Influence of a Set of Variables on a Boolean Function.

The influence of a set of variables on a Boolean function has three separate definitions in the literature, the first due to Ben-Or and Linial (1989), the second due to Fischer et al. (2002) and Blais (2009) and the third due to Tal (2017). The goal of the present work is to carry out a comprehensive study of the notion of influence of a set of variables on a Boolean function. To this end, we introduce a definition of this notion using the auto-correlation function. A modification of the definition leads to the notion of pseudo-influence. Somewhat surprisingly, it turns out that the auto-correlation based definition of influence is equivalent to the definition introduced by Fischer et al. (2002) and Blais (2009) and the notion of pseudo-influence is equivalent to the definition of influence considered by Tal (2017). Extensive analysis of influence and pseduo-influence as well as the Ben-Or and Linial notion of influence is carried out and the relations between these notions are established.