We consider extensions of monadic second order logic over $\omega$-words, which are obtained by adding one language that is not $\omega$-regular. We show that if the added language $L$ has a neutral letter, then the resulting logic is necessarily undecidable. A corollary is that the $\omega$-regular languages are the only decidable Boolean-closed full trio over $\omega$-words.

中文翻译：

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$\omega$-Regular Languages 的扩展

我们考虑在 $\omega$-words 上的一元二阶逻辑的扩展，这些扩展是通过添加一种不是 $\omega$-regular 的语言获得的。我们证明如果添加的语言 $L$ 有一个中性字母，那么结果逻辑必然是不可判定的。一个推论是 $\omega$-regular 语言是 $\omega$-words 上唯一可判定的布尔闭全三重奏。