One quantifier alternation in first-order logic with modular predicates

Adding modular predicates yields a generalization of first-order logic FO over words. The expressive power of FO[<,MOD] with order comparison x < y and predicates for x ≡ i mod n has been investigated by Barrington, Compton, Straubing and Thérien. The study of FO[<,MOD]fragments was initiated by Chaubard, Pin and Straubing. More recently, Dartois and Paperman showed that definability in the two-variable fragment FO[<,MOD] is decidable. In this paper we continue this line of work. We give an effective algebraic characterization of the word languages in Σ2[<,MOD]. The fragment Σ2 consists of first-order formulas in prenex normal form with two blocks of quantifiers starting with an existential block. In addition we show that ∆2[<,MOD], the largest subclass of Σ2[<,MOD] which is closed under negation, has the same expressive power as two-variable logic FO[<,MOD]. This generalizes the result FO[<] = ∆2[<] of Thérien and Wilke to modular predicates. As a byproduct, we obtain another decidable characterization of FO[<,MOD].