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Cellular automata: dynamics, simulations, traces

Understanding the emergence of complexity out of simple basic elements is a fundamental issue in various scientific fields: cellular biology, cognitive science, fluid mechanics, chemical turbulences, crystal formation, social dynamics, computer networks. . . These problems, abstracted from their particular modeled systems, were joined into what is now called the theory of complex systems. They led John von Neumann, motivated by the autoreproducibility question and inspired by Stanislaw Ulam, to define the first cellular automaton in the late forties. Merely formalized as a discrete space divided into cells whose states evolve in a discrete time according to their closest neighbors, it exhibits strange evolutions, such as patterns reproducing themselves indefinitely. This duality was popularized in the seventies by John Conway’s game of life. The emergence of computers would soon allow anyone to program it, and nevertheless admit that the overall behavior could turn out to be very complex, motivating Stephen Wolfram’s classification of the visual aspects of cellular automata in the eighties. But what does one exactly mean by complexity? This notion was the subject of many formalization attempts. First, the computing power, already suggested by von Neumann, was formalized in terms of Turing-equivalence, for instance for the game of life [BCG82], and gave rise to some algorithmic issues [Fis65], for language recognition [SI72], or for very peculiar problems [Moo64], which give evidence on what kind of processes could be performed with the model. Moreover, the computing power has also been studied for cellular automata with respect to each other. This approach has led to the notions of “cellular” simulations and “intrinsic” universality, whose premises could be seen in [Ban70, ACI87], before formalizations in [MI94] and mostly [Rap97, Oll02, The05]. The orders induced by the different kinds of simulations are still not very well understood. On the other hand, cellular automata have been studied in terms of predictability with respect to other computing models – Turing machines. Most of their long-term properties have been proved undecidable since the works of Jarkko Kari in the nineties [Kar90]. On top of that, cellular automata have joined the theory of dynamical systems thanks to the 1969 characterization by Hedlund, Curtis and Lyndon [Hed69] as continuous maps over configurations that commute with the translations of cells, the configuration space being endowed with the Cantor topology that makes it perfect, compact and totally disconnected. This branch of the study has resulted in many contributions, involving equicontinuity [Gil87], attractors [Hur90], measure [Ish92]. . . . In 1997, Petr Kůrka proposes the modifications of two topological classifications in [Ků97], and compares them both with a third one, based on the sequences of states that are successively taken by some single cell – or some finite group of cell – during the evolution of the cellular automaton. The principle of studying such “traces” of dynamical systems through a given partition of the space probably sprang from the study of geodesic streams by Hadamard at the end of the nineteenth century, and was given its name of “symbolic dynamics” by the eponymous book by Morse and Hedlund [MH38]. In the dawn of the Internet era, a reference book by Douglas Lind and Brian Marcus [LM95] stressed that symbolic dynamics find its most promising applications in code theory. If one sees the cellular automaton as modeling some physical phenomenon, then the letter sequences it produces may represent the measure of the phenomenon through some device with some precision. Hence, it seems relevant to study the behavior of these trace systems according to that of the global one. Topologically speaking, they are linked by a factorization, i.e. reading a letter of the infinite word corresponds to applying one step of the cellular automaton. To each trace can be associated the language