Towards a General Direct Product Testing Theorem
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The Direct Product encoding of a string a∈{0,1}^n on an underlying domain V⊆nk, is a function DP_V(a) which gets as input a set S∈ V and outputs a restricted to S. In the Direct Product Testing Problem, we are given a function F:V→{0,1}^k, and our goal is to test whether F is close to a direct product encoding, i.e., whether there exists some a∈{0,1}^n such that on most sets S, we have F(S)=DP_V(a)(S). A natural test is as follows: select a pair (S,S')∈ V according to some underlying distribution over V× V, query F on this pair, and check for consistency on their intersection. Note that the above distribution may be viewed as a weighted graph over the vertex set V and is referred to as a test graph. The testability of direct products was studied over various specific domains and test graphs (for example see Dinur-Steurer [CCC'14]; Dinur-Kaufman [FOCS'17]). In this paper, we study the testability of direct products in a general setting, addressing the question: what properties of the domain and the test graph allow one to prove a direct product testing theorem? Towards this goal we introduce the notion of coordinate expansion of a test graph. Roughly speaking a test graph is a coordinate expander if it has global and local expansion, and has certain nice intersection properties on sampling. We show that whenever the test graph has coordinate expansion then it admits a direct product testing theorem. Additionally, for every k and n we provide a direct product domain V⊆nk of size n, called the Sliding Window domain for which we prove direct product testability.
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