Ω(n log n) Lower Bounds on Length of Boolean Formulas

Abstract

A property of Boolean functions of n variables is described and shown to imply lower bounds as large as Ω(n log n) on the number of literals in any Boolean formula for any function with the property. Formulas over the full basis of binary operations (∧, ⊕, etc.) are considered. The lower bounds apply to all but the vanishing fraction of symmetric functions, in particular to all threshold functions with sufficiently large threshold and to the "congruent to zero modulo k" function for k>2. In the case k = 4 the bound is optimal.