We give new upper and lower bounds on the power of several restricted classes of arbitrary-order read-once branching programs (ROBPs) and standard-order ROBPs (SOBPs) that have received significant attention in the literature on pseudorandomness for space-bounded computation.

Regular SOBPs of length $n$ and width $\lfloor w(n+1)/2\rfloor$ can exactly simulate general SOBPs of length $n$ and width $w$, and moreover an $n/2-o(n)$ blow-up in width is necessary for such a simulation.
Our result extends and simplifies prior average-case simulations (Reingold, Trevisan, and Vadhan (STOC 2006), Bogdanov, Hoza, Prakriya, and Pyne (CCC 2022)), in particular implying that weighted pseudorandom generators (Braverman, Cohen, and Garg (SICOMP 2020)) for regular SOBPs of width $\text{poly}(n)$ or larger automatically extend to general SOBPs. Furthermore, our simulation also extends to general (even read-many) oblivious branching programs.

There exist natural functions computable by regular SOBPs of constant width that are average-case hard for permutation SOBPs of exponential width. Indeed, we show that Inner-Product mod $2$ is average-case hard for arbitrary-order permutation ROBPs of exponential width.

There exist functions computable by constant-width arbitrary-order permutation ROBPs that are worst-case hard for exponential-width SOBPs.

Read-twice permutation branching programs of subexponential width can simulate polynomial-width arbitrary-order ROBPs.