Pushing Boundaries in BPL Derandomization

We propose to study one of the most fundamental problems in computational complexity theory: the BPL vs. L problem. Specifically, we aim to construct improved pseudorandom generators (PRGs) and weighted PRGs—a notion introduced by the PI and coauthors (STOC 18, CCC 21). In recent years, weighted PRGs have gained significant attention and played a central role in numerous advancements. Notably, the PI and coauthors leveraged WPRG techniques to improve the 30-year-old Saks-Zhou algorithm for iterated matrix multiplication (STOC 23).Despite a flurry of progress in recent years, achieving breakthroughs on such fundamental problems will likely require fresh and innovative approaches. In this proposal, we present novel ideas drawing from spectral graph theory, computational methods, and combinatorial techniques. Some of these ideas build on the PI's analysis of rotating expanders (STOC 23), and of the Zig-Zag and other graph products (FOCS 24, ITCS 25). These efforts are further supported by promising preliminary results.We also introduce a novel computational model, termed input catalytic machines, along with new PRG variants. These models hold the potential to shed new light on classical questions and the barriers that hinder progress. This adds a conceptual dimension to the proposal, increasing its likelihood of success. Furthermore, we propose to study key algorithmic problems, specifically focusing on their space complexity.While challenging, we are well-positioned to tackle these longstanding open problems. The PI has made significant contributions, including introducing WPRGs, improving the Saks-Zhou algorithm, and achieving breakthroughs on foundational questions such as Ramsey graphs and tree code construction (STOC 16, STOC 18). Additionally, the PI's deep expertise in advanced mathematical theories, such as finite free probability, underpins the spectral methods integral to this proposal.