5 research outputs found
Autoreducibility of NP-Complete Sets
We study the polynomial-time autoreducibility of NP-complete sets and obtain
separations under strong hypotheses for NP. Assuming there is a p-generic set
in NP, we show the following:
- For every , there is a -T-complete set for NP that is -T
autoreducible, but is not -tt autoreducible or -T autoreducible.
- For every , there is a -tt-complete set for NP that is -tt
autoreducible, but is not -tt autoreducible or -T autoreducible.
- There is a tt-complete set for NP that is tt-autoreducible, but is not
btt-autoreducible.
Under the stronger assumption that there is a p-generic set in NP
coNP, we show:
- For every , there is a -tt-complete set for NP that is -tt
autoreducible, but is not -T autoreducible.
Our proofs are based on constructions from separating NP-completeness
notions. For example, the construction of a 2-T-complete set for NP that is not
2-tt-complete also separates 2-T-autoreducibility from 2-tt-autoreducibility
Non-autoreducible Sets for NEXP
We investigate autoreducibility properties of complete sets for NEXP under different polynomial-time reductions. Specifically, we show that under some polynomial-time reductions there are complete sets for NEXP that are not autoreducible.
We show that settling the question whether every complete set for NEXP under non-adaptative reduction is autoreducible under NOR-truth-table reduction either positively or negatively would lead to major results about the exponential time complexity classes
Autoreducibility, Mitoticity, and Immunity
We show the following results regarding complete sets. • NP-complete sets and PSPACE-complete sets are many-one autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are many-one autoreducible. • EXP-complete sets are many-one mitotic. • NEXP-complete sets are weakly many-one mitotic. • PSPACE-complete sets are weakly Turing-mitotic. • If one-way permutations and quick pseudo-random generators exist, then NP-complete languages are m-mitotic. • If there is a tally language in NP ∩ coNP − P, then, for every ɛ> 0, NP-complete sets are not 2 n(1+ɛ)-immune. These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets.