Using the renormalization group to classify Boolean functions

This paper argues that the renormalization group technique used to characterize phase transitions in condensed matter systems can be used to classify Boolean functions. A renormalization group transformation is presented that maps an arbitrary Boolean function ofN Boolean variables to one ofN−1 variables. Applying this transformation to a generic Boolean function (one whose output for each input is chosen randomly and independently to be one or zero with equal probability) yields another generic Boolean function. Moreover, applying the transformation to some other functions known to be non-generic, such as Boolean functions that can be written as polynomials of degree ξ with ξ N and functions that depend on composite variables such as the arithmetic sum of the inputs, yields non-generic results. One can thus define different phases of Boolean functions as classes of functions with different types of behavior upon repeated application of the renormalization transformation. Possible relationships between different phases of Boolean functions and computational complexity classes studied in computer science are discussed.