The number of Boolean functions computed by formulas of a given size

Abstract

Estimates are given of the number B(n, L) of distinct functions computed by propositional formulas of size L in n variables, constructed using only literals and ∧, ∨ connectives. (L is the number of occurrences of variables. L−1 is the number of binary ∧s and ∨s. B(n, L) is also the number of functions computed by two terminal series-parallel networks with L switches.) Enroute the read-once functions, which are closely related to Schroder numbers, are enumerated. Writing B(n, L)=b(n, L)L, we find that if L and β(n) go to infinity with increasing n and L≤2n/nβ(n), then b(n, L)∼cn, where c=2/(ln 4−1). Making a comparison with polynomial size Boolean circuits, this implies the following. For any constant α>1, almost all Boolean functions with formula complexity at most nα cannot be computed by any circuit constructed from literals and fewer than α−1nα two-input ∧, ∨ gates. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 349–382, 1998