Universality and Decidability of Number-Conserving Cellular Automata

Number-conserving cellular automata (NCCA) are particularly interesting, both because of their natural appearance as models of real systems, and because of the strong restrictions that number-conservation implies. Here we extend the definition of the property to include cellular automata with any set of states in \Zset, and show that they can be always extended to ``usual'' NCCA with contiguous states. We show a way to simulate any one dimensional CA through a one dimensional NCCA, proving the existence of intrinsically universal NCCA. Finally, we give an algorithm to decide, given a CA, if its states can be labeled with integers to produce a NCCA, and to find this relabeling if the answer is positive.Comment: 13 page

Probabilistic cellular automata with conserved quantities

We demonstrate that the concept of a conservation law can be naturally extended from deterministic to probabilistic cellular automata (PCA) rules. The local function for conservative PCA must satisfy conditions analogous to conservation conditions for deterministic cellular automata. Conservation condition for PCA can also be written in the form of a current conservation law. For deterministic nearest-neighbour CA the current can be computed exactly. Local structure approximation can partially predict the equilibrium current for non-deterministic cases. For linear segments of the fundamental diagram it actually produces exact results.Comment: 17 pages, 2 figure

Conservation Laws in Cellular Automata

If X is a discrete abelian group and B a finite set, then a cellular automaton (CA) is a continuous map F:B^X-->B^X that commutes with all X-shifts. If g is a real-valued function on B, then, for any b in B^X, we define G(b) to be the sum over all x in X of g(b_x) (if finite). We say g is `conserved' by F if G is constant under the action of F. We characterize such `conservation laws' in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.Comment: 19 pages, LaTeX 2E with one (1) Encapsulated PostScript figure. To appear in Nonlinearity. (v2) minor changes/corrections; new references added to bibliograph

Deterministic cellular automata resembling diffusion

We investigate number conserving cellular automata with up to five inputs and two states with the goal of comparing their dynamics with diffusion. For this purpose, we introduce the concept of decompression ratio describing expansion of configurations with finite support. We find that a large number of number-conserving rules exhibit abrupt change in the decompression ratio when the density of the initial pattern is increasing, somewhat analogous to the second order phase transition. The existence of this transition is formally proved for rule 184. Small number of rules exhibit infinite decompression ratio, and such rules may be useful for "engineering" of CA rules which are good models of diffusion, although they will most likely require more than two states.Comment: 13 pages 8 figure

Number-conserving cellular automata with a von Neumann neighborhood of range one

We present necessary and sufficient conditions for a cellular automaton with a von Neumann neighborhood of range one to be number-conserving. The conditions are formulated for any dimension and for any set of states containing zero. The use of the geometric structure of the von Neumann neighborhood allows for computationally tractable conditions even in higher dimensions.Comment: 15 pages, 3 figure

Universalities in cellular automata; a (short) survey

This reading guide aims to provide the reader with an easy access to the study of universality in the field of cellular automata. To fulfill this goal, the approach taken here is organized in three parts: a detailled chronology of seminal papers, a discussion of the definition and main properties of universal cellular automata, and a broad bibliography