Cellular Automata

In this chapter we introduce the notion of a cellular automaton. We fix a group and an arbitrary set which will be called the alphabet. A configuration is defined as being a map from the group into the alphabet. Thus, a configuration is a way of attaching an element of the alphabet to each element of the group. There is a. natural action of the group on the set of configurations which is called the shift action (see Sect. 1.1). A cellular automaton is a self-mapping of the set of configurations defined from a system of local rules commuting with the shift (see Definition 1.4.1). We equip the configuration set with the prodiscrete topology, that is, the topology of point-wise convergence associated with the discrete topology on the alphabet (see Sect. 1.2). It turns out that every cellular automaton is continuous with respect to the prodiscrete topology (Proposition 1.4.8) and commutes with the shift (Proposition 1.4.4). Conversely, when the alphabet is finite, every continuous self-mapping of the configuration space which commutes with the shift is a cellular automaton (Theorem 1.8.1). Another important fact in the finite alphabet case is that every bijective cellular automaton is invertible, in the sense that its inverse map is also a cellular automaton (Theorem 1.10.2). We give examples showing that, when the alphabet is infinite, a continuous self-mapping of the configuration space which commutes with the shift may fail to be a cellular automaton and a bijective cellular automaton may fail to be invertible. In Sect. 1.9, we introduce the prodiscrete uniform structure on the configuration space. We show that a self-mapping of the configuration space is a cellular automaton if and only if it is uniformly continuous and commutes with the shift (Theorem 1.9.1).