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On Local Testability in the Non-Signaling Setting

Authors Alessandro Chiesa, Peter Manohar, Igor Shinkar

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Author Details

Alessandro Chiesa
  • UC Berkeley, CA, USA
Peter Manohar
  • Carnegie Mellon University, Pittsburgh, PA, USA
Igor Shinkar
  • Simon Fraser University, Burnaby, Canada

Funding

  • Chiesa, Alessandro: Ethereum Foundation and the Interchain Foundation
  • Manohar, Peter: NSF Graduate Research Fellowship Program (under Grant No. DGE1745016); and the ARCS Foundation. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
  • Shinkar, Igor: NSERC Discovery Grant

Acknowledgements

We are grateful to Thomas Vidick for suggesting using irreducible curves to extend our initial non-testability result for axis-parallel lines to the case of general lines.

Cite AsGet BibTex

Alessandro Chiesa, Peter Manohar, and Igor Shinkar. On Local Testability in the Non-Signaling Setting. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 26:1-26:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.ITCS.2020.26

Abstract

Non-signaling strategies are a generalization of quantum strategies that have been studied in physics for decades, and have recently found applications in theoretical computer science. These applications motivate the study of local-to-global phenomena for non-signaling functions. We prove that low-degree testing in the non-signaling setting is possible, assuming that the locality of the non-signaling function exceeds a threshold. We additionally show that if the locality is below the threshold then the test fails spectacularly, in that there exists a non-signaling function which passes the test with probability 1 and yet is maximally far from being low-degree. Along the way, we present general results about the local testability of linear codes in the non-signaling setting. These include formulating natural definitions that capture the condition that a non-signaling function "belongs" to a given code, and characterizing the sets of local constraints that imply membership in the code. We prove these results by formulating a logical inference system for linear constraints on non-signaling functions that is complete and sound.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • non-signaling strategies
  • locally testable codes
  • low-degree testing
  • Fourier analysis

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