Context-free languages of infinite words have recently found increasing interest. Here, we will present a second-order logic with the same expressive power as Büchi or Muller pushdown automata for infinite words. This extends fundamental logical characterizations of Büchi, Elgot, Trakhtenbrot for regular languages of finite and infinite words and a more recent logical characterization of Lautemann, Schwentick and Thérien for context-free languages of finite words to ω-context-free languages. For our argument, we will investigate Greibach normal forms of ω-context-free grammars as well as a new type of Büchi pushdown automata which can alter their stack by at most one element and without ϵ-transitions. We show that they suffice to accept all ω-context-free languages. This enables us to use similar results recently developed for infinite nested words.

无限词的上下文无关语言最近引起了越来越多的兴趣。在这里，我们将展示具有与 Büchi 或 Muller 下推自动机相同的表达能力的二阶逻辑，用于无限单词。这将 Büchi、Elgot、Trakhtenbrot 对有限和无限词的常规语言的基本逻辑特征以及 Lautemann、Schwentick 和 Thérien 对有限词的上下文无关语言的更新逻辑特征扩展到ω -上下文无关语言。对于我们的论点，我们将研究ω -上下文无关文法的Greibach 范式，以及一种新型的 Büchi 下推自动机，该自动机最多可以通过一个元素改变它们的堆栈并且没有ϵ -transitions。我们证明它们足以接受所有ω- 无上下文语言。这使我们能够使用最近为无限嵌套词开发的类似结果。