While visibly pushdown languages properly generalise regular languages and are properly contained in deterministic context-free
languages, the complexity of their membership problem is equivalent to that of regular languages. However, the corresponding
counting problem could be harder than counting paths in a non-deterministic finite automaton: it is only known to be in
LogDCFL.
We investigate the membership and counting problems for generalisations of visibly pushdown automata, defined using the notion
of height-determinism. We show that, when the stack-height of a given PDA can be computed using a finite transducer, both problems have the same complexity as for visibly pushdown languages. We also
show that when allowing pushdown transducers instead of finite-state ones, both problems become LogDCFL-complete; this uses the fact that pushdown transducers are sufficient to compute the stack heights of all real-time height-deterministic
pushdown automata, and yields a candidate arithmetization of LogDCFL that is no harder than LogDCFL(our main result).