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

  • Giovanna J. Lavado
  • Giovanni Pighizzini
  • Shinnosuke Seki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7410)


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.


Finite automaton context-free grammar Parikh’s theorem descriptional complexity semilinear set Parikh equivalence 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Giovanna J. Lavado
    • 1
  • Giovanni Pighizzini
    • 1
  • Shinnosuke Seki
    • 2
  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoMilanoItaly
  2. 2.Department of Information and Computer ScienceAalto UniversityAaltoFinland

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