Partial tests, universal tests and decomposability
Fischer, E. and Goldhirsh, Y. and Lachish, Oded (2014) Partial tests, universal tests and decomposability. In: UNSPECIFIED (ed.) ITCS '14: Proceedings of the 5th conference on Innovations in theoretical computer science. New York, U.S.: ACM, pp. 483500. ISBN 9781450326988.

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Abstract
For a property P and a subproperty P', we say that P is P'partially testable with q queries} if there exists an algorithm that distinguishes, with high probability, inputs in P' from inputs εfar from P, using q queries. 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 MerlinArthur proofs of Proximity (MAPs) as defined independently in [14] a partition into r partiallytestable properties is the same as a MerlinArthur system where the proof consists of the identity of one of the r partiallytestable properties, giving a 2way 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 queryefficient partial test, in particular improving the lower bound set in [14]. For this we use neither the traditional Yaotype arguments nor the more recent communication complexity method, but open up a new approach for proving lower bounds. First, we use entropy analysis, which allows us to apply our arguments directly to 2sided tests, thus avoiding the cost of the conversion in [14] from 2sided to 1sided tests. Broadly speaking we use "distinguishing instances" of a supposed test to show that a uniformly random choice of a member of the subproperty has "low entropy areas", ultimately leading to it having a low total entropy and hence having a small base set. Additionally, to have our arguments apply to adaptive tests, we use a mechanism of "rearranging" the input bits (through a decision tree that adaptively reads the entire input) to expose the low entropy that would otherwise not be apparent. We also explore the possibility of a connection in the other direction, namely whether the existence of a good partition (or MAP) can lead to a relatively queryefficient standard property test. We provide some preliminary results concerning this question, including a simple lower bound on the possible tradeoff. Our second major result is a positive tradeoff result for the restricted framework of 1sided proximity oblivious tests. This is achieved through the construction of a "universal tester" that works the same for all properties admitting the restricted test. Our tester is very related to the notion of samplebased testing (for a nonconstant number of queries) as defined by Goldreich and Ron in [13]. In particular it partially resolves an open problem raised by [13].
Metadata
Item Type:  Book Section 

School:  Birkbeck Faculties and Schools > Faculty of Science > School of Computing and Mathematical Sciences 
Depositing User:  Dr Oded Lachish 
Date Deposited:  02 Nov 2015 15:11 
Last Modified:  09 Aug 2023 12:37 
URI:  https://eprints.bbk.ac.uk/id/eprint/13241 
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