Counting paths in planar width 2 branching programs

We revisit the problem of counting paths in width-2 planar branching programs. We show that this is hard for Boolean NC under ACC[5] reductions, completing a proof strategy outlined in [3]. On the other hand, for several restricted instances of width-2 planar branching programs, we show that the counting problem is TC-complete. We also show that nonplanar width-2 programs can be planarized in AC[2]. Using the equivalence of planar width-2 programs with the reduced-form representation of positive rationals, we show that the evaluation problem for this representation in the Stern-Brocot tree is also NC hard. In contrast, the evaluation problem in the continued fraction representation is in TC.