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Affine Extractors and AC0-Parity

Authors Xuangui Huang, Peter Ivanov, Emanuele Viola

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Author Details

Xuangui Huang
  • Northeastern University, Boston, MA, USA
Peter Ivanov
  • Northeastern University, Boston, MA, USA
Emanuele Viola
  • Northeastern University, Boston, MA, USA

Funding

Supported by NSF grants CCF-1813930 and CCF-2114116.

Acknowledgements

We are grateful to the anonymous reviewers for helpful feedback.

Cite As Get BibTex

Xuangui Huang, Peter Ivanov, and Emanuele Viola. Affine Extractors and AC0-Parity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.9

Abstract

We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
Keywords
  • affine extractor
  • resilient function
  • constant-depth circuit
  • parity gate
  • inner product

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