^) vvvvvvvv vvv vvvv (^) vvvv vvvvv (^) v vvvvv v (^) v v (^) v vvv vv (^) vv v (^) v v vvvv v (^) v v (^) vvvv (^) v vvvvvv v v vv v (^) v v (^) vv v (^) v vv vvvvv v v vvvvvvvv (^) vvvvvvvvv vvv (^) v vvvvvvvvvvvvvv vvv v v vvvv v (^) vvvvvvvvvvvvvvvvvvv vvvvv (^) vvvv vvv vvv vv (^) vvvvvvv (^) vv v (^) vvvvv vvvvvvv vv (^) vvvv (^) vvv (^) v vvvvvvv vvv (^) vvvvvvvvvvvvvvvvvvvvv vv vvvvvvvvv (^) vv vvv (^) vvvvvvvvvvvvvvvvvvvvvvvv v v (^) vv vv (^) vvvvvvvvv (^) vvvvvv (^) vv vv vvvvvv vvvv vv v vv (^) v vvvvvvvvvvvvvvvvvvvvvvvvvvvv v (^) vv v vv (^) vvvvvv vvvvvvvv (^) vv (^) vvvv v vv (^) vv vvvv (^) vvvvvvvvv vv vvvv v vvvvvvvvv (^) vvvvvvvv (^) vvvvvv vv (^) vvvv (^) v vvv vv vvvvvvvvvvvvvv vv (^) v (^) vvvvv (^) vv vv vv v vvvvv vvvvvvvv (^) v (^) vvv v (^) v (^) vv vvvvvvv v v vvvvvvv vvvvvv v v (^) v vvvvv v (^) v vv vv (^) v (^) vvvvvvvvvv vvvvv vvvvvvv vv (^) vvvv (^) vvvvvv vvv (^) vvvvvvv v vv (^) vvvvvvvvvvv v (^) vv v v vvv vvvvvvvvvvvvvvv vvvvv vv vv (^) vvv vvvvvv (^) 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v v (^) v v (^) v vvvvvvvvvvvvv vvvvvvvvvvv^ v^ v^ v v v vvvv vvvvv v v v v v v vv v v vv (^) vvvvvv v (^) vvvvvvvv^ v v (^) vvvvvvv v (^) vvvvvvvvvvvvvvvvvv^ v v (^) v v (^) vv v (^) vv vvv vvvvvvv vvvvvvvvvvvv v (^) v vvv vv vvvvvv vv vvvvvvv vv (^) vvvvv^ vv (^) v (^) vvvvvv vvvvvvvvvv vvv^ vvvvvvvvv v vvvvv (^) v (^) v vvvvvvvvvvvvvv vvvvvv^ vvvvvv (^) vvvvv vvvv^ v (^) vvvvvvvvvv vvv (^) vvv vv (^) vvv vvvvvvvvvvvvvvvvvvvvvvvv^ vvvvvvvvvvvvvvvvvv^ vvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvv (^) vvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v vv (^) vvv vvv v (^) vv vv (^) vv vvv vvv v v v vv v vv v v (^) v v v (^) v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v vv vv v v (^) v vvvvvvvvvvvvvv vvvvvv (^) vv (^) vvvvvvvv vv vv vv vv vvvv vvvvvvvvv vvvvvvvvv vvvv v v v (^) vv vvv v vvvv v (^) v (^) v (^) v 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vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v vv vv v v v (^) v vvvvvvvvvvvv vv vv vv vv v v vvvvv vvv v vvvvv v (^) vvvvv vv vv vv vv vvv v (^) vvvv (^) vvvv vv (^) vvvv v v vvvvv v vvvvv vv v v (^) vv vvvvv v (^) vv vv v vvvv vvv v vvvv (^) v (^) vv vv vvvvvv vvvvvv (^) vvvvvvvvvvvvvvvvvv (^) vv (^) v vv (^) v (^) v vvvv v vvvv (^) vvv vv vvvvvvvvvvv (^) vvvv v vv vvv (^) v vvvvvvvvvv vvvvvvvvvvvvvvvvv v (^) vvvvvvvvvvvvvv (^) v vv (^) vvvv v vvv vv (^) v (^) vvvv vv v (^) v vvv v (^) vvvvvv vvv (^) vvvvv vv v (^) v vv (^) v vvvvvvvvvvvvvvv v (^) vv v (^) vvvvvvvvvv (^) vvvvvvvv vvvvvv vvvvvvv vvv vvvv vv (^) v vvvv (^) vv (^) vvvv vvvv (^) vv vvv (^) vvvvv v vvvvvvvvvvvv (^) vv vvvv vv vv v (^) v v (^) v vv (^) vvvvv (^) vv (^) vv v (^) vvvvvv vvvvv vvvvvv v (^) v (^) v vvvvvv vv (^) v vv vvvvvvvvvvvvv (^) vv (^) vvvvvvvvvvv vvvv (^) vv vvvvvv (^) vvvv vvvvv vvv (^) vvvv (^) v v vvv v v vvvvvvvv vvvvvv (^) vv vvvvv v (^) v (^) vvvvvvv vvvv vv (^) vvvv v vvvvvvv vv vvvv (^) vvvv (^) vv vvvvvv vvvvv vvvv (^) vvvv (^) vvv v (^) vvvvv (^) vvvvvvvvv vvvvvvv (^) vvvvvvvvvv (^) v vvvvv (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvvv vv (^) vvv (^) vv vv (^) vv vvvvvvv vv vv vvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvv vv vvv vvv v vv (^) vv (^) vv vvv vvv v v v vv v vv v v v v (^) v v v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v vv vv v v (^) v vvvvvvvvv vvvvv v (^) vv vvvv (^) v vvvvvv vvv v (^) v vv (^) vvvv vvvvvvvv vvvvv v vvvvvvvvvvvvvvvvvvvvvvvvv vvv v vv vvv vvvv vvv vvv vvv vvvvvv (^) vvvv vvvv vv v (^) vv (^) vvvvv (^) vv (^) v vvvv v (^) v v vvvvvvvvvvvvvvvvvvvvvvv v (^) v v (^) vvvvv (^) vvvv vvvvvv v v (^) vvv vv vvvvvvvv vv vvvvv (^) vvvvvv v (^) vvvvvvvvvvvvvvv v (^) vvvvvvvvvv vv vv (^) vvvvvvvvvv vvvvvvvvvvv vv (^) vvv (^) vv vv vvvvvv (^) v v vvvvvvvvvvvvvvvvvvvvvvvvvv v vv vvvvvvv vv vvvv vvvvvv vvvvv v (^) vvv vv vvv (^) v (^) v vvvvvvvv vv v (^) vvvvvv (^) vvvvvvvv (^) vv v (^) vv v (^) vvvvvvvvvvvvv vvvvvvvvvv vv vvvvvvvvv (^) vvvvvvvvv (^) vvvvvv vvvvv vv vvvvvvvvvv (^) v vvvvvvvvvvvvvv vvvv (^) v vv (^) vvv (^) v vvvvvv (^) vvvvvvvvvv vvvv (^) v (^) vvvv (^) vvvvvvvvvvv vv (^) vvv (^) v v vvvvvv (^) vvvvvvv v (^) v vvv vv vvv vv v vvv v (^) vvvvvvvvvvvvvv v vv vvvvvvvvvvvvvvv (^) v (^) v vvv v vvvvvvvvvvvvvvvvvvv (^) vv (^) v (^) vv vvvvv v (^) vv vvv vvvvvv vvv vvvvvv vvvvvv vv (^) vvvvvvvv (^) vv vvvvv (^) v vvv vvv vvvvv vvvvvvvvvv vvv vvvvv vvv v vvvvv vv (^) v v v (^) vvvvvvvvv (^) vvvvv (^) vv vvvvv (^) vvvv vv (^) vvvv v vv vvvvvvv vvvvv vvvvvvvvv (^) v v (^) v vvv v vvvvvv vvvvv v (^) vvvvvvvvvvvvvvvv (^) v v vv vvv vvv (^) vvvv (^) v vvvvvvvvvv vvv (^) vvvv (^) vvv vvvvvvv vvvvv vv vvv vvv v vv vv (^) vv vvv vvv v v (^) v v v (^) v v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v vv vv v v v vvvvvvv v vvvvvvv (^) v vvvvvvvv vvvvv v vvvv vvvvvv (^) v v (^) v vvvvv vvvvvvvv vvvvv vvvvvvvvvvvv vvvvv (^) vvvvv (^) v vv vv vvvvvv v v (^) v (^) vvv vvvvvvvvvvvvvv vvvvvvvvv vvvvv (^) vv vv (^) vv vvv vv (^) v (^) vvvvvvv (^) vv vvvvvvvvvv (^) vvvv vv vv vvv v (^) vv v vvvvvvvvvvvvvv (^) vv vvvvvvv vvvvvvvvv vv vv vvvvvvvvvvv v (^) vv vvvvvvvvvvv v v vv v vv vv vvvvvvvvvvvvvvv vvvv (^) vv (^) vvvv vvvv v v vvv v vv (^) v (^) vv (^) vv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvv (^) vv vvv v vvv vvv vvvvvvvvv vvv vvvvvvv (^) vvvvvvvv (^) vvvvvvv vvvvvvvvv v (^) vvv vvvvvvvvv vvvv vv vvvvvvvv (^) v v (^) vvvvvvvvvv v v v vvvvvvvvvvvvvvvv vvvvvvvvvvv vvvvvv (^) v (^) vvvvvvvvvvvv (^) v vvvvvvv (^) vvvv v vvvvvvvvvvvvvv vv v (^) vvvvv vv vvvvvvvvvvvvv (^) vvv (^) v vvvvvvv v vvvvvvvvvvv vv v (^) v vvv vvvvvvvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvvvvv (^) vvvvvvv vvvvvvvvvvvv vv 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vvvv vvv v vvvvv (^) vvv (^) v vv (^) vv vvvv (^) vvvv vvv vv vvvvvvvv v v v vv vv v vvvv v (^) v (^) v (^) v v v vv v vvv (^) v (^) vvv (^) v vv (^) vvvv vvvvvv (^) v vvvvv (^) vvvvvv (^) v vv vvv (^) v vvv vvvvvvvv v vv vvvvvvvv vvvv v vvv v (^) v v vvvvv vvvv v (^) vvvvvvvv vv vvv v v (^) v vvvv (^) vvvv vvvvvv v v v (^) vvvvvv vvvvvv v (^) v v (^) vv vvv v vvv (^) vv vvvvvvvvvvvvvvvvv vvv vvvvvvv vvvvvvvvvvvvv v vv vvv vvvvvvvvvvv v v v v (^) v v (^) vvv vvv (^) vvv vvv vvvvvvvv vvv (^) v vvvv vvvvvvvvvvvvvvv v (^) v v vvv v vvvvv v vv (^) v (^) v v (^) vvv v vvvv (^) vvvv v (^) vv v (^) vvvv vvvvvvv vvvvvvvvvvvvvv vvvvvv (^) v vvvv vvvv vvvv vv vvvv v vvv vvv vv (^) vvvvvv vvvvv vv vv v v v v v v v v v v (^) v (^) v v v (^) v v v v vvvvv v vvv v vvvv v (^) vv vv vvvvv vv v v vvvv (^) vvv (^) vv (^) vv vvvvvvv v (^) v vv (^) v vvvvvvvvvvv (^) v vvvvvv (^) vvv vvvvvvvv vv (^) v (^) vv vv v vvv (^) v (^) v v v (^) v vv v vv vvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvv vv v vvvvv (^) vvvv vvv 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vvvvv (^) vv vvvvvv v v vvv vvv v vvvvv v vv v v vvv vvv v vvvvv v vv v vv vvvvvvvv vv vv (^) v vvvvv (^) vv vvvv v vvvvv vv vvvv (^) v vvvvv vv vvvvv v vv v (^) v vv vv v v (^) v v v v v v (^) v v vv vvvvvvv vvvv vvvv vvvv v (^) v v (^) v (^) vvvvvvvv vvv vvv (^) vvv (^) vvv vvv vvvvv v v v v (^) v v v vvv (^) v v v (^) v v (^) v v v vvvv vvvvv (^) vvv v v vv vv (^) v v v (^) v v (^) v v (^) v vv v v vvv (^) vvvvvvvvvvvvvvvvv vvv vvvv v (^) v (^) v v v v v v v v vvvvvvvvvvvvvvvvvv vvvv v v vv (^) vvv vvv v vvvvv (^) v v v vv v (^) v vv (^) vvvv (^) v v v (^) vvvvvvvvvvv vvv v vvvvvv v vvv vvvvvvvv (^) vvvv (^) v vvv (^) vv vvvvvvvvv vvvvvvv vvvvvvv (^) v (^) vv vv vvvvvv v (^) vv vv (^) v v (^) vv vv (^) vvvvv vv vvv (^) vvvv v (^) v v (^) v v (^) vvvv v (^) vvvv (^) vvvv vvvvv vvv vvvvvv v vvvvvvvvv (^) vvvvvvv vv (^) v v (^) vv v vv vvv (^) vvvv v v v (^) vvvvv vvvvvv (^) v vv vvvv (^) vvv v v vvvvv vvvv vvvvvvvvv (^) vvvvv v v vvvvvv (^) v v vvv vvvv vvvvvvv vv vvv v vv v v (^) v v vvv vv (^) v (^) vvvvv (^) vv vvv v v v (^) v vvvvvvvvv vvv v vvvv v vvvvvvv vvv vv (^) vv v vvv vv vvv (^) v vvv vv vvv v v v vvv v v v v v (^) v v v v v v (^) v v (^) v v vv (^) v v (^) v (^) v v (^) v (^) v vvvvvv (^) vvvvvvvvv vvvv vvvvvvvvvvv v vvvvv vv (^) v v vvv v v (^) v (^) vv (^) vvvvvv vvvvvvvvvvvv vvvvvvvv v v (^) v (^) v v vv v (^) v v (^) v v v vvvvvvvvv vvvvvvvv (^) vv v vvv (^) vvv v v vvvvv (^) v (^) vvvv (^) v vv (^) v vvvv v (^) vv vv (^) v vvvvvv vvvvvvv vvvv vvv vvvv vvv (^) v v v (^) v (^) v vvvvvvvvvvvv vvvvv (^) vvv vvvvvvvvvvvvvvvvvvv vvv (^) vv vvv (^) v v (^) vv v v (^) vvv vvvv v (^) vvvvv vv v (^) vv vv v vvv (^) v vv vvvv (^) v vv (^) vvv v v (^) vvv v v v vvvvvvvvvvv (^) vvv (^) v vv (^) vvvvvvvvvvvvvv v (^) vv vv (^) vv (^) v (^) v v v v vv (^) v vvvvvvvv vvvv (^) vvvv v (^) vv vv vvvvvvvvvv vv v v v v vvv (^) vvvvvvvv vv vv (^) v v v vv (^) vvv vvvvvvv (^) v vvvvvv v v (^) v vv (^) vv (^) vv vvvvvvvv (^) v (^) v vvv (^) vv (^) vv v (^) vvvv vv (^) 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v vv (^) v vv (^) v v v vvv v (^) vv v vv (^) v v (^) v vvv vvv vv vvv vv (^) vvvv (^) vvvvvv v (^) v v (^) v (^) vv vvvvvvvvvv vv (^) v (^) v vvvv vvvvvvvvvvvvvv vv vv v vv v vv v vvv vvv vvv vvv vvv vvv (^) v (^) v (^) v v (^) v vvvvvvv (^) vvvvvvvvvvvvvvvvvvvvvvvvvv (^) v (^) vvvvvvvvvvvvvvvvvv vv vvvvv (^) v vv v vvvvvv (^) vvvvv (^) v vvvvvvvvvvvvvvvvvvvvvvvvv v (^) vv (^) vvvvv v (^) vvv (^) v vv vv v v (^) v vvv vvv (^) vvvvvvvvvvvvv vv (^) vv (^) vvv vvvv (^) vvvvvvvvvvvvv vv (^) v (^) vvvv v (^) vvvvvv (^) v (^) vvvv vv v (^) vv (^) v vvvv (^) v v vv (^) v vv vvv (^) vvv vvvv v (^) vvvvv v (^) v vv (^) v (^) vv vv (^) v (^) v v vvv vvvvvvvvvvvvvvvvvv vv vvv (^) vv vvv vvvv (^) vvvvv vvvv (^) v vv v vvvvvvvvvv (^) vv v (^) vv vvv v v (^) vvv vvvvvvv (^) v vvvvvv vvvvvv v v vv vvv (^) v (^) v vvvv (^) v v vvv vvv v vvvv vvvvvv vvvv v (^) vv vvvvv vv v vvvvv vvvvvvvv vvv vvv v (^) v v (^) v (^) v v (^) v (^) v v vvvvvvvvvvvvvvv (^) vv vvv v vv vvvvv vvvvv vvv (^) v (^) v v vv v 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vvv (^) v (^) v vvvv (^) vv vvv v v vvv (^) v vv vv v vvvvvvvvvvvvvvvvvvvv (^) vvv (^) vvvvvvvvvv vv vv vvv (^) v (^) vvvv vvv v v v (^) vv v (^) vv (^) vv (^) vv vvvv v vvvvv (^) v v vv vv (^) vvvvvvvvv (^) vv v vvvvv (^) v vvvv (^) v vvv (^) vvv vvvv vvv v (^) vvv v vvvvvv vvv (^) vv (^) v vv vv vvvv vvvvvvvvvv (^) vvvvvvv (^) v v vvvvv (^) vvv (^) vv (^) vvvv vv vv (^) vv vvvv (^) v vv v v v vv vvvvv v vv (^) vvvv (^) vvvvv (^) v v vv vvvv (^) vvv vvvvvv v (^) v vvv v (^) v vvv v (^) vv vvvv (^) vv v vv vvvv vvvv vvvv (^) vv (^) v (^) vvv vv (^) vvvvvv vvvv vvvv v vv vvvv (^) vv vv (^) vv (^) v (^) vvv vvv vv vvvv (^) vvvv vv (^) v (^) vvv vvv vvvvvvvvvvvvvv vvv v vvvvvvvvvv (^) v v vvv v vvvvvvvvvvv (^) v vvvvvvvvvv v (^) v v v vvv (^) v vvv vvvv (^) v vvvvvvvvvvv vv vvvv (^) v vvvvvvvvv vvvvvv v vvvvvv (^) vvvvv v (^) vvvv vvvvv vvvv v vv (^) vvv vvvv (^) vvvvv (^) v (^) vvvv vvvvv vvvv vvvvvv v vvvvvvvvv vvvvvv (^) v vvvvv vvvvvvvvv v (^) vvvvvvvvv v v vvvv vvvvv (^) vvvv v (^) vvvv (^) vvvvv vvvvvvvvv (^) v vvvvvvvvv vvvvvv v vvvvvvvvv vvvvv v (^) vvvvvvvvv v vvvvvvvvvv vvvv vvvvv vvvv v (^) vvvv vvvvv (^) vvvv v v (^) vvvvv vvv (^) v vv vv vvvvvvvvvvv (^) v (^) vvv (^) vvv vv (^) v vv vvv (^) vvvv (^) vvv (^) v vvvvv (^) vvvv vvv (^) v v (^) vvvv (^) vvvvvvvvvvvv vv vvv (^) vvvvvv v vvvvvv RULE 124(01111100) v v v vv v v v v (^) v v v (^) v v vvvvv v v (^) v vv vvv v v vv (^) v vv v v (^) v v vvv v vvvv vv (^) v vv vvv vv v (^) v v v v v v vvvvvvv v vv (^) v v (^) v v (^) v v vv vvv vvvvvvv vv vvv (^) v v vvvvvvvvv v v v v v v v v (^) v (^) v (^) v v v v (^) v v v (^) v v vvvvvvvv vv (^) vvvvvv vvvv v (^) vvvv vvvvv vvv vvvv (^) v v v v v (^) vvvv vvv (^) vvv vv vvvvvv vvv vvvvvvvv v v (^) vv (^) vv (^) v v (^) vv (^) vv vvvvvvvvvvvv vvvvvvvvvvv vvv (^) vvvv vvvv v vvvvvvv (^) v vvv (^) vvvv v v (^) vvvv (^) v (^) v v vvvvv vvvvvvv vvv (^) v (^) vv vvvvvvvvvvvvvv vv vvvvvvvvvvv (^) vvvvvvv (^) v vv v vvv v v vv (^) vvv (^) vv vvvvv vvvvvv v vvvvvv (^) vv vv v v v v vvv vvvv (^) vv (^) v v (^) v vvvv v vv vvvvvvvvvvvvvvvvvv vvvvvv (^) vvv v (^) vvv (^) vv (^) v vvvv v (^) vvv vv vvv vvvvvvv (^) v vvvvvv (^) vvvvvv (^) vvvvv vvvvv v vvv v (^) vvvv vv vvvvv (^) v vv vvv vvvvvvv vvvvvv v (^) vvvvvvv vvvvvv (^) vvvvvvvvvv v (^) vvvv v (^) vvv vvv v vvvvvvvv vv vv vv vv vvvvvvvvv vvv vvv vvv vvv vvv v v (^) v (^) v (^) v v (^) v (^) v vvvvvv vvvvvvvvvvvv (^) v v vvvv (^) v (^) vv v vvvvvvvvvvvvv vvvvvvvv (^) vv vvvv vvvvvv (^) v (^) vv (^) v vvv (^) vv (^) v vvvvvv v vvv vvvvv vvvvv vvvvv vvv v (^) v (^) vvvv (^) vv (^) vvv (^) vvvv vvvvvvv (^) vvvv vv v (^) v v v (^) v (^) v v v (^) v (^) v v vvvvvvvvvvvvvvvvvv vvvv (^) vv v v vvvvvvvv vv vv vv vv (^) v vv v (^) v vvv vvvvv vv v vvvvvv (^) v v vvvvvv (^) v vvvvvvvv v (^) vvvvvvvvvvvvv v v (^) v (^) vvvvvv (^) v (^) vvvv v v (^) vvvvv (^) v v (^) vvvv v vv v vvvv vvv v (^) vvv (^) vvv v (^) vv vv vv vvvvvv vvvv (^) vvv (^) v vvvvvvvvvvvvvvvv (^) v (^) vvvvvvv vv v v (^) vvv vvv vvvv v (^) vv (^) vvvvvvv v (^) v vvvvv v vvvvvvvvvvv v vvvvvv v vvvvv vvvv vvv (^) v vvv v (^) vvvvv vvv vv (^) vv vvvvvvvvvvvvvvvvvvvvvv (^) v vv (^) vvvvvvvvvvvvv v vv v (^) v vvv (^) v v v vv (^) v (^) v vvvv vvvvvvvv (^) vvvvvvvvvvv vvvvvvv v vvvvv vvvv v vvv v (^) v (^) vv v vvv vvv (^) vvvv vvvvvvvv (^) v v v vvv vvvv vv (^) v vv vvv (^) v v vv (^) vvvvvv vvvvvvvvvv (^) vvvvvv vvv v (^) v (^) vv (^) vvvvvvvvvvvvv vv vvvvv vvvvvv vvv v vv vvv vvv vvv v v (^) vv vv vvv (^) v v (^) v (^) v (^) v v v vv vvvvv vvvvvv v v vvvv vv (^) v (^) vv (^) vv vvvvvvvvvvvv vvvvvvv (^) v vvvvvv vvvvvvv vv v (^) vv (^) vv (^) v vv (^) vv v (^) v v (^) v vvvvv v vvv v v (^) vvvvvvvv vv vvvvvvvvv v vvv v vvvv vv v vv v (^) vvv vvvvvvvvv vvvvvv v vvvv v (^) vv vvvv v (^) vv (^) vvvv v vv vvvv (^) vvv (^) v v vv vvvvvvv v (^) vv v v (^) vv v vvvvvvvvvvvvvvv (^) vvvvv vvvvvvv vvvvv vvvvvvvv vvv (^) vv v vv (^) v v (^) v (^) vvvvvvv vvvvv (^) v (^) vvvvvvvv (^) vv vvvvvvvvvvvvvvvvv v (^) vvvvv (^) v v vvvvv vvvvv (^) vvv v (^) v vvv v vvvvv (^) vv v vvvvv (^) v (^) vv (^) vvv v (^) vv vvvvvvv vvvvvvv v vvv vvvvvvvvvvvv vvvvvvvvvvvvvv vvvvv (^) vvvvvvvvvv (^) vv v vvvvv (^) v vvvvvv (^) v (^) v vv v vvv v (^) vvvv (^) v vv vvvvvv (^) v v (^) vvvvvvvvvvvvvvvv (^) vv vvvvv v (^) vv (^) v vv vvvvv vvvv vvv vv v (^) vvvvvvv vvvvv (^) v vv v (^) v vvv vvvv (^) v vvvv vvvvvvvv v v (^) v (^) vv v (^) vvv v (^) vv vv vvvvvvvvvvvv vv vvv (^) v v vvvvvvvvvvvvvvvvvvvvvv vv vvvv (^) vvvv vvvvvvv vv v vvv vvvv v v v (^) vvvvvvvv vv vvvvv (^) v vvvvv (^) vvv vv vvvv vvvv vvvvvvvvvvvvvv (^) vvvvvv (^) vv (^) v (^) vv (^) vvvvvvvv v (^) vvv (^) vvvv v vvvvv v vv (^) vv vvvvvvvvv vvvvvvvv (^) vvvvv (^) vvvvvvvvv (^) v v v vvv v v (^) v vv (^) v v (^) v vv v (^) v vvv (^) vv (^) v (^) vvv vv v vv v (^) vvvvvv (^) vv v vv vvvvv vv (^) v v v vv v vvv v (^) vv (^) v vvv vvvvvvvvvv vvvvvvvvv vvvv vvvvvvvvv v vv vvvvv v (^) vv vvv vvvv v (^) vv vvv vvvv v (^) vvv (^) vvvvvv (^) vvv vvvvvv vvvvvvvvvvvvvv v v vvv vv (^) vvv vv (^) v vvv vv vv vv vv vvvvv (^) vvvvv vvvvvv (^) vvv vv v (^) v (^) vvv vvv vv vvvv v (^) vvvvvvv vvvv (^) v (^) vvvv vvvvvvvvvvvv (^) vvvvvv vvv v vvv vv (^) v vv vvv vvv v (^) vvvvvv vvvvvv vvv vvvvv vvv vvv vvv vvv vvvv vvv v v (^) vvv v v (^) v (^) v v vv (^) vvvv vvvv vvvv vvv vv vv v vvvv v v vvvvvvvv (^) vvv (^) v vvvvvvvvvvvvvvvv vvvvvvvv vvvvvvvv (^) vv vv (^) vv vv vv v vvv (^) v (^) v vvv v (^) vvvvv (^) vvv vvvvvv (^) vvvvvvvvvvv vv (^) vvvvvv vvvv (^) vv (^) vvv vv v (^) vv v (^) v (^) v vv vvv (^) v v (^) v vv v vvv vvvvvv (^) v vv v (^) vvv vv vvv v (^) v (^) v v (^) vvvvvv (^) v (^) v v vvvvv vvvvv vvv v vvvvvvvv (^) vvvvvv (^) v v vvvvvvvv vvv vv (^) v (^) vvvvvv vv (^) vvvv vvvvvvv (^) vvvvvvvvvvv vvvv (^) vv vvv (^) v vvv (^) vvv (^) v vvvvvvvv v (^) vvvv vvvv v (^) v (^) vv vvvvvvvvv v vvvvv vvvv v vvvvv (^) v vvvv (^) v vvvvvvv vvvvvvvvvvvvvvvv v vvvvvvvvvvvvvv (^) vvvv vvv vv (^) v (^) v vvv v vvvv vv v v vvv (^) vv v (^) vvvv vv v (^) v (^) vvvvvv vvv (^) v vvvvvvvvvvvvv v v (^) vvvv vvvvvvvv vv (^) v vv v (^) vvvvvvvvvv (^) vvvv v vvvv vv vvvvvvvvv v (^) v (^) vv (^) vvvvvvvvvvvvvv vvvvvv vv vvv v (^) vvvvvv (^) v vvvvv (^) vvvv (^) v (^) vvv v v (^) v (^) vv vvvvvvv vvvvvvvvv v v v v vvv v vvvv (^) v vvv vvvvvvvvvvvvvvvvvvvv vvv (^) vvv (^) vvv vvvvvvvvvv (^) vvvvvvvvv vv vvvvvvvvvv (^) vv vvvvv (^) vvv vv v v vvvvvvv v vvvv (^) v vv vv v vvvv v vvvvvv v v (^) v vvvvvv v (^) v vvvv (^) v (^) vvv vv vvvvvvvvvvv (^) v vvv (^) v vvvvvvvvvv (^) vv (^) vvv vv vvvvv v vv vvvvvvv vvvv (^) vv vvvv vvvvvv v vv v (^) v (^) vv v v (^) vvvvvvvvv (^) v (^) v vvvvv (^) v vv (^) v (^) vv vvvvvvvvvvvvv vv v vvvv (^) vvvvvvvvvvv (^) vvvv vv v vvvvv (^) vv v vv v (^) v (^) v (^) v v v vv v (^) vvv v (^) v vv (^) v vvvvvv vvvvvvvvv (^) vvvv vvvvvvvvv vvvv v (^) vv vvv vvvv (^) v vv vvv vvv v vvv (^) vvvv (^) vvv vvv vv v vvv (^) vv v vvvv v (^) vvvvv vvvvvvvv v (^) vvvv (^) v v (^) vvvv v (^) vvvvv vvvvvvv (^) v vvvv (^) vv v (^) vvvvvvvv vv v v (^) v vvvvv v (^) vv vv vv vv vvvv (^) vvvvvvvvvv vvvvvvvv vvvvv v (^) v vv v vvvvvvvvvv v vv v vv v v (^) vvvvvvv vvvv (^) vvvvvv (^) v vvv (^) vvvvvvv (^) v (^) vvvvvvvv vvvvvvvvvvvv v vv v v v vv (^) vvv vv (^) vv vv vv v v v (^) v vvvvvvvv vvvv vv v (^) vvvv vv v (^) v vvvvvv vv (^) vv vv v (^) vv vv (^) v (^) v v v v (^) v vvvvv v vvv vvvvvv vvv vvvvvvvvvv vv v (^) vvvv v vvvv vvvvv (^) v (^) vv vv v v (^) v v v (^) v v (^) vvvvv v (^) v vvv vvvvv (^) vvv vvvv vvvvvvv (^) v v v (^) v vvv v vv vvvvvvvv (^) v vvvvvvvvvv (^) v vvv (^) v v vvv (^) v (^) v (^) vvvvvvvvvvv v vvvvvvvvvvvvvvvvvvvvvvvv (^) vvv vv (^) v v vv (^) vvvvvv vvvvv v v vv v (^) vv (^) v v (^) v (^) v v vvvvvv (^) vvv (^) v vvvvvvvvv (^) vv v v v (^) vvvvvvvvvvvv (^) vv (^) vv v v v vvvv vvv (^) vv vv vvvv (^) vv vv (^) vvvvvv v (^) v v (^) v v (^) v (^) vv vvvv v (^) v vvv v vv vvv vv vvvvvvvvv (^) vvvvvvvvvvvv vvv (^) vv vv v (^) vv v (^) vv vvvv v (^) vvvvvvvvvvvvv vvvvvv vv (^) vv v (^) v (^) vv (^) v vv vvv (^) vvvvvv vvvvv (^) vv (^) vvvvvvvvv vvvvvvvvvvvvvv v vv (^) v vvv (^) v (^) v (^) vvv vvv (^) v vvvv vv vvv (^) vvv vv (^) v vvvvv v vvv vvvvvvvvvvvvv (^) vv vvvvvvv vvv (^) vvvvvv vvv (^) v v (^) vvvv (^) v vvv vvvvvvv (^) vv (^) v v (^) vvv (^) vv vv v (^) vvv v vvvvvvvvvvvvv vvv vv (^) vvvv (^) v v vvvvvv v (^) vvvvvvvv vvv v v (^) v vvvv vv v (^) vvv vvv vv (^) vv vvv (^) vvv (^) vvvvv v v v vvv vvvvv vv v v vvvvvvvvvv vv v v (^) vvv (^) vvvvv v (^) v v (^) vv (^) v vvv (^) vv vvvvvvvvvvvvvvv v (^) v vv v vvvvvvvvvvvvvv vvv vvvvvvvvv (^) vv v vvvvvv v v v v (^) v vv (^) vvv (^) v vvv (^) v vvvvvv v vvvvvvvvv vvvvvv (^) v v vvvvvv vv v vvv vv v vvv vvvvvvvv (^) vvvvvvv vvvvvvvvv vvvvv (^) vvv vv (^) vvvvvvvv v (^) vvvvvv (^) v vvvvv (^) v v vv (^) vv vvv vvv v (^) v vvvvv (^) v vv vvvvvvv (^) v v (^) vv v vvvv vvvvv (^) vvvvvv vvvv v (^) vvvvvvv (^) vv v vvvv vv vvv (^) vvv (^) vvvvv v (^) vvv vvvvv (^) vvvvvvvvv v v v (^) vv vvvvvv vvvv vvv vvvvvv vvv vvvv vvvv (^) v (^) v (^) v vvvvv v vv (^) v vv vvvv vvv (^) v vvv v v (^) vvvvvvvvvv vv vvvvvvvv vvvv vvvv v (^) v vvv vvv (^) v (^) vvv vvv v (^) v vvvvvvvvvv (^) vvvvvvvv (^) vv vvvvvv vvvvvvv (^) v (^) vv (^) vvvv vvvvvvvv (^) vvvv vvvvvvvvvvvv v (^) v vvvv vvvv vvvv (^) vv v vv vvv vv vvvvvv vv vvvvv v v v vvvv (^) vv vvvvvvvv (^) v vvvv v (^) vvvv (^) vvv vvvvvvvvv vv vv (^) vvvvv (^) v (^) vvvvv (^) v v v vvvv (^) vv v vvvvvvvvvvvv vvv vv vvvv vvv vvv v v vvv v vvvv v vvvvv vvvv (^) vv vvvvvvvvvvv vvvv RULE 137(10001001) v v (^) vv vvv v (^) v v (^) vvv (^) v vvvv (^) vvv v (^) v v (^) vvvv vvv v v vvvvv (^) vvvv vvvvv v v v vvvvv v vvvv vvvvv vvvvvv vvvvvvv v v v (^) v vvvvvvvvvvv vvvv vvvv vvvvv v v v v vvv vvv vvv vvv v vvv v vvv v vvv v vvv v v v (^) v (^) v v (^) vvvvvvvvvvvvvvvvv v vvvvvvvvvvvvvvv (^) v (^) vvvv (^) vvvvv (^) vvvvvvvvvv vvvvv (^) vvvvvv vvvvvvvv vvvvvvvvvvvv vvv v (^) v v (^) vv vvvvvvvvvvvvv (^) v vvv vvvvv vvvv vvvv vv vvvvv vvv vvvvvv (^) v (^) v (^) vvvvv v (^) v v vvvv vv (^) vv v (^) v v (^) vv vvvvvvv vvvv (^) vvvv vvv (^) vvv v v (^) v (^) v vvvv v vvv (^) vvv vvvv vv (^) vv vvvvvvvvvvv (^) vvv (^) vvvvvvvvvvvvvvvvvv (^) v vvv v (^) v (^) v (^) vvv (^) v vv vvvvvvvvvvv v vvvvvvvvv (^) vvvvv (^) v v (^) v vv vvvvvv v vvvvvvvvvvvvvvvvvvvvvv vv v (^) v vv vv (^) vv vvvvv vvv v v vvv (^) v (^) v (^) v vv (^) v vvv v vvvvvvvvvvv (^) vv vvvvvvvvvvv vvvvv (^) vvvvvv (^) v vvvvv vvvvvvvv vvvvvv vvvvv vv vv (^) vvvv vvv vvvvv vv (^) v (^) vvv (^) vvvvv vv (^) vvvvvvv (^) vvvv (^) vv (^) v v (^) vv vv v v vvv v v (^) vvvvv v vvvv (^) vvv vv (^) vvvv vvvv vvvv (^) vvvv vvvv v vvvv v (^) vvvvv v vvv vvvvvvvv v (^) vvvv vvv v vvv v vvv v vvv v vvv v vvv v vvv v vvv v (^) vvvvvvvvvv vvvvvvvvv vvvv vvvvvv vv v (^) vvvvv vvvvv vvvv (^) v vvv vvvvvvv vv (^) vvvv vvvv vvvv vvvvv v vvv vv (^) vv v vvv vvv vv v vv (^) v v (^) v v vvv (^) vv v vvvvvv vvvvvvvvvvvvvvvvvvv (^) v v vv vvv vv v v v (^) v (^) vvv vvvvv (^) vvv (^) vvv vvv (^) v v (^) vv vvv vv v vv (^) v vvvv v vvvvvvv vvvv vvv (^) v vvvv vv v (^) v (^) v vvvvvvvv (^) vvvv (^) v (^) v (^) vv (^) vv vvvvvvvvvvvvvv vv (^) vvvvvv v (^) vvvvvvvvvvvvvvvvvvvvvvvv vv (^) vvvv vvv v (^) v (^) vvv vvv (^) v vvv (^) vv v (^) vvvv (^) v vv v v v vvvvv (^) vvvvv (^) v (^) vvvv v vvv vvv v (^) v vvvv (^) vvvvvvvv vvvvvvvv vv vv (^) vvv (^) v (^) vvvvvv v vv (^) vvvvvvvvv vv (^) vvv vvvv (^) v vvvv vvvvvv vvvvvvvvv vvv (^) v (^) v vvvvv v (^) v vvvv vvv (^) vvvvvvvvvvvvv vv v (^) v (^) vv v (^) v (^) vv (^) vv v vvvvvvv vv vvvvv vvvvvvvvvvvvvvvvvvv (^) v vvvvvv (^) vv (^) v v vv (^) vvvvv (^) v v (^) vvv v vvv vvvvvvv (^) v vv v (^) v (^) vv v (^) vv v vvvvvv (^) v v (^) v vvv vv vvv v v vvvvv v (^) v vvvvvv v vv (^) v v vvv vv (^) vvv (^) vv vvv vv vv vvvvvv (^) vvvvv v vv vvvvvv (^) vvvvv v (^) vvv vvvvv v v (^) vv vvvv vvv v (^) v vv (^) vvvvv vv vvvvvv vvvv (^) vvvvv vvvv vv (^) vvvv vvvvvvv v (^) v vv (^) vv vv vvvv vv (^) vvv v (^) vvvvvvvv (^) vvvvvvvv vvvvvvvv vv (^) vvvv vv (^) vvvvvv vv vvvv (^) 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v vvvvvv v vv v vvvv (^) v vvvvv (^) v v (^) v v v vv (^) vvvvvvvv vvvv vv (^) v v vvvv vv vvvvvvv v (^) vvv (^) vvv (^) vv v vvvvvv v v v (^) v (^) vv v (^) vvvv (^) v vvv vvvvvvvvv vvvvvvvvvvvvvv vv vvv vvv vvv vvv vvv vvv vvv v (^) v v (^) v (^) v v (^) v (^) v vv vvvvvv vvvv vvvv (^) vvvv vvv vv (^) vvvv (^) vvvv vv (^) vv vv vv (^) vv vv vv vv v vvvv v (^) v vvvv v vvvv v vv (^) v (^) v vvv vvvvvvvvvvvvvvvvvvvv vvvv (^) v vvvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvvvvvv (^) v vv (^) vv vv vvvv (^) vvvv vvvv vvvv vv v (^) vv (^) v v (^) v (^) v (^) vv v (^) v vvvvvv (^) v vvvvv (^) vvvvvv (^) vvv v (^) vvvv (^) vv v v (^) v (^) vvv v vvv (^) vvv (^) v vvvvvv vvvvvv vv vv vv vv (^) vvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvv (^) vv vvvv vv (^) v vv (^) v (^) v v (^) vvvvvvvvvvvvvvvv vvvv vvvvvvv vv vvvvvvvvvvvv vvvvvvvvv vvvvv vvvvvv (^) v v (^) v v vvvvv vvv vv (^) v (^) vv vvv (^) vvvvvv (^) v (^) vv v vv (^) v v v (^) vvvvv vvvvv vvv vvvvv vvv vvvvvvv vvvvvvvvvv vvvv vvvvv (^) v (^) vvv v v (^) v vvvvvvvvv v (^) vvvv vvvvvvvv (^) v vvvvvv vvvvvvvv (^) v v v v (^) vv vvvvvvv v (^) vvv v v (^) vv v vvvv vvvvvvvvvvvv (^) v vvvvvvvvv vvvvvv v vvvvvvv vvvvvv vvv vvv vvvvvvvvv (^) vv (^) vvv v (^) vv v (^) v vvvv vvvvv vvvvv (^) vv vvvvv v (^) vvvvvvvv (^) vvvvv v (^) vvvvv (^) v (^) vvvv v vvvvv v v (^) vv v vvvv (^) v vvv v v vvvv (^) vvvv (^) vvvvvvvv (^) vvvv (^) vvvvvvvv v v (^) vvv v vvvvv (^) vvv vvvvvv vvv v v (^) vvvvvv (^) v vv vv (^) vv (^) v (^) vvvvv vv (^) vvvvvvv vvvvvvv (^) vvvvvvv vv (^) vv vvvvvvvv vv vvv vvvvvvv vvv v vv vvvvv vvvvvvvvvvv (^) v (^) vvvv (^) vvvvv (^) vv vvvvv (^) vv vvvvv vv v (^) vvv (^) vv vv (^) vvvv (^) v (^) v v vv vvv vvvvvvvv vv v (^) v vvvv vv (^) v v vvvvvvvvvvvvvvvvv (^) vvvvvv (^) vv v v (^) vvvvvv (^) v v (^) v (^) vvv vv v (^) vvvv vvvvvvv v vv vvvv (^) v (^) vvvvvvvvvv (^) v (^) vvvvv v (^) v (^) vvvv vv vvvvv vvvvvv v (^) vvv vvvv v vvvv vvvvv (^) v vvvv (^) v vvvv v v vvvvvvvv vv vvvvv (^) vvvvvvvv vv vvvvv vvvvv vvvv (^) vv vvvvv (^) vvvvvv (^) v vvvv vvv (^) vv v v (^) v vv (^) v v (^) vv (^) v vvvvv vv vvvvv (^) vv vvvvv vvvvvv vv (^) vv vvvvvvvvvv v (^) v vvv vvvvvvv v vvv vv vvvvv v (^) vvv vvvvvvv v (^) v vvvvvv vvv vvv vvvvvvvvv (^) v vvvvvvvvv vv vv vvv (^) vvvvvv vv (^) vvv v vvvvv (^) vvvvvvvvvvvv v (^) vvv vvvvvvvv (^) vv v (^) v vvvvv v (^) vvvv vvvvvv (^) v v (^) vv v v (^) vvvvv (^) v vvvvv vvvvvvvv vvvvvvvvvvv vvvvvvvvvv RULE 184(10111000) v v v v (^) v v vv vvv vvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvv (^) vvvvvv (^) v vvvvvv v vvvvvv v vvvvvv vvvvvvvvvvvvvvvv [(-1,1),(1,-1)] vvvvvvvv (^) v vvvvvv v vvvvv vvvvvvvv vvvvv vvvvvvvv (^) vvvvvvvvv (^) vv vv vvvvvvvv vvvvvv (^) vvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvv (^) vvvvvv (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v v v v v (^) v v v v v v (^) v v (^) v v vvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvv (^) vvvvvvvv vvvvvv (^) v vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvvvv vvvv vv vv (^) vv (^) vv vv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvv (^) vvvvv (^) vv (^) vvvv vv vvvvvvvvvv (^) v vvvvvvvvvvvvvvvvvvv vvvvvvvvv vvvvvvvvvv vvvvvvv vvvvvvvvvvvvvv vvvvvvvvvv vvvvvv v v v vvvvvvv vv vv v v (^) v v v (^) v v v (^) v v v v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v v v (^) v v v v v v v v (^) v v v v v v vvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvv vv vv v vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvv vvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvv vv vv v vvvvvvvvvv (^) vvvvvvvv vvvvvv vvvv vv (^) vv vv vv vvvvvvvvvvvvvvvvvvvvvvvv (^) v v v v v v v v v (^) v v v v v v v v v v v v v (^) v v v v v v v vv vv vv vv (^) vv v (^) v (^) v (^) v v vvv vv v vv (^) vv v v v v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vv v v (^) v v v v v v (^) v v v v vv vv v v (^) v v v (^) v vv vv v v v (^) v (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v v (^) v v v v v v (^) v v v v v v v v vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vv (^) vv vv vv (^) vv vv vv vv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vv (^) vv vv v vv (^) vv v v (^) vvv vvvvvvvvvvvvvvvvvvv (^) vv (^) vv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) v v v v v v v v v v v v vv (^) vv v v (^) v v v v vv (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v v v v v v (^) v v (^) v v v v v v (^) v v vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vv vv vv (^) vv (^) v vvvvvvvvvv v vvvvvvvv v vvvvvv (^) v vvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v v vv v v v v v v (^) v v v v (^) v v (^) v v v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv (^) vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v v v v v v (^) v v v v v v (^) v v (^) v vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvvvvvvv vvv v vv v vv v vv (^) v vv v vv v vv (^) v vv (^) v v v v v v (^) v v v v v v v v v (^) v v v v v v v v v v vv (^) vv vv vv (^) vv vv vv v v v v v v v v v (^) v v v v v v v v v (^) v v v (^) v v v (^) v v (^) v (^) v v (^) v vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv RULE 226(11100010) v v v (^) vv vv vv vv vv vv vv vv vv vvv (^) vvvvvvvvvvvvvvvvvv v (^) v v (^) vv vv vv (^) vv vv vv (^) vv vv vv v (^) v v (^) vv vv vv (^) vv vv vv (^) vv vv vv vvvvvvvvvvv (^) v vvvvvvvvvvvv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vv vv (^) vv vv vv (^) vv vv vv (^) vv vv vv (^) vv vv (^) vv vv v v vv vv (^) vv v vvv (^) v vvv v vv v vv (^) v vv (^) vv v v (^) v v v v (^) v v v v v v v v v v v v v (^) vv vv vv vv v v (^) v v v (^) v v v v v v (^) vv v vv vv vv (^) vv (^) v v v v v v v v (^) v vv v vv vv v (^) v vv (^) vv vv vv vv v (^) v v v v (^) v v v (^) v v v v v v (^) v v v v v v v v vv v vvvvvvvvvvvvvvvv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv vv v vvv (^) v vvv v vvv v vvv (^) v vv (^) v vv v vv v vv v vv (^) vv vv vv (^) vv (^) vv vv vv v (^) v v v v (^) v v v v v v v v v v v v (^) v v v v v v (^) vv vv vv vv (^) vv v v (^) v v (^) v v v (^) v v v v vvvv v v v v v v v v (^) v vv v vv v vv vv (^) vv vv vv vv (^) vv vv vv vv (^) v v v v v v v v (^) v v v v v v v v v (^) v v v (^) v v v v v v v v v v vv (^) vv vv vv (^) vv (^) vv vv vv v v (^) v v (^) v v v (^) v v v v v v v (^) v v v v v (^) v v vv v v v v v (^) v vv vv v v v v v v v v (^) v vv v vv vv vv (^) vv vv v v v v v (^) v v v v v v v v v (^) v v v (^) v vv v v (^) vv vv vv (^) vv (^) vv vv v v v (^) v v v v v v v v v v v v (^) v v v v v v v (^) v v v (^) v v v v v v v v v v (^) vv vv vv v v (^) v v v v v v v vvvvvvvvvvvvvvvvv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv v vv v vv vv vv (^) vv vv vv (^) v vv vv vv vv (^) vv vv vv (^) vv v v v v (^) v v v (^) v v v v v v v v v (^) v v v v v v v vv vv (^) vv (^) vv vv vv (^) vv vv vv v v (^) v v v v v v v (^) v v v v v v v v v (^) v v v v v v vv (^) v vv (^) vv vv vvvv v v v v v v v v v v v (^) v v v v v v v v v (^) v vv vv vv vv vv (^) vv vv vv (^) vv v v v v (^) v v v (^) v v v v v v v v v (^) v v v v v v v vv v (^) vv (^) vv vv vv (^) vv vv v (^) v v (^) v v (^) v v v vv v v v vv v vv vv vv v v v v v (^) v v v (^) v v v v vv vv vv (^) vv (^) vv vv vv (^) vv v v v v v v v v v v (^) v v v v v v v vv (^) v (^) v v (^) v v v v v v v v v (^) v v v v v v v v v vv vv (^) vv vv v (^) v v v v v v v v (^) v v v v vv (^) vv v (^) v v v vvvvvv vvvvvvvvvv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vv vvvvvvvvvvvvv vvv (^) vvv (^) vvv vvv vvv v v v v v v v v v v v (^) v v v v v (^) v vv vv (^) vv v (^) v v v v v v v v v v vvvvv vv vv vv (^) vv vv vv vv (^) vv v v v v v (^) v v v v v v v v v (^) v v v (^) v v v v v v v v v v vv (^) vv vv vv (^) vv (^) vv vv vv v v v v v v v v v (^) v v v v v v v v v v v v (^) v v v v v v (^) v v (^) v (^) v v (^) v v v (^) v v v v vv v v v v v (^) v vv vv vv (^) vv vv vv v v v v v v v (^) v v v v vv v (^) v (^) v (^) v v vvvvvv (^) v (^) v vv vv vv (^) v (^) v v (^) v v v v vv vv (^) vv v v v v v v vv (^) v vv vv vv (^) vv v v v v v (^) v v v v v v v v v (^) v vv (^) v vvvvvvvvvvvvvvvvv vv vv vv vv vv vv vv vv vv vv vv vv vvv vvv vvv vvv (^) vvv vvv v v v v (^) v v v (^) v v v v v v v v v (^) v vv vv (^) vv (^) vv vv vv (^) vv v v v v v v v v v v v (^) v v v v v v v (^) v (^) v v (^) v (^) v v vv (^) vvvvvv v v v v v v v v v v v (^) v v v v v v vv vv vv vv (^) vv vv v v v v (^) v v v (^) v v v v v v v v v (^) v vv v v (^) v v v v v v v (^) v v v v v v v vvvvvvvvvvvv
Figure 4. ( continued)
118 R. Alonso-Sanz and M. Mart´ın
[58]: to decide whether an arbitrary initial configuration contains a density of ones above or below Ρ c. 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:
m ( iT a ( i ,^ a
( i ,^...^ ,^ a ( T i ^ ^
Tt 1 exp Β( T t ) a ( t i Tt 1 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:
m ( iT ( a (1 i , a (2 i ,... , a ( iT ) Tt 1 Α t^^1 a ( it Tt 1 Α t^^1
f (^) i ( T
1 if m ( iT > 0_._ 5 1 a ( iT if m ( iT 0_._ 5 0 if m ( iT < 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, Β 1 1 rule will be again, as at T 2, a 3 " 3 square. After T 3 these cells are featured as live:
