Deterministic Parikh automata on infinite words

Various variants of Parikh automata on infinite words have recently been introduced and studied in the literature. However, with some exceptions only their non-deterministic versions have been studied. In this paper we study the deterministic versions of all variants of Parikh automata on infinite words that have not yet been studied in the literature. We compare the expressiveness of the deterministic models, study their closure properties and decision problems with applications to model checking. The model of deterministic limit Parikh automata turns out to be most interesting, as it is the only deterministic Parikh model generalizing the ω-regular languages, the only deterministic Parikh model closed under the Boolean operations and the only deterministic Parikh model for which all common decision problems are decidable.