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... In this paper we describe research on using genetic algorithms (GAs) to evolve CAs to perform computations. GAs are search and optimization methods based on ideas from natural genetics and evolution =-=[19, 31, 42, 56]-=-. A GA works on populations of “chromosomes” that represent candidate solutions to a given problem, applying “genetic” operators such as fitness-based reproduction, crossover, and mutation to members ...

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...es of application areas include engineering design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., =-=[40, 46]-=-), neural-network design (e.g., [8, 12, 38, 58, 62, 33]), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GA...

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... In this paper we describe research on using genetic algorithms (GAs) to evolve CAs to perform computations. GAs are search and optimization methods based on ideas from natural genetics and evolution =-=[19, 31, 42, 56]-=-. A GA works on populations of “chromosomes” that represent candidate solutions to a given problem, applying “genetic” operators such as fitness-based reproduction, crossover, and mutation to members ...

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... In this paper we describe research on using genetic algorithms (GAs) to evolve CAs to perform computations. GAs are search and optimization methods based on ideas from natural genetics and evolution =-=[19, 31, 42, 56]-=-. A GA works on populations of “chromosomes” that represent candidate solutions to a given problem, applying “genetic” operators such as fitness-based reproduction, crossover, and mutation to members ...

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... In this paper we describe research on using genetic algorithms (GAs) to evolve CAs to perform computations. GAs are search and optimization methods based on ideas from natural genetics and evolution =-=[19, 31, 42, 56]-=-. A GA works on populations of “chromosomes” that represent candidate solutions to a given problem, applying “genetic” operators such as fitness-based reproduction, crossover, and mutation to members ...

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...nlikely to be adjacent in a lexicographic ordering, and thus cannot be moved together from a parent to an offspring via simple crossover. More disruptive crossover operators such as uniform crossover =-=[84]-=- run the risk of destroying the necessary structures. The problem of designing a representation that will work well with genetic operators is a general one for GAs. One solution that has been explored...

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...es of application areas include engineering design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., =-=[40, 46]-=-), neural-network design (e.g., [8, 12, 38, 58, 62, 33]), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GA...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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... of designing a representation that will work well with genetic operators is a general one for GAs. One solution that has been explored is adapting the representation to suit the operators (e.g., see =-=[32]-=-). Phenomena in natural genetics such as inversion and jumping genes may be a form of representation adaptation. These have inspired some work 38sin GAs along these lines (e.g., see [31, 42]). Loss of...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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... design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., [8, 12, 38, 58, 62, 33]), robot control (e.g., =-=[22, 39, 18]-=-), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolutionary processes in natural systems. Ex...

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...t up and parcel out. Recent work on CAs has yielded much new insight into the mechanisms by which complex behavior can arise in nonlinear spatially-extended systems with local interactions (e.g., see =-=[23, 34, 88, 91]-=-). However, little is known about how to harness this complex behavior to perform useful computation, since in general it is hard to predict, much less design, the behavior of such systems. The study ...

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...ing design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., =-=[8, 12, 38, 58, 62, 33]-=-), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolution...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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...ing design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., =-=[8, 12, 38, 58, 62, 33]-=-), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolution...

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...t up and parcel out. Recent work on CAs has yielded much new insight into the mechanisms by which complex behavior can arise in nonlinear spatially-extended systems with local interactions (e.g., see =-=[23, 34, 88, 91]-=-). However, little is known about how to harness this complex behavior to perform useful computation, since in general it is hard to predict, much less design, the behavior of such systems. The study ...

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...e of a binary CA is defined as the fraction of 1s in the output bits of its rule φ. For example, the λ value of the majority rule is 1/2. The λ parameter was originally used in studies of CA behavior =-=[47]-=-, but, as we will show, it turns out to be useful in understanding the GA’s behavior. (There is a simple interpretation of how λ is related to a CA’s behavior. λ gives the density of 1s in the first i...

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...as many desirable features for a large class of parallel computations. In scientific modeling applications, CAs have been used to simulate, for example, magnetic spin systems [15, 90], fluid dynamics =-=[21, 27]-=-, chemical oscillations [53, 64], crystal growth [52, 65], galaxy formation [30], stellar accretion disks [77], dynamics in cytoskeletal lattices [82], and the formation of biological patterns (e.g., ...

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...d to balance the level of population diversity with the need to quickly propagate newly discovered innovations to the rest of the population is discussed in several places in GA literature (e.g., see =-=[60]-=-). As was noted above, most of these impediments are also forces that help the GA in the initial stages of its search. None are specific to the ρc = 1/2 task or even to the problem of evolving CAs. Ra...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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...mbly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolutionary processes in natural systems. Examples include models of economic systems (e.g., =-=[3, 44]-=-), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., [5, 6, 13, 14, 4...

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...As with k = 2 and r = 3, and with spatially periodic boundary conditions: si t = s i+N t . We most often set N to 149, but also look at the behavior of CA on larger N (up to 999). 3. Previous Work In =-=[61]-=- we reported results of evolving one-dimensional CAs to perform a particular density classification task: the “ρc = 1/2” task. This work was a re-examination of an experiment performed by Packard [66]...

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...ing design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., =-=[8, 12, 38, 58, 62, 33]-=-), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolution...

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...[61] we reported results of evolving one-dimensional CAs to perform a particular density classification task: the “ρc = 1/2” task. This work was a re-examination of an experiment performed by Packard =-=[66]-=-, meant to test the hypothesis that a GA evolving CA rules to perform a difficult computational task will tend to select rules close to conjectured phase transitions in rule space between ordered and ...

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...7, 79, 89]). GAs have also been used as scientific models of evolutionary processes in natural systems. Examples include models of economic systems (e.g., [3, 44]), models of the immune system (e.g., =-=[26]-=-), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., [5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]), m...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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...tems whose architecture has many desirable features for a large class of parallel computations. In scientific modeling applications, CAs have been used to simulate, for example, magnetic spin systems =-=[15, 90]-=-, fluid dynamics [21, 27], chemical oscillations [53, 64], crystal growth [52, 65], galaxy formation [30], stellar accretion disks [77], dynamics in cytoskeletal lattices [82], and the formation of bi...

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...ing design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., =-=[8, 12, 38, 58, 62, 33]-=-), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolution...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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...ion, symbiosis, and resource flow in ecologies (e.g., [5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., =-=[9, 24, 54, 78]-=-), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, 57, 68, 86, 87]). The goals of our research are: (1) to better understand the ways in which CAs can perform...

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...k. But how is the strategy related to the CA’s dynamical behavior? The overall strategy can be decomposed into the CA’s intrinsic computational elements: domains, particles, and particle interactions =-=[37]-=-. There are three time-invariant spatially-homogeneous domains: (i) all white, W = 0 ∗ , (ii) all black, B = 1 ∗ , and (iii) checkerboard, # = (10) ∗ � (01) ∗ . The results in [20] establish that W an...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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...lity in a range of applications. Examples of application areas include engineering design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., =-=[4, 36]-=-), automatic programming (e.g., [40, 46]), neural-network design (e.g., [8, 12, 38, 58, 62, 33]), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protei...

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..., B = 1 ∗ , and (iii) checkerboard, # = (10) ∗ � (01) ∗ . The results in [20] establish that W and B are regular attractors, as defined in [37]. When the domains are filtered out using the methods of =-=[16]-=-, one finds that the domain boundaries form six particles, the first five of which are time-invariant. These are listed in Table 1. The types of interactions between particles are also evident when th...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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...ars in machine learning and other disciplines because of their utility in a range of applications. Examples of application areas include engineering design (e.g., aircraft design [10], circuit design =-=[80]-=-, and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., [8, 12, 38, 58, 62, 33]), robot control (e.g., [22, 39, 18]...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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...lity in a range of applications. Examples of application areas include engineering design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., =-=[4, 36]-=-), automatic programming (e.g., [40, 46]), neural-network design (e.g., [8, 12, 38, 58, 62, 33]), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protei...

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...s (e.g., [3, 44]), models of the immune system (e.g., [26]), models of ecological phenomena such as biological arms races, host-parasite co-evolution, symbiosis, and resource flow in ecologies (e.g., =-=[5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]-=-), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, ...

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...ion, symbiosis, and resource flow in ecologies (e.g., [5, 6, 13, 14, 40, 42, 43, 45, 50, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., =-=[9, 24, 54, 78]-=-), and models of the interaction between evolution and learning (e.g., [1, 2, 7, 25, 41, 63, 57, 68, 86, 87]). The goals of our research are: (1) to better understand the ways in which CAs can perform...

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...m the collective effect of large numbers of locally interacting simple components. In engineering applications, CAs have been used to perform, among other things, parallel formal-language recognition =-=[70, 81]-=- and a range of image-processing tasks [55, 72, 73, 76, 83, 93]. There are many other potential engineering applications of CAs, such as forecasting, spatio-temporal noise-reduction, the discovery of ...

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...ing design (e.g., aircraft design [10], circuit design [80], and engine-turbine design [71]), operations research (e.g., [4, 36]), automatic programming (e.g., [40, 46]), neural-network design (e.g., =-=[8, 12, 38, 58, 62, 33]-=-), robot control (e.g., [22, 39, 18]), and molecular biology (e.g., DNA sequence assembly [69] and protein-structure prediction [17, 79, 89]). GAs have also been used as scientific models of evolution...

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...0, 51, 67, 74, 75, 85]), models of phenomena in population genetics such as the evolution of recombination (e.g., [9, 24, 54, 78]), and models of the interaction between evolution and learning (e.g., =-=[1, 2, 7, 25, 41, 63, 57, 68, 86, 87]-=-). The goals of our research are: (1) to better understand the ways in which CAs can perform computations; (2) to learn how best to use GAs to evolve computationally useful CAs; and (3) to understand ...

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...ghtly peaked about λ = 1/2 with variance 2 −n /4. Thus, the vast majority of rules is found at λ = 1/2. Using Sanov’s theorem, for example, with r = 3 there are about 10 −16 fewer rules at λc ≈ 0.146 =-=[49]-=- than at λ = 1/2. The steepness of the binomial distribution near its maximum gives an indication of the magnitude of the drift “force”. Note that the last histogram in Figure 12 gives the GA’s rough ...

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... is no direct experimental evidence linking computational capability with λ in cellular automata. For a review of these issues and of relations among computation, dynamics, and cellular automata, see =-=[59]-=-. Although the results in [66] were not replicated in our experiments, we did observe several interesting phenomena in the GA’s behavior. In [61] we qualitatively described the “epochs of innovation” ...

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...e-dimensional CA is often presented as a “space-time diagram”, a plot of st over a range of time steps. Two examples are given in Figure 1. These show the actions of the Gacs-Kurdyumov-Levin (GKL) CA =-=[29]-=- on two random ICs; one with ρ0 > 1/2 and the other with ρ0 < 1/2 . (Here and later on we use the shorthand ρ0 for the density ρ(s0) of an IC.) In both cases, the CA relaxes to a fixed pattern—in one ...