Number of Restrictions of a Boolean Function act as a Complexity Measure

Abstract

Complexity of boolean functions can be computed in many ways. Various complexity measures exist which are based on different models of representation of the boolean function. The complexity measures range from very coarse and simple to very fine and hard to compute. The introduced complexity meausre called Restriction complexity is a measure based on number of restrictions of a boolean function. In this paper we define restriction complexity, compute its values for some functions, and determine its range in the form of upper and lower bounds on it. We also show that restriction complexity is close to the cardinality of support measure in an almost all sense as defined by Abu-Mostafa.