Division in logspace-uniform<i>NC</i><sup>1</sup>

Chiu, Andrew; Davida, George I.; Litow, Bruce

doi:10.1051/ita:2001119

Cited by 74 publications

(59 citation statements)

References 17 publications

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“…We can do so by replacing each addition and multiplication gate with logtime-uniform constant-depth threshold circuitry. The existence of such circuitry is folklore in the case of addition gates; for multiplication gates this was shown by Hesse [Hes01,HAB02], building on earlier work by Chiu [Chi95,CDL01]. AND, OR, and negation gates can be easily transformed into threshold gates, and the latter into majority gates.…”

Section: Randomized Algorithms With Unbounded Errormentioning

confidence: 99%

A Survey of Lower Bounds for Satisfiability and Related Problems

Melkebeek¹

2006

FNT in Theoretical Computer Science

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Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a linear-time, logarithmic-space algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving non-trivial lower bounds on the computational complexity of satisfiability.In this article we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the state-of-the-art results and present the underlying arguments in a unified framework.

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Section: Randomized Algorithms With Unbounded Errormentioning

confidence: 99%

A Survey of Lower Bounds for Satisfiability and Related Problems

Melkebeek¹

2006

FNT in Theoretical Computer Science

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“…Hence there exists exactly one such α ′ in [0, a), namely α ′ = α rem a. Moreover, integer division belongs to N C [BCH86,CDL01]. …”

Section: Proofsmentioning

confidence: 99%

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Ziegler

2009

Applied Mathematics and Computation

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Abstract. We turn 'the' Church-Turing Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and well-defined scientific problem(s):Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory-rather than vaguely referring to "nature" in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability and complexity in computational physics.

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“…To compute the desired coefficient in PSPACE, we use the Chinese Remaindering technique; See [CDL01] for more details. Since symmetric polynomials are easy to compute (e.g.…”

Section: Monomialsmentioning

confidence: 99%

Small Space Analogues of Valiant’s Classes and the Limitations of Skew Formulas

Mahajan

Rao

2011

comput. complex.

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In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the "space" complexity of the computed function. Looking for a similar relationship in Valiant's algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. We introduce the class VL as an algebraic variant of deterministic log-space L. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width.Further, to define algebraic variants of non-deterministic space-bounded classes, we introduce the notion of "read-once" certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs can be expressed as a read-once exponential sum over polynomials in VL, i.e. VBP ∈ Σ R · VL. We also show that Σ R · VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sums over a restricted class of constant-width arithmetic circuits are within VQP, and this is the largest known such subclass of poly-log-width circuits with this property.We also study the power of skew formulas and show that exponential sums of a skew formula cannot represent the determinant polynomial.

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Division in logspace-uniformNC¹

Cited by 74 publications

References 17 publications

A Survey of Lower Bounds for Satisfiability and Related Problems

A Survey of Lower Bounds for Satisfiability and Related Problems

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Small Space Analogues of Valiant’s Classes and the Limitations of Skew Formulas

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Division in logspace-uniformNC1

Cited by 74 publications

References 17 publications

A Survey of Lower Bounds for Satisfiability and Related Problems

A Survey of Lower Bounds for Satisfiability and Related Problems

Physically-relativized Church–Turing Hypotheses: Physical foundations of computing and complexity theory of computational physics

Small Space Analogues of Valiant’s Classes and the Limitations of Skew Formulas

Contact Info

Product

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Division in logspace-uniformNC¹