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Universalité et complexité des automates cellulaires coagulants

Abstract : Cellular automata are a well know family of discrete dynamic systems, defined by S. Ulam and J. von Neumannin the 40s. The have been successfully studied from the point of view of modeling, dynamics and computational complexity. In this work, we adopt this last point of view to study the family of freezing cellular automata, those where the state of a cell can only evolve following an order relation on the set of states. We study the complexity of these cellular automata from two points of view, the ability of some freezing cellular automata to simulate every other freezing cellular automata, called intrinsic universality, and the time complexity to predict the evolution of a cell starting from a given finite configuration, called prediction complexity. We show that despite the severe restriction of the ordering of states, freezing cellular automata can still exhibit highly complex behaviors.On the one hand, we show that in two or more dimensions there exists an intrinsically universal freezing cellular automaton, able to simulate any other freezing cellular automaton by encoding its states into blocks of cells, where each cell can change at most twice. This result is minimal in dimension two and can be even simplified to one change per cell in higher dimensions.On the other hand, we extensively study the computational complexity of the prediction problem for totalistic freezing cellular automata with two states and von Neumann neighborhood in dimension two. In this family of 32 cellular automata, we find two automata with the maximum complexity for classical synchronous cellular automata, while in the case of asynchronous evolution, the maximum complexity can only be achived in dimension three. For most of the other automata of this family, we show that they have a lower complexity (assuming P 6≠NP).
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Submitted on : Monday, April 15, 2019 - 11:29:07 AM
Last modification on : Wednesday, November 20, 2019 - 1:42:38 AM


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Diego Maldonado. Universalité et complexité des automates cellulaires coagulants. Autre. Université d'Orléans, 2018. Français. ⟨NNT : 2018ORLE2027⟩. ⟨tel-02099753⟩



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