Some 3CNF Properties Are Hard to Test

• Authors: Eli Ben-Sasson;Prahladh Harsha;Sofya Raskhodnikova
• Venue: SIAM Journal on Computing
• Year: 2005

For a Boolean formula $\phi$ on n variables, the associated property $P_\phi$ is the collection of n-bit strings that satisfy $\phi$. We study the query complexity of tests that distinguish (with high probability) between strings in $P_\phi$ and strings that are far from $P_\phi$ in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires $\Omega(n)$ queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with $O(\sqrt{n})$ queries [E. Fischer et al., Monotonicity testing over general poset domains, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 474--483]. Notice that for every negative instance (i.e., an assignment that does not satisfy $\phi$) there are three bit queries that witness this fact. Nevertheless, finding such a short witness requires reading a constant fraction of the input, even when the input is very far from satisfying the formula that is associated with the property.A property is linear if its elements form a linear space. We provide sufficient conditions for linear properties to be hard to test, and in the course of the proof include the following observations which are of independent interest: In the context of testing for linear properties, adaptive two-sided error tests have no more power than nonadaptive one-sided error tests. Moreover, without loss of generality, any test for a linear property is a linear test. A linear test verifies that a portion of the input satisfies a set of linear constraints, which define the property, and rejects if and only if it finds a falsified constraint. A linear test is by definition nonadaptive and, when applied to linear properties, has a one-sided error. Random low density parity check codes (which are known to have linear distance and constant rate) are not locally testable. In fact, testing such a code of length n requires $\Omega(n)$ queries.