Decidability of Sensitivity and Equicontinuity for Linear Higher-Order Cellular Automata

Abstract

We study the dynamical behavior of linear higher-order cellular automata (HOCA) over (Formula Presented). In standard cellular automata the global state of the system at time t only depends on the state at time (Formula Presented), while in HOCA it is a function of the states at time (Formula Presented),.., (Formula Presented), where (Formula Presented) is the memory size. In particular, we provide easy-to-check necessary and sufficient conditions for a linear HOCA over (Formula Presented) of memory size n to be sensitive to the initial conditions or equicontinuous. Our characterizations of sensitivity and equicontinuity extend the ones shown in [23] for linear cellular automata (LCA) over (Formula Presented) in the case (Formula Presented). We also prove that linear HOCA over (Formula Presented) of memory size n are indistinguishable from a subclass of LCA over (Formula Presented). This enables to decide injectivity and surjectivity for linear HOCA over(Formula Presented) of memory size n by means of the decidable characterizations of injectivity and surjectivity provided in [2] and [20] for LCA over (Formula Presented).