Computational Aspects of Cellular Automata on Countable Sofic Shifts

Abstract

We investigate the computational properties of cellular automata on countable (equivalently, zero entropy) sofic shifts with an emphasis on nilpotency, periodicity, and asymptotic behavior. As a tool for proving decidability results, we prove the Starfleet Lemma, which is of independent interest. We present computational results including the decidability of nilpotency and periodicity, the undecidability of stability of the limit set, and the existence of a $\mathrm{\Pi}^0_1$-complete limit set and a $\mathrm{\Sigma}^0_3$-complete asymptotic set.

Research supported by the Academy of Finland Grant 131558.