2001 (English)In: Journal of computer and system sciences (Print), ISSN 0022-0000, E-ISSN 1090-2724, Vol. 62, no 4, 589-607 p.Article in journal (Refereed) Published
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 M. Blum et al. (J. Comput. System Sci. 47 (1993), 549-595) 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 is an element of 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-2(n) and H = F-2. Questions bearing close relationship to linear-consistency testing seem to hav e been implicitly considered in recent work on the construction of PCPs and in particular in the work of J. Hastad  (in Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing. El Paso. Texas, 4-6 May 1997, pp. 1-10). 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 For All epsilon > 0, NP = MIP1-epsilon 1/2 [O(log n), 3, 1]. That is, 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 - epsilon and rejects noninstances with probability at least;. Such a result is of some interest in the study of probabilistically checkable proofs.
Place, publisher, year, edition, pages
2001. Vol. 62, no 4, 589-607 p.
IdentifiersURN: urn:nbn:se:kth:diva-20781ISI: 000169803100003OAI: oai:DiVA.org:kth-20781DiVA: diva2:339478
QC 201005252010-08-102010-08-10Bibliographically approved