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On Local Testability in the Non-Signaling Setting

Authors Alessandro Chiesa, Peter Manohar, Igor Shinkar



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LIPIcs.ITCS.2020.26.pdf
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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

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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