Extensions of $\omega$-Regular Languages
Mikołaj Bojańczyk, Edon Kelmendi, Rafał Stefański, Georg Zetzsche · arXiv (Cornell University) · 2020
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.