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

Authors Lijie Chen , Xin Lyu, Avishay Tal, Hongxun Wu

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Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • pseudorandom generators
  • derandomization
  • read-once branching programs

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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.

Cite As Get BibTex

Lijie Chen, Xin Lyu, Avishay Tal, and Hongxun Wu. New PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 39:1-39:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.ICALP.2023.39

Author Details

Lijie Chen
  • Miller Institute for Basic Research in Science at University of California at Berkeley, CA, USA
Xin Lyu
  • University of California at Berkeley, CA, USA
Avishay Tal
  • University of California at Berkeley, CA, USA
Hongxun Wu
  • University of California at Berkeley, CA, USA

Funding

  • Chen, Lijie: Supported by a Miller Research Fellowship.
  • Tal, Avishay: Supported by a Sloan Research Fellowship and NSF CAREER Award CCF-2145474.

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