Counting Paths in VPA Is Complete for #NC

We give a #NC (1) upper bound for the problem of counting accepting paths in any fixed visibly pushdown automaton. Our algorithm involves a non-trivial adaptation of the arithmetic formula evaluation algorithm of Buss, Cook, Gupta and Ramachandran (SIAM J. Comput. 21:755-780, 1992). We also show that the problem is #NC (1) hard. Our results show that the difference between #BWBP and #NC (1) is captured exactly by the addition of a visible stack to a nondeterministic finite-state automaton