Abstract
This work shows that for each i ∈ ω there exists a $\Sigma ^1_i$-hard ω-word language definable in Monadic Second Order Logic extended with the unbounding quantifier (MSO+U). This quantifier was introduced by Bojańczyk to express some asymptotic properties. Since it is not hard to see that each language expressible in MSO+U is projective, our finding solves the topological complexity of MSO+U. The result can immediately be transferred from ω-words to infinite labelled trees. As a consequence of the topological hardness we note that no alternating automaton with a Borel acceptance condition — or even with an acceptance condition of a bounded projective complexity — can capture all of MSO+U. The same holds for deterministic and nondeterministic automata since they are special cases of alternating ones. We also give exact topological complexities of related classes of languages recognized by nondeterministic ωB-, ωS- and ωBS-automata studied by Bojańczyk and Colcombet. Furthermore, we show that corresponding alternating automata have higher topological complexity than nondeterministic ones — they inhabit all finite levels of the Borel hierarchy. The paper is an extended journal version of [8]. The main theorem of that article is strengthened here.