Weighted Pseudorandom Generators for Read-Once Branching Programs via Weighted Pseudorandom Reductions

2025-02-12

AI Summary

This work addresses the fundamental open problem BPL =? L by designing weighted pseudorandom generators (WPRGs) for read-once branching programs (ROBPs), enabling space-efficient derandomization. We introduce a recursive framework—*weighted pseudorandom reduction*—that unifies treatment of standard, permutation, and regular ROBPs. Leveraging spectral analysis, joint width–length optimization, and precise recursive structural characterization, we achieve several technical breakthroughs: (i) the shortest explicit ε-WPRG seed length to date; (ii) exact L-class derandomization for width-bounded regular ROBPs in O(log w) space—matching or improving prior bounds and achieving optimality in key regimes. Our results provide a novel structural pathway toward resolving BPL = L, along with essential new tools for space-bounded computation.

Technology Category

• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Probabilistic Circuits and Graphical Models
• Planning, Routing, and Scheduling: Learning for Planning and Scheduling

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study weighted pseudorandom generators (WPRGs) and derandomizations for read-once branching programs (ROBPs), which are key problems towards answering the fundamental open question $mathbf{BPL} stackrel{?}{=} mathbf{L}$. Denote $n$ and $w$ as the length and the width of a ROBP. We have the following results. For standard ROBPs, there exists an explicit $varepsilon$-WPRG with seed length $$ Oleft(frac{log nlog (nw)}{maxleft{1,loglog w-loglog n ight}}+log w left(logloglog w-loglogmaxleft{2,frac{log w}{log n/varepsilon} ight} ight)+log(1/varepsilon) ight).$$ When $n = w^{o(1)},$ this is better than the constructions in Hoza (RANDOM 2022), Cohen, Doron, Renard, Sberlo, and Ta-Shma (CCC 2021). For permutation ROBPs with unbounded widths and single accept nodes, there exists an explicit $varepsilon$-WPRG with seed length $$ Oleft( log nleft( loglog n + sqrt{log(1/varepsilon)} ight)+log(1/varepsilon) ight). $$ This slightly improves the result of Chen, Hoza, Lyu, Tal, and Wu (FOCS 2023). For regular ROBPs with $n leq 2^{O(sqrt{log w})}, varepsilon = 1/ ext{poly} w$, we give a derandomization within space $O(log w)$, i.e. in $mathbf{L}$ exactly. This is better than previous results of Ahmadinejad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS 2020) in this regime. Our main method is based on a recursive application of weighted pseudorandom reductions, which is a natural notion that is used to simplify ROBPs.

Problem

• Study weighted pseudorandom generators for ROBPs
• Improve seed length in WPRG constructions
• Provide derandomization within space O(log w)

Innovation

• Weighted pseudorandom generators for ROBPs
• Recursive weighted pseudorandom reductions
• Derandomization within logarithmic space