Bulking I: An abstract theory of bulking

Abstract

a b s t r a c tThis 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. This 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 network is regular; interactions are local, uniform and synchronous. A spacetime diagram is the geometrical representation of an orbit obtained by piling up the successive configurations. The present paper aims at structuring the sets of space-time diagrams generated by cellular automata.To deal with the richness of these objects, some families of space-time diagrams are identified through spatio-temporal transforms preserving the notion of cellular automata. In particular, one type of transform, commonly used in algorithmic constructions in the cellular automata literature, is to be taken into account: cells grouping.A very common grouping transform appears early in algorithmic constructions on cellular automata as a tool to simplify the description of the algorithm. A typical use of this tool appear in the work of Fischer [6]. To recognize prime numbers in real time, a first construction is given to recognize the primality of n at time 3n, the cells are then packed in 3 × 3 blocks defining a new cellular automaton achieving real-time recognition, as depicted in Fig. 1.Rescaling also appears when comparing neighborhoods, in particular the relation between first neighbors and the oneway neighborhood. To simulate, up to a translation, a first neighbors cellular automaton by a one-way cellular automaton, Choffrut and Čulik II [3] propose to add to the set of states every pair of original states and compute a transition in two time ✩ The re...

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