A Note on Limited Pushdown Alphabets in Stateless Deterministic Pushdown Automata

TL;DR

This paper demonstrates that the hierarchy of languages accepted by stateless deterministic pushdown automata can be established over a binary alphabet, improving previous results that required larger alphabets, and extends some results to m-state automata.

Contribution

The paper shows that the hierarchy can be established with a binary alphabet, solving an open problem and extending results to m-state automata.

Findings

Hierarchy established over binary alphabet

Solved open problem by Meduna et al.

Extended results to m-state automata

Abstract

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.