Majority gates vs. general weighted threshold gates

Goldmann, Mikael; Håstad, Johan; Razborov, Alexander A.

doi:10.1007/bf01200426

Cited by 138 publications

(31 citation statements)

References 12 publications

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“…Было разработано множество аналитиче-ских и комбинаторных методов для доказательства нижних оценок на пороговый вес [24]- [29], [17]. Весьма активно изучалось также понятие пороговой длины [24], [27], [30], [28], [17].…”

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“…Более точно, изучался наименьший вес, обозначаемый через W (f, d), порогового элемента степени d для функции f , где f фиксирована, а d изменяется. Типы булевых функций, изучав-шихся с этой точки зрения, включают пороговые элементы [25], [27], [31], списки разрешения [21], ДНФ-формулы [22] и другие функции [32], [33]. В последних ре-зультатах [32], [33] доказаны первые дважды-экспоненциальные нижние оценки на W (f, d) для большого числа различных d.…”

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Небольшое Уменьшение Степени Многочлена С Заданной Знаковой Функцией Может Экспоненциально Увеличить Его Вес И Длину

Podolskii¹,

Шерстов²,

Sherstov³

2010

Mat. Zametki

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Небольшое Уменьшение Степени Многочлена С Заданной Знаковой Функцией Может Экспоненциально Увеличить Его Вес И Длину

Podolskii¹,

Шерстов²,

Sherstov³

2010

Mat. Zametki

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“…The latter is a fundamental family of functions that has been a topic of study since the 1960s and that plays an important role in numerous areas. LTGs occur in the study of neural networks where these functions serve as elementary models of neurons (Rosenblatt 1958;Block 1962;Minsky & Papert 1988) and in circuit complexity theory (Goldmann et al 1992;Hajnal et al 1993;Goldmann & Karpinski 1998);cf. Feldman et al (2006), Sherstov (2008), Kalai et al (2008), Khot & Saket (2008), O'Donnell & Servedio (2008), and references within for a number of recent investigations on LTGs.…”

Section: Introductionmentioning

confidence: 99%

Quantum matchgate computations and linear threshold gates

Nest

2010

Proc. R. Soc. A.

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The theory of matchgates is of interest in various areas in physics and computer science. Matchgates occur, for example, in the study of fermions and spin chains, in the theory of holographic algorithms and in several recent works in quantum computation. In this paper, we completely characterize the class of Boolean functions computable by unitary two-qubit matchgate circuits with some probability of success. We show that this class precisely coincides with that of the linear threshold gates. The latter is a fundamental family that appears in several fields, such as the study of neural networks. Using the above characterization, we further show that the power of matchgate circuits is surprisingly trivial in those cases where the computation is to succeed with high probability. In particular, the only functions that are matchgate-computable with success probability greater than 3/4 are functions depending on only a single bit of the input.

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“…It computes a monotone Boolean function where f (x) = 1 iff n i=1 x i w i ≥ T . Weighted threshold functions play an important role in complexity theory (e.g., [21,9,10,2]) and learning theory(e.g., [17,15,16]). The (unweighted) threshold function has a polynomial size monotone formula [1,30].…”

Section: Introductionmentioning

confidence: 99%

“…The question if every monotone weighted threshold function can be exactly computed by a monotone formula remains open. The question of constructing monotone formulae for weighted threshold functions is closely related to an open problem of Goldman and Karpinski [10]; they ask if every weighted threshold function has a monotone constant depth polynomial size circuit with unweighted threshold gates (such simulations are known in the non-monotone case [9,10,2]). If such a monotone simulation is possible then using the formulae of [1,30] we get an efficient monotone formulae for weighted threshold functions.…”

Section: Introductionmentioning

confidence: 99%

Monotone Circuits for Weighted Threshold Functions

Beimel

Weinreb

20th Annual IEEE Conference on Computational Complexity (CCC'05)

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Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently non-monotone since addition is a non-monotone function. In this work we bypass this addition step and construct a polynomial size logarithmic depth unbounded fan-in monotone circuit for every weighted threshold function, i.e., we show that weighted threshold functions are in mAC 1 . (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions.)Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. Using general results for compiling monotone circuits (Yao, 1989) and monotone formulae (Benaloh and Leichter, 1990) into secret sharing schemes, we get secret sharing schemes for every weighted threshold access structure. Specifically, we get: (1) information-theoretic secret sharing schemes where the size of each share is quasi-polynomial in the number of users, and (2) computational secret sharing schemes where the size of each share is polynomial in the number of users.

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Majority gates vs. general weighted threshold gates

Cited by 138 publications

References 12 publications

Небольшое Уменьшение Степени Многочлена С Заданной Знаковой Функцией Может Экспоненциально Увеличить Его Вес И Длину

Небольшое Уменьшение Степени Многочлена С Заданной Знаковой Функцией Может Экспоненциально Увеличить Его Вес И Длину

Quantum matchgate computations and linear threshold gates

Monotone Circuits for Weighted Threshold Functions

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