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REPORTS > KEYWORD > SYMMETRIC BOOLEAN FUNCTIONS:
Reports tagged with symmetric Boolean functions:
TR13-032 | 26th February 2013
Mark Bun, Justin Thaler

Dual Lower Bounds for Approximate Degree and Markov-Bernstein Inequalities

Revisions: 2

The $\epsilon$-approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within $\epsilon$ in the $\ell_\infty$ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an ... more >>>


TR19-138 | 6th October 2019
Srikanth Srinivasan, Utkarsh Tripathi, S Venkitesh

On the Probabilistic Degrees of Symmetric Boolean functions

The probabilistic degree of a Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is defined to be the smallest $d$ such that there is a random polynomial $\mathbf{P}$ of degree at most $d$ that agrees with $f$ at each point with high probability. Introduced by Razborov (1987), upper and lower bounds on probabilistic degrees ... more >>>


TR20-046 | 13th April 2020
Srikanth Srinivasan

A Robust Version of Heged\H{u}s's Lemma, with Applications

Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of ... more >>>


TR23-044 | 28th March 2023
Sourav Chakraborty, Chandrima Kayal, Manaswi Paraashar

Separations between Combinatorial Measures for Transitive Functions

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory.
For example, symmetric functions, that is, functions that are invariant under the action of $S_n$ is an important class of functions in the study of Boolean functions.
A function $f:\{0,1\}^n \to \{0,1\}$ ... more >>>


TR23-099 | 8th July 2023
Sourav Chakraborty, Chandrima Kayal, Rajat Mittal, Manaswi Paraashar, Swagato Sanyal, Nitin Saurabh

On the Composition of Randomized Query Complexity and Approximate Degree

Revisions: 1

For any Boolean functions $f$ and $g$, the question whether $\text{R}(f\circ g) = \tilde{\Theta}(\text{R}(f) \cdot \text{R}(g))$, is known as the composition question for the randomized query complexity. Similarly, the composition question for the approximate degree asks whether $\widetilde{\text{deg}}(f\circ g) = \tilde{\Theta}(\widetilde{\text{deg}}(f)\cdot\widetilde{\text{deg}}(g))$. These questions are two of the most important and ... more >>>




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