Lower bounds for testing triangle-freeness in Boolean functions

Abstract

Given a Boolean function $${f : \mathbb{F}_{2}^{n} \to \{0,1\}}$$ , we say a triple (x, y, x + y) is a triangle in f if $${f(x)\!=\!f(y)\!=\!f(x+y)\!=\!1}$$ . A triangle-free function contains no triangle. If f differs from every triangle-free function on at least $${\epsilon \cdot 2^n}$$ points, then f is said to be $${\epsilon}$$ -far from triangle-free. In this work, we analyze the query complexity of testers that, with constant probability, distinguish triangle-free functions from those $${\epsilon}$$ -far from triangle-free. Let the canonical tester for triangle-freeness denotes the algorithm that repeatedly picks x and y uniformly and independently at random from $${\mathbb{F}_2^n}$$ , queries f(x), f(y) and f(x + y), and checks whether f(x) = f(y) = f(x + y) = 1. Green showed that the canonical tester rejects functions $${\epsilon}$$ -far from triangle-free with constant probability if its query complexity is a tower of 2’s whose height is polynomial in $${1/\epsilon}$$ . Fox later improved the height of the tower in Green’s upper bound to $${O(\log{1/\epsilon})}$$ . A trivial lower bound of $${\Omega(1/\epsilon)}$$ on the query complexity is immediate. In this paper, we give the first non-trivial lower bound for the number of queries needed. We show that, for every small enough $${\epsilon}$$ , there exists an integer $${n_{0}(\epsilon)}$$ such that for all $${n\geq n_{0}}$$ there exists a function $${f :\mathbb{F}_{2}^{n} \to \{0,1\}}$$ depending on all n variables which is $${\epsilon}$$ -far from being triangle-free and requires $${\Omega \left((\frac{1}{\epsilon})^{4.847\cdots}\right)}$$ queries for the canonical tester. We also show that the query complexity of any general (possibly adaptive) one-sided tester for triangle-freeness is at least square root of the query complexity of the corresponding canonical tester. Consequently, this means that any one-sided tester for triangle-freeness must make at least $${\Omega\left((\frac{1}{\epsilon})^{2.423\cdots}\right)}$$ queries.