Some 3CNF properties are hard to test

For a Boolean formula on n variables, the associated property P is the collection of n-bit strings that satisfy . We study the query complexity of tests that distinguish (with high probability) between strings in P and strings that are far from P in Hamming distance. We prove that there are 3CNF formulae (with O(n) clauses) such that testing for the associated property requires (n) queries, even with adaptive tests. This contrasts with 2CNF formulae, whose associated properties are always testable with O( 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 ) 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 a...

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Eli Ben-Sasson, Prahladh Harsha, Sofya Raskhodnikova
• Keywords: 34th Annual Acm, Algorithms, Associated Property, General Poset Domains, STOC 2003