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 $\mathbb {Z}_m$. In standard cellular automata the global state of the system at time t only depends on the state at time $t-1$, while in HOCA it is a function of the states at time $t-1$, ..., $t-n$, where $n\ge 1$ is the memory size. In particular, we provide easy-to-check necessary and sufficient conditions for a linear HOCA over $\mathbb {Z}_m$ 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 $\mathbb {Z}_m^n$ in the case $n=1$. We also prove that linear HOCA over $\mathbb {Z}_m$ of memory size n are indistinguishable from a subclass of LCA over $\mathbb {Z}_m^n$. This enables to decide injectivity and surjectivity for linear HOCA over $\mathbb {Z}_m$ of memory size n by means of the decidable characterizations of injectivity and surjectivity provided in [2] and [20] for LCA over $\mathbb {Z}^n_m$.