Certificates of Non-Membership for Classes of Read-Once Functions

A certificate of non-membership for a Boolean function f with respect to a class C, f 6∈ C, is a set S of input strings such that the values of f on strings from S are inconsistent with any function h ∈ C. We study certificates of non-membership with respect to several classes of read-once functions, generated by their bases. For the basis {&,∨,¬}, we determine the optimal certificate size for every function outside the class and deduce that 6 strings always suffice. For the same basis augmented with a function x1 . . . xs ∨ x1 . . . xs, we show that there exist n-variable functions requiring Ω(ns−1) strings in a certificate as n → ∞. For s = 2, we show that this bound is tight by constructing certificates of size O(n) for all functions outside the class.