Influence of a Set of Variables on a Boolean Function
This work addresses a foundational issue in theoretical computer science for researchers analyzing Boolean functions, but it is incremental as it builds on prior definitions and focuses on theoretical properties.
The authors tackled the problem of defining the influence of a set of variables on Boolean functions by introducing a new definition based on the auto-correlation function and developing its basic theory, resulting in generalizations of the Poincaré inequality and edge expansion property, as well as new characterizations of resilient and bent functions.
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 Poincaré 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.