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From Local to Robust Testing via Agreement Testing

Authors Irit Dinur, Prahladh Harsha, Tali Kaufman, Noga Ron-Zewi



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Author Details

Irit Dinur
  • Department of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
Prahladh Harsha
  • Tata Institute of Fundamental Research, India
Tali Kaufman
  • Department of Computer Science, Bar-Ilan University, Ramat Gan, Israel
Noga Ron-Zewi
  • Department of Computer Science, University of Haifa, Haifa, Israel

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Irit Dinur, Prahladh Harsha, Tali Kaufman, and Noga Ron-Zewi. From Local to Robust Testing via Agreement Testing. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 29:1-29:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.29

Abstract

A local tester for an error-correcting code is a probabilistic procedure that queries a small subset of coordinates, accepts codewords with probability one, and rejects non-codewords with probability proportional to their distance from the code. The local tester is robust if for non-codewords it satisfies the stronger property that the average distance of local views from accepting views is proportional to the distance from the code. Robust testing is an important component in constructions of locally testable codes and probabilistically checkable proofs as it allows for composition of local tests. In this work we show that for certain codes, any (natural) local tester can be converted to a roubst tester with roughly the same number of queries. Our result holds for the class of affine-invariant lifted codes which is a broad class of codes that includes Reed-Muller codes, as well as recent constructions of high-rate locally testable codes (Guo, Kopparty, and Sudan, ITCS 2013). Instantiating this with known local testing results for lifted codes gives a more direct proof that improves some of the parameters of the main result of Guo, Haramaty, and Sudan (FOCS 2015), showing robustness of lifted codes. To obtain the above transformation we relate the notions of local testing and robust testing to the notion of agreement testing that attempts to find out whether valid partial assignments can be stitched together to a global codeword. We first show that agreement testing implies robust testing, and then show that local testing implies agreement testing. Our proof is combinatorial, and is based on expansion / sampling properties of the collection of local views of local testers. Thus, it immediately applies to local testers of lifted codes that query random affine subspaces in F_q^m, and moreover seems amenable to extension to other families of locally testable codes with expanding families of local views.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Local testing
  • Robust testing
  • Agreement testing
  • Affine-invariant codes
  • Lifted codes

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