Abstract
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:
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For every \({k \geq 2}\), there is a k-T-complete set for NP that is k-T-autoreducible, but is not k-tt-autoreducible or (k − 1)-T-autoreducible.
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For every \({k \geq 3}\), there is a k-tt-complete set for NP that is k-tt-autoreducible, but is not (k − 1)-tt-autoreducible or (k − 2)-T-autoreducible.
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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 \({\cap}\) coNP, we show:
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For every \({k \geq 2}\), there is a k-tt-complete set for NP that is k-tt-autoreducible, but is not (k − 1)-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.
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Hitchcock, J.M., Shafei, H. Autoreducibility of NP-Complete Sets under Strong Hypotheses. comput. complex. 27, 63–97 (2018). https://doi.org/10.1007/s00037-017-0157-z
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DOI: https://doi.org/10.1007/s00037-017-0157-z