Dimension, Halfspaces, and the Density of Hard Sets

Harkins, Ryan C.; Hitchcock, John M.

doi:10.1007/978-3-540-73545-8_15

Cited by 4 publications

References 18 publications

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Weights of Exact Threshold Functions

Babai

Hansen

Podolskii

et al. 2010

Mathematical Foundations of Computer Science 2010

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We consider Boolean exact threshold functions defined by linear equations, and in general degree d polynomials. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close. In the linear case in particular they are almost matching. This quantity is the same as the maximum magnitude of integer coefficients of linear equations required to express every possible intersection of a hyperplane in R n and the Boolean cube {0, 1} n , or in the general case intersections of hypersurfaces of degree d in R n and the Boolean cube {0, 1} n . In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension k of the affine subspace spanned by the solutions, and give upper and lower bounds in this case as well. Our bounds here in terms of k leave a substantial gap, a challenge for future work.

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Weights of Exact Threshold Functions

Babai

Hansen

Podolskii

et al. 2010

Mathematical Foundations of Computer Science 2010

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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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NE Is Not NP Turing Reducible to Nonexponentially Dense NP Sets

Fu¹

2012

LATIN 2012: Theoretical Informatics

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A long standing open problem in the computational complexity theory is to separate NE from BPP, which is a subclass of NPT(NP) ∩ P/Poly. In this paper, we show that NE ⊆ NPT(NP ∩ Nonexponentially-Dense-Class), where Nonexponentially-Dense-Class is the class of languages A without exponential density (for each constant c > 0, |A ≤n | ≤ 2 n c for infinitely many integers n). Our result implies NE ⊆ NPT(padding(NP, g(n))) for every time constructible super-polynomial function g(n) such as g(n) = n ⌈log⌈log n⌉⌉ , where Padding (NP, g(n)) is class of all languages LB = {s10 g(|s|)−|s|−1 : s ∈ B} for B ∈ NP. We also show NE ⊆ NPT(Ptt(NP) ∩ TALLY).

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Dimension, Halfspaces, and the Density of Hard Sets

Cited by 4 publications

References 18 publications

Weights of Exact Threshold Functions

Weights of Exact Threshold Functions

Axiomatizing Resource Bounds for Measure

NE Is Not NP Turing Reducible to Nonexponentially Dense NP Sets

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