  For a property $P$ and a sub-property $P'$, 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 $\epsilon$-far from $P$ by using
$q$ queries. There are natural properties that 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.
  We prove the existence of a property $P$ such that the only subsets $P'$ for
which $P$ is $P'$-partially testable are very small. To prove this we introduce
new techniques for proving property testing lower bounds. In addition to
obtaining some broad-brush criteria for non-testability, this implies a lower
bound on the possibility of PCPPs with a sublinear proof. This also implies
lower bounds on MAPs, a notion newly defined by Gur and Rothblum.
  The new techniques rely on analyzing a proposed partial tester. We show that
the queries performed by a tester must, with high probability, query indexes
where a uniformly random member of the sub-property has low entropy. We then
show how one can aggregate the "entropy loss" to deduce that a random choice in
the sub-property must have low entropy, and therefore the sub-property must be
small.
  We develop two techniques for aggregating the entropy loss. A simpler
technique that applies to non-adaptive testers is based on partitioning the
input bits into high query probability parts and parts where there is an
entropy loss when conditioned on the high probability parts. Against adaptive
testers we develop a technique based on constructing a decision tree. The
root-to-leaf paths in this tree rearrange the input into parts where each
exhibits entropy loss when conditioned on the path prefix. This decision tree
is constructed by combining carefully selected decision trees from those used
by the adaptive testing algorithm.
