# Parikh Image of Pushdown Automata

## TL;DR

This paper analyzes the complexity of converting pushdown automata to context-free grammars and finite automata, establishing optimal bounds and exploring Parikh equivalence, especially for unary automata and deterministic cases.

## Contribution

It proves the optimality of classical PDA-to-grammar conversion algorithms for unary automata and introduces a new efficient conversion for unary deterministic PDAs.

## Key findings

- Conversion algorithm is optimal for unary PDAs.
- Parikh equivalence simplifies automata comparison.
- New bounds for unary deterministic PDA to grammar conversion.

## Abstract

We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with $n$ states and $p \geq 2n + 4$ stack symbols that accepts one single long word for which every equivalent context-free grammar needs $\Omega(n^2(p-2n-4))$ variables. This family shows that the classical algorithm for converting a PDA to an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family. We conclude that, when assuming this weaker equivalence, the conversion algorithm is also optimal. Second, Parikh's theorem motivates the comparison of PDAs against finite state automata. In particular, the same family of unary PDAs gives a lower bound on the number of states of every Parikh-equivalent finite state automaton. Finally, we look into the case of unary deterministic PDAs. We show a new construction converting a unary deterministic PDA into an equivalent context-free grammar that achieves best known bounds.