Converting nondeterministic automata and context-free grammars into Parikh equivalent one-way and two-way deterministic automata

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 eO(n⋅lnn) and p(n)p(n) states, respectively, where p(n)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 2O(h2)2O(h2) and 2O(h)2O(h) states, respectively. Even these bounds are tight.