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Testing Linearity against Non-Signaling Strategies

Authors Alessandro Chiesa, Peter Manohar, Igor Shinkar

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Author Details

Alessandro Chiesa
  • UC Berkeley, Berkeley (CA), USA
Peter Manohar
  • UC Berkeley, Berkeley (CA), USA
Igor Shinkar
  • UC Berkeley, Berkeley (CA), USA

Funding

This work was supported by the UC Berkeley Center for Long-Term Cybersecurity.

Cite AsGet BibTex

Alessandro Chiesa, Peter Manohar, and Igor Shinkar. Testing Linearity against Non-Signaling Strategies. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 17:1-17:37, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.CCC.2018.17

Abstract

Non-signaling strategies are collections of distributions with certain non-local correlations. They have been studied in Physics as a strict generalization of quantum strategies to understand the power and limitations of Nature's apparent non-locality. Recently, they have received attention in Theoretical Computer Science due to connections to Complexity and Cryptography. We initiate the study of Property Testing against non-signaling strategies, focusing first on the classical problem of linearity testing (Blum, Luby, and Rubinfeld; JCSS 1993). We prove that any non-signaling strategy that passes the linearity test with high probability must be close to a quasi-distribution over linear functions. Quasi-distributions generalize the notion of probability distributions over global objects (such as functions) by allowing negative probabilities, while at the same time requiring that "local views" follow standard distributions (with non-negative probabilities). Quasi-distributions arise naturally in the study of Quantum Mechanics as a tool to describe various non-local phenomena. Our analysis of the linearity test relies on Fourier analytic techniques applied to quasi-distributions. Along the way, we also establish general equivalences between non-signaling strategies and quasi-distributions, which we believe will provide a useful perspective on the study of Property Testing against non-signaling strategies beyond linearity testing.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • property testing
  • linearity testing
  • non-signaling strategies
  • quasi-distributions

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