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 \(\mathbb {Z}^n_m\).