Some 3CNF properties are hard to test

For a boolean formula ψ on n variables, the associated property P_ψ is the collection of n-bit strings that satisfy ψ. We prove that there are 3CNF properties that require a linear number of queries, even for adaptive tests. This contrasts with 2CNF properties that are testable with O(√n) queries [7]. Notice that for every bad instance (i.e. an assignment that does not satisfy ψ) there is a 3-bit query that witnesses this fact. Nevertheless, finding such a short witness requires a linear number of queries, even for assignments that are very far from satisfying. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include a couple of observations which are of independent interest. 1. In the context of linear property testing, adaptive 2-sided error tests have no more power than non-adaptive 1-sided error tests. 2. Random linear LDPC codes with linear distance and constant rate are very far from being locally testable.