Redundancy in Complete Sets

Christian Glaßer, Aduri Pavan, Alan L. Selman, Liyu Zhang

Abstract: We show that a set is m-autoreducible if and only if it is m-mitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al. [12], complete sets for all of the following complexity classes are m-mitotic: NP, coNP, $\bigoplus$ P, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the cases of NP, PSPACE, NEXP, and PH, this at once answers several well-studied open questions. These results tell us that complete sets share a redundancy that was not known before. In particular, every NP-complete set A splits into two NP-complete sets A 1 and A 2.