Abstract
We show that every set in P is strongly testable under a suitable encoding. By "strongly testable" we mean having a (proximity oblivious) tester that makes a constant number of queries and rejects with probability that is proportional to the distance of the tested object from the property. By a "suitable encoding" we mean one that is polynomialtime computable and invertible. This result stands in contrast to the known fact that some sets in P are extremely hard to test, providing another demonstration of the crucial role of representation in the context of property testing.
The testing result is proved by showing that any set in P has a strong canonical PCP, where canonical means that (for yesinstances) there exists a single proof that is accepted with probability 1 by the system, whereas all other potential proofs are rejected with probability proportional to their distance from this proof. In fact, we show that UP equals the class of sets having strong canonical PCPs (of logarithmic randomness), whereas the class of sets having strong canonical PCPs with polynomial proof length equals "unambiguous MA". Actually, for the testing result, we use a PCPofProximity version of the foregoing notion and an analogous positive result (i.e., strong canonical PCPPs of logarithmic randomness for any set in UP).
BibTeX  Entry
@InProceedings{dinur_et_al:LIPIcs:2018:10123,
author = {Irit Dinur and Oded Goldreich and Tom Gur},
title = {{Every Set in P Is Strongly Testable Under a Suitable Encoding}},
booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
pages = {30:130:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10123},
URN = {urn:nbn:de:0030drops101234},
doi = {10.4230/LIPIcs.ITCS.2019.30},
annote = {Keywords: Probabilistically checkable proofs, property testing}
}
Keywords: 

Probabilistically checkable proofs, property testing 
Collection: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

08.01.2019 