Denseness of volatile and nonvolatile sequences of functions

TL;DR

This paper investigates the properties of volatile and non-volatile sequences of Boolean functions, showing that volatile sequences are densely distributed among all sequences, while non-volatile sequences are not dense among noise-stable sequences.

Contribution

It establishes the density of volatile sequences within all Boolean function sequences and the non-density of non-volatile sequences among noise-stable sequences.

Findings

Volatile sequences are dense in the set of all Boolean sequences.

Non-volatile sequences are not dense among noise-stable sequences.

The study extends the understanding of volatility and stability in Boolean functions.

Abstract

In a recent paper by Jonasson and Steif, definitions to describe the volatility of sequences of Boolean functions, \( f_n \colon \{ -1,1 \}^n \to \{ -1,1 \} \) were introduced. We continue their study of how these definitions relate to noise stability and noise sensitivity. Our main results are that the set of volatile sequences of Boolean functions is a natural way "dense" in the set of all sequences of Boolean functions, and that the set of non-volatile Boolean sequences is not "dense" in the set of noise stable sequences of Boolean functions.