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 f1,f2,f3 mapping a finite Abelian group G to an Abelian group H: Pick x, y ∈ G uniformly and independently at random and check if f1(x)+f2(y) =f3(x + y). We analyze this test for two cases: (1) G and H are arbitrary Abelian groups and (2) G = Fn 2 and H = F2. 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 H˚astad [9]. It is abstracted explicitly for the first time here. As an application of our results we give yet another new and tight characterization of NP, namely ∀ɛ>0, NP = MIP 1 1−ɛ, [O(log n), 3, 1]. 2 I.e., every language in NP has 3-prover 1-round proof systems in which the verifier tosses O(log n) coins and asks each of the three provers one question each. The provers respond with one bit

Citations