Deciding Linear Height and Linear Size-To-Height Increase of Macro Tree Transducers

Authors Paul Gallot , Sebastian Maneth, Keisuke Nakano, Charles Peyrat

Thumbnail PDF


  • Filesize: 0.7 MB
  • 20 pages

Document Identifiers

Author Details

Paul Gallot
  • Universität Bremen, Germany
Sebastian Maneth
  • Universität Bremen, Germany
Keisuke Nakano
  • Tohoku University, Sendai, Japan
Charles Peyrat
  • ENS Paris-Saclay, France


We thank the anonymous reviewers of a previous version of this paper for their critical comments.

Cite AsGet BibTex

Paul Gallot, Sebastian Maneth, Keisuke Nakano, and Charles Peyrat. Deciding Linear Height and Linear Size-To-Height Increase of Macro Tree Transducers. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 138:1-138:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We present a novel normal form for (total deterministic) macro tree transducers (mtts), called "depth proper normal form". If an mtt is in this normal form, then it is guaranteed that each parameter of each state appears at arbitrary depths in the output trees of that state. Intuitively, if some parameter only appears at certain bounded depths in the output trees of a state, then this parameter can be eliminated by in-lining the corresponding output paths at each call site of that state. We use regular look-ahead in order to determine which of the paths should be in-lined. As a consequence of changing the look-ahead, a parameter that was previously appearing at unbounded depths, may be appearing at bounded depths for some new look-ahead; for this reason, our construction has to be iterated to obtain an mtt in depth-normal form. Using the normal form, we can decide whether the translation of an mtt has linear height increase or has linear size-to-height increase.

Subject Classification

ACM Subject Classification
  • Theory of computation → Transducers
  • automata
  • formal language theory
  • macro tree transducer
  • normal form


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Alfred V. Aho and Jeffrey D. Ullman. Translations on a context-free grammar. Inf. Control., 19(5):439-475, 1971. URL:
  2. Rajeev Alur and Loris D'Antoni. Streaming tree transducers. J. ACM, 64(5):31:1-31:55, 2017. URL:
  3. Mikolaj Bojanczyk and Amina Doumane. First-order tree-to-tree functions. In Holger Hermanns, Lijun Zhang, Naoki Kobayashi, and Dale Miller, editors, LICS '20: 35th Annual ACM/IEEE Symposium on Logic in Computer Science, Saarbrücken, Germany, July 8-11, 2020, pages 252-265. ACM, 2020. URL:
  4. Hubert Comon-Lundh, Max Dauchet, Rémi Gilleron, Cristof Löding, Florent Jacquemard, Denis Lugiez, Sophie Tison, and Marc Tommasi. Tree Automata Techniques and Applications. URL:, November 2007.
  5. Frank Drewes and Joost Engelfriet. Decidability of the finiteness of ranges of tree transductions. Inf. Comput., 145(1):1-50, 1998. URL:
  6. Joost Engelfriet. Bottom-up and top-down tree transformations - A comparison. Math. Syst. Theory, 9(3):198-231, 1975. URL:
  7. Joost Engelfriet. Context-free grammars with storage. CoRR, abs/1408.0683, 2014. URL:
  8. Joost Engelfriet, Kazuhiro Inaba, and Sebastian Maneth. Linear-bounded composition of tree-walking tree transducers: linear size increase and complexity. Acta Informatica, 58(1-2):95-152, 2021. URL:
  9. Joost Engelfriet and Sebastian Maneth. Macro tree transducers, attribute grammars, and MSO definable tree translations. Inf. Comput., 154(1):34-91, 1999. URL:
  10. Joost Engelfriet and Sebastian Maneth. Macro tree translations of linear size increase are MSO definable. SIAM J. Comput., 32(4):950-1006, 2003. URL:
  11. Joost Engelfriet and Erik Meineche Schmidt. IO and OI. I. J. Comput. Syst. Sci., 15(3):328-353, 1977. URL:
  12. Joost Engelfriet and Erik Meineche Schmidt. IO and OI. II. J. Comput. Syst. Sci., 16(1):67-99, 1978. URL:
  13. Joost Engelfriet and Heiko Vogler. Macro tree transducers. J. Comput. Syst. Sci., 31(1):71-146, 1985. URL:
  14. Emmanuel Filiot, Sebastian Maneth, Pierre-Alain Reynier, and Jean-Marc Talbot. Decision problems of tree transducers with origin. Inf. Comput., 261:311-335, 2018. URL:
  15. M. J. Fischer. Grammars with Macro like Productions. PhD thesis, Harvard University, 1968. See also Proc. 9th Sympos. on SWAT, pp. 131-142, 1968. Google Scholar
  16. Zoltán Fülöp. On attributed tree transducers. Acta Cybern., 5(3):261-279, 1981. URL:
  17. Zoltán Fülöp and Heiko Vogler. Syntax-Directed Semantics - Formal Models Based on Tree Transducers. Monographs in Theoretical Computer Science. An EATCS Series. Springer, 1998. URL:
  18. Paul Gallot, Sebastian Maneth, Keisuke Nakano, and Charles Peyrat. Deciding linear height and linear size-to-height increase for macro tree transducers, 2024. URL:
  19. Ferenc Gécseg and Magnus Steinby. Tree languages. In Grzegorz Rozenberg and Arto Salomaa, editors, Handbook of Formal Languages, Volume 3: Beyond Words, pages 1-68. Springer, 1997. URL:
  20. Donald E. Knuth. Semantics of context-free languages. Math. Syst. Theory, 2(2):127-145, 1968. URL:
  21. Donald E. Knuth. Correction: Semantics of context-free languages. Math. Syst. Theory, 5(1):95-96, 1971. URL:
  22. Thomas Perst and Helmut Seidl. Macro forest transducers. Inf. Process. Lett., 89(3):141-149, 2004. URL:
  23. William C. Rounds. Mappings and grammars on trees. Math. Syst. Theory, 4(3):257-287, 1970. URL:
  24. James W. Thatcher. Generalized sequential machine maps. J. Comput. Syst. Sci., 4(4):339-367, 1970. URL:
  25. James W. Thatcher. There’s a lot more to finite automata theory than you would have thought. In Proc. 4th Ann. Princeton Conf. on Informations Sciences and Systems, pages 263-276, 1970. Also published in revised form under the title "Tree automata: an informal survey" in Currents in the Theory of Computing (ed. A. V. Aho), Prentice-Hall, 1973, 143-172. Google Scholar