Phenomenology of Non-local Cellular Automata

The dynamical systems with non-local connections have potential applications to economic and biological systems. This paper studies the dynamics of non-local cellular automata. In particular, all two-state three-input non-local cellular automata are classified according to the dynamical behaviors starting from random initial configurations and random wirings. The rule space is studied with the mean-field parameterization which provides an improvement over the previous used "A parameterization". The concept of robust universal computation, concerning the ability for a system to do universal computation with random setups of initial condition, is introduced. It is argued that since non-local connections provide a handy way for information transmission, it is much easier for·a non-local cellular automaton to be a universal computer than for a local one, though it may not be robust. A particularly interesting "edge of chaos" nonlocal cellular automaton, the rule 184, is studied in detail. It exhibits irregular fluctuations of the density, large coherent structures, and long. transient times.