How a SIMD machine can implement a complex cellular automata? a case study: von Neumann's 29-state cellular automaton

This study is a part of an effort to simulate the 29-state self-reproducing cellular automaton described by John von Neumann in a manuscript that dates back to 1952. We are interested in the programming of very large SIMD arrays which, as a consequence of scaling them up, incorporate some features of cellular automata. Designing tools for programming them requires an experimental ground: considering that von Neumann's 29-state is the only known very large and complex cellular automaton, its simulation is a necessary first step. Embedded in a two-dimensional cellular array, using 29 states per cell and 5-cell neighborhood, this automaton exhibits the capabilities of universal computation and universal construction. This paper concentrates on the transition rule that governs the complex behavior of the 29-state automaton. We give a detailed presentation of its transition rule, with illustrative examples to ease its comprehension. We then discuss its implementation on a SIMD machine, using only 13 bits per processing element to encode the rule, each processing element corresponding to a cell. Finally, we present experimental results based upon the simulation of general-purpose components of the automaton: pulser, decoder, periodic pulser on the SIMD machine.