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Volume 5 (2009) Article 10 pp. 191-216
Distribution-Free Testing Lower Bound for Basic Boolean Functions
Received: September 18, 2008
Published: October 17, 2009
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Keywords: property testing, distribution-free testing, decision list, conjunction, linear threshold function
ACM Classification: F.2.2, G.3
AMS Classification: 68Q99, 69W20

Abstract: [Plain Text Version]

$ \newcommand{\D}{{\cal D}} $

In the distribution-free property testing model, the distance between functions is measured with respect to an arbitrary and unknown probability distribution $\D$ over the input domain. We consider distribution-free testing of several basic Boolean function classes over $\{0,1\}^n$, namely monotone conjunctions, general conjunctions, decision lists, and linear threshold functions. We prove that for each of these function classes, $\Omega((n/\log n)^{1/5})$ oracle calls are required for any distribution-free testing algorithm. Since each of these function classes is known to be distribution-free properly learnable (and hence testable) using $\Theta(n)$ oracle calls, our lower bounds are polynomially related to the best possible.