Perfect correspondences between dot-depth and polynomial-time hierarchies.

We introduce the polynomial-time tree reducibility (ptt-reducibility). Our main result establishes a one-one correspondence between this reducibility and inclusions between complexity classes. More precisely, for languages B and C it holds that B ptt-reduces to C if and only if the unbalanced leaf-language class of B is robustly contained in the unbalanced leaf-language class of C. Formerly, such correspondence was only known for balanced leaf-language classes [Bovet, Crescenzi, Silvestri, Vereshchagin]. We show that restricted to regular languages, the levels 0, 1/2, 1, and 3/2 of the dot-depth hierarchy (DDH) are closed under ptt-reducibility. This gives evidence that the correspondence between the dot-depth hierarchy and the polynomial-time hierarchy is closer than formerly known. Our results also have applications in complexity theory: We obtain the first gap theorem of leaf-language definability above the Boolean closure of NP. Previously, such gap theorems were only known for P,NP, and Σ$^{\rm P}_{\rm 2}$ [Borchert, Kuske, Stephan, and Schmitz].