Decidable and Computational properties of Cellular Automata (PhD Thesis)

In this thesis we investigate decidable and computational properties of Cellular Automata. This investigation is intended to be a contribute to the study of the more general theory of Complex Systems. A central interest in the sciences of complex systems is to understand the laws by which a global complex behavior can emerge for the collective interaction of simple components. Computation Theory and Dynamical System Theory provide a general framework for understanding and describing the behavior of such systems. Since Cellular Automata offer a very large and diverse dynamical behavior as well as a wide variety of possible computational models, they represent an ideal subject to investigate the possible relations between dynamics and computation. In the first part of the dissertation, we investigate the class of regular Cellular Automata. We are mostly interested in decidable properties of regular Cellular Automata. We show that regularity is an undecidable property, i.e. there is no algorithm which can decide if some cellular automaton is regular. Despite this negative result, the dynamics of regular Cellular Automata is, in some sense, predictable. A fact which supports this argument is that some of the topological properties which are in general undecidable for general Cellular Automata are decidable if we restrict only to the class of regular Cellular Automata. This suggests that regularity is a property which cannot be related to computational universality. In the second part of the dissertation, we introduce a measure of computational complexity for Cellular Automata. We consider the process of computation in Cellular Automata as a a flow toward a subshift attractor. The basins of attraction of subshift attractors are dense open sets. We characterize such basins of attraction by using formal language theory and we show that deciding whether some Turing machine halts on some input word is equivalent to decide if some basin of attraction contains some open set. We can then have arbitrarly high basin languages complexity. We introduce a classification of Cellular Automata related to such basin languages complexity. In our classification the computational power of Cellular Automata is explicitly related to a topological property. We can then explore the intersection classes between our classification and other topological classification of Cellular Automata. From the emptiness of some intersection classes we can easily derive some necessary dynamical conditions for the universality. In particular we show that, according to our model, regular Cellular Automata cannot be universal.