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State Complexity of Projection and Quotient on Unranked Trees

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Descriptional Complexity of Formal Systems (DCFS 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7386))

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Abstract

We consider projection and quotient operations on unranked tree languages, and investigate their state complexities on deterministic unranked tree automata. We give a tight upper bound on the number of vertical states which is different than the known state complexity of projection for string language. Since there are two ways to define concatenation on trees, we define four different quotient operations on trees and obtain tight bounds for each operation. The state complexity of sequential bottom-quotient differs by a multiplicative factor (n+1) from the corresponding result of left-quotient on ordinary finite automata.

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References

  1. Bach, E., Shallit, J.: Algorithmic number theory, vol. I. MIT Press (1996)

    Google Scholar 

  2. Brüggemann-Klein, A., Murata, M., Wood, D.: Regular tree and regular hedge languages over unranked alphabets. HKUST Technical report (2001)

    Google Scholar 

  3. Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Löding, C., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications (2007), electronic book, tata.gforge.inria.fr

  4. Cristau, J., Löding, C., Thomas, W.: Deterministic Automata on Unranked Trees. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 68–79. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  5. Gécseg, F., Steinby, M.: Tree languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 1–68. Springer (1997)

    Google Scholar 

  6. Gruber, H., Holzer, M.: Tight Bounds on the Descriptional Complexity of Regular Expressions. In: Diekert, V., Nowotka, D. (eds.) DLT 2009. LNCS, vol. 5583, pp. 276–287. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  7. Holzer, M., Kutrib, M.: Nondeterministic finite automata – Recent results on the descriptional and computational complexity. Internat. J. Foundations Comput. Sci. 20, 563–580 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Holzer, M., Kutrib, M.: Descriptional and computational complexity of finite automata — A survey. Inf. Comput. 209, 456–470 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jirásková, G., Masopust, T.: State Complexity of Projected Languages. In: Holzer, M., Kutrib, M., Pighizzini, G. (eds.) DCFS 2011. LNCS, vol. 6808, pp. 198–211. Springer, Heidelberg (2011)

    Google Scholar 

  10. Leung, H.: Descriptional complexity of NFA of different ambiguity. International Journal of Foundations of Computer Science (16), 975–984 (2005)

    Google Scholar 

  11. Martens, W., Niehren, J.: On the minimization of XML schemas and tree automata for unranked trees. J. Comput. System Sci. 73, 550–583 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Neven, F.: Automata theory for XML researchers. SIGMOD Record (31), 39–46 (2002)

    Google Scholar 

  13. Okhotin, A.: Unambiguous Finite Automata over a Unary Alphabet. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 556–567. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  14. Piao, X.: State complexity of tree automata. PhD thesis, School of Computing, Queen’s University (2011)

    Google Scholar 

  15. Piao, X., Salomaa, K.: Transformations between different models of unranked bottom-up tree automata. Fundamenta Informaticae (109), 405–424 (2011)

    Google Scholar 

  16. Piao, X., Salomaa, K.: State complexity of concatenation of regular tree languages. Theoretical Computer Science (accepted for publication), doi:10.1016/j.tcs.2011.12.048

    Google Scholar 

  17. Piao, X., Salomaa, K.: State trade-offs in unranked tree automata. In: Holzer, M., Kutrib, M., Pighizzini, G. (eds.) DCFS 2011. LNCS, vol. 6808, pp. 261–274. Springer, Heidelberg (2011)

    Google Scholar 

  18. Piao, X., Salomaa, K.: Lower bounds for the size of deterministic unranked tree automata. Theoretical Computer Science, doi:10.1016/j.tcs.2012.03.043

    Google Scholar 

  19. Raeymaekers, S., Bruynooghe, M.: Minimization of finite unranked tree automata (2004) (manuscript)

    Google Scholar 

  20. Rozenberg, G., Salomaa, A.: Handbook of Formal Languages, vol. I–III. Springer (1997)

    Google Scholar 

  21. Schwentick, T.: Automata for XML. J. Comput. System Sci. (73), 289–315 (2007)

    Google Scholar 

  22. Schmidt, E.M.: Succinctness of descriptions of context-free, regular and finite languages. PhD thesis, Cornell University (1978)

    Google Scholar 

  23. Wong, K.: On the complexity of projections of discrete-event systems. In: Proc. of WSDES, Cagliari, Italy, pp. 201–206 (1998)

    Google Scholar 

  24. Yu, S.: Regular languages. In: [20], vol. I, pp. 41–110 (1997)

    Google Scholar 

  25. Yu, S.: State complexity: Recent results and open problems. Fundamenta Informaticae (64), 471–481 (2005)

    Google Scholar 

  26. Yu, S., Zhuang, Q., Salomaa, K.: The state complexity of some basic operations on regular languages. Theoretical Computer Science (125), 315–328 (1994)

    Google Scholar 

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Authors and Affiliations

  1. School of Computing, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

    Xiaoxue Piao & Kai Salomaa

Authors
  1. Xiaoxue Piao
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  2. Kai Salomaa
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität Giessen, Arndtstraße 2, 35392, Giessen, Germany

    Martin Kutrib

  2. Departamento de Ciência de Computadores, Universidade do Porto, Faculdade de Ciências, Rua do Campo Alegre 1021-1055, 4169-007, Porto, Portugal

    Nelma Moreira  & Rogério Reis  & 

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Piao, X., Salomaa, K. (2012). State Complexity of Projection and Quotient on Unranked Trees. In: Kutrib, M., Moreira, N., Reis, R. (eds) Descriptional Complexity of Formal Systems. DCFS 2012. Lecture Notes in Computer Science, vol 7386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31623-4_22

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  • DOI: https://doi.org/10.1007/978-3-642-31623-4_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31622-7

  • Online ISBN: 978-3-642-31623-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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