On The Influences of Variables on Boolean Functions in Product Spaces

TL;DR

This paper introduces a unified framework for defining variable influences on Boolean functions in general product spaces, generalizes the BKKKL theorem, and extends known influence results.

Contribution

It proposes a family of influence definitions encompassing existing ones and proves a generalized, tight BKKKL theorem applicable to these definitions.

Findings

Unified influence definitions for Boolean functions in product spaces

Generalized BKKKL theorem applicable to all influence definitions

Extended known influence results to broader settings

Abstract

In this paper we consider the influences of variables on Boolean functions in general product spaces. Unlike the case of functions on the discrete cube where there is a clear definition of influence, in the general case at least three definitions were presented in different papers. We propose a family of definitions for the influence, that contains all the known definitions, as well as other natural definitions, as special cases. We prove a generalization of the BKKKL theorem, which is tight in terms of the definition of influence used in the assertion, and use it to generalize several known results on influences in general product spaces.