Abstract
Automata, Logic and Semantics One of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure.
Highlights
Cellular automata (CA) are local descriptions of global dynamics
The Garden-of-Eden theorem by Moore [19] and its converse by Myhill [20], which link surjectivity of the global map of 2D CA to pre-injectivity have the distinction of being the first rigorous results of cellular automata theory
We include several properties studied in topological dynamics: CA with any of these properties are surjective, and we show that the converse implications holds precisely for CA on amenable groups
Summary
Cellular automata (CA) are local descriptions of global dynamics. Given an underlying uniform graph (e.g., the square grid on the plane) a CA is defined by a finite alphabet, a finite neighborhood for the nodes of the graph, and a local function that maps states of a neighborhood into states of a point. Preservation of the uniform product measure by surjective CA characterizes amenable groups: it fails catastrophically for non-amenable ones, in the sense given by the following statement. For finitely generated groups with decidable word problem, Martin-Lof randomness can be defined: such definition depends on the measure defined on the Borel σ-algebra, which for our aims will be the product measure. Under these additional hypotheses, we show that the result by Calude et al [3] about surjective CA preserving Martin-Lof randomness, holds precisely for amenable groups.
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