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The Chomsky-Schützenberger Theorem for Quantitative Context-Free Languages

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Developments in Language Theory (DLT 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7907))

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Abstract

Weighted automata model quantitative aspects of systems like the consumption of resources during executions. Traditionally, the weights are assumed to form the algebraic structure of a semiring, but recently also other weight computations like average have been considered. Here, we investigate quantitative context-free languages over very general weight structures incorporating all semirings, average computations, lattices. In our main result, we derive the Chomsky-Schützenberger Theorem for such quantitative context-free languages, showing that each arises as the image of the intersection of a Dyck language and a recognizable language under a suitable morphism. Moreover, we show that quantitative context-free languages are expressively equivalent to a model of weighted pushdown automata. This generalizes results previously known only for semirings.

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References

  1. Autebert, J., Berstel, J., Boasson, L.: Context-free languages and pushdown automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. Word, Language, Grammar, vol. 1, pp. 111–174. Springer (1997)

    Google Scholar 

  2. Bar–Hillel, Y., Perles, M., Shamir, E.: On formal properties of simple phrase structure grammars. Z. Phonetik. Sprach. Komm. 14, 143–172 (1961)

    MathSciNet  MATH  Google Scholar 

  3. Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs on Theoretical Computer Science, vol. 12. Springer (1988)

    Google Scholar 

  4. Birkhoff, G.: Lattice Theory. AMS (1967)

    Google Scholar 

  5. Chatterjee, K., Doyen, L., Henzinger, T.: Quantitative languages. ACM Transactions on Computational Logic 11(4), Article 23 (2010)

    Google Scholar 

  6. Chatterjee, K., Doyen, L., Henzinger, T.: Expressiveness and closure properties for quantitative languages. In: LICS 2009, pp. 199–208. IEEE Comp. Soc. (2009)

    Google Scholar 

  7. Chatterjee, K., Doyen, L., Henzinger, T.A.: Probabilistic weighted automata. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 244–258. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  8. Chatterjee, K., Velner, Y.: Mean-payoff pushdown games. In: 27th Annual ACM/IEEE Symposium on Logic in Computer Science, pp. 195–204 (2012)

    Google Scholar 

  9. Chomsky, N., Schützenberger, M.P.: The algebraic theory of context-free languages. In: Computer Programming and Formal Systems, pp. 118–161. North-Holland (1963)

    Google Scholar 

  10. Droste, M., Gastin, P.: Weighted automata and weighted logics. Theor. Comput. Sci. 380(1-2), 69–86 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Droste, M., Meinecke, I.: Describing average- and longtime-behavior by weighted MSO logics. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 537–548. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  12. Droste, M., Meinecke, I.: Weighted automata and regular expressions over valuation monoid. Intern. J. of Foundations of Comp. Science 22, 1829–1844 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Droste, M., Stüber, T., Vogler, H.: Weighted finite automata over strong bimonoids. Information Sciences 180, 156–166 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Droste, M., Vogler, H.: Weighted automata and multi-valued logics over arbitrary bounded lattices. Theoretical Computer Science 418, 14–36 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  15. Eilenberg, S.: Automata, Languages, and Machines – Volume A. Pure and Applied Mathematics, vol. 59. Academic Press (1974)

    Google Scholar 

  16. Golan, J.S.: Semirings and their Applications. Kluwer Acad. Publ. (1999)

    Google Scholar 

  17. Grätzer, G.: General Lattice Theory. Birkhäuser, Basel (2003)

    MATH  Google Scholar 

  18. Hulden, M.: Parsing CFGs and PCFGs with a Chomsky-Schützenberger representation. In: Vetulani, Z. (ed.) LTC 2009. LNCS, vol. 6562, pp. 151–160. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  19. Kozen, D.: Automata and Computability. Springer (1997)

    Google Scholar 

  20. Kuich, W., Salomaa, A.: Semirings, Automata, Languages. Monogr. Theoret. Comput. Sci. EATCS Ser., vol. 5. Springer (1986)

    Google Scholar 

  21. Meinecke, I.: Valuations of weighted automata: Doing it in a rational way. In: Kuich, W., Rahonis, G. (eds.) Algebraic Foundations in Computer Science. LNCS, vol. 7020, pp. 309–346. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  22. Okhotin, A.: Non-erasing variants of the Chomsky-Schützenberger theorem. In: Yen, H.-C., Ibarra, O.H. (eds.) DLT 2012. LNCS, vol. 7410, pp. 121–129. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  23. Petre, I., Salomaa, A.: Algebraic systems and pushdown automata. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, ch. 7, pp. 257–311. Springer (2009)

    Google Scholar 

  24. Reps, T., Schwoon, S., Jha, S., Melski, D.: Weighted pushdown systems and their application to interprocedural dataflow analysis. Science of Programming 58, 206–263 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science. Springer (1978)

    Google Scholar 

  26. Schützenberger, M.P.: On the definition of a family of automata. Inf. and Control 4, 245–270 (1961)

    Article  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Computer Science, Leipzig University, D-04109, Leipzig, Germany

    Manfred Droste

  2. Department of Computer Science, Technische Universität Dresden, D-01062, Dresden, Germany

    Heiko Vogler

Authors
  1. Manfred Droste
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  2. Heiko Vogler
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Editor information

Editors and Affiliations

  1. LIGM, University Paris-Est Marne-la-Vallée, 5 Bd Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Marie-Pierre Béal

  2. LIAFA, UMR 7089, University Paris Diderot, 75205, Paris cedex 13, France

    Olivier Carton

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Droste, M., Vogler, H. (2013). The Chomsky-Schützenberger Theorem for Quantitative Context-Free Languages. In: Béal, MP., Carton, O. (eds) Developments in Language Theory. DLT 2013. Lecture Notes in Computer Science, vol 7907. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38771-5_19

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  • DOI: https://doi.org/10.1007/978-3-642-38771-5_19

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38770-8

  • Online ISBN: 978-3-642-38771-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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