State complexity of the root operation for transitive regular languages

Ascertain whether the worst-case deterministic state complexity bounds for the root operation on regular languages can be improved when the input language is restricted to be transitive; specifically, determine if computing sqrt(U) for transitive regular languages requires fewer states than the known general bounds.

Background

The paper uses the root and intersection operations within the construction that relates rational ω-expressions to rational lasso expressions. While the general state complexities for root and intersection are known, the authors note that the languages for which they compute the root are always transitive in their setting. This raises the possibility that tighter state complexity bounds might hold under the transitivity restriction.