# Information theory and the complexity of boolean functions

## Abstract

This paper explores the connections between two areas pioneered by Shannon: the transmission of information with a fidelity criterion, and the realization of Boolean functions by networks and formulae. We study three phenomena: 1. The effect of the relative number of O's and l's in a function's table on its complexity. 2. The effect of the number of unspecified entries in a partially specified function's table on its complexity. 3. The effect of the number of errors allowed in the realization of a function on its complexity. Our main result is a precise version of the following statement: The complexity of approximately realizing a partially specified Boolean function, in whose table a fraction d of the entries are unspecified and a fraction p of the specified entries are l'swith errors allowed in a fraction not more than e of the specified entries, is less by the factor (1 -d) [H(p) - H(e)] (where H(z) = -z log2z -(1 -z) log2 (1 -z) is the binary entropy function) than the complexity of exactly realizing an arbitrary fully specified Boolean function. We also give an intuitively appealing information-theoretic interpretation of the result. © 1977 Springer-Verlag New York Inc.