Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent Deterministic Automata

We investigate the conversion of nondeterministic finite automata and context-free grammars into Parikh equivalent deterministic finite automata, from a descriptional complexity point of view. We prove that for each nondeterministic automaton with n states there exists a Parikh equivalent deterministic automaton with e √ n·lnn) states. Furthermore, this cost is tight. In contrast, if all the strings accepted by the given automaton contain at least two different letters, then a Parikh equivalent 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 n variables there exists a Parikh equivalent deterministic automaton with 2 ) states. Even this bound is tight.