On the power of small-depth threshold circuits

Håstad, Johan; Goldmann, Mikael

doi:10.1109/fscs.1990.89582

Cited by 79 publications

(85 citation statements)

References 14 publications

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“…Showing NP is not in ACC 0 is one of the frontiers in complexity theory. It is well known that a function in ACC 0 has a polylog(n) k-party deterministic communication complexity, where k is polylog(n) [17,7]. In fact the protocol is simultaneous where all the players, without interacting, speak once to an external referee who determines the output based only on the messages she receives.…”

Section: Introductionmentioning

confidence: 99%

The NOF Multiparty Communication Complexity of Composed Functions

et al. 2014

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Section: Introductionmentioning

confidence: 99%

The NOF Multiparty Communication Complexity of Composed Functions

et al. 2014

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“…This model is harder to analyze than the 2-party model, and very few lower bounds are known. On the other hand, lower bounds in this model have many applications in complexity theory, including constructions of pseudorandom generators for space bounded computation, universal traversal sequences, and time-space tradeoffs [2], as well as circuit complexity lower bounds [13,16,18]. vious lower bounds fit into a "nearly" Hadamard framework.…”

Section: Introductionmentioning

confidence: 99%

Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

Ford

Gál

2005

Automata, Languages and Programming

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Abstract. We develop a new method for estimating the discrepancy of tensors associated with multiparty communication problems in the "Number on the Forehead" model of Chandra, Furst and Lipton. We define an analogue of the Hadamard property of matrices for tensors in multiple dimensions and show that any k-party communication problem represented by a Hadamard tensor must have Ω(n/2 k ) multiparty communication complexity. We also exhibit constructions of Hadamard tensors, giving Ω(n/2 k ) lower bounds on multiparty communication complexity for a new class of explicitly defined Boolean functions.

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“…Moreover, any explicit lower bound on the multiparty communication complexity in the NOF model with (log n) ω (1) players would yield to superpolynomial circuit lower bounds for the class ACC 0 of bounded-depth circuits with modulo gates [79,5,32]. Thus proving lower bounds in the NOF model is a well-motivated problem.…”

Section: Problem 5 What Is the "Right" Definition Of The Multiparty mentioning

confidence: 99%

Interactive Information Complexity

Braverman¹

2017

SIAM Rev.

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Abstract. The primary goal of this paper is to define and study the interactive information complexity of functions. Let f (x, y) be a function, and suppose Alice is given x and Bob is given y. Informally, the interactive information complexity IC(f ) of f is the least amount of information Alice and Bob need to reveal to each other to compute f . Previously, information complexity has been defined with respect to a prior distribution on the input pairs (x, y). Our first goal is to give a definition that is independent of the prior distribution. We show that several possible definitions are essentially equivalent. We establish some basic properties of the interactive information complexity IC(f ). In particular, we show that IC(f ) is equal to the amortized (randomized) communication complexity of f . We also show a direct sum theorem for IC(f ) and give the first general connection between information complexity and (nonamortized) communication complexity. This connection implies that a nontrivial exchange of information is required when solving problems that have nontrivial communication complexity. We explore the information complexity of two specific problems: equality and disjointness. We show that only a constant amount of information needs to be exchanged when solving equality with no errors, while solving disjointness with a constant error probability requires the parties to reveal a linear amount of information to each other. This paper originally appeared in [SIAM J. Comput., 44 (2015), pp. 1698-1739]. The present version has been slightly updated with some recent developments.

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On the power of small-depth threshold circuits

Cited by 79 publications

References 14 publications

The NOF Multiparty Communication Complexity of Composed Functions

The NOF Multiparty Communication Complexity of Composed Functions

Hadamard Tensors and Lower Bounds on Multiparty Communication Complexity

Interactive Information Complexity

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