Computational mechanics of cellular automata: An example

We illustrate and extend the techniques of computational mechanics in explicating the structures that emerge in the space-time behavior of elementary one-dimensional cellular automaton rule 54. The dominant regular domain of the cellular automation is identified and a domain filter is constructed to locate and classify defects in the domain. The primary particles are identified and a range of interparticle interactions is studied. The deterministic equation of motion of the filtered space-time behavior is derived. Filters of increasing sophistication are constructed for the efficient gathering of particle statistics and for the identification of higher-level defects, particle interactions, and secondary domains. We define the emergence time at which the space-time behavior condenses into configurations consisting only of domains, particles, and particle interactions. Taken together, these techniques serve as the basis for the investigation of pattern evolution and self-organization in this representative system.