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.
@cite_9: 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).
@cite_3: 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...
@cite_5: The current proof of the probabilistically checkable proofs (PCP) theorem (i.e., @math ) is very complicated. One source of difficulty is the technically involved analysis of low-degree tests. Here, we refer to the difficulty of obtaining strong results regarding low-degree tests; namely, results of the type obtained and used by Arora and Safra [J. ACM, 45 (1998), pp. 70--122] and [J. ACM, 45 (1998), pp. 501--555]. In this paper, we eliminate the need to obtain such strong results on low-degree tests when proving the PCP theorem. Although we do not remove the need for low-degree tests altogether, using our results it is now possible to prove the PCP theorem using a simpler analysis of low-degree tests (which yields weaker bounds). In other words, we replace the strong algebraic analysis of low-degree tests presented by Arora and Safra and by a combinatorial lemma (which does not refer to low-degree tests or polynomials).
@cite_10: In this work we look back into the proof of the PCP (probabilistically checkable proofs) theorem, with the goal of finding new proofs that are “more combinatorial” and arguably simpler. For that we introduce the notion of an assignment tester, which is a strengthening of the standard PCP verifier, in the following sense. Given a statement and an alleged proof for it, while the PCP verifier checks correctness of the statement, the assignment tester checks correctness of the statement and the proof. This notion enables composition that is truly modular; i.e., one can compose two assignment testers without any assumptions on how they are constructed. A related notion called PCPs of proximity was independently introduced in [E. Ben-, Proceedings of the 36th Annual ACM Symposium on Theory of Computing, Chicago, IL, 2004, ACM, New York, 2004, pp. 1-10]. We provide a toolkit of (nontrivial) generic transformations on assignment testers. These transformations may be interesting in their own right, and allow us to present the following two main results: 1. A new proof of the PCP theorem. This proof relies on a rather weak assignment tester given as a “black box.” From this, we construct combinatorially the full PCP. An important component of this proof is a new combinatorial aggregation technique (i.e., a new transformation that allows the verifier to read fewer, though possibly longer, “pieces” of the proof). An implementation of the black-box tester can be obtained from the algebraic proof techniques that already appear in [L. , Proceedings of the 23rd ACM Symposium on Theory of Computing, New Orleans, LA, 1991, ACM, New York, 1991, pp. 21-31; U. , J. ACM, 43 (1996), pp. 268-292]. 2. Our second construction is a “standalone” combinatorial construction showing @math . This implies, for example, that approximating max-SAT is quasi-NP-hard. This construction relies on a transformation that makes an assignment tester “oblivious,” so that the proof locations read are independent of the statement that is being proven. This eliminates, in a rather surprising manner, the need for aggregation in a crucial point in the proof.
@cite_12: Given a function f : X rarr Sigma, its lscr-wise direct product is the function F = flscr : Xlscr rarr Sigmalscr defined by: F(x1,...,xlscr) = (f(x1),...,f(xlscr)). We are interested in the local testability of the direct product encoding (mapping f rarr flscr). Namely, given an arbitrary function F : Xlscr rarr Sigmalscr, we wish to determine how close it is to flscr for some f : X rarr Sigma, by making two random queries into F. In this work we analyze the case of low acceptance probability of the test. We show that even if the test passes with small probability, epsiv>0, already F must have a non-trivial structure and in particular must agree with some flscr on nearly epsiv of the domain. Moreover, we give a structural characterization of all functions F on which the test passes with probability epsiv. Our results can be viewed as a combinatorial analog of the low error dasialow degree testpsila, that is used in PCP constructions.
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.