Weighted Logics for Nested Words and Algebraic Formal Power Series

  • Christian Mathissen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5126)

Abstract

Nested words, a model for recursive programs proposed by Alur and Madhusudan, have recently gained much interest. In this paper we introduce quantitative extensions and study nested word series which assign to nested words elements of a semiring. We show that regular nested word series coincide with series definable in weighted logics as introduced by Droste and Gastin. For this, we establish a connection between nested words and series-parallel-biposets. Applying our result, we obtain a characterization of algebraic formal power series in terms of weighted logics. This generalizes a result of Lautemann, Schwentick and Thérien on context-free languages.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alur, R., Kumar, V., Madhusudan, P., Viswanathan, M.: Congruences for visibly pushdown languages. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1102–1114. Springer, Heidelberg (2005)Google Scholar
  2. 2.
    Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Proc. of the 36th STOC, Chicago, pp. 202–211. ACM, New York (2004)Google Scholar
  3. 3.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 1–13. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Arenas, M., Barceló, P., Libkin, L.: Regular languages of nested words: Fixed points, automata, and synchronization. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 888–900. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  5. 5.
    Chomsky, N., Schützenberger, M.P.: The algebraic theory of context-free languages. In: Computer Programming and Formal Systems, pp. 118–161. North-Holland, Amsterdam (1963)Google Scholar
  6. 6.
    Courcelle, B.: Monadic second-order definable graph transductions: a survey. Theoretical Computer Science 126, 53–75 (1994)CrossRefMathSciNetMATHGoogle Scholar
  7. 7.
    Droste, M., Gastin, P.: Weighted automata and weighted logics. Theoretical Computer Science 380, 69–86 (2007)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Droste, M., Gastin, P.: Weighted automata and weighted logics. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata, ch.5. Springer, Heidelberg (to appear, 2008)Google Scholar
  9. 9.
    Droste, M., Rahonis, G.: Weighted automata and weighted logics on infinite words. In: H. Ibarra, O., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 49–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Droste, M., Vogler, H.: Weighted tree automata and weighted logics. Theoretical Computer Science 366, 228–247 (2006)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Ésik, Z., Németh, Z.L.: Higher dimensional automata. Journal of Automata, Languages and Combinatorics 9(1), 3–29 (2004)MathSciNetMATHGoogle Scholar
  12. 12.
    Hashiguchi, K., Ichihara, S., Jimbo, S.: Formal languages over free binoids. Journal of Automata, Languages and Combinatorics 5(3), 219–234 (2000)MathSciNetMATHGoogle Scholar
  13. 13.
    Kuich, W.: Word, Language, Grammar. In: Handbook of Formal Languages, ch.9. Semirings and formal power series, vol. 1, pp. 609–677. Springer, Heidelberg (1997)Google Scholar
  14. 14.
    Kuich, W., Salomaa, A.: Semirings, Automata, Languages. Springer, Heidelberg (1986)MATHGoogle Scholar
  15. 15.
    Lautemann, C., Schwentick, T., Thérien, D.: Logics for context-free languages. In: CSL 1994. LNCS, vol. 933, pp. 205–216. Springer, Heidelberg (1994)Google Scholar
  16. 16.
    Mathissen, C.: Definable transductions and weighted logics for texts. In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 324–336. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Mäurer, I.: Weighted picture automata and weighted logics. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 313–324. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  18. 18.
    Meinecke, I.: Weighted logics for traces. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 235–246. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Schwarz, S.: Łukasiewicz logics and weighted logics over MV-semirings. Journal of Automata, Languages and Combinatorics 12(4), 485–499 (2007)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Christian Mathissen
    • 1
  1. 1.Institut für InformatikUniversität LeipzigLeipzigGermany

Personalised recommendations