Algorithmic Information Theory and Cellular Automata Dynamics Julien Cervelle, Bruno Durand, and Enrico Formenti LIM - CMI 39 rue Joliot-Curie, 13453 Marseille Cedex 13, France {cervelle,bdurand,eforment}@cmi.univ-mrs.fr Abstract. We study the ability of discrete dynamical systems to trans- form/generate randomness in cellular spaces. Thus, we endow the space of bi-infinite sequences by a metric inspired by information distance (de- fined in the context of Kolmogorov complexity or algorithmic information theory). We prove structural properties of this space (non-separability, completeness, perfectness and infinite topological dimension), which turn out to be useful to understand the transformation of information per- formed by dynamical systems evolving on it. Finally, we focus on cellular automata and prove a dichotomy theorem: continuous cellular automata are either equivalent to the identity or to a constant one. This means that they cannot produce any amount of randomness. Keywords: Kolmogorov complexity, topology, cellular automata, discrete dy- namical systems 1 Introduction Cellular automata are formal models for complex systems. They were introduced by J. von Neumann for modeling cellular growth and self-replication. Afterwards, they have been successfully applied in a large number of scientific disciplines ranging from mathematics to computer science, from physics to chemistry and geology. They are essentially a massive parallel model made of a lattice of elementary identical finite state machines (automata), usually called cells. Cells are updated synchronously according to a finite set of local rules that compute the new state of the cell from the states of a finite set of neighboring cells (eventually including the cell itself). A snapshot of the states of the cells is called a configuration.A stack of configurations in which each one is obtained from the preceding, applying the updating rule, is called an evolution. The success of the model relies essentially on its simple definition coupled with the rich variety of different evolutions. Many evolutions display a complex or random behavior. It is a recurring question in the community to understand if the complexity or randomness in such evolutions is an intrinsic property of the model or if it is a side effect due to inappropriate topological context, or, J. Sgall, A. Pultr, and P. Kolman (Eds.): MFCS 2001, LNCS 2136, pp. 248–260, 2001. c  Springer-Verlag Berlin Heidelberg 2001