We study the ability of a genetic algorithm to design cellular automata ... interpreted as implementing (or approximating) a computation 1]. ...... Arti cial Life. II.
Emergent Computation in Cellular Automata Rajarshi Das Center for Nonlinear Studies, MS{B258 Los Alamos National Laboratory Los Alamos, NM 87545 October 1998 Abstract
We study the ability of a genetic algorithm to design cellular automata exhibiting spontaneous global synchronization. We show that in order to produce emergent coordination, the evolutionary process can take advantage of the underlying medium's potential to form embedded particles. The particles, typically phase defects between synchronous regions, are designed by the evolutionary process to resolve frustrations in the global phase. We describe in detail typical solutions discovered by the global algorithm, delineating the discovered synchronization algorithm in terms of embedded particles and their interactions. While distinct in detail, we nd that the evolved cellular automata embody similar computational strategies for establishing global synchrony. We also use the particle-level description to analyze the evolutionary sequence by which this solution was discovered. These results have implications both for understanding emergent collective behavior in natural systems and for the automatic programming of decentralized spatially extended multiprocessor systems.
1 Introduction Natural evolution has created many systems in which the actions of simple, locallyinteracting components give rise to coordinated global information processing. Insect colonies, economic systems, the immune system, and the brain have all been cited as examples of systems in which \emergent computation" occurs (e.g., see [5, 11]). In the following, \emergent computation" refers to the appearance in a system's temporal behavior of information-processing capabilities that are neither explicitly represented in the system's elementary components or their couplings nor in the system's initial and boundary conditions. Our interest is in phenomena in which many locally-interacting processors, unguided by a central control, result in globally-coordinated information processing that is more powerful than can be done by individual components or linear combinations of components. More precisely, \emergent computation" signi es that the global information processing can be interpreted as implementing (or approximating) a computation [1]. While observations of the behavior of such a decentralized, multicomponent system might suggest that a computation is taking place, understanding the emergent logic by which the computation is performed is typically very dicult. It is also not well understood how evolution creates the capacity for emergent computation in such systems. In this paper we explore these questions in a simple framework by coupling an evolutionary process|a genetic algorithm (GA)|to a population of behaviorally rich dynamical systems|one-dimensional cellular automata (CAs). In this scheme, survival of an individual CA is determined by its ability to perform a synchronization task. We apply a framework, called \computational mechanics", for analyzing the emergent logic embedded in the spatiotemporal behavior of spatially-extended systems such as cellular automata [2]. The results demonstrate how globally-coordinated information processing is mediated by \particles" and \particle interactions" in space-time. We also analyze in detail the evolutionary history over which a GA evolved such emergent computation in cellular automata. Though our model is simpli ed, the results are relevant to understanding emergent computation in more complicated systems.
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2 Cellular Automata CAs are arguably the simplest example of decentralized, spatially extended systems. In spite of their simple de nition they exhibit rich dynamics which over the last decade have come to be widely appreciated [7, 12, 13]. An one-dimensional CA consists of a collection of time-dependent variables sit, called the local states, arrayed on a lattice of N sites (or cells), i = 0; 1; :::; N ? 1. We will take each to be a Boolean variable: sit 2 f0; 1g. The collection of all local states is called the con guration: st = s0t s1t sNt ?1. s0 denotes an initial con guration (IC). Typically, the equation of motion for a CA is speci ed by a lookup table that maps a site's neighborhood ti to a new local state for that site at the next time step : sit+1 = (ti ), where ti = sit?r sit sit+r and r is called the CA's radius. The global equation of motion maps a con guration at one time step to the next: st+1 = (st ), where it is understood that the local function is applied simultaneously to all lattice sites. It is also useful an operator that operates on a set of con gurations or substrings of con gurations|that is, on a formal language|by applying separately to each member of the set. The behavior of one-dimensional CA is often illustrated by a \space-time diagram" as shown in Figure 1. The CAs in the GA experiments reported below had r = 3 with spatially periodic boundary conditions: sit = sit+N .
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Figure 1: A space-time diagram for a randomly generated r = 3 binary-state cellular automaton. elementary CA (ECA) 110. The lattice of 149 sites, displayed horizontally at the top, starts with s0 being an arbitrary initial con guration. Cells in state 1 are displayed as black and cells in state 0 are displayed as white. Time increases down the page.
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3 The Synchronization Task
Our goal is to nd a CA that, given any initial con guration s0, within M time steps reaches a nal con guration that oscillates between all 0s and all 1s on successive time steps: 0N = (1N ) and 1N = (0N ). M , the desired upper bound on the synchronization time, is a parameter of the task that depends on the lattice size N . This is perhaps the simplest non-trivial synchronization task for a CA. The task is nontrivial since synchronous oscillation is a global property of a con guration, whereas a small-radius (e.g., r = 3) CA employs only local interactions mediated by the sites' neighborhoods. Thus, while the locality of interaction can directly lead to regions of local synchrony, it is more dicult to design a CA that will guarantee that spatially distant regions are in phase. Since regions that are not in synchrony can be distributed throughout the lattice, a successful CA must transfer information over large space-time distances ( N ) to remove phase defects separating regions that are locally synchronous, in order to produce a globally synchronous con guration. For reference, we consider a simple benchmark radius 3 CA osc , which is a naive candidate solution with osc (1N ) = 0N and osc (0N ) = 1N . Its look-up table is de ned by: osc () = 1 if = 07; osc () = 0 otherwise. Typical behavior of osc starting from a random IC is shown in Figure 2. We de ned the performance PKN ( ) of a given CA on a lattice of size N to be the fraction of K randomly chosen initial con gurations on which produces correct nal behavior. We then measured P10N 4 ( osc ) to be 0.54, 0.09, and 0.02, for N = 149, 599, and 999, respectively. (The behavior of a CA on these three values of N give a good idea of how the behavior scales with lattice size.) osc is not a successful solution precisely because it is unable to remove phase defects. A
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Figure 2: Typical space-time diagram produced by osc . It should be noted that osc is unable to remove the two phase defects|borders separating regions that are locally synchronous. 4
more sophisticated CA must be found to produce the desired collective behavior. It turned out that the successful solutions discovered by our GA were surprisingly interesting and complex.
4 Genetic Algorithms The search for high-performance CAs for the task above is dicult for a number of reasons. First, the number of possible CAs grows very rapidly with increasing number of states per site and with increasing neighborhood size. Even with two state per site, exhaustive search is impractical when the neighborhood size of CAs is more than ve. Calculus based methods which require continuity and derivative information are also not applicable because of the discrete nature of the CA search space. These observations suggest the need for a di erent technique which can search through a large number of possibilities for solutions in an ecient fashion and which does not require derivative or other auxillary problem-speci c knowledge. In this work, we investigate whether an evolutionary process can discover high-performance CAs for the computational task discussed above. Our choice of an evolutionary process as the search engine is not accidental. An evolutionary process can be viewed as a massively parallel method to search among a large number of possible \solutions." Moreover, at an abstract level, the underlying mechanisms of an evolutionary process can be remarkably simple: natural selection of entities that reproduce with variation. These features provided us the main inspiration to employ an evolutionary approach in this work. As an abstract computational model of an evolutionary process, a genetic algorithm (GA) was used to evolve CAs. GAs [10] are a class of computational models of evolution which have gained popularity as ecient stochastic search algorithms [6]. In our experiments, a GA starts with a randomly generated population of CA lookup tables, and evaluates the CA rules in the population by estimating each rule's success in performing a given task. CAs which are more successful in performing the task are preferentially selected, and a new population of CAs is created by applying the genetic operators of crossover and mutation on the genomic representation of the more successful CAs in the existing population. This process, when repeated for a number of iterations, often results in the discovery of high-performance CAs. The experimental design (Figure 3) used in this work for evolving CAs is interesting for a number of reasons. First, it consists of two nonlinear dynamical systems, a GA and a CA, operating at di erent time scales, which are coupled to each other. Due to the coupling, the dynamics in one system in uences the dynamics of the other. Thus, a complete delineation of the results obtained from the experimental setup not only requires an in-depth knowledge of how each of the components work, but also how they interact with each other. Although the operators in the GA act on the CA look-up tables (the genotype), the tness of a look-up table is determined by the spatio-temporal behavior of the corresponding CA (its phenotype). Since the mapping from the genotype to the phenotype is typically nonlinear in a CA, and because the elements in the genotype often display complex nonlinear interactions, 5
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the process through which a GA nds high-performance CAs is not readily apparent. In addition, since a CA's behavior (i.e., its phenotype) is often very complex and dicult to characterize rigorously, the problems above are seriously compounded.
5 Evolved CAs In a majority of the GA runs for the synchronization task, the best CAs in the nal generation had performance P10N 4 between 0.65 and 0.99. Space-time diagrams from two such rules are illustrated in Figure 4. Starting from the randomly generated ICs shown in the gures, these CAs are able to reach global synchronization. However, the space-time diagrams contain stable phase-defects|borders separating regions that are out of phase with respect to each other. Although the phase-defects have annihilated each other in these two diagrams, they may exist in the lattice inde ntely and prevent these CAs from achieving maximal performance. Moreover, when such a CA is tested on larger lattice sizes, the presence of phase defects leads to drastically inferior performance. 0
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Figure 4: Typical space-time diagrams from two di erent rules with P101494 0:9. Several di erent types of phase defects can be observed. Randomly generated ICs have been used. In some of the runs however, the GA discovered successful CAs with P10N 4 = 1:0. The performances of the more sophisticated CAs remained xed at 1.0 even when tested on larger lattices of size 599 and 999. Typical space-time diagrams from four sophisticated rules (each from a di erent run) are illustrated in Figures 5(a)-(d). Unlike the space-time diagrams of the less sophisticated rules, we do not notice any stable phase defects occurring between regions that are locally synchronous. Instead, we observe an abundance of \organized patterns" in the space-time which grow and shrink over time until the CA reaches global synchronization. 7
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Figure 5: Typical space-time diagrams from four sophisticated rules (each from a di erent GA run) are illustrated. Randomly generated ICs have been used. All these rules demonstrated performance P10N 4 = 1:0 for lattice sizes N = 149, 599, and 999.
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The fundamental question that we want to answer is: How do these patterns and their dynamics aid the CA in performing the computational tasks? In other words, what is the emergent logic through which a CA is performing the desired computation?
6 Computational Mechanics Analysis of Evolved CAs In order to understand the computation performed by the successful CAs, this work adopts the \computational mechanics" framework for CAs developed by Crutch eld and Hanson [2, 9, 8]. This framework describes the \intrinsic computation" embedded in the temporal development of the spatial con gurations in terms of domains, particles, and particle interactions without any functionality or utility attached to the descriptions. A domain is, roughly, a homogeneous region of space-time in which the same \pattern" appears. The notion of a domain can be formalized by describing the domain's regularities using the minimal deterministic nite automaton (DFA) that accepts all and only those spatial con gurations that are consistent with the regularity. Once the domains have been detected in the spatio-temporal behavior of a high- tness CA, nonlinear transducers can be constructed to lter the domains out of the con guration, leaving just the deviations from those regularities. In each of the high-performance CAs, the resulting ltered space-time diagram reveals the propagation of domain boundaries. These boundaries are called particles when they remain spatially localized over time. These \embedded" particles are one of the main mechanisms for carrying information over long space-time distances. This information might indicate, for example, the partial result of some local processing which has occurred elsewhere at an earlier time. Operations on the information carried by the particles are performed when the particles interact. The collection of domains, domain boundaries, particles, and particle interactions for a CA represents the basic information-processing elements embedded in the CA's behavior.
7 Analysis of Evolved Rules The aim here is to delineate the pattern dynamics in these CAs in terms of the embedded domains, particles and particle interactions, and to show how they give rise to high performance in the CAs. The second goal is to determine the common theme underlying the particle-level strategies used by di erent high- tness CAs to perform the synchronization task. Finally, once the computational structures have been identi ed, the objective is to describe the temporal stages in the evolutionary process which lead to the discovery of a high-performance CA. In this section, two representative CA rules with di erent look-up tables have been selected from the set of rules with high performance for the synchronization task. These rules are then analyzed in detail. The two rules are: (i) A = 0x FEB1C6EA B8E0C4DA 6484A5AA F410C8A0, and (ii) B = 0x FE5CEEF4 EDD2E0AA 60F1EE40 FCB6B280. 9
(Each CA is given as a hexadecimal string which, when translated to a binary string, gives the output bits of in lexicographic order ( = 07 on the left).) While the analysis presented here focuses on these two rules, the main conclusions drawn from the analysis are general in their scope: in most cases the analysis can be easily extended to other high-performance rules without much diculty or modi cation.
7.1 Analysis of A
Figure 6(a) gives a space-time diagram for one of the GA-discovered CAs with 100% performance, here called A . In the example given in the gure, global synchronization occurs at time step 58. The gure also shows that the space-time behavior of A is dominated by two types of distinct patterns: regions where the all-1s pattern alternates with the all-0s pattern; and regions of jagged black diagonal lines alternating with jagged white diagonal lines. Both these patterns can be characterized as regular domains since each satis es the temporal invariance and spatial homogeneity conditions. One of the domains, denoted here as 0A, consists of two languages, 0000+ and 1111+, such that (0000+) = 1111+ and (1111+) = 0000+. 0A is the synchronous domain, characterizing regions in the con gurations where the cells are oscillating in synchrony. 0A will also be denoted as S . For the CA to reach global synchrony, the entire lattice has to be occupied by the S domain. The second domain, denoted here as 1A , also has a temporal periodicity of two, but has a spatial periodicity of four. 1A consists of two languages (0001)+ and (1110)+, such that ((0001)+) = (1110)+ and ((1110)+) = (0001)+. In subsequent discussions, 1A will also 0
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Figure 6: (a) Space-time diagram of A starting with a random initial condition. (b) The same space-time diagram after ltering with the transducer TA which maps all domains to white and all defects to black. Greek letters label particles described in the text. 10
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Figure 7: The spatial transducer TA which accepts words in domains 0A and 1A.
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be denoted as D. Unlike the languages in 0A , the con gurations in 1A have a temporal periodicity of four. Figure 7 shows a spatial transducer TA, which accepts words in domain 0A and 1A . When TA is applied to the space-time diagram in Figure 6(a), it produces the plot shown in Figure 6(b). In the gure, the ltering not only determines the location of the particles in the space-time diagram, but it also helps in readily identifying the spatial and temporal features of the particles. In addition, various types of particle interactions can be distinguished. As shown in Table 1, A supports ve stable particles, and one unstable \particle", , which occurs at SS boundaries1. \lives" for only one time step, after which it decays into two other particles, and , respectively occurring at SD and DS boundaries. moves to the right with velocity 1, while moves to the left at the same speed. S
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D alternates between (0001)+ and (1110)+ , i.e.
Particles
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Domain Temporal Spatial Velocity Boundary Periodicity Displacement 0 SS DS 2 2 1 SD 2 -2 -1 DS 4 -12 -3 SD 2 6 3 2 -2 -1 DD
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+ !; + !; + ! , if d mod 4 = 1 + ! + ! + ! + , if d mod 4 6= 1 + ! +
Table 1: The domains, particles, and particle interactions that dominate the spatio-temporal behavior of A . SS and DD represent phase defects within the S and D domains respectively. The reactions between particles and depend on the relative distance d between them, where 0 d 2r. How does A perform the synchronization task? From any random initial con guration, A produces local regions of synchronization, i.e., regions which are in the S domain. However, in many cases, adjacent S domains are out-of-phase with respect to each other. Wherever such phase defects occur, A resolves them by propagating particles|the boundaries between the S domains and the D domain|in opposite directions. Encoded in A 's 1 Due to the temporal periodicity of the S domains, the two adjacent S domains at any boundary can be
either in-phase or out-of-phase with each other. The in-phase and the out-of-phase defects between two S domains will be represented as SS and SS respectively.
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look-up table are interactions involving these particles that allow one or the other competing synchronized region to annihilate the other and to itself expand. Similar sets of interactions continue to take place among the remaining synchronized regions until the entire con guration has one coherent phase. The following simple scenario illustrates the role of the unstable particle in A 's synchronization strategy. Let A start from a simple IC consisting of a pair of SS domain boundaries which are at a small distance from one another. An example of such an initial condition is shown in Figure 8(a). Each SS domain boundary forms the particle , which exists for only one time step and then decays into a - pair, with and traveling at equal and opposite velocities. In this example, two such pairs are formed, and the rst interaction is between the two interior particles: the from the left pair and the from the right pair. As a result of this interaction, the two interior particles are replaced by and , which have velocities of -3 and 3, respectively. Due to their greater speed, the new interior particles can intercept the remaining and particles. Since the pair of interactions + ! ; and + ! ; are annihilative, and because the resulting domain is S , the con guration is now globally synchronized (Figure 8(b)). One necessary re nement to this explanation comes from noticing that the - interaction depends on the inter-particle distance d, where 0 d 2r. If d mod 4 6= 1, then the interaction + ! + takes place. But if d mod 4 = 1, then the reaction + ! occurs. The particle is essentially a defect in the D domain, which regenerates and when it reacts with any other particle. α 0
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Figure 8: (a) Space-time diagram of A starting with two S domains which are out of phase with each other. A 's strategy can be described as a competition between the two S domains according to their relative size, with the D domain acting as the mediator. (b) The same space-time diagram in (a) after ltering with the transducer TA which maps all domains to white and all defects to black. 13
The fundamental innovation of A is the formation of the D domain, which allows two globally out-of-phase S domains to compete according to their relative size and so allows for the resolution of global phase frustration. The particle interactions in the ltered space-time diagram in Figure 6(b) (starting from a random IC) are somewhat more complicated than in this simple example, but it is still possible to identify essentially the same set of particle interactions ( + ! + , + ! , and + ! ;) that e ect the global synchronization in the CA.
7.2 Analysis of B
Figure 9(a) gives a space-time diagram for another GA-discovered CA with 100% performance, here called B . Analysis of Figure 9(a) shows that two di erent regular domains dominate the space-time behavior of B . One of the domains, denoted here as 0B , is the familiar synchronous domain S described in the previous section. The second domain, denoted here as 1B , has a temporal periodicity of two and a spatial periodicity of four. 1A consists of two languages, (0001)+ and (0011)+, such that ((0001)+) = (0011)+ and ((0011)+) = (0001)+. The con gurations in 1B however, have a spatial and temporal periodicity of four. Both 0B and 1B are regular domains since each satis es the temporal invariance and spatial homogeneity conditions. As in the previous section, 0B and 1B will also be represented as S and D respectively. The transducer TB , which accepts words in domains 0B and 1B , has been applied to the space-time diagram in Figure 9(a), the resulting plot is shown in Figure 9(b). The domains, particles and particle interactions observed in the ltered space-time plots of B 0
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Figure 9: (a) Space-time diagram of B starting with a random initial condition. (b) The same space-time diagram after ltering with the transducer TB which maps all domains to white and all defects to black. Greek letters label particles described in the text. 14
are presented in Table 2. S
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D alternates between (0001)+ and (0011)+, i.e.
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Domain Temporal Spatial Velocity Boundary Periodicity Displacement 0 SS DS 2 2 1 SD 4 0 0 DS 2 -6 -3 SD 2 6 3 8 0 0 DD
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! +
+ !; + !; + ! , if d mod 4 = 3 + ! + ! + ! + , if d mod 4 6= 3 + ! +
Table 2: The domains, particles, and particle interactions that dominate the spatio-temporal behavior of B . The reactions between particles and depend on the relative distance d between them, where 0 d 2r. The most striking feature to be observed in Table 2 is its similarity to the domain, particle, and particle interaction table for rule A . Indeed the number of domains, the number of particles, and the number and types of particle interactions are identical in the two tables. Most important of all, as in A, the out-of-phase defect SS is unstable in rule B , and the defect spontaneously gives rise to the D domain along with the particles and . This allows two adjacent out-of-phase synchronous domains to compete for space in the lattice with D acting as the mediating domain. In short, the strategy used by B to perform the synchronization task is very similar to the strategy used by A. Indeed, our analysis shows that in spite the di erences in spatio-temporal behavior, the four highperformance CAs depicted in Figure 5 make use of the same particle logic to achieve global synchronization.
7.3 The Evolution to Synchronization
Once the computational structures and their emergent logic in a high- tness CA has been identi ed one may also ask: How are such computational structures discovered by the GA in the rst place? Computational mechanics analysis of the behavior of the ancestors of the high-performance CAs gives a detailed picture of the interplay between the evolutionary dynamics and the computational structures that are embedded in the CA's behavior. We 15
show that although the existence of some computational structures in the behavior of a high- tness CA can be directly attributed to the ampli cation process resulting from the selection pressures in evolution, the discovery of some other computational structure is often due to pure chance. Figure 10(a) plots the best tness in the population versus generation for the rst 30 generations of the run in which A was evolved. The gure shows that, over time, the best tness in the population is marked by periods of sharp increases. Qualitatively, the overall increase in tness can be divided into ve \epochs". The rst epoch starts at generation 0 and each of the following epochs corresponds to the discovery of a new, signi cantly improved strategy for performing the synchronization task. Similar epochs were seen in most of the runs resulting in CAs with 100% performance. In Figure 10(a), the beginning of each epoch is labeled with the best CA in the population at that generation. Epoch 0: Growth of Disordered Regions. To perform the synchronization task, a CA must have (07) = 1 and (17) = 0. These mappings insure that local regions will have the desired oscillation. Since the existence of the S domain is guaranteed by xing just two bits in the chromosome, approximately 1=4 of the CAs in a random initial population have S . However, S 's stability under small perturbations depends on other output bits. For example, 0 is a generation 0 CA with these two bits set correctly, but under 0 a small perturbation in S leads to the creation of a disordered region. This is shown in Figure 10(b), where the IC contains a single 1 at the center site. In the gure, the disordered region grows until it occupies the whole lattice. Epoch 1: Stabilization of the Synchronous Domain. Unlike 0, 1 eliminates disordered regions by expanding (and thereby stabilizing) local synchronous domains. The stability of the synchronous regions comes about because 1 maps all the eight neighborhoods with six or more 0s to 1, and seven out of eight neighborhoods with six or more 1s to 0. Under our lexicographic ordering, most of these bits are clustered at the left and right ends of the chromosome. This means it is easy for the crossover operator to bring them together from two separate CAs to create CAs like 1. Figure 10(c) shows that under 1, the synchronous regions fail to occupy the entire lattice. A signi cant number of constant-velocity particles (here, boundaries between adjacent S domains) persist inde nitely and prevent global synchronization from being reached. A more detailed analysis of 1's space-time behavior shows that it supports one type of stable SS particle, , and three di erent types of stable SS particles: , , and , each with di erent velocities. Examples of these particles are labeled in Figure 10(c), and their properties and interactions are summarized in Table 3. A survey of their interactions indicates that the particle dominates: it persists after collision with any of the SS particles. None of the interactions are annihilative: particles are produced in all interactions. As a result, once a set of particles comes into existence in the space-time diagram, one can guarantee that at least one particle persists in the nal con guration. 16
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Figure 10: Evolutionary history of A: (a) Fitness versus generation for the most t CA in each population. The arrows indicate the generations in which the GA discovered each new signi cantly improved strategy. (b){(f) Space-time diagrams illustrating the behavior of the best at each of the ve generations marked in (a). The ICs are disordered except for (b), which consists of a single 1 in the center of a eld of 0s. The same Greek letters in di erent gures represent di erent types of particles.
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Cellular Automata Particles and Interactions Chromosome Generation Label Domain Temporal Velocity 149 599 999 (P104 ; P104 ; P104 ) Boundary Periodicity 1 = 1 2 -1/2 SS F8A19CE6 SS 4 -1/4 B65848EA (0.00,
SS 8 -1/8 D26CB24A 0.00, SS 2 0 EB51C4A0 0.00) + ! , + ! , + ! 2 = 5 2 -1/2 SS F8A1AE2F 6 0 SS CF6BC1E2 (0.33, D26CB24C 0.07, + !; 3C266E20 0.03) 3 = 13 4 -3/4 SS F8A1AE2F 6 0 SS CE6BC1E2 (0.57,
12 1/4 SS C26CB24E 0.33, 2 1/2 SS 3C226CA0 0.27) + ! ;, + ! ;, + ! ; A = 100 0 SS FEB1C6EA DS 2 1 B8E0C4DA (1.00,
SD 2 -1 6484A5AA 1.00, DS 4 -3 F410C8A0 1.00) SD 2 3 2 -1 DD Decay: ! + Annihilative: + ! ;, + ! ; Reactive: + ! (d mod 4 = 1), + ! , + ! Reversible: + ! + (d mod 4 6= 1), + ! + Table 3: A and its ancestors: Particles and their dynamics for the best CAs in early generations of the run that found A. The table shows only the common particles and common two-particle interactions that play a signi cant role in determining tness. ; indicates a domain with no particles. Each CA is given as a hexadecimal string which, when translated to a binary string, gives the output bits of in lexicographic order ( = 07 on the left).
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Epoch 2: Suppression of In-Phase Defects. The rise in tness can be attributed to
2 's ability to suppress in-phase (SS ) defects. In addition, and annihilate each other; and 2 is able to reach synchronous con gurations due these annihilations. However, since the velocity di erence between and is only 1=2, the two particles might fail to annihilate each other before the maximum of M time steps have elapsed. Epoch 3: Re nement of Particle Velocities. A much improved CA, 3, is found in generation 13. Its typical behavior is illustrated in Figure 10(e). 3 di ers from 2 in
two respects, both of which result in improved performance. First, as noted in Table 3, the velocity di erence between and , the two most commonly occurring particles produced by 3 , is larger (1 as compared to 1=2 in 2 ), so their annihilative interaction typically occurs more quickly. This means 3 has a better chance of reaching a synchronized state within M time steps. Second, the probabilities of occurrence of and are almost equal, meaning that there is a greater likelihood they will pairwise annihilate, leaving only a single synchronized domain. In spite of these improvements, it is easy to determine that 1e 's strategy will ultimately fail to synchronize on a signi cant fraction of ICs. As long as SS particles exist in the space-time diagram, there is a non-zero probability that a pair SS defect sites would be occupied by a pair of identical particles moving in parallel. In the absence of other particles in the lattice such a particle pair could exist inde nitely, preventing global synchrony. Thus a completely new strategy is required to overcome persistent parallel-traveling particles. Epoch 4: The Final Innovation. In the 20th generation a nal dramatic increase in tness is observed when 4 is discovered. All SS phase-defects are unstable and result in the D domain which allows two globally out-of-phase S domains to compete according to their relative size and so allows for the resolution of global phase frustration.
8 Conclusions Recent progress in understanding the intrinsic information processing in spatially extended systems such as CAs has provided a new set of tools for the analysis of temporally and evolutionarily emergent behavior [2, 9, 8]. In this paper we use these tools to analyze in some detail the individual CA behavioral mechanisms responsible for increased tness. The identi cation of the computational structures embedded in a CA's behavior makes it possible to delineate the particle-level emergent logic or \strategy" which allows the successful CAs to perform the computational tasks. Since a high- tness CA for a given computational task is performing some \useful" computation, i.e., accomplishing a desired mapping from input to output, we can assign speci c functionality to each of the discovered computational structures according to the role they play in performing the computational task. It is in this light that the work presented here on CAs goes beyond earlier work by Crutch eld and Hanson. The results from the analysis based on the computational mechanics framework also facilitates a comparison between di erent CAs at the functional level. Di erent high-performance 19
CAs use di erent look-up tables and exhibit very dissimilar spatio-temporal behavior. However, careful analysis of space-time behavior shows that, at the particle-level, the di erent GA-discovered strategies used by these rules to perform a given computational task fall under an equivalence class. The results presented here also demonstrate how evolution can engender emergent computation in a spatially-extended system. For the synchronization task and other computational tasks reported elsewhere [4, 3] evolution was able to take advantage of the underlying medium's potential to form embedded particles and in the end was able to design highperformance CAs. These discoveries are encouraging, both for using GAs to model the evolution of emergent computation in nature and for the automatic engineering of emergent computation in decentralized multicomponent systems. Acknowledgement This work was done in collaboration with J. P. Crutch eld and M. Mitchell at the Santa Fe Institute. I would like to express my deepest gratitude for their insights, guidance, and encouragement throughout the course of this work.
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