NDLI: Direct Product Testing

Content Provider	IEEE Xplore Digital Library
Author	Dinur, I. Steurer, D.
Copyright Year	2014
Abstract	A direct product function is a function of the form $g(x_{1},$ ⋯, $x_{k})=(g_{1}(x_{1}),$ ⋯, $g(x_{k})).$ We show that the direct product property is locally testable with two queries, that is, a canonical two-query test distinguishes between direct product functions and functions that are far from direct products with constant probability. This local testing question comes up naturally in the context of PCPs, where direct products play a prominent role for gap amplification. We consider the following natural two query test for a given function $f:[N]^{k}→[M]^{k}$ Two query direct product test: Choose x, y that agree on a random set A of t coordinates and accept if $f(x)_{A}=f(y)_{A}.$ We provide a comprehensive analysis of this test for all parameters N, M, k, t≤O(k) and success probability δ>0. Our main result is that if a given function $f:[N]^{k}→[M]^{k}$ passes the test with probability δ≥1-ε then there is a direct product function g such that P[f(x)=g(x)]≥1-O(ε). This is the first result relating success in the above (or any) test to the fraction of the domain on which f is equal to a direct product function. This test has been analyzed in previous works for the case t≪k≪N, and results show closeness of f to a direct product under a less natural measure of "approximate agreement". In the small soundness regime, we prove that if the test above passes with probability δ ≥ exp(-k), then the function agrees with a direct product function on local parts of the domain. This extends the previous range of parameters of δ≥exp(-3√k) to the entire meaningful range of δ>exp(-k).
Starting Page	188
Ending Page	196
File Size	277028
Page Count	9
File Format	PDF
ISBN	9781479936267
ISSN	10930159
DOI	10.1109/CCC.2014.27
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2014-06-11
Publisher Place	Canada
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Games Testing Context Decoding Polynomials Conferences Computational complexity probabilistically checkable proofs parallel repetition property testing
Content Type	Text
Resource Type	Article

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