Operational State Complexity under Parikh Equivalence

  • Giovanna J. Lavado
  • Giovanni Pighizzini
  • Shinnosuke Seki
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8614)


We investigate, under Parikh equivalence, the state complexity of some language operations which preserve regularity. For union, concatenation, Kleene star, complement, intersection, shuffle, and reversal, we obtain a polynomial state complexity over any fixed alphabet, in contrast to the intrinsic exponential state complexity of some of these operations in the classical version. For projection we prove a superpolynomial state complexity, which is lower than the exponential one of the corresponding classical operation. We also prove that for each two deterministic automata A and B it is possible to obtain a deterministic automaton with a polynomial number of states whose accepted language has as Parikh image the intersection of the Parikh images of the languages accepted by A and B. Finally, we prove that for each finite set there exists a small context-free grammar defining a language with the same Parikh image.


state complexity regular languages Parikh equivalence context-free grammars semilinear sets 


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Giovanna J. Lavado
    • 1
  • Giovanni Pighizzini
    • 1
  • Shinnosuke Seki
    • 2
    • 3
  1. 1.Dipartimento di InformaticaUniversità degli Studi di MilanoItaly
  2. 2.Helsinki Institute for Information Technology (HIIT)AaltoFinland
  3. 3.Department of Information and Computer ScienceAalto UniversityAaltoFinland

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