Composition Closure of Linear Extended Top-down Tree Transducers

Abstract

Linear extended top-down tree transducers (or synchronous tree-substitution grammars) are popular formal models of tree transformations that are extensively used in syntax-based statistical machine translation. The expressive power of compositions of such transducers with and without regular look-ahead is investigated. In particular, the restrictions of ε-freeness, strictness, and nondeletion are considered. The composition hierarchy turns out to be finite for all ε-free (all rules consume input) variants of these transducers except for the nondeleting ε-free transducers. The least number of transducers needed for the full expressive power of arbitrary compositions is presented. In all remaining cases (incl. the nondeleting ε-free transducers) the composition hierarchy does not collapse.

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References

  1. 1.

    Arnold, A., Dauchet, M.: Transductions inversibles de forêts. Thèse 3ème cycle M. Dauchet, Université de Lille (1975)

    Google Scholar 

  2. 2.

    Arnold, A., Dauchet, M.: Bi-transductions de forêts. In: ICALP, pp 74–86. Edinburgh University Press (1976)

  3. 3.

    Arnold, A., Dauchet, M.: Morphismes et bimorphismes d’arbres. Theoret. Comput. Sci. 20(1), 33–93 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Chiang, D.: An introduction to synchronous grammars. In: ACL. ACL. Part of a tutorial given with K. Knight (2006)

  5. 5.

    Dauchet, M.: Transductions de forêts — bimorphismes de magmoïdes. Première thèse, Université de Lille (1977)

  6. 6.

    Engelfriet, J.: Bottom-up and top-down tree transformations — a comparison. Math. Systems Theory 9(3), 198–231 (1975)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Engelfriet, J.: Top-down tree transducers with regular look-ahead. Math. Systems Theory 10(1), 289–303 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Engelfriet, J.: Three hierarchies of transducers. Math. Systems Theory 15(2), 95–125 (1982)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Engelfriet, J., Maneth, S.: Macro tree translations of linear size increase are MSO definable. SIAM J. Comput. 32(4), 950–1006 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Engelfriet, J., Schmidt, E.M.: IO and OI I. J. Comput. System Sci. 15(3), 328–353 (1977)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Fülöp, Z., Maletti, A.: Linking theorems for tree transducers. Submitted manuscript; available at: http://www.inf.u-szeged.hu/fulop/publ/linking.pdf (2014)

  12. 12.

    Fülöp, Z., Maletti, A., Vogler, H.: Preservation of recognizability for synchronous tree substitution grammars. In: ATANLP. ACL, pp 1–9 (2010)

  13. 13.

    Fülöp, Z., Maletti, A., Vogler, H.: Weighted extended tree transducers. Fundam. Inform. 111(2), 163–202 (2011)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Fülöp, Z., Vogler, H.: Syntax-Directed Semantics—Formal Models Based on Tree Transducers. EATCS Monographs on Theoret. Comput. Sci. Springer (1998)

  15. 15.

    Gécseg, F., Steinby, M.: Tree Automata. Akadémiai Kiadó, Budapest (1984) 2nd edition availble at. arXiv:1509.06233

  16. 16.

    Gécseg, F., Steinby, M.: Tree languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages. chap. 1, vol. 3, pp 1–68. Springer (1997)

  17. 17.

    Graehl, J., Hopkins, M., Knight, K., Maletti, A.: The power of extended top-down tree transducers. SIAM J. Comput. 39(2), 410–430 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Graehl, J., Knight, K., May, J.: Training tree transducers. Comput. Linguist. 34(3), 391–427 (2008)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Knight, K., Graehl, J.: An overview of probabilistic tree transducers for natural language processing. In: CICLing, LNCS, vol. 3406, pp 1–24, Springer (2005)

  20. 20.

    Lemay, A., Maneth, S., Niehren, J.: A learning algorithm for top-down XML transformations. In: PODS. ACM, pp 285–296 (2010)

  21. 21.

    Maletti, A.: Compositions of extended top-down tree transducers. Inf. Comput. 206(9–10), 1187–1196 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    May, J., Knight, K., Vogler, H.: Efficient inference through cascades of weighted tree transducers. In: ACL, pp 1058–1066 (2010)

  23. 23.

    Rounds, W.C.: Mappings and grammars on trees. Math. Systems Theory 4 (3), 257–287 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Thatcher, J.W.: Generalized2 sequential machine maps. J. Comput. System Sci. 4(4), 339–367 (1970)

    Article  MATH  Google Scholar 

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Correspondence to Andreas Maletti.

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This is a revised and extended version of [Z. Fülöp and A. Maletti: Composition closure of ε -free linear extended top-down tree transducers. In Proc. 17th DLT, volume 7907 of LNCS, pages 239–251. Springer-Verlag, 2013].

This work was partially supported by the exchange project 55 657 of the German Academic Exchange Service (DAAD) and Hungarian Scholarship Board Office (MÖB). Z. Fülöp was partially supported by the NKFI grant K 108 448, and A. Maletti was partially supported by the German Research Foundation (DFG) grant MA / 4959 / 1-1.

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Engelfriet, J., Fülöp, Z. & Maletti, A. Composition Closure of Linear Extended Top-down Tree Transducers. Theory Comput Syst 60, 129–171 (2017). https://doi.org/10.1007/s00224-015-9660-2

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Keywords

  • Extended top-down tree transducer
  • Composition hierarchy
  • Bimorphism