Information Theory and the Complexity of Boolean Functions

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: The effect of the relative number of O\u27s and l\u27s in a function\u27s table on its complexity. The effect of the number of unspecified entries in a partially specified function\u27s table on its complexity. 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 fractiond of the entries are unspecified and a fractionp of the specified entries are l\u27swith errors allowed in a fraction not more thane of the specified entries, is less by the factor (1 −d) [H(p) − H(e)] (whereH(z) = −z log2 z −(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