Converting nondeterministic automata and context-free grammars into parikh equivalent deterministic automata

  • Authors:

  • Giovanna J. Lavado;Giovanni Pighizzini;Shinnosuke Seki

  • Affiliations:

  • Dipartimento di Informatica, Università degli Studi di Milano, Milano, Italy;Dipartimento di Informatica, Università degli Studi di Milano, Milano, Italy;Department of Information and Computer Science, Aalto University, Aalto, Finland

  • Venue:
  • DLT'12 Proceedings of the 16th international conference on Developments in Language Theory
  • Year:
  • 2012

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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.