In [EGZ09] it has been shown that infinitary strong normalization (SNi) is Pi-1-1-complete. Suprisingly, it turns out that infinitary weak normalization (WNi) is a harder problem, being Pi-1-2-complete, and thereby strictly higher in the analytical hierarchy.
@article{arxiv.1003.1057,
title = {Levels of Undecidability in Infinitary Rewriting: Normalization and Reachability},
author = {Joerg Endrullis},
journal= {arXiv preprint arXiv:1003.1057},
year = {2010}
}