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Abstract:

We extend the notion of linearity testing to the task of checking linear-consistency of multiple functions. Informally, functions are "linear " if their graphs form straight lines on the plane. Two such functions are "consistent " if the lines have the same slope. We propose a variant of a test of Blum, Luby and Rubinfeld [8] to check the linear-consistency of three functions f 1; f 2; f 3 mapping a finite Abelian group G to an Abelian group H: Pick x; y 2 G uniformly and independently at random and check if f 1 (x) + f 2 (y) = f 3 (x + y). We analyze this test for two cases: (1) G and H are arbitrary Abelian groups and (2) G = F n 2 and H = F 2. Questions bearing close relationship to linear-consistency testing seem to have been implicitly considered in recent work on the construction of PCPs (and in particular in the work of Hastad [9]). It is abstracted explicitly for the first time here. We give an application of this problem (and of our results): A (yet another) new and tight characterization of NP, namely 8ffl? 0; NP = MIP 1\Gammaffl;

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