Linear Consistency Testing

• Yonatan Aumann
• Michael O. Rabin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1671)

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,yG 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.

Keywords

Abelian Group Proof System Linearity Testing Rejection Probability Parallel Repetition
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

1. 1.
Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)
2. 2.
Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)
3. 3.
Aumann, Y., Rabin, M.O.: Manuscript (1999)Google Scholar
4. 4.
Bellare, M., Coppersmith, D., Håstad, J., Kiwi, M., Sudan, M.: Linearity testing in characteristic two. IEEE Transactions on Information Theory 42(6), 1781–1795 (1996)
5. 5.
Bellare, M., Goldreich, O., Sudan, M.: Free bits, PCPs, and non- approximability-towards tight results. SIAM Journal on Computing 27(3), 804–915 (1998)
6. 6.
Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistically checkable proofs and applications to approximation. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on the Theory of Computing, San Diego, California, May 16-18, pp. 294–304 (1993)Google Scholar
7. 7.
Blum, M., Kannan, S.: Designing programs that check their work. Journal of the ACM 42(1), 269–291 (1995)
8. 8.
Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences 47(3), 549–595 (1993)
9. 9.
Håstad, J.: Some optimal inapproximability results. In: Proceedings of the Twenty- Ninth Annual ACM Symposium on Theory of Computing, El Paso, Texas, May 4-6, pp. 1–10 (1997)Google Scholar
10. 10.
Raz, R.: A parallel repetition theorem. SIAM Journal on Computing 27(3), 763–803 (1998)
11. 11.
Trevisan, L.: Recycling queries in PCPs and in linearity tests. In: STOC (1998)Google Scholar

Authors and Affiliations

• Yonatan Aumann
• 1
• 2
• Michael O. Rabin
• 3
• 4
• 5
1. 1.Department of Mathematics and Computer ScienceBar-Ilan UniversityRamat-GanIsrael
2. 2.Department of Numerical Analysis and Computing ScienceRoyal Institute of TechnologyStockholmSweden
3. 3.DEASHarvard UniversityCambridgeUSA
4. 4.Institute of Computer ScienceHebrew UniversityJerusalemIsrael
5. 5.Department of Electrical Engineering and Computer Science, MITCambridgeUSA