Linear Consistency Testing

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 ∈ 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) $G = \mathbb{F}^n_2$ and $H = \mathbb{F}_2$.

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 Håstad [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 ∀ ε > 0. $\mathrm{NP} = \mathrm{MIP}_{1-\epsilon,\frac{1}{2}}[=(\log n),3,1]$ 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 each such that the verifier accepts instance of the language with probability 1– ε and rejects non-instances with probability at least $\frac{1}{2}$. Such a result is of some interest in the study of probabilistically checkable proofs.