Abstract
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].
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Glaßer, C., Travers, S., Wagner, K.W. (2006). Perfect Correspondences Between Dot-Depth and Polynomial-Time Hierarchy. In: Ibarra, O.H., Dang, Z. (eds) Developments in Language Theory. DLT 2006. Lecture Notes in Computer Science, vol 4036. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11779148_37
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DOI: https://doi.org/10.1007/11779148_37
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