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Genetic Approaches to Search for Computing Patterns in Cellular Automata

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Abstract

The emergence of computation in complex systems "with simple components is a hot topic in the science of complexity. A uniform framework to study emergent computation in complex systems are cellular automata. They are discrete systems in which an array of cells evolves from generation to generation on the basis of local transition rules. The well-established problem of emergent computation and universality in cellular automata has been tackled by a number of people in the last thirty years and remains an area "where amazing phenomena at the edge of theoretical computer science and nonlinear science can be discovered. Future work could also evaluate all discovered cellular automata and calculate for each cellular automaton some rule-based parameters, e.g., Langtons lamda. All cellular automata simulating an AND gate may have similar values for these parameters that could lead to answer the question Where are the edges of computational universality? and may therefore lead to a better understanding of the emergence of computation in complex systems with local interactions.

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... Surprising computational tasks could result from interactions of independent agents in complex systems as emergence of computation is a hot topic in the science of complexity [33]. A promising environment to study emergent computation is cellular automata [16] which are the simplest mathematical representation of complex systems [7] and an important modelling paradigm in the natural sciences and an extremely useful approach in the study of complex systems [20]. They are uniform frameworks in which the simple agents are cells evolving through time on the basis of a local function, called the transition rules [31]. ...
... To our knowledge only few searches for automata simulating glider guns are known. Sapin et al. have searched for glider guns emitting the gliders they have discovered [27], [21], [20]. ...
... An AND gate was manually built in [23] with a discovered gun and the emitting glider simulated by an automaton demonstrated universal in [25], [26]. In [20], simulation of AND gates were searched for a specific period-2 glider and the glider guns found in [27] with an even period that emit this glider. ...
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We aim to construct an automatic system for the discovery of Turing-universal cellular automata. A new step toward this automatic system, presented in this paper, is an automatic method of detection of cellular automata simulating AND gates. The heuristics of the search for these cellular automata is to search for a glider, then to search for a gun emitting this glider and then to search for a simulation of an AND gate. Results show how a large number of simulations of logic gates can be discovered.
... Surprising computational tasks could result from interactions of independent agents in complex systems as emergence of computation is a hot topic in the science of complexity [26]. A promising environment to study emergent computation is cellular automata [11] which are the simplest mathematical representation of complex systems [5] and an important modelling paradigm in natural sciences and an extremely useful approach in the study of complex systems [15]. They are uniform frameworks in which the simple agents are cells evolving through time on the basis of a local function, called the transition rules [24]. ...
... This search for AND gates is inspired by a previous search described in [15]. A glider gun G is ramdomly chosen among the 20383 guns that were found in [16] with a period p that is even. ...
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We aim to construct an automatic system for the discovery of Turing-universal cellular automata. A new step toward this automatic system, presented in this paper, is an automatic method of detection of cellular automata simulating NAND gates. The heuristics of the search for these cellular automata is to pick a existing generator of mobile self-localized patterns of non-resting states and to search for a simulation of AND gate, then to search for an element able to redirect a stream of such patterns. Results show how simulations of NAND gates can be discovered.
... It holds that D[4] = I[2] and D[5] = I[3]. We know that A(D[4]) = D[5], since D is the space-time diagram of a CA A, and so A(I [2]) = I[3]. ...
... It holds that D[4] = I[2] and D[5] = I[3]. We know that A(D[4]) = D[5], since D is the space-time diagram of a CA A, and so A(I [2]) = I[3]. Additionally, it holds thatA 3 (D[1]) = D[4] = I [2]. ...
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... The same CA definition as above was used by Sapin et al. in their approach to automatically discover lattice structures that simulate the operation of an AND gate [109,111]. Genetic algorithms and Tabu search were used to find gliders, guns, and spatial configurations of these two structures. The local interactions among the discovered structures represent the operation of an AND gate. ...
... The contribution of this approach is in its progressive refinement of partial results that evolves the overall lattice configuration towards a desired solution. The broader impact of this research might lead collision-based computing towards automatic design of a universal cellular automaton [20,110,111]. In this dissertation I propose several novel tasks that challenge the ability of 2DCA to solve a problem by emergent system behavior. ...
... A fourth category consists of studies involved in systematic searches by using different types of heuristic approaches to find complex cellular automata. For instance, in [12], behavioral metrics were employed in a genetic algorithm to select cellular automata similar to the Game of Life, while the spontaneous emergence of glider guns in cellular automata with two states in two dimensions was developed in [40,41]. These works applied an evolutionary search for new glider guns, and an automatic process was provided to classify glider guns that could implement collision-based universal cellular automata. ...
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... Currently, much research still focuses on analysing CAs with known or designed evolution rules and using them in particular applications such as urban modelling and image processing. However, in many applications, formulating suitable rules is not easy[6],[7],[8]: often, only the desired initial and final patterns, or the evolution processes, are known. To be able to apply a CA, underlying rules for the CA must be identified. ...
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Cellular automata (CA) with given evolution rules have been widely investigated, but the inverse problem of extracting CA rules from observed data is less studied. Current CA rule extraction approaches are both time consuming and inefficient when selecting neighborhoods. We give a novel approach to identifying CA rules from observed data and selecting CA neighborhoods based on the identified CA model. Our identification algorithm uses a model linear in its parameters and gives a unified framework for representing the identification problem for both deterministic and probabilistic CA. Parameters are estimated based on a minimum variance criterion. An incremental procedure is applied during CA identification to select an initial coarse neighborhood. Redundant cells in the neighborhood are then removed based on parameter estimates, and the neighborhood size is determined using the Bayesian information criterion. Experimental results show the effectiveness of our algorithm and that it outperforms other leading CA identification algorithms.
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... Promising environments in which to study complex systems are the universes of cellular automata [12,18] which are the simplest mathematical representations of complex systems [7], and important modelling paradigms in natural sciences. They are also uniform frameworks in which cells evolve through time on the basis of a local functions, called the transition rules [26]. ...
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We aim to construct an automatic system for the discovery of Turing-universal cellular automata. In this chapter, steps towards this automatic system are presented. These steps will lead to the discovery of thousands of computing patterns in cellular automata.
... Within the nature, intradisciplinary research of merging the different biological systems is coming up on the scene to further improve the solutions presented earlier by single biological systems. Ideas from cellular automata are merging to genetic algorithms to present new and improved solution to variety of problems [90]. is synergistic mating of biological systems is further needed to �gure out the consequences of individual arti�cial system concocted with other arti�cial system. ...
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... For instance, behavioral metrics are employed in a genetic algorithm to select cellular automata similar to the Game of Life [30]. The spontaneous emergence of glider guns in cellular automata with two states in two dimensions is developed in [47] and [46]. Those works apply an evolutionary search for new glider guns, and an automatic process is provided to classify of glider guns that can implement collision-based universal cellular automata. ...
Preprint
The emergence of complex behaviors in cellular automata is an area that has been widely developed in recent years with the intention to generate and analyze automata that produce space-moving patterns or gliders that interact in a periodic background. Frequently, this type of automata has been found through either an exhaustive search or a meticulous construction of the evolution rule. In this study, the specification of cellular automata with complex behaviors was obtained by utilizing randomly generated specimens. In particular, it proposed that a cellular automaton of $n$ states should be specified at random and then extended to another automaton with a higher number of states so that the original automaton operates as a periodic background where the additional states serve to define the gliders. Moreover, this study presented an explanation of this method. Furthermore, the random way of defining complex cellular automata was studied by using mean-field approximations for various states and local entropy measures. This specification was refined with a genetic algorithm to obtain specimens with a higher degree of complexity. With this methodology, it was possible to generate complex automata with hundreds of states, demonstrating that randomly defined local interactions with multiple states can construct complexity.
... Since t 1 = 1 we simply apply the rule to the first row of I, i.e.Ī A (ti) [2, 2] = f 150 (I[1, 1], I[1, 2], I[1, 3]) = 1. Since t 2 = 2, to findĪ A (ti)[3,3], we first need to compute one additional configuration by evaluating the rule on configuration I A (ti)[2]. It is easy to check that A(Ī A (ti) [2]) = (0, 0, 0), and thus I A (ti)[3,3] = f 150 (0, 0, 0) = 0. ...
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... Uno de los aspectos más interesantes en los ACE es encontrar una representación algebraica o matricial general para las reglas de evolución. Algunos métodos han sido propuestos para realizar o estudiar la evolución de los ACE, como diagramas [27]- [29], álgebra matricial [30]- [32], ecuaciones diferenciales [33], métodos matriciales [34] y algoritmos genéticos [35] [36]. Sin embargo, los diagramas sólo permiten determinar características de la evolución del ACE, pero no realizan dicha evolución. ...
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... Gliders in cellular automata are discrete phenomenological analogies of solitons [Park et al., 1986;Fokas et al., 1990;Takahashi & Satsuma, 1990;Krész, 2008;Martínez et al., 2011;Zhang & Adamatzky, 2009;Martinez et al., 2013]. Therefore, most designs of glider-based computing devices implemented in cellular automata [Squier & Steiglitz, 1994;Adamatzky et al., 2006;Martínez et al., 2006;Sapin et al., 2009] could be also considered in a context of soliton-based computing. ...
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Coherent space-time structures (gliders) which emerge in class IV cellular automata (CA) can be seen as "vehicles" that move information, as they are common in CA which support universal computation. A technique for evolving gliders from heterogeneous 2D CA is described. Rather than use a genetic algorithm on a population of rules, and image-filtering to detect structures for measuring fitness, a particle swarm is employed which interacts intimately with the CA and performs genetic operators locally on the heterogeneous rules, as the dynamics emerge. The swarm selects for coherent motion. These particles do not fly on a search mission – instead, they "ride" on the backs of clusters of emerging structures, due to attractive forces. In exchange for a "good ride", they reward local dynamics with more coherent motion by performing genetic operators of selection and reproduction. This technique not only demonstrates an efficient way to evolve a large variety of gliders: it also simulates emergent complexity through co-adaptation.
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A largely phenomenological study of two-dimensional cellular automata is reported. Qualitative classes of behavior similar to those in one-dimensional cellular automata are found. Growth from simple seeds in two-dimensiona! cellular automata can produce patterns with complicated boundaries, characterized by a variety of growth dimensions. Evolution from disordered states can give domains with boundaries that execute effectively continuous motions. Some global properties of cellular automata can be described by entropies and Lyapunov exponents. Others are undecidable.
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We present results from experiments in which a genetic algorithm (GA) was used to evolve cellular automata (CAs) to perform a particular computational task - one-dimensional density classification. We look in detail at the evolutionary mechanisms producing the GA's behavior on this task and the impediments faced by the GA. In particular, we identify four “epochs of innovation” in which new CA strategies for solving the problem are discovered by the GA, describe how these strategies are implemented in CA rule tables, and identify the GA mechanisms underlying their discovery. The epochs are characterized by a breaking of the task's symmetries on the part of the GA. The symmetry breaking results in a short-term fitness gain but ultimately prevents the discovery of the most highly fit strategies. We discuss the extent to which symmetry breaking and other impediments are general phenomena in any GA search.
Article
In order for computation to emerge spontaneously and become an important factor in the dynamics of a system, the material substrate must support the primitive functions required for computation: the transmission, storage, and modification of information. Under what conditions might we expect physical systems to support such computational primitives?This paper presents research on cellular automata which suggests that the optimal conditions for the support of information transmission, storage, and modification, are achieved in the vicinity of a phase transition. We observe surprising similarities between the behaviors of computations and systems near phase transitions, finding analogs of computational complexity classes and the halting problem within the phenomenology of phase transitions.We conclude that there is a fundamental connection between computation and phase transitions, especially second-order or “critical” transitions, and discuss some of the implications for our understanding of nature if such a connection is borne out.
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Cellular automata are discrete dynamical systems with simple construction but complex self-organizing behaviour. Evidence is presented that all one-dimensional cellular automata fall into four distinct universality classes. Characterizations of the structures generated in these classes are discussed. Three classes exhibit behaviour analogous to limit points, limit cycles and chaotic attractors. The fourth class is probably capable of universal computation, so that properties of its infinite time behaviour are undecidable.
Article
The main benchmark problems in the cellular automata literature related to discovering rules able to exhibit given target behaviours have been the density classification task and the parity problem, in which the rule must decide about global properties of the number of bits in the initial configuration of binary cellular automata. Here a suitable combination of evolutionary computation techniques is presented and discussed that led to unparalleled good results in the standard formulation of the density classification, with the discovery of a few thousand rules with higher efficacy than the best currently known rule. Furthermore, the same basic techniques were also very successfully applied to the parity problem and to density classification in two and three dimensions, so that, for all these cases, the quality of the results achieved also seem to constitute, by far, the best ones currently available for all these computational tasks.
Article
One of the major challenges in the field of integrated circuit design is coping with difficult combinatorial optimization problems. Simply finding the minimum length of wire needed to connect a block of transistors is NP-hard. When factoring in a handful of other simultaneous optimization dimensions such as connections to other blocks of components and area minimization, it is easy to see the difficulty facing circuit designers. Since many of the problems encountered do not have polynomial-time solutions, very large scale integration (VLSI) algorithm designers experiment with various optimization techniques such as integer linear programming and simulated annealing. Optimization methods for VLSI computer-aided design (CAD) incorporating evolutionary search began appearing in research articles in the late 1980’s. As the first body of work devoted exclusively to evolutionary algorithms (EA’s) in VLSI CAD, this monograph attempts to fill a noticeable void in the literature. The book is divided into two parts. Part I “Basic Principles” provides an overview of the book and discusses the underlying principles of EA’s and algorithm performance issues. The treatment of EA’s here is cursory at best and concentrates mainly on the classic genetic algorithm. An overview of some aspects of VLSI CAD is also included; however, only the latter phase of the circuit design process is covered, and analog integrated circuit design is not touched upon. The second part entitled “Practice” comprises roughly threequarters of the book and deals with tools and applications. A software tool called the Genetic Algorithm Managing Environment (GAME) is described in one chapter. It provides a flexible environment in which an algorithm designer can easily interface EA’s to VLSI CAD tools to facilitate experimentation and production runs. One of the author’s themes is that problem-specific knowledge is necessary for EA’s to be competitive with other optimization approaches. Support for this argument is provided by the numerous case studies examined in a lengthy chapter concerning applications of EA’s to logic synthesis, mapping, and testing. In logic synthesis, one wishes to implement a Boolean function in hardware while satisfying certain constraints (e.g., power, delay area). A category of logic realization called Fixed Polarity Reed Muller expressions is described. Such expressions are difficult to minimize and an approach is described that uses a hybrid EA (one that incorporates problem-specific heuristics). For large Boolean functions, it is shown that the EA presented can achieve smaller expressions as compared to standard tools, although the EA required more computer time. The section on mapping includes EA applications in partitioning, floorplanning, and placement and routing problems. Partitioning consists of mapping blocks of circuit components to two-dimensional
Article
This paper presents the synthesis and analysis of a special class of non-uniform cellular automata (CAs) based associative memory, termed as generalized multiple attractor CAs (GMACAs). A reverse engineering technique is presented for synthesis of the GMACAs. The desired CAs are evolved through an efficient formulation of genetic algorithm coupled with the reverse engineering technique. This has resulted in significant reduction of the search space of the desired GMACAs. Characterization of the basins of attraction of the proposed model establishes the sparse network of GMACAs as a powerful pattern recognizer for memorizing unbiased patterns. Theoretical analysis also provides an estimate of the noise accommodating capability of the proposed GMACA based associative memory. An in-depth analysis of the GMACA rule space establishes the fact that more heterogeneous CA rules are capable of executing complex computation like pattern recognition.
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Theory of Self-Reproducing Automata
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 1971. Vita. Bibliography: leaves 100-101.
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Previous computational models of self-replication using cellular automata (CA) have been manually designed, a difficult and time-consuming process. We show here how genetic algorithms can be applied to automatically discover rules governing self-replicating structures. The main difficulty in this problem lies in the choice of the fitness evaluation technique. The solution we present is based on a multiobjective fitness function consisting of three independent measures: growth in number of components, relative positioning of components, and the multiplicity of replicants. We introduce a new paradigm for CA models with weak rotational symmetry, called orientation-insensitive input, and hypothesize that it facilitates discovery of self-replicating structures by reducing search-space sizes. Experimental yields of self-replicating structures discovered using our technique are shown to be statistically significant. The discovered self-replicating structures compare favorably in terms of simplicity with those generated manually in the past, but differ in unexpected ways. These results suggest that further exploration in the space of possible self-replicating structures will yield additional new structures. Furthermore, this research sheds light on the process of creating self-replicating structures, opening the door to future studies on the discovery of novel self-replicating molecules and self-replicating assemblers in nanotechnology
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Finding good random number generators (RNGs) is a hard problem that is of crucial import in several fields, ranging from large-scale statistical physics simulations to hardware self-test. In this paper, we employ the cellular programming evolutionary algorithm to automatically generate two-dimensional cellular automata (CA) RNGs. Applying an extensive suite of randomness tests to the evolved CAs, we demonstrate that they rapidly produce high-quality random-number sequences. Moreover, based on observations of the evolved CAs, we are able to handcraft even better RNGs, which not only outperform previously demonstrated high-quality RNGs, but can be potentially tailored to satisfy given hardware constraints
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The von Neumann architecture-which is based upon the principle of one he von Neumann architecture-which is based upon the principle of one complex processor that sequentially performs a single complex task at a given moment-has dominated computing technology for the past 50 years. Recently, however, researchers have begun exploring alternative computational systems based on entirely different principles. Although emerging from disparate domains, the work behind these systems shares a common computational philosophy, which the author calls cellular computing. This philosophy promises to provide new means for doing computation more efficiently-in terms of speed, cost, power dissipation, information storage, and solution quality. Simultaneously, cellular computing offers the potential of addressing much larger problem instances than previously possible, at least for some application domains. Cellular computing has attracted increasing research interest. Work in this field has produced results that hold prospects for a bright future. Yet questions must be answered before cellular computing can become a mainstream paradigm. What classes of computational tasks are most suited to it? How do we match the specific properties and behaviors of a given model to a suitable class of problems?
Discrete dinamics lab (ddlab)," {Online}
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On spiral gliderguns in hexagonal cellular automata: Activator inhibitor paradigm
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Schwefel Parallel Problem Solving from
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Adamatzky Collision-Based Computing
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