The Direct Product encoding of a string @math on an underlying domain @math , is a function DP @math which gets as input a set @math and outputs @math restricted to @math . In the Direct Product Testing Problem, we are given a function @math , and our goal is to test whether @math is close to a direct product encoding, i.e., whether there exists some @math such that on most sets @math , we have @math DP @math . A natural test is as follows: select a pair @math according to some underlying distribution over @math , query @math 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 @math 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 @math and @math we provide a direct product domain @math of size @math , called the Sliding Window domain for which we prove direct product testability.
The question of testing the direct product was studied extensively when the underlying domain @math @cite_5 @cite_10 @cite_12 @cite_9 @cite_3 . In this setting, Goldreich and Safra @cite_5 proposed a constant query test. Dinur and Reingold @cite_10 suggested the two-query test mentioned above and analyzed it in the high acceptance regime but with a relaxed distance measure.
@cite_9 @cite_3 @cite_5 @cite_10 @cite_12
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We consider monotonicity testing of functions f:[n]d→ 0,1 , in the property testing framework of Rubinfeld and Sudan [23] and Goldreich, Goldwasser and Ron [14]. Specifically, we consider the framework of distribution-free property testing, where the distance between functions is measured with respect to a fixed but unknown distribution D on the domain and the testing algorithms have an oracle access to random sampling from the domain according to this distribution D. We show that, though in the uniform distribution case, testing of boolean functions defined over the boolean hypercube can be done using query complexity that is polynomial in @math and in the dimension d, in the distribution-free setting such testing requires a number of queries that is exponential in d. Therefore, in the high-dimensional case (in oppose to the low-dimensional case), the gap between the query complexity for the uniform and the distribution-free settings is exponential. The “direct-product code” of a function @math gives its values on all @math -tuples @math . This basic construct underlies “hardness amplification” in cryptography, circuit complexity, and probabilistically checkable proofs (PCPs). Goldreich and Safra [SIAM J. Comput., 29 (2000), pp. 1132--1154] pioneered its local testing and its PCP application. A recent result by Dinur and Goldenberg [Proceedings of the Forty-Ninth Annual IEEE Symposium on Foundations of Computer Science, 2008, pp. 613--622] enabled for the first time testing proximity to this important code in the “list-decoding” regime. In particular, they give a @math -query test which works for polynomially small success probability @math and show that no such test works below success probability @math . Our main result is a @math -query test which works for exponentially small success probability @math . Our techniques (based on recent simplified decoding algorithms for the same code [R. , Proceeding... Combinatorial property testing, initiated by Rubinfeld and Sudan [23] and formally defined by Goldreich, Goldwasser and Ron in [18], deals with the following relaxation of decision problems: Given a fixed property P and an input f, distinguish between the case that f satisfies P, and the case that no input that differs from f in less than some fixed fraction of the places satisfies P. An (e, q)-test for P is a randomized algorithm that queries at most q places of an input f and distinguishes with probability 2 3 between the case that f has the property and the case that at least an e-fraction of the places of f need to be changed in order for it to have the property. Here we concentrate on labeled, d-dimensional grids, where the grid is viewed as a partially ordered set (poset) in the standard way (i.e. as a product order of total orders). The main result here presents an (e, poly(1 e))-test for every property of 0 1 labeled, d-dimensional grids that is characterized by a finite collection of forbidden induced posets. Such properties include the “monotonicity” property studied in [9,8,13], other more complicated forbidden chain patterns, and general forbidden poset patterns. We also present a (less efficient) test for such properties of labeled grids with larger fixed size alphabets. All the above tests have in addition a 1-sided error probability. This class of properties is related to properties that are defined by certain first order formulae with no quantifier alternation over the syntax containing the grid order relations. We also show that with one quantifier alternation, a certain property can be defined, for which no test with query complexity of O(n 1 4) (for a small enough fixed e) exists. The above results identify new classes of properties that are defined by means of restricted logics, and that are efficiently testable. They also lay out a platform that bridges some previous results. A direct product function is a function of the form g(x_1, ldots, x_k)=(g_1(x_1), ldots, g_k(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]^kto[M]^k begin quote bf 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. End quote We provide a comprehensive analysis of this test for all parameters N, M, k, tle O(k) and success probability delta>0. Our main result is that if a given function f:[N]^kto[M]^k passes the test with probability delta ge 1-eps then there is a direct product function g such that Pr[f(x)=g(x)]ge 1-O(eps). 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 ll k ll 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 delta ge 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 delta ge exp(-sqrt[3]k) to the entire meaningful range of delta>exp(-k).