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Towards a General Direct Product Testing Theorem Goldenberg, Elazar C. S., Karthik Property Testing Direct Product PCP Johnson graph Ramanujan Complex Derandomization The Direct Product encoding of a string a in {0,1}^n on an underlying domain V subseteq ([n] choose k), is a function DP_V(a) which gets as input a set S in 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 in {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')in V according to some underlying distribution over V x 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 domains and test graphs: Dinur and Steurer (CCC '14) analyzed it when V equals the k-th slice of the Boolean hypercube and the test graph is a member of the Johnson graph family. Dinur and Kaufman (FOCS '17) analyzed it for the case where V is the set of faces of a Ramanujan complex, where in this case V=O_k(n). 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 subseteq (n choose k) of size n, called the Sliding Window domain for which we prove direct product testability. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Elazar Goldenberg and Karthik C. S. 2018 Is Part Of LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.FSTTCS.2018.11 urn:nbn:de:0030-drops-99105 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.11 eng https://creativecommons.org/licenses/by/3.0/legalcode