A Note on Limited Pushdown Alphabets in Stateless Deterministic Pushdown Automata

Recently, an infinite hierarchy of languages accepted by stateless deterministic pushdown automata has been established based on the number of pushdown symbols. However, the witness language for the n-th level of the hierarchy is over an input alphabet with 2(n-1) elements. In this paper, we improve this result by showing that a binary alphabet is sufficient to establish this hierarchy. As a consequence of our construction, we solve the open problem formulated by Meduna et al. Then we extend these results to m-state realtime deterministic pushdown automata, for all m at least 1. The existence of such a hierarchy for m-state deterministic pushdown automata is left open.