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REPORTS > KEYWORD > THRESHOLD DEGREE:
Reports tagged with threshold degree:
TR13-110 | 12th August 2013
Xiaoming Sun, Marcos Villagra

#### Exponential Quantum-Classical Gaps in Multiparty Nondeterministic Communication Complexity

There are three different types of nondeterminism in quantum communication: i) $\nqp$-communication, ii) $\qma$-communication, and iii) $\qcma$-communication. In this \redout{paper} we show that multiparty $\nqp$-communication can be exponentially stronger than $\qcma$-communication. This also implies an exponential separation with respect to classical multiparty nondeterministic communication complexity. We argue that there exists ... more >>>

TR14-009 | 21st January 2014
Alexander A. Sherstov

#### Breaking the Minsky-Papert Barrier for Constant-Depth Circuits

The threshold degree of a Boolean function $f$ is the minimum degree of
a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv\mathrm{sgn}\; p(x)$. In a seminal 1969
monograph, Minsky and Papert constructed a polynomial-size constant-depth
$\{\wedge,\vee\}$-circuit in $n$ variables with threshold degree $\Omega(n^{1/3}).$ This bound underlies ... more >>>

TR15-147 | 8th September 2015
Alexander A. Sherstov

#### The Power of Asymmetry in Constant-Depth Circuits

The threshold degree of a Boolean function $f$ is the minimum degree of
a real polynomial $p$ that represents $f$ in sign: $f(x)\equiv\mathrm{sgn}\; p(x)$. Introduced
in the seminal work of Minsky and Papert (1969), this notion is central to
some of the strongest algorithmic and complexity-theoretic results for
more >>>

TR16-140 | 9th September 2016
Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, Prashant Nalini Vasudevan

#### On SZK and PP

Revisions: 3

In both query and communication complexity, we give separations between the class NISZK, containing those problems with non-interactive statistical zero knowledge proof systems, and the class UPP, containing those problems with randomized algorithms with unbounded error. These results significantly improve on earlier query separations of Vereschagin [Ver95] and Aaronson [Aar12] ... more >>>

TR19-003 | 2nd January 2019
Alexander A. Sherstov, Pei Wu

#### Near-Optimal Lower Bounds on the Threshold Degree and Sign-Rank of AC^0

The threshold degree of a Boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is the minimum degree of a real polynomial $p$ that represents $f$ in sign: $\mathrm{sgn}\; p(x)=(-1)^{f(x)}.$ A related notion is sign-rank, defined for a Boolean matrix $F=[F_{ij}]$ as the minimum rank of a real matrix $M$ with $\mathrm{sgn}\; M_{ij}=(-1)^{F_{ij}}$. Determining the maximum ... more >>>

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