We show that some natural refinements of the Straubing and Brzozowski hierarchies correspond (via the so called leaf-languages) step by step to similar refinements of the polynomial-time hierarchy. This extends a result of H.-J. Burtschik and H. Vollmer on relationship between the Straubing and the polynomial hierarchies. In particular, this applies to the boolean hierarchy and the plus–hierarchy.
Victor L. Selivanov