Testing for Concise Representations
Ilias Diakonikolas,
Homin Lee, Kevin Matulef, Krzysztof Onak, Ronitt Rubinfeld, Rocco Servedio, Andrew Wan
48th Annual Symposium on Foundations of Computer Science (FOCS), 2007
Abstract:
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. \cite{FKR+:04} with ideas from learning theory, and yields property testers that make poly(s/epsilon) queries (independent of n) for Boolean function classes such as s-term DNF formulas (answering a question posed by Parnas et al. \cite{PRS02}), 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/epsilon)-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 Ω-tilde (sqrt(s)) query lower bound for non-adaptively 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 non-adaptive algorithms) up to a polynomial factor.
Conference version: [PDF]
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ECCC report: [ECCC]