A reducibility for the dot-depth hierarchy

Hierarchies considered in computability theory and in complexity theory are related to some reducibilities in the sense that levels of the hierarchies are downward closed and have complete sets. In this paper we propose a reducibility having similar relationship to the Brzozowski's dot-depth hierarchy and some its refinements. We prove some basic facts on the corresponding degree structure and discuss relationships of the reducibility to complexity theory (via the leaf-language approach).