%AChen, Lijie%ALyu, Xin%ATal, Avishay%AWu, Hongxun%AEtessami, Kousha Ed.%AFeige, Uriel Ed.%APuppis, Gabriele Ed.%D2023%ISchloss Dagstuhl – Leibniz-Zentrum für Informatik
%Kpseudorandom generators; derandomization; read-once branching programs; Theory of computation → Pseudorandomness and derandomization
%MOSTI ID: 10484211
%PMedium: X
%TNew PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs
%XWe 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-Kelley PRG (with the right parameters) from [Forbes-Kelley 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-Kelley PRG.
Country unknown/Code not availablehttps://doi.org/10.4230/LIPIcs.ICALP.2023.39OSTI-MSA