# Lower bounds for testing triangle-freeness in Boolean functions

Bhattacharyya, Arnab and Xie, Ning (2015) Lower bounds for testing triangle-freeness in Boolean functions. In: COMPUTATIONAL COMPLEXITY, 24 (1). pp. 65-101.

 PDF com_com-24_1_65_2015.pdf - Published Version Restricted to Registered users only Download (345kB) | Request a copy
Official URL: http://dx.doi.org/10.1007/s00037-014-0092-1

## Abstract

Given a Boolean function , we say a triple (x, y, x + y) is a triangle in f if . A triangle-free function contains no triangle. If f differs from every triangle-free function on at least points, then f is said to be -far from triangle-free. In this work, we analyze the query complexity of testers that, with constant probability, distinguish triangle-free functions from those -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 , 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 -far from triangle-free with constant probability if its query complexity is a tower of 2's whose height is polynomial in . Fox later improved the height of the tower in Green's upper bound to . A trivial lower bound of 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 , there exists an integer such that for all there exists a function depending on all n variables which is -far from being triangle-free and requires 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 queries.

Item Type: Journal Article Copy right for this article belongs to the SPRINGER BASEL AG, PICASSOPLATZ 4, BASEL, 4052, SWITZERLAND Property testing; query lower bounds; Boolean function triangles Division of Electrical Sciences > Computer Science & Automation Id for Latest eprints 24 Apr 2015 05:45 24 Apr 2015 05:45 http://eprints.iisc.ac.in/id/eprint/51376