# $RL=L$ Progress Since 2006

Reingold, Trevisan, and Vadhan's breakthrough 2006 paper (http://dl.acm.org/citation.cfm?id=1132583) reduced the problem of showing $RL=L$ to producing pseudorandom walks on regular digraphs that are not consistently labeled.

Has any further progress on this problem been made since then?

• One line of work has shown how to improve the analysis of the classic INW PRG for the special cases of fooling regular and permutation branching programs. The seed length is only $$\widetilde{O}(\log n)$$ or $$O(\log n)$$ in some cases. [LRTV09] [BV10] [De11] [KNP11] [Ste12] [BRRY14] [HPV21]
• Another line of work has developed a PRG framework based on iterated pseudorandom restrictions. This framework has been used to fool various classes of functions that can be computed by "arbitrary-order" read-once branching programs. Again, in some cases the seed length is $$\widetilde{O}(\log n)$$ or even $$O(\log n)$$. [GMRTV12] [GLS12] [RSV13] [CSV15] [SVW17] [CHRT18] [HLV18] [FK18] [MRT19] [DHH19] [Lee19] [LV20] [DHH20] [DMRTV21]