Testing Assignments of Boolean CSPs

Given an instance I of a CSP, a tester for I distinguishes assignments satisfying I fromthose which are far from any assignment satisfying I. The efficiency of a tester is measuredby its query complexity, the number of variable assignments queried by the algorithm. In thispaper, we characterize the hardness of testing Boolean CSPs according to the relations usedto form constraints. In terms of computational complexity, we show that if a Boolean CSPis sublinear-query testable (resp., not sublinear-query testable), then the CSP is in NL (resp.,P-complete, ⊕L-complete or NP-complete) and that if a sublinear-query testable Boolean CSPis constant-query testable (resp., not constant-query testable), then counting the number ofsolutions of the CSP is in P (resp., #P-complete). The classification is done by showing an Ω(n)lower bound for testing Horn 3-SAT and investigating Post’s lattice, the inclusion structure ofBoolean algebras associated with CSPs.