Influence of a Set of Variables on a Boolean Function

TL;DR

This paper introduces a new auto-correlation based definition of influence for sets of variables on Boolean functions, generalizing existing concepts and providing new characterizations and inequalities.

Contribution

It proposes a novel influence measure based on auto-correlation, extending theoretical understanding and characterizations of Boolean functions.

Findings

New influence measure generalizes previous definitions

Derived Poincaré inequality and edge expansion properties

Characterized resilient and bent functions using influence

Abstract

The influence of a variable is an important concept in the analysis of Boolean functions. The more general notion of influence of a set of variables on a Boolean function has four separate definitions in the literature. In the present work, we introduce a new definition of influence of a set of variables which is based on the auto-correlation function and develop its basic theory. Among the new results that we obtain are generalisations of the Poincar\'e inequality and the edge expansion property of the influence of a single variable. Further, we obtain new characterisations of resilient and bent functions using the notion of influence. We show that the previous definition of influence due to Fischer et. al. (2002) and Blais (2009) is half the value of the auto-correlation based influence that we introduce. Regarding the other prior notions of influence, we make a detailed study of these and show that each of these definitions do not satisfy one or more desirable properties that a notion of influence may be expected to satisfy.