## 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 ϵ 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 = {formula presened} and H = 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 8∀ ϵ> 0; NP = MIP{formula presnted}. 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 {formula presented}. Such a result is of some interest in the study of probabilistically checkable proofs.

Original language | English |
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Title of host publication | Randomization, Approximation, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999, Proceedings |

Editors | Jose D. P. Rolim, Alistair Sinclair, Dorit Hochbaum, Klaus Jansen |

Publisher | Springer Verlag |

Pages | 109-120 |

Number of pages | 12 |

ISBN (Print) | 3540663290, 9783540663294 |

DOIs | |

State | Published - 1999 |

Event | 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999 - Berkeley, United States Duration: 8 Aug 1999 → 11 Aug 1999 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 1671 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 3rd International Workshop on Randomization and Approximation Techniques in Computer Science and 2nd International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, RANDOM-APPROX 1999 |
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Country/Territory | United States |

City | Berkeley |

Period | 8/08/99 → 11/08/99 |

### Bibliographical note

Publisher Copyright:© Springer-Verlag Berlin Heidelberg 1999.

### Funding

We thank the reviewers of RANDOM’99 as well as the referee of the current paper for numerous comments and corrections. M.O.R.’s research was supported in part by NSF grant NSF-CCR-97-00365. M.S.’s research was supported in part by a Sloan Foundation Fellowship, and MIT-NEC Research Initiation Grant, and NSF Career Award CCR-9875511.

Funders | Funder number |
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MIT-NEC | CCR-9875511 |

Alfred P. Sloan Foundation |