Weighted Automata and Logics for Infinite Nested Words

  • Manfred Droste
  • Stefan Dück
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)


Nested words introduced by Alur and Madhusudan are used to capture structures with both linear and hierarchical order, e.g. XML documents, without losing valuable closure properties. Furthermore, Alur and Madhusudan introduced automata and equivalent logics for both finite and infinite nested words, thus extending Büchi’s theorem to nested words. Recently, average and discounted computations of weights in quantitative systems found much interest. Here, we will introduce and investigate weighted automata models and weighted MSO logics for infinite nested words. As weight structures we consider valuation monoids which incorporate average and discounted computations of weights as well as the classical semirings. We show that under suitable assumptions, two resp. three fragments of our weighted logics can be transformed into each other. Moreover, we show that the logic fragments have the same expressive power as weighted nested word automata.


nested words weighted automata weighted logics quantitative automata valuation monoids 


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  1. 1.
    Alur, R., Arenas, M., Barceló, P., Etessami, K., Immerman, N., Libkin, L.: First-order and temporal logics for nested words. Logical Methods in Computer Science 4(4), 1–44 (2008)CrossRefGoogle Scholar
  2. 2.
    Alur, R., Madhusudan, P.: Adding nesting structure to words. Journal of the ACM 56(3), 16:1–16:43 (2009)Google Scholar
  3. 3.
    Berstel, J., Reutenauer, C.: Rational Series and Their Languages. EATCS Monographs in Theoretical Computer Science, vol. 12. Springer (1988)Google Scholar
  4. 4.
    Bollig, B., Gastin, P.: Weighted versus probabilistic logics. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 18–38. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  5. 5.
    Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik und Grundlagen Math. 6, 66–92 (1960)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative languages. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 385–400. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Chatterjee, K., Doyen, L., Henzinger, T.A.: Expressiveness and closure properties for quantitative languages. In: LICS, pp. 199–208. IEEE Computer Society (2009)Google Scholar
  8. 8.
    Droste, M., Kuich, W., Vogler, H. (eds.): Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science. Springer (2009)Google Scholar
  9. 9.
    Droste, M., Gastin, P.: Weighted automata and weighted logics. Theor. Comput. Sci. 380(1-2), 69–86 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Droste, M., Götze, D., Märcker, S., Meinecke, I.: Weighted tree automata over valuation monoids and their characterization by weighted logics. In: Kuich, W., Rahonis, G. (eds.) Algebraic Foundations in Computer Science. LNCS, vol. 7020, pp. 30–55. Springer, Heidelberg (2011)Google Scholar
  11. 11.
    Droste, M., Meinecke, I.: Weighted automata and weighted MSO logics for average and long-time behaviors. Inf. Comput. 220, 44–59 (2012)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Droste, M., Pibaljommee, B.: Weighted nested word automata and logics over strong bimonoids. In: Moreira, N., Reis, R. (eds.) CIAA 2012. LNCS, vol. 7381, pp. 138–148. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  13. 13.
    Droste, M., Rahonis, G.: Weighted automata and weighted logics on infinite words. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 49–58. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Eilenberg, S.: Automata, Languages, and Machines, Volume A, Pure and Applied Mathematics, vol. 59. Academic Press (1974)Google Scholar
  15. 15.
    Elgot, C.C.: Decision problems of finite automata design and related arithmetics. Transactions of the American Mathematical Society 98(1), 21–52 (1961)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Kuich, W., Salomaa, A.: Semirings, Automata, Languages. EATCS Monographs in Theoretical Computer Science, vol. 6. Springer (1986)Google Scholar
  17. 17.
    Löding, C., Madhusudan, P., Serre, O.: Visibly pushdown games. In: Lodaya, K., Mahajan, M. (eds.) FSTTCS 2004. LNCS, vol. 3328, pp. 408–420. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Mathissen, C.: Weighted logics for nested words and algebraic formal power series. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part II. LNCS, vol. 5126, pp. 221–232. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  19. 19.
    Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science. Springer (1978)Google Scholar
  20. 20.
    Schützenberger, M.P.: On the definition of a family of automata. Information and Control 4(2-3), 245–270 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Trakhtenbrot, B.A.: Finite automata and logic of monadic predicates. Doklady Akademii Nauk SSR 140, 326–329 (1961) (in Russian)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Manfred Droste
    • 1
  • Stefan Dück
    • 1
  1. 1.Institut für InformatikUniversity LeipzigLeipzigGermany

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