Untitled

Bun, Mark; Thaler, Justin

doi:10.4086/toc.2016.v012a016

Cited by 8 publications

(1 citation statement)

References 47 publications

(67 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The following standard dual formulation of this first variant of approximate degree can be found in, e. g., [31]. for every polynomial p :…”

Section: Two Variants Of Approximate Degree and Their Dual Formulationsmentioning

confidence: 99%

Untitled

Bun¹,

Kothari²,

Thaler³

2020

Theory of Comput.

Self Cite

View full text Add to dashboard Cite

The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. • k-Distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness function is Ω(n 3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of k-distinctness is O(n 3/4−1/(2 k+2 −4)). • Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n] → [n] isΩ(n 1/2). This proves a conjecture of

show abstract

“…The following standard dual formulation of this first variant of approximate degree can be found in, e. g., [31]. for every polynomial p :…”

Section: Two Variants Of Approximate Degree and Their Dual Formulationsmentioning

confidence: 99%

Untitled

Bun¹,

Kothari²,

Thaler³

2020

Theory of Comput.

Self Cite

View full text Add to dashboard Cite

show abstract

A Nearly Optimal Lower Bound on the Approximate Degree of AC^0

Bun

Thaler

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

View full text Add to dashboard Cite

The approximate degree of a Boolean function f : {−1, 1} n → {−1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constantdepth circuits.Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n · polylog(n)) variables with approximate degree at least, then D is polynomially larger than d. Moreover, if f is computed by a polynomial-size Boolean circuit of constant depth, then so is F .By recursively applying our transformation, for any constant δ > 0 we exhibit an AC 0 function of approximate degree Ω(n 1−δ ). This improves over the best previous lower bound of Ω(n 2/3 ) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width.We describe several applications of these results. We give:• For any constant δ > 0, an Ω(n 1−δ ) lower bound on the quantum communication complexity of a function in AC 0 .• A Boolean function f with approximate degree at least C(f ) 2−o(1) , where C(f ) is the certificate complexity of f . This separation is optimal up to the o(1) term in the exponent.• Improved secret sharing schemes with reconstruction procedures in AC 0 .

show abstract

On the Power of Statistical Zero Knowledge

Bouland¹,

Chen

Holden³

et al. 2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

Self Cite

View full text Add to dashboard Cite

We examine the power of statistical zero knowledge proofs (captured by the complexity class SZK) and their variants. First, we give the strongest known relativized evidence that SZK contains hard problems, by exhibiting an oracle relative to which SZK (indeed, even NISZK) is not contained in the class UPP, containing those problems solvable by randomized algorithms with unbounded error. This answers an open question of Watrous from 2002 [Aar]. Second, we "lift" this oracle separation to the setting of communication complexity, thereby answering a question of Göös et al. (ICALP 2016). Third, we give relativized evidence that perfect zero knowledge proofs (captured by the class PZK) are weaker than general zero knowledge proofs. Specifically, we exhibit oracles relative to which SZK ⊆ PZK, NISZK ⊆ NIPZK, and PZK = coPZK. The first of these results answers a question raised in 1991 by Aiello and Håstad (Information and Computation), and the second answers a question of Lovett and Zhang (2016). We also describe additional applications of these results outside of structural complexity.The technical core of our results is a stronger hardness amplification theorem for approximate degree, which roughly says that composing the gapped-majority function with any function of high approximate degree yields a function with high threshold degree.

show abstract

Untitled

Cited by 8 publications

References 47 publications

Untitled

Untitled

A Nearly Optimal Lower Bound on the Approximate Degree of AC^0

On the Power of Statistical Zero Knowledge

Contact Info

Product

Resources

About