Grammatical characterizations of NPDAs and VPDAs with counters

We give a characterization of NPDAs with reversal-bounded counters (NPCMs) in terms of context-free grammars with monotonic counters. We show that the grammar characterization can be used to give simple proofs of previously known results such as the semilinearity of the Parikh map of any language accepted by an NPCM. We prove a Chomsky-Schutzenberger-like theorem: A language L is accepted by an NPCM if and only if there is a k≥1 and an alphabet Σ containing at least k distinguished symbols, p1,...,pk, such that L=h(D∩Ek(R)) for some homomorphism h, Dyck language D⊆Σ⁎, and regular set R⊆Σ⁎, where Ek(R)={w|w∈R,|w|p1=⋯=|w|pk}. We also give characterizations of other machine models, such as visibly pushdown automata with reversal-bounded counters (VPCMs). We then investigate the complexity of some decision problems concerning these grammatical models. Finally, we introduce other grammatical models equivalent to NPCM.