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., complete sets for all of the following complexity classes are m-mitotic: $\mathrm{NP}$, $\mathrm{coNP}$, $\oplus\mathrm{P}$, $\mathrm{PSPACE}$, and $\mathrm{NEXP}$, as well as all levels of $\mathrm{PH}$, $\mathrm{MODPH}$, and the Boolean hierarchy over $\mathrm{NP}$. In the cases of $\mathrm{NP}$, $\mathrm{PSPACE}$, $\mathrm{NEXP}$, and $\mathrm{PH}$, 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 $\mathrm{NP}$-complete set A splits into two $\mathrm{NP}$-complete sets $A_1$ and $A_2$. We disprove the equivalence between autoreducibility and mitoticity for all polynomial-time-bounded reducibilities between 3-tt-reducibility and Turing-reducibility: There exists a sparse set in $\mathrm{EXP}$ that is polynomial-time 3-tt-autoreducible, but not weakly polynomial-time T-mitotic. In particular, polynomial-time T-autoreducibility does not imply polynomial-time weak T-mitoticity, which solves an open question by Buhrman and Torenvliet.

We study the question of whether the class DisjNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provides additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair; an oracle relative to which optimal proof systems exist, and hence complete pairs exist, but no pair is NP-hard; and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.

We study the question of whether the class DisNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provide additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair, an oracle relative to which optimal proof systems exist, hence complete pairs exist, but no pair is NP-hard, and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.