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, and in fact the minimal $O(1/\epsilon)$.

We prove that this is not always the case. More than that, 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 size. 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 more intricate technique based on constructing a decision tree. The root-to-leaf paths in this tree rearrange the input into parts where each part 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.