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 $\omega$-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.

中文翻译：

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无限字上的确定性 Parikh 自动机

最近文献中介绍并研究了关于无限字的 Parikh 自动机的各种变体。然而，除了一些例外，仅研究了它们的非确定性版本。在本文中，我们研究了文献中尚未研究过的无限词上 Parikh 自动机所有变体的确定性版本。我们比较确定性模型的表达能力，研究它们的封闭特性和决策问题及其在模型检查中的应用。确定性极限 Parikh 自动机模型被证明是最有趣的，因为它是泛化 $\omega$-正则语言的唯一确定性 Parikh 模型，唯一在布尔运算下封闭的确定性 Parikh 模型，也是唯一一个在布尔运算下封闭的确定性 Parikh 模型。所有常见的决策问题都是可判定的。