On Decidability of Intermediate Levels of Concatenation Hierarchies

It is proved that if definability of regular languages in the Sigma(n) fragment of the first-order logic on finite words is decidable, then it is decidable also for the Delta(n+1) fragment. In particular, the decidability for Delta(5) is obtained. More generally, for every concatenation hierarchy of regular languages, it is proved that decidability of one of its half levels implies decidability of the intersection of the following half level with its complement.