Call $\mathsf{PPBP}$ the class of languages decided by poly-size families of permutation branching programs, which are layered branching programs (i.e., the ones defined here) whose transitions functions are permutations (without any restriction on the width).  We know from Barrington's famous result and from the fact that arbitrary poly-size branching programs capture exactly logspace that$$\mathsf{NC}^1\subseteq\mathsf{PPBP}\subseteq\mathsf{L}/\mathsf{poly}.$$In the closing section of his 1989 paper, Barrington briefly considers this question and observes that $\mathsf{PPBP}$ is known to contain $\mathsf{L}$-complete problems under $\mathsf{NC}^1$ reductions. This suggests that the first inclusion above is strict but fails to yield a characterization of $\mathsf{PPBP}$ in terms of a well-known class, because it is not known whether $\mathsf{PPBP}$ is closed under $\mathsf{NC}^1$ reductions (right?).Has any improvement on understanding the power of permutation branching programs been made since Barrington's work?Edit: Just for completeness, this is Barrington on what I call here $\mathsf{PPBP}$ (second paragraph of Sect. 8 of his paper Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in $\mathsf{NC}^1$, Journal of Computer and System Sciences, 38(1):150-164, 1989):The power of general poly-size permutation BPs (no restriction on width) was mentioned as an open problem in [Barrington's Ph.D. thesis].  Cook and McKenzie [a 1986 tech report] have just shown that [the word problem for permutation groups] is complete for log space under $\mathsf{NC}^1$ reductions [...]  A poly-size PBP can be constructed to solve this problem, given an appropriate definition of recognition of a language by a PBP.The fact that he never explicitly says that poly-size permutation BP are equivalent to poly-size BP led me to assume that having an $\mathsf{L}$-complete problem in $\mathsf{PPBP}$ was not known to be enough to get $\mathsf{PPBP}=\mathsf{L}/\mathsf{poly}$ (e.g., lack of closure under reductions).
What is the power of general poly-size permutation branching programs?
cc.complexity theory;complexity classes;circuit complexity;logspace;structural complexity