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Parikh’s Theorem and Descriptional Complexity

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SOFSEM 2012: Theory and Practice of Computer Science (SOFSEM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7147))

Abstract

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.

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References

  1. Parikh, R.J.: On context-free languages. J. ACM 13(4), 570–581 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  2. Goldstine, J.: A simplified proof of Parikh’s theorem. Discrete Mathematics 19(3), 235–239 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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)

    Chapter  Google Scholar 

  4. Aceto, L., Ésik, Z., Ingólfsdóttir, A.: A fully equational proof of Parikh’s theorem. RAIRO - Theoretical Informatics and Applications 36(2), 129–153 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ginsburg, S., Spanier, E.H.: Semigroups, Presburger formulas, and languages. Pacific J. Math. 16(2), 285–296 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  6. Esparza, J.: Petri nets, commutative context-free grammars, and basic parallel processes. Fundam. Inform. 31(1), 13–25 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Verma, K.N., 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)

    Chapter  Google Scholar 

  8. Kopczyński, E., To, A.W.: Parikh images of grammars: Complexity and applications. In: Symposium on Logic in Computer Science, pp. 80–89 (2010)

    Google Scholar 

  9. Esparza, J., Ganty, P., Kiefer, S., Luttenberger, M.: Parikh’s theorem: A simple and direct automaton construction. Information Processing Letters 111(12), 614–619 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pighizzini, G., Shallit, J., Wang, M.: Unary context-free grammars and pushdown automata, descriptional complexity and auxiliary space lower bounds. Journal of Computer and System Sciences 65(2), 393–414 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ginsburg, S., Rice, H.G.: Two families of languages related to ALGOL. J. ACM 9, 350–371 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gruska, J.: Descriptional complexity of context-free languages. In: MFCS, Mathematical Institute of the Slovak Academy of Sciences, pp. 71–83 (1973)

    Google Scholar 

  13. Ginsburg, S., Spanier, E.H.: Bounded Algol-like languages. Transactions of the American Mathematical Society 113(2), 333–368 (1964)

    MathSciNet  MATH  Google Scholar 

  14. Malcher, A., Pighizzini, G.: Descriptional Complexity of Bounded Context-Free Languages. In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 312–323. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  15. Hopcroft, J.E., Ullman, J.D.: Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979)

    Google Scholar 

  16. Bader, C., Moura, A.: A generalization of Ogden’s lemma. J. ACM 29, 404–407 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dömösi, P., Kudlek, M.: Strong Iteration Lemmata for Regular, Linear, Context-Free, and Linear Indexed Languages. In: Ciobanu, G., Păun, G. (eds.) FCT 1999. LNCS, vol. 1684, pp. 226–233. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Informatica e Comunicazione, Università degli Studi di Milano, via Comelico 39, I-20135, Milano, Italy

    Giovanna J. Lavado & Giovanni Pighizzini

Authors
  1. Giovanna J. Lavado
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  2. Giovanni Pighizzini
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Editor information

Editors and Affiliations

  1. Faculty of Informatics and Information Technologies, Institute of Informatics and Software Engineering, Slovak University of Technology in Bratislava, Ilkovičova 3, 842 16, Bratislava 4, Slovakia

    Mária Bieliková

  2. Dept. of Intelligent Systems and Business Informatics, Alpen-Adria-Universität Klagenfurt, Universitätsstr. 65-57, 9020, Klagenfurt, Austria

    Gerhard Friedrich

  3. Department of Computer Science, University of Oxford, UK

    Georg Gottlob

  4. Security Engineering Group, Technische Universität Darmstadt, Hochschulstr. 10, 64289, Darmstadt, Germany

    Stefan Katzenbeisser

  5. University of Illinois at Chicago, Dept. of Math., Stat. and Comp. Sci, 851 S. Morgan Street, Chicago, IL 60607-7045, USA; and University of Szeged, Research Group on Artificial Intelligence of the Hungarian Academy of Sciences, 6701 Szeged, Postafiók, Hungary

    György Turán

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Lavado, G.J., Pighizzini, G. (2012). Parikh’s Theorem and Descriptional Complexity. In: Bieliková, M., Friedrich, G., Gottlob, G., Katzenbeisser, S., Turán, G. (eds) SOFSEM 2012: Theory and Practice of Computer Science. SOFSEM 2012. Lecture Notes in Computer Science, vol 7147. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27660-6_30

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  • DOI: https://doi.org/10.1007/978-3-642-27660-6_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-27659-0

  • Online ISBN: 978-3-642-27660-6

  • eBook Packages: Computer ScienceComputer Science (R0)

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