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

Abstract

A certificate of non-membership for a Boolean function f with respect to a class 𝒞, f ∉ 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 {&, ∨, $\neg$}, 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 ... x_s ∨ ${\bar x}_1$ ... ${\bar x}_s$, we show that there exist n-variable functions requiring Ω(n^s−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.