License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2023.39
URN: urn:nbn:de:0030-drops-180916
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18091/
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Chen, Lijie ; Lyu, Xin ; Tal, Avishay ; Wu, Hongxun

New PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs

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LIPIcs-ICALP-2023-39.pdf (0.8 MB)

Abstract

We give the first pseudorandom generators with sub-linear seed length for the following variants of read-once branching programs (roBPs):
1) First, we show there is an explicit PRG of seed length O(log²(n/ε)log(n)) fooling unbounded-width unordered permutation branching programs with a single accept state, where n is the length of the program. Previously, [Lee-Pyne-Vadhan RANDOM 2022] gave a PRG with seed length Ω(n) for this class. For the ordered case, [Hoza-Pyne-Vadhan ITCS 2021] gave a PRG with seed length Õ(log n ⋅ log 1/ε).
2) Second, we show there is an explicit PRG fooling unbounded-width unordered regular branching programs with a single accept state with seed length Õ(√{n ⋅ log 1/ε} + log 1/ε). Previously, no non-trivial PRG (with seed length less than n) was known for this class (even in the ordered setting). For the ordered case, [Bogdanov-Hoza-Prakriya-Pyne CCC 2022] gave an HSG with seed length Õ(log n ⋅ log 1/ε).
3) Third, we show there is an explicit PRG fooling width w adaptive branching programs with seed length O(log n ⋅ log² (nw/ε)). Here, the branching program can choose an input bit to read depending on its current state, while it is guaranteed that on any input x ∈ {0,1}ⁿ, the branching program reads each input bit exactly once. Previously, no PRG with a non-trivial seed length is known for this class.
We remark that there are some functions computable by constant-width adaptive branching programs but not by sub-exponential-width unordered branching programs.
In terms of techniques, we indeed show that the Forbes-Kelly PRG (with the right parameters) from [Forbes-Kelly FOCS 2018] already fools all variants of roBPs above. Our proof adds several new ideas to the original analysis of Forbes-Kelly, and we believe it further demonstrates the versatility of the Forbes-Kelly PRG.

BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs.ICALP.2023.39,
  author =	{Chen, Lijie and Lyu, Xin and Tal, Avishay and Wu, Hongxun},
  title =	{{New PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{39:1--39:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18091},
  URN =		{urn:nbn:de:0030-drops-180916},
  doi =		{10.4230/LIPIcs.ICALP.2023.39},
  annote =	{Keywords: pseudorandom generators, derandomization, read-once branching programs}
}

Keywords: pseudorandom generators, derandomization, read-once branching programs
Collection: 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)
Issue Date: 2023
Date of publication: 05.07.2023

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