Rankers over Infinite Words

We consider the fragments FO, Σ2 ∩ FO, Π2 ∩ FO, and ∆2 of first-order logic FO[<] over finite and infinite words. For all four fragments, we give characterizations in terms of rankers. In particular, we generalize the notion of a ranker to infinite words in two possible ways. Both extensions are natural in the sense that over finite words, they coincide with classical rankers and over infinite words, they both have the full expressive power of FO. Moreover, the first extension of rankers admits a characterization of Σ2 ∩ FO while the other leads to a characterization of Π2 ∩ FO. Both versions of rankers yield characterizations of the fragment ∆2 = Σ2 ∩ Π2. As a byproduct, we also obtain characterizations based on unambiguous temporal logic and unambiguous interval temporal logic.