Some pointwise and decidable properties of non-uniform cellular automata
This work addresses theoretical computer science problems for researchers studying cellular automata, providing incremental extensions to known results in higher-dimensional settings.
The paper tackles the problem of characterizing dynamical properties like nilpotency and periodicity in non-uniform cellular automata (NUCA) with finite memory, showing that pointwise properties are equivalent to global ones and establishing decidability results for these properties and injectivity in certain perturbed cases.
For non-uniform cellular automata (NUCA) with finite memory over an arbitrary universe with multiple local transition rules, we show that pointwise nilpotency, pointwise periodicity, and pointwise eventual periodicity properties are respectively equivalent to nilpotency, periodicity, and eventual periodicity. Moreover, we prove that every linear NUCA which satisfies pointwise a polynomial equation (which may depend on the configuration) must be an eventually periodic linear NUCA. Generalizing results for higher dimensional group and linear CA, we also establish the decidability results of the above dynamical properties as well as the injectivity for arbitrary NUCA with finite memory which are local perturbations of higher dimensional linear and group CA. Some generalizations to the case of sparse global perturbations of higher dimensional linear and group CA are also obtained.