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Summary: SPLITTING NP-COMPLETE SETS
CHRISTIAN GLAßER, A. PAVAN, ALAN L. SELMAN§, AND 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., complete sets for all of the following complexity classes are m-mitotic: 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 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 A1 and A2.
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 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.
Key words. computational and structural complexity, NP-complete sets, autoreducibility, mi-
toticity
AMS subject classifications. 68Q17,68Q15
1. Introduction. We show in this paper that NP-complete sets split into two
equivalent parts. Let L be an NP-complete 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
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