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

Issue title: Russian-Finnish Symposium in Discrete Mathematics

Article type: Research Article

Authors: Chistikov, Dmitry | Fedorova, Valentina | Voronenko, Andrey

Affiliations: Max Planck Institute for Software Systems (MPI-SWS), Germany. [email protected] | Moscow State University, Russia. [email protected] | Moscow State University, Russia. [email protected]

Note: [] Part of this research was done while Dmitry Chistikov was affiliated with Moscow State University. Address for correspondence: MPI-SWS, Paul-Ehrlich-Str. 26, 67663 Kaiserslautern, Germany

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 ... xs ∨ ${\bar x}_1$ ... ${\bar x}_s$, 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.

Keywords: certificate of non-membership, read-once function

DOI: 10.3233/FI-2014-1032

Journal: Fundamenta Informaticae, vol. 132, no. 1, pp. 63-77, 2014