On Some Dynamical Properties of Monotone Networks
Cellular automata were first introduced in the 1950’s by John Von Neumann. Informally, a cellular automaton can be viewed as a discrete dynamical system whose global behaviour is generated by the (simple) local interactions of its elementary cells, hereafter called the sites of the automaton. As pointed out by Wolfram (1984), cellular automata have arisen in several disciplines, because they provide examples in which the generation of complex behaviour by the cooperative effects of simple components may be studied. No unified mathematical framework has been yet developped to modelize the iterative behaviour of general networks of automata, but some tools such as algebraic operators and Lyapounov functions (Goles (1985), Fogelman (1985)), modular arithmetic and polynomial algebra (Martin (1983)), arithmetic in finite fields (Gill (1966), Tchuente (1982)) have been introduced to analyze special classes of cellular automata. In this paper, we characterize the dynamics of some monotone cellular automata. First we use a morphism technique to derive the iterative behaviour of automata whose transition functions are generalized majority functions. Then, we study some relationships between the connection-graph and the iteration-graph of boolean automata with memory, assuming that the transition function of each site is monotone with regard to its inputs.
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