We initiate a study of doubly-efficient interactive proofs of proximity, while focusing on properties that can be tested within query-complexity that is significantly sub-linear, and seeking interactive proofs of proximity in which 1. The query-complexity of verification is significantly smaller than the query-complexity of testing. 2. The query-complexity of the honest prover strategy is not much larger than the query-complexity of testing. We call such proof systems doubly-sublinear IPPs (dsIPPs). We present a few doubly-sublinear IPPs. A salient feature of these IPPs is that the honest prover does not employ an optimal strategy (i.e. a strategy that maximizes the verifier's acceptance probability). In particular, the honest prover in our IPP for sets recognizable by constant-width read-once oblivious branching programs uses a distance-approximator for such sets.

We initiate a study of a new model of property testing that is a hybrid of testing properties of distributions and testing properties of strings. Specifically, the new model refers to testing properties of distributions, but these are distributions over huge objects (i.e., very long strings). Accordingly, the model accounts for the total number of local probes into these objects (resp., queries to the strings) as well as for the distance between objects (resp., strings), and the distance between distributions is defined as the earth mover's distance with respect to the relative Hamming distance between strings. We study the query complexity of testing in this new model, focusing on three directions. First, we try to relate the query complexity of testing properties in the new model to the sample complexity of testing these properties in the standard distribution testing model. Second, we consider the complexity of testing properties that arise naturally in the new model (e.g., distributions that capture random variations of fixed strings). Third, we consider the complexity of testing properties that were extensively studied in the standard distribution testing model: Two such cases are uniform distributions and pairs of identical distributions.

We consider the complexity of enumerating ordered sets, defined as solving the following type of a computational problem: For a predetermined ordered set, given i ∈ N , one is required to answer with the i th member of the set (according to the predetermined order). Our focus is on countable sets such as the primes and the rational numbers, although in these cases we provide no decisive answers. In general, we do not report of any exciting results, but rather make a few observations and suggest some open problems.

For a natural number m , a distribution is called m -grained, if each element appears with probability that is an integer multiple of 1/m . We prove that, for any constant c<1 , testing whether a distribution over [Θ(m)] is m -grained requires Ω(m^c) samples, where testing a property of distributions means distinguishing between distributions that have the property and distributions that are far (in total variation distance) from any distribution that has the property.

We consider the problem of testing isomorphism to a fixed graph in the bounded-degree graph model. Our main result is that, for almost all d -regular n -vertex graphs H , testing isomorphism to H can be done using (cid:101) O ( √ n ) queries. This result is shown to be optimal (up to a polylog factor) by a matching lower bound, which also holds for almost all graphs H . The performance of our tester depends on natural graph parameters of the fixed ( n -vertex) graph H such as its diameter and the minimum radius of “distinguishing neighborhoods” (i.e., the minimum r = r ( n ) such that the “ r -neighborhoods” of the n different vertices are pairwise non-isomorphic)

Interactive proofs of proximity (IPPs) offer ultra-fast approximate verification of assertions regarding their input, where ultra-fast means that only a small portion of the input is read and approximate verification is analogous to the notion of approximate decision that underlies property testing. Specifically, in an IPP, the prover can make the verifier accept each input in the property, but cannot fool the verifier into accepting an input that is far from the property (except for with small probability). The verifier in an IPP system engages in two very different types of activities: interacting with an untrusted prover, and querying its input. The definition allows for arbitrary coordination between these two activities, but keeping them separate is both conceptually interesting and necessary for important applications such as addressing temporal considerations (i.e., at what time is each of the services available) and facilitating the construction of zero-knowledge schemes. In this work we embark on a systematic study of IPPs with proof-oblivious queries, where the queries should not be affected by the interaction with the prover. We assign the query and interaction activities to separate modules, and consider different limitations on their coordination. The most strict limitation requires these activities to be totally isolated from one another; they just feed their views to a separate deciding module. We show that such systems can be efficiently emulated by standard testers. Going to the other extreme, we only disallow information to flow from the interacting module to the querying module, but allow free information flow in the other direction. We show that extremely efficient one-round (i.e., two-message) systems of such type can be used to verify properties that are extremely hard to test (without the help of a prover). That is, the complexity of verifying can be polylogarithmic in the complexity of testing. This stands in contrast the MAPs (viewed as 1 / 2-round systems) in which proof-oblivious queries are as limited as our isolated model. Our focus is on an intermediate model that allows shared randomness between the querying and interacting modules but no information flow between them. In this case we show that 1-round systems are efficiently emulated by standard testers but 3 / 2-round systems of extremely low complexity exist for properties that are extremely hard to test. One additional result about this model is that it can efficiently emulate any IPP for any property of low-degree polynomials.

We consider the problem of efficiently constructing an as large as possible family of permutations such that each pair of permutations are far part (i.e., disagree on a constant fraction of their inputs). Specifically, for every n ∈ ℕ, we present a collection of N = N ( n ) = ( n !) Ω(1) pairwise far apart permutations { π i : [ n ] → [ n ]} i ∈[ N ] and a polynomial-time algorithm that on input i ∈ [ N ] outputs an explicit description of π i . From a coding theoretic perspective, we construct permutation codes of constant relative distance and constant rate along with efficient encoding (and decoding) algorithms. This construction is easily extended to produce constant composition codes on smaller alphabets, where in these codes every codeword is balanced; namely, each symbol appears the same number of times. Our construction combines routing on the Shuffle-Exchange network with any good binary error correcting code. Specifically, we uses codewords of a good binary code in order to determine the switching instructions in the Shuffle-Exchange network.