Complete Sets and Structure in Subrecursive Classes

Buhrman, Harry; Torenvliet, Leen

doi:10.1007/978-3-662-22110-5_2

Cited by 9 publications

(6 citation statements)

References 65 publications

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“…It has not been known whether NP-complete sets are m-autoreducible. Buhrman and Torenvliet raised this question in their survey papers [16,17]. Below, we resolve this question.…”

Section: Autoreducibilitymentioning

confidence: 91%

Autoreducibility, mitoticity, and immunity

Glaßer

Ogihara

Pavan

et al. 2007

Journal of Computer and System Sciences

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We show the following results regarding complete sets.• NP-complete sets and PSPACE-complete sets are polynomial-time many-one autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are polynomial-time many-one autoreducible.• EXP-complete sets are polynomial-time many-one mitotic.• If there is a tally language in NP ∩ coNP − P, then, for every > 0, NP-complete sets are not 2 n(1+ ) -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.

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“…It has not been known whether NP-complete sets are m-autoreducible. Buhrman and Torenvliet raised this question in their survey papers [16,17]. Below, we resolve this question.…”

Section: Autoreducibilitymentioning

confidence: 91%

Autoreducibility, mitoticity, and immunity

Glaßer

Ogihara

Pavan

et al. 2007

Journal of Computer and System Sciences

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“…As indicated by the survey papers [1,3,10,12], this line of inquiry has shed light on a wide variety of topics in computational complexity. The ongoing fruitfulness of this research is not surprising because resource-bounded measure is a complexity-theoretic generalization of classical Lebesgue measure, which was one of the most powerful quantitative tools of twentiethcentury mathematics.…”

Section: Introductionmentioning

confidence: 99%

Dimension in Complexity Classes

Lutz¹

2003

SIAM J. Comput.

121

130

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A theory of resource-bounded dimension is developed using gales, which are natural generalizations of martingales. When the resource bound ∆ (a parameter of the theory) is unrestricted, the resulting dimension is precisely the classical Hausdorff dimension (sometimes called "fractal dimension"). Other choices of the parameter ∆ yield internal dimension theories in E, E 2 , ESPACE, and other complexity classes, and in the class of all decidable problems. In general, if C is such a class, then every set X of languages has a dimension in C, which is a real number dim(X | C) ∈ [0, 1]. Along with the elements of this theory, two preliminary applications are presented:1. For every real number 0 ≤ α ≤ 1 2 , the set FREQ(≤ α), consisting of all languages that asymptotically contain at most α of all strings, has dimension H(α) -the binary entropy of α -in E and in E 2 .2. For every real number 0 ≤ α ≤ 1, the set SIZE(α 2 n n ), consisting of all languages decidable by Boolean circuits of at most α 2 n n gates, has dimension α in ESPACE.

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“…Resource-bounded measure has given us a generalization of the probabilistic method that works inside complexity classes (leading, for example, to improved lower bounds on Boolean circuit size [14] and the densities of complete problems [16]) and new complexity-theoretic hypotheses (e.g., the hypothesis that NP is a non-measure 0 subset of exponential time) with many plausible consequences, i.e., significant explanatory power. The somewhat outdated survey papers [4,2,2,5,17,22] and more recent papers in the bibliography [10] give a more detailed account of the scope of resource-bounded measure and its applications.…”

Section: Introductionmentioning

confidence: 99%

Axiomatizing Resource Bounds for Measure

Lutz

Nandakumar

et al. 2011

Models of Computation in Context

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Resource-bounded measure is a generalization of classical Lebesgue measure that is useful in computational complexity. The central parameter of resource-bounded measure is the resource bound ∆, which is a class of functions. When ∆ is unrestricted, i.e., contains all functions with the specified domains and codomains, resource-bounded measure coincides with classical Lebesgue measure. On the other hand, when ∆ contains functions satisfying some complexity constraint, resource-bounded measure imposes internal measure structure on a corresponding complexity class. Most applications of resource-bounded measure use only the "measure-zero/measure-one fragment" of the theory. For this fragment, ∆ can be taken to be a class of type-one functions (e.g., from strings to rationals). However, in the full theory of resource-bounded measurability and measure, the resource bound ∆ also contains type-two functionals. To date, both the full theory and its zero-one fragment have been developed in terms of a list of example resource bounds chosen for their apparent utility. This paper replaces this list-of-examples approach with a careful investigation of the conditions that suffice for a class ∆ to be a resource bound. Our main theorem says that every class ∆ that has the closure properties of Mehlhorn's basic feasible functionals is a resource bound for measure. We also prove that the type-2 versions of the time and space hierarchies that have been extensively used in resource-bounded measure have these closure properties. In the course of doing this, we prove theorems establishing that these time and space resource bounds are all robust.Keywords: basic feasible functionals, computational complexity, resource-bounded measure, type-two functionals IntroductionResource-bounded measure is a generalization of classical Lebesgue measure theory that allows us to quantify the "sizes" (measures) of interesting subsets of various complexity classes. This quantitative capability has been useful in computational complexity because it has intersected informatively with reducibilities, completeness, randomization, circuit-size, and many other central ideas of complexity theory. Resource-bounded ⋆ A preliminary version of the paper was presented in the Workshop on Logic in Computational Complexity, 2009.

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Complete Sets and Structure in Subrecursive Classes

Abstract: In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions.

Cited by 9 publications

References 65 publications

Autoreducibility, mitoticity, and immunity

Autoreducibility, mitoticity, and immunity

Dimension in Complexity Classes

Axiomatizing Resource Bounds for Measure

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