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

We study the dynamical behavior of linear higher-order cellular automata (HOCA) over ℤ_ . In standard cellular automata the global state of the system at time t only depends on the state at time −1, while in HOCA it is a function of the states at time −1 , ..., −, where ≥1 is the memory size. In particular, we provide easy-to-check necessary and sufficient conditions for a linear HOCA over ℤ_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 ℤ^_ in the case =1. We also prove that linear HOCA over ℤ_of memory size n are indistinguishable from a subclass of LCA over ℤ^_. This enables to decide injectivity and surjectivity for linear HOCA over ℤ_ of memory size n by means of the decidable characterizations of injectivity and surjectivity provided in [2] and [20] for LCA over ℤ^_.