Non-Uniform Reductions

Buhrman, Harry; Hescott, Benjamin; Homer, Steven; Torenvliet, Leen

doi:10.1007/s00224-008-9163-5

Cited by 9 publications

(8 citation statements)

References 33 publications

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“…The answer to this question is yes for EXP [7], so in this case a negative answer for NP would imply NP = EXP. We believe that it may be possible to show the 2-tt-complete sets are nonuniformly 2-tt-autoreducible under the Measure Hypothesis -first show they are nonuniformly 2-tt-honest complete as an extension of [9,6].…”

Section: Discussionmentioning

confidence: 99%

Autoreducibility of NP-Complete Sets under Strong Hypotheses

Hitchcock

Shafei

2017

comput. complex.

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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 k ≥ 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.• For every k ≥ 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.• 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 k ≥ 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-Tautoreducibility from 2-tt-autoreducibility.

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Section: Discussionmentioning

confidence: 99%

Autoreducibility of NP-Complete Sets under Strong Hypotheses

Hitchcock

Shafei

2017

comput. complex.

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“…The µ p (NP ∩ coNP) = 0 hypothesis is admittedly strong. However, we note that strong hypotheses on NP ∩ coNP have been used in some prior investigations [29,18,13].…”

Section: Separating Uniform Completeness From Nonuniform Completenessmentioning

confidence: 97%

“…This result has been known for larger classes like EXP and NEXP without using any advice. They also proved a separation between uniform and nonuniform reductions in EXP by showing that there exists a language that is complete in EXP under many-one reductions that use one bit of advice, but is not 2-tt-complete [13]. They also proved that a nonuniform reduction can be turned into a uniform one by increasing the number of queries.…”

Section: :2mentioning

confidence: 99%

“…Buhrman et al [13] began a systematic study of nonuniform completeness. They proved, under a strong hypothesis on NP, that every 1-tt-complete set for NP is many-one complete with 1 bit of advice.…”

Section: :2mentioning

confidence: 99%

“…While Buhrman et al [13] have some results about nonuniform reductions in NP, most of their results are focused on larger complexity classes like EXP. Inspired by their results on EXP, we work toward a similarly solid understanding of NP-completeness under nonuniform reductions.…”

Section: :2mentioning

confidence: 99%

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Nonuniform Reductions and NP-Completeness

Hitchcock

Shafei

2022

Theory Comput Syst

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Nonuniformity is a central concept in computational complexity with powerful connections to circuit complexity and randomness. Nonuniform reductions have been used to study the isomorphism conjecture for NP and completeness for larger complexity classes. We study the power of nonuniform reductions for NP-completeness, obtaining both separations and upper bounds for nonuniform completeness vs uniform complessness in NP.Under various hypotheses, we obtain the following separations: 1. There is a set complete for NP under nonuniform many-one reductions, but not under uniform many-one reductions. This is true even with a single bit of nonuniform advice. 2. There is a set complete for NP under nonuniform many-one reductions with polynomialsize advice, but not under uniform Turing reductions. That is, polynomial nonuniformity is stronger than a polynomial number of queries. 3. For any fixed polynomial p(n), there is a set complete for NP under uniform 2-truth-table reductions, but not under nonuniform many-one reductions that use p(n) advice. That is, giving a uniform reduction a second query makes it more powerful than a nonuniform reduction with fixed polynomial advice. 4. There is a set complete for NP under nonuniform many-one reductions with polynomial advice, but not under nonuniform many-one reductions with logarithmic advice. This hierarchy theorem also holds for other reducibilities, such as truth-table and Turing.We also consider uniform upper bounds on nonuniform completeness. Hirahara (2015) showed that unconditionally every set that is complete for NP under nonuniform truth-table reductions that use logarithmic advice is also uniformly Turing-complete. We show that under a derandomization hypothesis, the same statement for truth-table reductions and truth-table completeness also holds.

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A Thirty Year Old Conjecture about Promise Problems

Hughes

Pavan

Russell

et al. 2012

Automata, Languages, and Programming

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Abstract. Even, Selman, and Yacobi [ESY84,SY82] formulated a conjecture that in current terminology asserts that there do not exist disjoint NP-pairs all of whose separators are NP-hard viaTuring reductions. In this paper we consider a variant of this conjecture-there do not exist disjoint NP-pairs all of whose separators are NP-hard via bounded-truthtable reductions. We provide evidence for this conjecture. We also observe that if the original conjecture holds, then some of the known probabilistic public-key cryptosystems are not NP-hard to crack.

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Non-Uniform Reductions

Cited by 9 publications

References 33 publications

Autoreducibility of NP-Complete Sets under Strong Hypotheses

Autoreducibility of NP-Complete Sets under Strong Hypotheses

Nonuniform Reductions and NP-Completeness

A Thirty Year Old Conjecture about Promise Problems

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