Bulking I: An abstract theory of bulking

This paper is the first part of a series of two papers dealing with bulking: a quasi-order on cellular automata comparing space-time diagrams up to some rescaling. Bulking is a generalization of grouping taking into account universality phenomena, giving rise to a maximal equivalence class. In the present paper, we discuss the proper components of grouping and study the most general extensions. We identify the most general space-time transforms and give an axiomatization of bulking quasi-order. Finally, we study some properties of intrinsically universal cellular automata obtained by comparing grouping to bulking. Bulking is introduced as a tool to structure cellular automata, considered as the sets of their orbits. To achieve this goal, sets of orbits are considered up to spatio-temporal transforms. Such quotients are then compared according to algebraic relations to obtain quasi-orders on the set of cellular automata, in a way similar to reductions in the case of recursive functions. It turns out that the obtained equivalence classes tend to capture relevant properties: in particular, the greatest element, when it exists, corresponds to a notion of intrinsic universality. The first and present paper is concerned with the choice of the main ingredients to define an interesting bulking. The second paper, Bulking II: Classifications of Cellular Automata [4], studies the structure of the main three varieties of bulking. A cellular automaton is a discrete dynamical system consisting of a network of cells fulfilling the following properties: each cell acts as a finite state machine; The results presented here first appeared to a great extent in French in the PhD theses of Ollinger [13] and Theyssier [14]