New Approach to Search for Gliders in Cellular Automata

New Approach to Search for Gliders in Cellular Automata E. Sapin Faculty of Computing, Engineering and Mathematical Sciences, University of the West of England, Bristol BS16 1QY, UK emmanuelsapin@hotmail.com Abstract. This paper deals with a new method to search for mobile self-localized patterns of non-resting states in cellular automata. The spontaneous emergence of such patterns called gliders have been searched througout all the automata of a specific space. As a result, thousands of novel gliders were discovered through the subsequent experiments and studied in terms of speed and period thanks to an automatic process of identification of gliders. 1 Introduction The emergence of computation in complex systems with simple components is a hot topic in the science of complexity [23]. A uniform framework to study this emergent computation in complex systems is cellular automata [15]. They are discrete systems in which an array of cells evolve from generation to generation on the basis of local transition rules [22]. The well-established problems of emergent computation and universality in cellular automata has been tackled by a number of people in the last thirty years [4, 12, 10, 14, 1] and remains an area where amazing phenomena at the edge of theoretical computer science and non-linear science can be discovered. The most well-known universal automaton is the Game of Life [7]. It was shown to be universal by Conway [5] who employed gliders. The latter are mobile self-localized patterns of non-resting states, used by Conway to carry information. Such gliders are the basic element for showing the universality in Conway s demonstration. The search for gliders is then a new study of science of complexity and was notably explored by Adamatzky et al. with a phenomenological search [13], Wuensche, who used his Z-parameter and entropy [25] and Eppstein [6]. Lohn et al. [11] and Ventrella [21] have searched for gliders using genetic algorithms [8], while, Sapin et al. [18, 19] used genetic algorithms to evolve transition rules of cellular automata then, searched for gliders exhibited by the evolved automata. In this work, a new method of research for gliders is presented. Instead of using any method to chose transition rules of cellular automata, the challenge here is to search for gliders in all the automata of a space. As the ones use by adamtzky et al. [2] as models of reaction-diffusion excitable chemical systems,

ternary totalistic cellular automata are chosen. Then, in order to limited the search space, cellular automata using the four nearest neibours to updates their cell states are considered. Following on from this, the space called V, of 2D ternary totalistic cellular automata using the four nearest neibours to updates their cell states at the next generation are chosen. The first section presentes notations and definitions, the second describes the search for gliders, while those which are found are described in the next section. Finally, the last section summarizes the presented results and discusses directions for future research. 2 Definitions and Notations In this section, some definitions and notations about cellular automata are presented. 2.1 Cellular Automata A local transition rule of a 2D ternary cellular automata using the four nearest neighbours to updates their cell states at the next generation is a function φ : {0, 1, 2} 5 {0, 1, 2}. Among these automata, totalistic automata are studied. An automaton is totalistic iff a cell-state is updated depending on just the numbers, not positions of different cell-states in its neighbourhoods. The functions of the local transition rule are then: φ(q 1, q 2, q 3, q 4, q 5 ) = f(σ1(x) t, σ2(x) t ) (1) where f : {0,..., 5} 2 {0, 1, 2} and σp(x) t is the number of cell x s neighbours with cell-state p {0, 1, 2} at time step t. To give a compact representation of the cell-state transition rule, the formalism in [3] is adopted. The cell-state transition rule is represented as a matrix M = (M ij ), where 0 i j 7, 0 i + j 5, and M ij {0, 1, 2}. The output state of each neighbourhood is given by the row-index i (the number of neighbours in cell-state 1) and column-index j (the number of neighbours in cell-state 2). We do not have to count the number of neighbours in cell-state 0, because it is given by 7 - (i + j). The matrix M contains then 21 elements. Ternary automata are considered, in which the state 0 is a dedicated substrate state. A state 0 is dedicated substrate state iff a cell in state 0, whose neighbourhood is filled only with states 0, does not change its state, 0 is analogue of quiescent state in cellular automaton models [24]. Then M 00 is equal 0 in all automata so the studied space V contains 3 20. A glider, called G and exhibited by some automata of V, is presented generation after generation as the one from figure 1 while the set of automata of V that exhibit this glider is determined by the process used in [20]. Finally, the elements of M that can vary without changing the evolution of G is represented by the letter X on table 1, that shows the transition rules of the automata that exhibit G. In turn, fourteen elements of M can change their value without changing the evolution of G so 3 14 of the 3 20 automata of V exhibit the glider G.

Fig.1. Glider of period 1 exhibited by some automata of V shown generation after generation along its period. 0 1 X X X X 0 2 X X X 0 X X X 0 0 X X X X Table 1. Transition rule of automata that exhibits the glider G. 3 Search for Gliders 3.1 Search Method Gliders are searched for in all automata of V. The search method is inspired by the one use in [17]. A random configuration of cells is generated in a square of 40 40 centered in a 200 200 space. These cells evolve during three hundred generations with the transition rule of the tested automaton. At each generation, each group of connected cells, figure 2, is isolated in an empty space and evolves during 30 generations. For each generation, the original pattern is searched in the test universe. Three situations can arrise: The initial pattern has reappeared at its first location, it is then periodic and the simulation do not need to continue. It has reappeared at another location, it is then a glider and the current generation is the period of the glider. It has not reappeared. Fig. 2. Groups of isolated cells.

3.2 Number of Gliders In order to determine how many different gliders were found, an automatic system that determines if a glider is new is required. At least two criteria can be used: Shapes of gliders. Sets of used neighbourhood states. The Shapes of gliders are not taken into account because the gliders can be found in different orientations and in different generations. So, in order to determine if a glider is new, the set of neighbourhood states used by the given gun are compared to the ones of the other guns. For each gun and each neighbourhood state, four cases exist: The neighbourhood state is not used by this gun. The neighbourhood state is used by this gun and the value of the central cell at the next generation is 0, 1 or 2. Two gliders are different iff at least one neighbourhood state is not in the same case for the two gliders. Thanks to this qualification of different gliders through the experimentations, 5186 different ones were discovered, all of them emerging spontaneously from random configurations of cells. The 5186 gliders can be found in [16] in mcl format. 4 Description of Gliders The period and the speed of the found gliders are described in the first subsection below then some specific gliders are shown in the last part. 4.1 Caracteristic of the Gliders The distribution of the periods of the found gliders is shown figure 3. Fig.3. Distribution of the periods of the found gliders.

There are very few gliders of period 1 and no gliders beyond period 4 were found. The distribution of the speeds of the found gliders is shown in figure 4. In order to ascertain this, the position after a period is compared to the intial position, with the speed being in cells per generation and only the four direct neighbours of a cell C are spaces by 1 while the diagonal cells are spaced by 2. Fig.4. Distribution of the speeds of the found gliders. All the gliders with speeds of 1 3 and 2 3 have a period of 3 and the other gliders of period 3 have a speed of 1. Only the gliders of period 4 are capable of a speed of 1 4,and finally, the gliders of period 1 have a speed of 1. 3267 gliders move horizontally or vertically and the others move diagonnaly. 4.2 Specific Gliders The glider accepts by the highest number of automata is the one in figure 1. It is also one with the highest speed of 1. A glider with the slowest speed of 1 4 is shown in figure 5, and is exhibited by the automata of table 2. Fig. 5. Glider of period 4. This glider has a period of 4 and moves by 1 cell per period. A period 2 diagonal glider of speed 1, exhibited by automata of table 3, is shown figure 6.

0 2 1 2 X X 0 0 2 1 0 0 1 1 X 0 1 X 0 0 0 Table 2. Transition rule of automata that exhibits the glider of figure 5 Fig. 6. Diagonal glider. 0 2 2 2 X 2 0 1 2 1 1 0 1 X X 0 0 0 0 1 0 Table 3. Transition rule of automata that exhibits the glider of figure 6. 5 Synthesis and Perspectives This paper deals with the emergence of computation in complex systems with local interactions, while more particularly, this paper presents a new approach to searching for gliders in cellular automata. The presented search method was to look for the appearence of gliders from the evolution of random configuration of cells for all automata of a space. 2D ternary totalistic automata using the four nearest neighbours to updates their cell states were chosen, 5186 gliders exhibited by automata of this space were found. These gliders were not known before and they have been studied in terms of speed and period. Further goals can be to find glider guns by the same method in this space of automata or to extend the method to other spaces of automata. All these automata may be potential candidates for being shown as universal automata and future work could be aimed at proving the universality of some of them. Future work could also be to calculate for each automata some rule-based parameters, e.g., Langton s lamda [9]. All automata exhibing gliders may have similar values for these parameters, which could lead to a better understanding of the link between the rule transition and the emergence of computation in cellular

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