  We investigate the conversion of one-way nondeterministic finite automata and
context-free grammars into Parikh equivalent one-way and two-way deterministic
finite automata, from a descriptional complexity point of view.
  We prove that for each one-way nondeterministic automaton with $n$ states
there exist Parikh equivalent one-way and two-way deterministic automata with
$e^{O(\sqrt{n \ln n})}$ and $p(n)$ states, respectively, where $p(n)$ is a
polynomial. Furthermore, these costs are tight. In contrast, if all the words
accepted by the given automaton contain at least two different letters, then a
Parikh equivalent one-way deterministic automaton with a polynomial number of
states can be found.
  Concerning context-free grammars, we prove that for each grammar in Chomsky
normal form with h variables there exist Parikh equivalent one-way and two-way
deterministic automata with $2^{O(h^2)}$ and $2^{O(h)}$ states, respectively.
Even these bounds are tight.
