Algumas propriedades de autômatos celulares unidimensionais conservativos e reversíveis

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Oliveira, Angelo Schranko de
Oliveira, Pedro Paulo Balbi de
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Silva, Leandro Nunes de Castro
Macau, Elbert Einstein Nehrer
Engenharia Elétrica
Cellular automata (CAs) can be defined as discrete dynamical systems over n-dimensional networks of locally connected components, whose evolution occur in a discrete, synchronous and homogeneous fashion. Among their several applications, they have been used as a tool for complex systems modeling governed by fundamental laws of conservation (number-conserving cellular automata) or reversibility (reversible cellular automata). Another fundamental property that can be observed in CAs is regarding to their linearity (linear cellular automata) or nonlinearity. Usually, linear phenomena present low dynamic complexity, however, nonlinear phenoma can present complex behaviours like sensitive dependence on initial conditions and routes to chaos. This work focuses on investigating properties of cellular automata belonging to the intersection of those four classes, namely, reversible, number-conserving, and linear or nonlinear cellular automata. After presenting basic definitions, the notions of number-conserving cellular automata, conservation degree and reversibility are reviewed. Following, a dynamical characterisation parameter which relates the reversibility property of a onedimensional cellular automaton and the pre-images of their basic blocks is introduced, and some proofs of its general properties are given. Empirical observations herein suggest that a cellular automaton is reversible and number-conserving if, and only if, its local transition function is a composition of the local transition functions of the reversible, number-conserving cellular automata with neighbourhood size n=2; such an observation was drawn for neighbourhood sizes n∈{2, 3, 4, 5, 6} and number of states q=2; n∈{2, 3} and q=3; n∈{2, 3} and q=4. A proof for such a conjecture would allow the enumeration between neighbourhood lengths and the quantity of reversible, numberconserving cellular automata in the corresponding space, which can be easily identified by working out the compositions of the local transition functions with n=2. Finally, some relationships between reversible, number-conserving, linear and nonlinear CA rules, their spatio-temporal diagrams and basin of attraction fields are presented.
autômato celular , conservabilidade , reversibilidade , não-linearidade , sistema dinâmico discreto , NKS , cellular automaton , conservativity , reversibility , nonlinearity , discrete dynamical system , NKS
OLIVEIRA, Angelo Schranko de. Algumas propriedades de autômatos celulares unidimensionais conservativos e reversíveis. 2009. 71 f. Dissertação (Mestrado em Engenharia Elétrica) - Universidade Presbiteriana Mackenzie, São Paulo, 2009.