论文标题

传统布尔函数的温和序列

A tame sequence of transitive Boolean functions

论文作者

Forsström, Malin Palö

论文摘要

给定一系列布尔函数$(f_n)_ {n \ geq 1} $,$ f_n \ colon \ {0,1 \}^{n}^{n} \ to \ to \ {0,1 \} $ x^{(n)} =(x_t^{(n)})_ {t \ geq 0} $上的$ \ {0,1 \}^{n} $,让$ c_n $为$(0,1)$ in the Process $(f_n(x_t)$ geq $ geq 0 progence $(0,1)$(0,1)$(0,1)的次数。 In \cite{js2006}, the authors conjectured that if $ (f_n)_{n \geq 1} $ is non-degenerate, transitive and satisfies $ \lim_{n \to \infty} \mathbb{E}[C_n] = \infty$, then $ (C_n)_{n \geq 1} $ is not tight.我们给出了一系列布尔函数序列的明确示例,该函数反驳了这种猜想。

Given a sequence of Boolean functions $(f_n)_{n \geq 1}$, $f_n \colon \{ 0,1 \}^{n} \to \{ 0,1 \}$, and a sequence $(X^{(n)})_{n\geq 1} $ of continuous time $p_n $-biased random walks $ X^{(n)} = (X_t^{(n)})_{t \geq 0}$ on $ \{ 0,1 \}^{n}$, let $ C_n $ be the (random) number of times in $(0,1) $ at which the process $ (f_n(X_t))_{t \geq 0} $ changes its value. In \cite{js2006}, the authors conjectured that if $ (f_n)_{n \geq 1} $ is non-degenerate, transitive and satisfies $ \lim_{n \to \infty} \mathbb{E}[C_n] = \infty$, then $ (C_n)_{n \geq 1} $ is not tight. We give an explicit example of a sequence of Boolean functions which disproves this conjecture.

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