Testing for concise representations

We describe a general method for testing whether a function on n input variables has a concise representation. The approach combines ideas from the junta test of Fischer et al. [6] with ideas from learning theory, and yields property testers that make poly(s/t) queries (independent of n) for Boolean function classes such as s-term DNF formulas (answering a question posed by Parnas et al. [12]), size-s decision trees, size-s Boolean formulas, and size-s Boolean circuits. The method can be applied to non-Boolean valued function classes as well. This is achieved via a generalization of the notion of variation from Fischer et al. to non-Boolean functions. Using this generalization we extend the original junta test of Fischer et al. to work for non-Boolean functions, and give poly(s/ε)-query testing algorithms for non-Boolean valued function classes such as size-s algebraic circuits and s-sparse polynomials over finite fields. We also prove an ω̃(√ s) query lower bound for nonadaptively testing s-sparse polynomials over finite fields of constant size. This shows that in some instances, our general method yields a property tester with query complexity that is optimal (for nonadaptive algorithms) up to a polynomial factor.