On the dynamical behaviour of linear higher-order cellular automata and its decidability

Higher-order cellular automata (HOCA) are a variant of cellular automata (CA) used in many applications (ranging, for instance, from the design of secret sharing schemes to data compression and image processing), and in which the global state of the system at time $t$ depends not only on the state at time $t-1$, as in the original model, but also on the states at time $t-2, \ldots, t-n$, where $n$ is the memory size of the HOCA. We provide decidable characterizations of two important dynamical properties, namely, sensitivity to the initial conditions and equicontinuity, for linear HOCA over the alphabet $\mathbb{Z}m$. Such characterizations extend the ones shown in [23] for linear CA (LCA) over the alphabet $\mathbb{Z}^{n}m$ in the case $n=1$. We also prove that linear HOCA of size memory $n$ over $\mathbb{Z}m$ form a class that is indistinguishable from a specific subclass of LCA over $\mathbb{Z}m^n$. This enables to decide injectivity and surjectivity for linear HOCA of size memory $n$ over $\mathbb{Z}m$ using the decidable characterization provided in [2] and [19] for injectivity and surjectivity of LCA over $\mathbb{Z}ⁿm$. Finally, we prove an equivalence between LCA over $\mathbb{Z}_m^n$ and an important class of non-uniform CA, another variant of CA used in many applications.