  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.
