On the complexity of linear boolean operators with thin matrices

Gashkov, S. B.; Sergeev, I. S.

doi:10.1134/s1990478911020074

Cited by 7 publications

(7 citation statements)

References 10 publications

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“…Separations between ∨-and ⊕-circuits have also been considered by Sergeev et al [11,12] who proved the slightly weaker bound Gap ∨/⊕ (n) = Ω(n/(log 6 n log log n)). Furthermore, Jukna [17] has informed us that the bound in Corollary 3 can actually be proved more directly using existing methods [15,28].…”

Section: Our Resultsmentioning

confidence: 93%

Separating OR, SUM, and XOR circuits

Göös

Järvisalo

Kaski

et al. 2016

Journal of Computer and System Sciences

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Given a boolean n × n matrix A we consider arithmetic circuits for computing the transformation x ↦ Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative integers), and non-monotone XOR-circuits (addition modulo 2). Our focus is on separating OR-circuits from the two other models in terms of circuit complexity: We show how to obtain matrices that admit OR-circuits of size O(n), but require SUM-circuits of size Ω(n3/2/log2n).We consider the task of rewriting a given OR-circuit as a XOR-circuit and prove that any subquadratic-time algorithm for this task violates the strong exponential time hypothesis.

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Section: Our Resultsmentioning

confidence: 93%

Separating OR, SUM, and XOR circuits

Göös

Järvisalo

Kaski

et al. 2016

Journal of Computer and System Sciences

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“…The following theorem allows to extend results [1,2] on comparison of complexity of implementation of some circulant matrices via rectifier circuits and modulo 2 rectifier circuits to bounded depth circuits. The proof is by induction.…”

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confidence: 90%

“…Следующая теорема позволяет распространить результаты [1,2] о сравнении сложности реализации некоторых циркулянтных матриц вентильными схемами и вентильными схемами по модулю 2 на схемы ограниченной глубины.…”

Section: Compute Coefficients Of C(x)unclassified

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Implementation of linear maps with circulant matrices via modulo 2 rectifier circuits of bounded depth

Sergeev

2013

Preprint

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In the present note we show that for any constant k ∈ N an arbitrary Boolean circulant matrix can be implemented via modulo 2 rectifier circuit of depth 2k − 1 and complexity O n 1+1/k , and also via circuit of depth 2k and complexity O n 1+1/k log −1/k n . * Research supported in part by RFBR, grants 11-01-00508, 11-01-00792, and OMN RAS "Algebraic and combinatorial methods of mathematical cybernetics and information systems of new generation" program (project "Problems of optimal synthesis of control systems").

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“…Данный факт является некоторым аналогом одного результата из [64]. Из результата [23] (который, впрочем, может быть доказан методом Лупанова) вытекает верхняя оценка L + (A n )/L(A n ) O(n/ log n) для произвольной булевой матрицы.…”

Section: лемма 14 (I) пустьunclassified

Об Одном Методе Получения Нижних Оценок Сложности Монотонных Арифметических Схем, Вычисляющих Действительные Многочлены

Gashkov¹,

Sergeev²

2012

Матем. сб.

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On the complexity of linear boolean operators with thin matrices

Cited by 7 publications

References 10 publications

Separating OR, SUM, and XOR circuits

Separating OR, SUM, and XOR circuits

Implementation of linear maps with circulant matrices via modulo 2 rectifier circuits of bounded depth

Об Одном Методе Получения Нижних Оценок Сложности Монотонных Арифметических Схем, Вычисляющих Действительные Многочлены

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