ω-rational Languages: High Complexity Classes vs. Borel Hierarchy

The paper investigates classes of languages of infinite words with respect to the acceptance conditions of the finite automata recognizing them. Some new natural classes are compared with the Borel hierachy. In particular, it is proved that fin,= is as high as ${\textsf{F}}^R_{\sigma}$ and ${\textsf{G}}^R_{\delta}$ . As a side effect, it is also proved that in this last case, considering or not considering the initial state of the FA makes a substantial difference.