On active and passive testing

TL;DR

This paper investigates the query complexity of active and passive testing for Boolean functions, establishing tight bounds for k-linear functions and exploring other function classes using advanced combinatorial and algebraic methods.

Contribution

It extends the analysis of testing complexity from dictator functions to k-linear functions and other classes, providing tight bounds and novel methodological insights.

Findings

Passive and active testing of k-linear functions requires Theta(k*log n) queries.

The results generalize previous work on dictator functions to broader classes.

The paper introduces new techniques combining algebraic, combinatorial, and probabilistic methods.

Abstract

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have recently suggested to restrict the tester to take its queries from a smaller random subset of polynomial size of the inputs. This model is called active testing, and in the extreme case when the size of the set we can query from is exactly the number of queries performed it is known as passive testing. We prove that passive or active testing of k-linear functions (that is, sums of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is not too large. This extends the case k=1, (that is, dictator functions), analyzed by Balcan et. al. We also consider other classes of functions including low degree polynomials, juntas, and partially symmetric functions. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdos--Rado theorem on Delta-systems.