TITLE: Complementation of $\omega$-regular languages in reverse mathematics
QUESTION [14 upvotes]: Does anyone know where Büchi's theorem that $\omega$-regular languages are closed under complementation fits into the reverse-mathematics classification scheme? That is, is it equivalent over $\mathrm{RCA}_0$ to one of the usual subsystems of second-order arithmetic? Or, if not known to be equivalent, what is known about where it fits in?
The formulation I have in mind is, for a fixed finite signature $\Sigma$, the statement: for every finite automaton $M$ (over $\Sigma$), there exists a finite automaton $M^c$, such that, for every $\omega$-word $\alpha$ over $\Sigma$, it holds that $\alpha$ is (Büchi-)accepted by $M$ if and only if $\alpha$ is not accepted by $M^c$.

REPLY [4 votes]: There is a new paper addressing exactly this question:
L. A. Kołodziejczyk, H. Michalewski, P. Pradic, M. Skrzypczak, The logical strength of Büchi's decidability theorem
accepted to CSL 2016. The paper is available at the first author's website:
http://www.mimuw.edu.pl/~lak/buchi_strength.pdf.
The abstract states:

We prove that the following are equivalent over the weak second-order 
  arithmetic theory $\text{RCA}_0$:

Büchi’s complementation theorem for nondeterministic automata on infinite words, 
the decidability of the depth-n fragment of the MSO theory of (N, ≤), for each n ≥ 5, 
the induction scheme for $\Sigma^0_2$ formulae of arithmetic.