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].
Work partially funded by the AutoMathA programme of the ESF, the PHC programme Polonium, and by the Polish government grant no. N206 008 32/0810.
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Bojańczyk, M., Segoufin, L. (2008). Tree Languages Defined in First-Order Logic with One Quantifier Alternation. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5126. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70583-3_20
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DOI: https://doi.org/10.1007/978-3-540-70583-3_20
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