Abstract
Interactive proofs of proximity allow a sublineartime verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof.
The first variant, known as MAproofs of Proximity (MAP), is fully noninteractive, meaning that the proof is a function of the input only. The second variant, known as AMproofs of Proximity (AMP), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both MAPs and AMPs is the total number of bits that the verifier observes  namely, the sum of the proof length and query complexity.
Our main result is an exponential separation between the power of MAPs and AMPs. Specifically, we exhibit an explicit and natural property Pi that admits an AMP with complexity O(log n), whereas any MAP for Pi has complexity Omega~(n^{1/4}), where n denotes the length of the input in bits. Our MAP lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014).
Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting we show that AMPs can only be quadratically stronger than MAPs. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.
BibTeX  Entry
@InProceedings{gur_et_al:LIPIcs:2018:9077,
author = {Tom Gur and Yang P. Liu and Ron D. Rothblum},
title = {{An Exponential Separation Between MA and AM Proofs of Proximity}},
booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
pages = {73:173:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770767},
ISSN = {18688969},
year = {2018},
volume = {107},
editor = {Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/9077},
URN = {urn:nbn:de:0030drops90772},
doi = {10.4230/LIPIcs.ICALP.2018.73},
annote = {Keywords: Property testing, Probabilistic proof systems, Proofs of proximity}
}
Keywords: 

Property testing, Probabilistic proof systems, Proofs of proximity 
Collection: 

45th International Colloquium on Automata, Languages, and Programming (ICALP 2018) 
Issue Date: 

2018 
Date of publication: 

04.07.2018 