Abstract
We study tree languages that can be defined in Δ 2 . These are tree languages definable by a first-order formula whose quantifier prefix is $\exists^*\forall^*$, and simultaneously by a first-order formula whose quantifier prefix is $\forall^*\exists^*$, both formulas over the signature with the descendant relation. We provide an effective characterization of tree languages definable in Δ 2 . This characterization is in terms of algebraic equations. Over words, the class of word languages definable in Δ 2 forms a robust class, which was given an effective algebraic characterization by Pin and Weil [11].
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