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

In the parity problem, a given cellular automaton has to classify any initial configuration into two classes according to its parity. Elementary cellular automaton rule 60 can solve the parity problem in periodic boundary conditions with array size of the power of two. The spectral analysis of the configurations of rule 60 at each time step in the evolution reveals that spatial periodicity emerges as the evolution proceeds and the patterns with longer period split into the ones with shorter period. This phenomenon is analogous to the cascade process in which large scale eddies split into smaller ones in turbulence. By measuring the Lempel-Ziv complexity of configuration, we found the stepping decrease of the complexity during the evolution. This result might imply that a decision problem solving process is accompanied with the decline of complexity of configuration

Complexity Analysis in Cyclic Tag System Emulated by Rule 110

It is known that elementary cellular automaton rule 110 is capable of supporting universal computation by emulating cyclic tag system. Since the whole information necessary to perform computation is stored in the configuration, it is reasonable to investigate the complexity of configuration for the analysis of computing process. In this research we employed Lempel-Ziv complexity as a measure of complexity and calculated it during the evolution of emulating cyclic tag system by rule 110. As a result, we observed the stepwise decline of complexity during the evolution. That is caused by the transformation from table data to moving data and the elimination of table data by a rejector.Comment: AUTOMATA 2013: 19th International Workshop on Cellular Automata and Discrete Complex Systems, Universit\"at Giessen, Germany, September 17-19, 201