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)

Abstract

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.

Keywords

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

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References

  1. 1.
    Aceto, L., Ésik, Z., Ingólfsdóttir, A.: A fully equational proof of Parikh’s theorem. RAIRO-Theor. Inf. Appl. 36(2), 129–153 (2002)CrossRefMATHGoogle Scholar
  2. 2.
    Câmpeanu, C., Salomaa, K., Yu, S.: Tight lower bound for the state complexity of shuffle of regular languages. Journal of Automata, Languages and Combinatorics 7(3), 303–310 (2002)MATHMathSciNetGoogle Scholar
  3. 3.
    Domaratzki, M., Pighizzini, G., Shallit, J.: Simulating finite automata with context-free grammars. Inform. Process. Lett. 84(6), 339–344 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fund. Inform. 31(1), 13–25 (1997)MATHMathSciNetGoogle Scholar
  5. 5.
    Ginsburg, S., Spanier, E.H.: Bounded ALGOL-like languages. T. Am. Math. Soc. 113, 333–368 (1964)MATHMathSciNetGoogle Scholar
  6. 6.
    Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas, and languages. Pac. J. Math. 16(2), 285–296 (1966)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Goldstine, J.: A simplified proof of Parikh’s theorem. Discrete Math 19(3), 235–239 (1977)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979)Google Scholar
  9. 9.
    Huynh, D.T.: The complexity of semilinear sets. In: de Bakker, J.W., van Leeuwen, J. (eds.) ICALP 1980. LNCS, vol. 85, pp. 324–337. Springer, Heidelberg (1980)CrossRefGoogle Scholar
  10. 10.
    Jirásková, G., Masopust, T.: On a structural property in the state complexity of projected regular languages. Theor. Comput. Sci. 449, 93–105 (2012)CrossRefMATHGoogle Scholar
  11. 11.
    Lavado, G.J., Pighizzini, G., Seki, S.: Converting nondeterministic automata and context-free grammars into Parikh equivalent one-way and two-way deterministic automata. Inform. Comput. 228, 1–15 (2013)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Parikh, R.J.: On context-free grammars. J. ACM 13(4), 570–581 (1966)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Pighizzini, G., Shallit, J.: Unary language operations, state complexity and jacobsthal’s function. Int. J. Found. Comput. S. 13(1), 145–159 (2002)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Shallit, J.O.: A Second Course in Formal Languages and Automata Theory. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  15. 15.
    To, A.W.: Model Checking Infinite-State Systems: Generic and Specific Approaches. Ph.D. thesis, School of Informatics, University of Edinburgh (August 2010)Google Scholar
  16. 16.
    Verma, K., Seidl, H., Schwentick, T.: On the complexity of equational Horn clauses. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 337–352. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. 17.
    Yu, S.: State complexity of regular languages. Journal of Automata, Languages and Combinatorics 6, 221–234 (2000)Google Scholar
  18. 18.
    Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theor 125, 315–328 (1994)MATHMathSciNetGoogle Scholar

Copyright information

© 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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