Self-Predicting Boolean Functions

TL;DR

This paper studies self-predicting Boolean functions, which are functions that perfectly predict themselves as optimal predictors under noisy conditions, providing insights into their properties and optimality.

Contribution

It introduces the concept of self-predicting functions and characterizes their properties within the context of optimal prediction under noise.

Findings

Identification of conditions for self-predicting functions

Characterization of optimal predictors in noisy settings

Insights into the structure of functions that predict themselves

Abstract

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $\delta$. This paper is about self-predicting functions, which are those that coincide with their optimal predictor.