TY - GEN
T1 - Hard properties with (very) short Pcpps and their applications
AU - Ben-Eliezer, Omri
AU - Fischer, Eldar
AU - Levi, Amit
AU - Rothblum, Ron D.
N1 - Publisher Copyright: © Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Ron D. Rothblum.
PY - 2020/1
Y1 - 2020/1
N2 - We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed `, we construct a property P(`) ⊆ {0, 1}n satisfying the following: Any testing algorithm for P(`) requires Ω(n) many queries, and yet P(`) has a constant query PCPP whose proof size is O(n · log(`) n), where log(`) denotes the ` times iterated log function (e.g., log(2) n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n · polylog n). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed `, we construct a property that has a constant-query tester, but requires Ω(n/log(`)(n)) queries for every tolerant or erasure-resilient tester.
AB - We show that there exist properties that are maximally hard for testing, while still admitting PCPPs with a proof size very close to linear. Specifically, for every fixed `, we construct a property P(`) ⊆ {0, 1}n satisfying the following: Any testing algorithm for P(`) requires Ω(n) many queries, and yet P(`) has a constant query PCPP whose proof size is O(n · log(`) n), where log(`) denotes the ` times iterated log function (e.g., log(2) n = log log n). The best previously known upper bound on the PCPP proof size for a maximally hard to test property was O(n · polylog n). As an immediate application, we obtain stronger separations between the standard testing model and both the tolerant testing model and the erasure-resilient testing model: for every fixed `, we construct a property that has a constant-query tester, but requires Ω(n/log(`)(n)) queries for every tolerant or erasure-resilient tester.
KW - Coding theory
KW - Erasure resilient testing
KW - PCPP
KW - Property testing
KW - Randomized encoding
KW - Tolerant testing
UR - https://www.scopus.com/pages/publications/85078040384
U2 - 10.4230/LIPIcs.ITCS.2020.9
DO - 10.4230/LIPIcs.ITCS.2020.9
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020
A2 - Vidick, Thomas
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 11th Innovations in Theoretical Computer Science Conference, ITCS 2020
Y2 - 12 January 2020 through 14 January 2020
ER -