Non-Interactive Proofs of Proximity

Abstract

We initiate a study of non-interactive proofs of proximity. These proof-systems consist of a verifier that wishes to ascertain the validity of a given statement, using a short (sublinear length) explicitly given proof, and a sublinear number of queries to its input. Since the verifier cannot even read the entire input, we only require it to reject inputs that are far from being valid. Thus, the verifier is only assured of the proximity of the statement to a correct one. Such proofsystems can be viewed as the N P (or more accurately MA) analogue of property testing.We explore both the power and limitations of non interactive proofs of proximity. We show that such proof-systems can be exponentially stronger than property testers, but are exponentially weaker than the interactive proofs of proximity studied by Rothblum, Vadhan and Wigderson (STOC 2013). In addition, we show a natural problem that has a full and (almost) tight multiplicative trade-off between the length of the proof and the verifier's query complexity. On the negative side, we also show that there exist properties for which even a linearly-long (non-interactive) proof of proximity cannot significantly reduce the query complexity.

Trading Query Complexity for Sample-Based Testing and Multi-testing Scalability

“…Since r is at most 2 (10α 2 ) −1 n (q −2 −γ) , this gives the 2-sided (ǫ, 1/10)-test with query complexity O(n 1−γ ). Definition 6.3 (following Definition 2.1 of [11]). A Merlin-Arthur proof of proximity (MAP) for a property L ⊆ Ξ n , with proximity parameter ǫ, query complexity q and proof complexity p, consists of a probabilistic algorithm V , called the verifier, that is given a proof string π ∈ Ξ p ; in addition, it is given oracle access to a word w ∈ Ξ n , to which it is allowed to make up to q queries.…”

“…This in turn allows to convert in certain cases tests requiring proofs as per the MAP scenario (defined in [11] and also developed in [6]) to tests that still have a sublinear query complexity but do not require such proofs. In this setting we deploy the generalization of our result to partial testing, as a MAP scenario converts to a union of partial testing problems.…”

“…We define partial tests (of which tests are a special case), because we would like our main result to also have applications in the realm of MAPs as defined in [11].…”

Partial tests, universal tests and decomposability

“…Some natural properties require many queries to test, but can be partitioned into a small number of subsets for which they are partially testable with very few queries, sometimes even a number independent of the input size.For properties over 0, 1, the notion of being thus partitionable ties in closely with Merlin-Arthur proofs of Proximity (MAPs) as defined independently in [14]; a partition into r partially-testable properties is the same as a Merlin-Arthur system where the proof consists of the identity of one of the r partially-testable properties, giving a 2-way translation to an O(log r) size proof.Our main result is that for some low complexity properties a partition as above cannot exist, and moreover that for each of our properties there does not exist even a single subproperty featuring both a large size and a query-efficient partial test, in particular improving the lower bound set in [14]. For this we use neither the traditional Yao-type arguments nor the more recent communication complexity method, but open up a new approach for proving lower bounds.…”

“…For properties over 0, 1, the notion of being thus partitionable ties in closely with Merlin-Arthur proofs of Proximity (MAPs) as defined independently in [14]; a partition into r partially-testable properties is the same as a Merlin-Arthur system where the proof consists of the identity of one of the r partially-testable properties, giving a 2-way translation to an O(log r) size proof.…”