Abstract. It is proved that in every concatenation hierarchy of regular languages, decidability of one of its half levels, obtained by polynomial closure, implies decidability of the intersection of the following half level with its complement. In terms of the quantifier-alternation hierarchy of sentences in the first-order logic of finite words, this means that decidability of (definability in) the Σn fragment implies that of ∆n+1. In particular, the decidability of ∆5 is obtained.

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