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BPL ⊆ L-AC¹

Authors Kuan Cheng , Yichuan Wang

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LIPIcs.CCC.2024.32.pdf
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ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Randomized Space Complexity
  • Circuit Complexity
  • Derandomization

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Abstract

Whether BPL = 𝖫 (which is conjectured to be equal) or even whether BPL ⊆ NL, is a big open problem in theoretical computer science. It is well known that 𝖫 ⊆ NL ⊆ L-AC¹. In this work we show that BPL ⊆ L-AC¹ also holds. Our proof is based on a new iteration method for boosting precision in approximating matrix powering, which is inspired by the Richardson Iteration method developed in a recent line of work [AmirMahdi Ahmadinejad et al., 2020; Edward Pyne and Salil P. Vadhan, 2021; Gil Cohen et al., 2021; William M. Hoza, 2021; Gil Cohen et al., 2023; Aaron (Louie) Putterman and Edward Pyne, 2023; Lijie Chen et al., 2023]. We also improve the algorithm for approximate counting in low-depth L-AC circuits from an additive error setting to a multiplicative error setting.

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Kuan Cheng and Yichuan Wang. BPL ⊆ L-AC¹. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 32:1-32:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.CCC.2024.32

Author Details

Kuan Cheng
  • Center on Frontiers of Computing Studies, School of Computer Science, Peking University, Beijing, China
Yichuan Wang
  • Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China

Acknowledgements

We thank anonymous reviewers for helpful comments.

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