Upper Bound on the Products of Particle Interactions in Cellular Automata

Wim Hordijk, Cosma Rohilla Shalizi, and James P. Crutchfield

00-09-052

Particlelike objects are observed to propagate and interact in many spatially extended dynamical systems. For one of the simplest classes of such systems, one-dimensional cellular automata, we establish a rigorous upper bound on the number of distinct products that these interactions can generate. The upper bound is controlled by the structural complexity of the interacting particles--a quantity which is defined here and which measures the amount of spatiotemporal information that a particle stores. Along the way we establish a number of properties of domains and particles that follow from the computational mechanics analysis of cellular automata; thereby eludicating why that approach is of general utility. The upper bound is tested against several relatively complex domain-particle cellular automata and found to be tight. PACS: 45.70.Qj, 05.45, 05.65+b