Linear Consistency Testing

Yonatan Aumann, Johan Håstad, Michael O. Rabin, Madhu Sudan

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 \(H = \mathbb{F}_2\).