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

Abstract

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, …,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 Zm. These characterizations have an impact in applications since the involved linear HOCA are usually required to exhibit a chaotic or stable behaviour. Moreover, they extend the ones shown in [28] for linear CA (LCA) over the alphabet Zmn in the case n=1. We also show that linear HOCA of memory size n over Zm form a class that is indistinguishable from a specific subclass of LCA over Zmn. This enables to decide injectivity and surjectivity for linear HOCA of memory size n over Zm using the decidable characterization provided in [2] and [25] for injectivity and surjectivity of LCA over Zmn. Finally, we prove an equivalence between LCA over Zmn and an important class of non-uniform CA, another variant of CA used in many applications.