Quantifier Alternation for Infinite Words

We investigate the expressive power of quantifier alternation hierarchy of first-order logic over words. This hierarchy includes the classes $Σ_i$ (sentences having at most $i$ blocks of quantifiers starting with an $\exists$) and $\mathcal{B}Σ_i$ (Boolean combinations of $Σ_i$ sentences). So far, this expressive power has been effectively characterized for the lower levels only. Recently, a breakthrough was made over finite words, and decidable characterizations were obtained for $\mathcal{B}Σ_2$ and $Σ_3$, by relying on a decision problem called separation, and solving it for $Σ_2$. The contribution of this paper is a generalization of these results to the setting of infinite words: we solve separation for $Σ_2$ and $Σ_3$, and obtain decidable characterizations of $\mathcal{B}Σ_2$ and $Σ_3$ as consequences.

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