Optimal Error Pseudodistributions for Read-Once Branching Programs

摘要

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length n and width w read-once branching programs with seed length O(log na. log(nw)+log na.log(1/?μ)) and error ?μ. It remains a central question to reduce the seed length to O(log (nw/?μ)), which would prove that e???e???e?? = e??. However, there has been no improvement on Nisana??s construction for the case n = w, which is most relevant to space-bounded derandomization. Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length Oìf(log na. log(nw)+log(1/?μ)). A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing e???e???e?? and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length O(log na. log(nw)+log(1/?μ)), but their techniques are restricted to hitting sets. In this work, we construct a PRPD with seed length O(log na. log (nw)a. log log(nw)+log(1/?μ)). This improves upon the construction by Braverman, Cogen and Garg by a O(log log(1/?μ)) factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.