Given an instance I of a CSP, a tester for I distinguishes assignments satisfying X from those which are far from any assignment satisfying I. The efficiency of a tester is measured by its query complexity, the number of variable assignments queried by the algorithm. In this paper, we characterize the hardness of testing Boolean CSPs in terms of the algebra generated by the relations used to form constraints. In terms of computational complexity, we show that if a non-trivial Boolean CSP is 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 CSP is constant-query testable (resp., not constant-query testable), then counting the number of solutions of the CSP is in P (resp., #P-complete). Also, we conjecture that a CSP instance is testable in sublinear time if its Gaifman graph has bounded treewidth. We confirm the conjecture when a near-unanimity operation is a polymorphism of the CSP.
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