For a property P and a sub-property P , we say that P is Ppartially testable with q queries if there exists an algorithm that distinguishes, with high probability, inputs in P from inputs -far 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 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. First, we use entropy analysis, which allows us to apply our arguments directly to 2-sided tests, thus avoiding the cost of the conversion in [14] from 2-sided to 1-sided tests. Broadly speaking we use "distinguishing instances" of a supposed test to show that a uniformly random choice of a member of the sub-property 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. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from permissions@acm.org. 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 query-efficient standard property test. We provide some preliminary results concerning this question, including a simple lower bound on the possible trade-off. Our second major result is a positive trade-off result for the restricted framework of 1-sided 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 sample-based testing (for a non-constant number of queries) as defined by Goldreich and Ron in [13] . In particular it partially resolves an open problem raised by [13] .
Independently of this work, Fischer, Goldhirsh and Lachish REF introduced the notion of partial testing, which is closely related to the notion of MAP.
Partial tests, universal tests and decomposability
ITCS '14
Computer Science Mathematics