Comparison of polynomial-time reducibilities

Ladner, Richard E.; Lynch, Nancy; Selman, Alan L.

doi:10.1145/800119.803891

Cited by 62 publications

(61 citation statements)

References 8 publications

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“…Theorem 4.9 can be also viewed as demonstrating that SZK is closed under a type of polynomial-time reducibility, which is formalized by the following two definitions. [Ladner et al 1975]). We say a promise problem truth-table reduces to a promise problem if there exists a (deterministic) polynomial-time computable function f , which on input x produces a tuple (y 1 , .…”

Section: Proof By Corollary 42mentioning

confidence: 99%

A complete problem for statistical zero knowledge

Sahai

Vadhan

2003

J. ACM

139

143

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We present the first complete problem for SZK, the class of promise problems possessing statistical zero-knowledge proofs (against an honest verifier). The problem, called STATISTICAL DIFFERENCE, is to decide whether two efficiently samplable distributions are either statistically close or far apart. This gives a new characterization of SZK that makes no reference to interaction or zero knowledge.We propose the use of complete problems to unify and extend the study of statistical zero knowledge. To this end, we examine several consequences of our Completeness Theorem and its proof, such as:-A way to make every (honest-verifier) statistical zero-knowledge proof very communication efficient, with the prover sending only one bit to the verifier (to achieve soundness error 1/2).Preliminary versions of this work appeared as SAHAI, A., AND VADHAN, S. P. 1997. A complete promise problem for statistical zero-knowledge, In

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Section: Proof By Corollary 42mentioning

confidence: 99%

A complete problem for statistical zero knowledge

Sahai

Vadhan

2003

J. ACM

139

143

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“…In terms of generalized use, a concept which we consider to be of independent interest in complexity theory, we have given a formal account of the fact that standard diagonalization arguments employed for separating reducibilities ultimately do not rely on the effectivity of the reductions involved but instead exploit differences in the cardinality of the generalized use. This remark applies to separations with respect to single oracles in the style of Ladner, Lynch and Selman [10], Proposition 11 but also to separations by random oracles.…”

Section: Resultsmentioning

confidence: 98%

“…In the proof of Proposition 11 we use an abstract version of techniques employed by Ladner, Lynch and Selman [10] for separating several variants of polynomial time bounded reducibilities. In fact we obtain many of their separation results as immediate corollaries to Proposition 11.…”

Section: Separating Generalized Reducibilitiesmentioning

confidence: 99%

Separations by Random Oracles and “Almost” Classes for Generalized Reducibilities

Merkle

Wang²

2001

MLQ - Math. Log. Quart.

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Let ≤r and ≤s be two binary relations on 2 N which are meant as reducibilities. Let both relations be closed under finite variation (of their set arguments) and consider the uniform distribution on 2 N , which is obtained by choosing elements of 2 N by independent tosses of a fair coin. Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤r X, differs from the lower ≤s-cone of X. By closure under finite variation, the Kolmogorov 0-1 law yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles. Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2 N which can be defined by a countable set of total continuous functionals on 2 N in the same way as the usual resourcebounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises all natural resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations.Mathematics Subject Classification: 03D15, 03D30, 03D75, 68Q10, 68Q15.

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“…(ii) (iii) interacting facets, including a variety of reducibilities [31], complete languages under these reducibilities [64, 681, measure structure [35], and category structure [34, 181. E2 is the smallest deterministic time complexity class known to contain NP.…”

Section: Introductionmentioning

confidence: 99%

The quantitative structure of exponential time

Lutz

[1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference

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Recent results on the internal, measure-theoretic structure of the exponential time complexity classes linear polynomial E = DTIME(2 ) and E = DTIME(2 ) 2 are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time many-one reducibility, circuit-size complexity, Kolmogorov complexity, and the density of hard languages. Possible implications for the structure of NP are also discussed. ABSTRACT Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative w ays with bi-immunity, complexity cores, polynomial-time reductions, completeness, circuit-size complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. Disciplines Theory and Algorithms

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Comparison of polynomial-time reducibilities

Cited by 62 publications

References 8 publications

A complete problem for statistical zero knowledge

A complete problem for statistical zero knowledge

Separations by Random Oracles and “Almost” Classes for Generalized Reducibilities

The quantitative structure of exponential time

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