Shigeru Ninagawa

1/ f noise in elementary cellular automaton rule 110

Cellular Automata are considered to be discrete dynamical systems as well as computing systems. Spectral analysis has been employed to investigate the behavior of dynamical systems. We calculated the power spectra from the evolutions starting from a random initial configuration to analyze the temporal behavior in elementary cellular automata. As a result, rule 110 has 1/f spectrum for the longest time steps. Rule 110 alone has proved to be capable of supporting universal computation in elementary cellular automata. These results suggest that there is a relationship between computational universality and 1/f noise in cellular automata.

Evolving cellular automata by 1/ f noise

It is speculated that there is a relationship between 1/f noise and computational universality in two-dimensional cellular automata. We use genetic algorithms to find two-dimensional cellular automata which have 1/f spectrum. Spectrum is calculated from the evolution of the state of cell from a random initial configuration. The fitness function is constructed in consideration of the spectral similarity to 1/f spectrum. The result shows that the rule with the third highest fitness in the experiment has 1/f spectrum and it behaves like the Game of Life, although two rules with the highest and the second highest fitness do not have 1/f spectrum.

Evolution of one-dimensional cellular automata by 1/f noise

It is speculated that there is a relationship between 1/f noise and computational universality in cellular automata. We use genetic algorithms to search for one-dimensional and two-state, five-neighbor cellular automata which have 1/f-type spectrum. A power spectrum is calculated from the evolution starting from a random initial configuration. The fitness is estimated from the power spectrum in consideration of the similarity to 1/f-type spectrum. The result shows that the rule with the highest average fitness has a propagating structure like other computationally universal cellular automata, although computational universality of the rule has not been proved yet.

Computational universality and 1/f noise in elementary cellular automata

Elementary (one-dimensional two-state three-neighbor) cellular automaton rule 110 is capable of supporting universal computation and exhibits 1/f noise. However, the power spectra of rule 110 do not always exhibit 1/f-type spectrum because of periodic background particular to rule 110. Since periodic background in rule 110 does not play any role in supporting universal computation, we can expect that the removal of periodic background does not change the behavior essential to supporting universal computation. As a result of removing periodic background, we found that the power spectra fit better to 1/f noise. This result suggests 1/f noise is an intrinsic property of computational universality in cellular automata.