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

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## 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 |
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Publication: | COMPUTATIONAL COMPLEXITY |

Publisher: | SPRINGER BASEL AG |

Additional Information: | Copy right for this article belongs to the SPRINGER BASEL AG, PICASSOPLATZ 4, BASEL, 4052, SWITZERLAND |

Keywords: | Property testing; query lower bounds; Boolean function triangles |

Department/Centre: | Division of Electrical Sciences > Computer Science & Automation |

Date Deposited: | 24 Apr 2015 05:45 |

Last Modified: | 24 Apr 2015 05:45 |

URI: | http://eprints.iisc.ac.in/id/eprint/51376 |

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