A tame sequence of transitive Boolean functions

TL;DR

This paper constructs a specific sequence of transitive Boolean functions that counter the previous conjecture, showing that the number of function value changes can be unbounded yet still tight.

Contribution

It provides an explicit example of a transitive Boolean function sequence that disproves the conjecture linking non-degeneracy, transitivity, and unbounded expected change count.

Findings

Counterexample to the conjecture.

Transitive Boolean functions can have unbounded expected change counts while remaining tight.

Challenges previous assumptions about function sequence behavior.

Abstract

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.