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In the parity problem, a given cellular automaton has to classify any initialconfiguration into two classes according to its parity. Elementary cellularautomaton rule 60 can solve the parity problem in periodic boundary conditionswith array size of the power of two. The spectral analysis of theconfigurations of rule 60 at each time step in the evolution reveals thatspatial periodicity emerges as the evolution proceeds and the patterns withlonger period split into the ones with shorter period. This phenomenon isanalogous to the cascade process in which large scale eddies split into smallerones in turbulence. By measuring the Lempel-Ziv complexity of configuration, wefound the stepping decrease of the complexity during the evolution. This resultmight imply that a decision problem solving process is accompanied with thedecline of complexity of configuration.

Solving the Parity Problem with Rule 60 in Array Size of the Power of Two