 
Summary: SPLITTING NPCOMPLETE SETS
CHRISTIAN GLAßER, A. PAVAN, ALAN L. SELMAN§, AND LIYU ZHANG¶
Abstract. We show that a set is mautoreducible if and only if it is mmitotic. 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 mmitotic: NP,
coNP, 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 wellstudied
open questions. These results tell us that complete sets share a redundancy that was not known
before. In particular, every NPcomplete set A splits into two NPcomplete sets A1 and A2.
We disprove the equivalence between autoreducibility and mitoticity for all polynomialtime
bounded reducibilities between 3ttreducibility and Turingreducibility: There exists a sparse set
in EXP that is polynomialtime 3ttautoreducible, but not weakly polynomialtime Tmitotic. In
particular, polynomialtime Tautoreducibility does not imply polynomialtime weak Tmitoticity,
which solves an open question by Buhrman and Torenvliet.
Key words. computational and structural complexity, NPcomplete sets, autoreducibility, mi
toticity
AMS subject classifications. 68Q17,68Q15
1. Introduction. We show in this paper that NPcomplete sets split into two
equivalent parts. Let L be an NPcomplete set containing an infinite number of
strings. Then there is a set S P such that the sets L1 = S L and L2 = S L are
