Parikh’s Theorem and Descriptional Complexity

  • Giovanna J. Lavado
  • Giovanni Pighizzini
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7147)


It is well known that for each context-free language there exists a regular language with the same Parikh image. We investigate this result from a descriptional complexity point of view, by proving tight bounds for the size of deterministic automata accepting regular languages Parikh equivalent to some kinds of context-free languages. First, we prove that for each context-free grammar in Chomsky normal form with a fixed terminal alphabet and h variables, generating a bounded language L, there exists a deterministic automaton with at most \(2^{h^{O(1)}}\) states accepting a regular language Parikh equivalent to L. This bound, which generalizes a previous result for languages defined over a one letter alphabet, is optimal. Subsequently, we consider the case of arbitrary context-free languages defined over a two letter alphabet. Even in this case we are able to obtain a similar bound. For alphabets of at least three letters the best known upper bound is a double exponential in h.


finite automata formal languages context-free languages descriptional complexity Parikh’s theorem bounded languages 


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© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Giovanna J. Lavado
    • 1
  • Giovanni Pighizzini
    • 1
  1. 1.Dipartimento di Informatica e ComunicazioneUniversità degli Studi di MilanoMilanoItaly

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