Abstract
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 \(e^{O(\sqrt{n \cdot \ln n})}\) 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 \(2^{O(n^2)}\) states. Even this bound is tight.
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Lavado, G.J., Pighizzini, G., Seki, S. (2012). Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent Deterministic Automata. In: Yen, HC., Ibarra, O.H. (eds) Developments in Language Theory. DLT 2012. Lecture Notes in Computer Science, vol 7410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31653-1_26
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