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Uniformization Problems for Tree-Automatic Relations and Top-Down Tree Transducers

Authors Christof Löding, Sarah Winter

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Christof Löding
Sarah Winter

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Christof Löding and Sarah Winter. Uniformization Problems for Tree-Automatic Relations and Top-Down Tree Transducers. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 65:1-65:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)
https://doi.org/10.4230/LIPIcs.MFCS.2016.65

Abstract

For a given binary relation of finite trees, we consider the synthesis problem of deciding whether there is a deterministic top-down tree transducer that uniformizes the relation, and constructing such a transducer if it exists. A uniformization of a relation is a function that is contained in the relation and has the same domain as the relation. It is known that this problem is decidable if the relation is a deterministic top-down tree-automatic relation. We show that it becomes undecidable for general tree-automatic relations (specified by non-deterministic top-down tree automata). We also exhibit two cases for which the problem remains decidable. If we restrict the transducers to be path-preserving, which is a subclass of linear transducers, then the synthesis problem is decidable for general tree-automatic relations. If we consider relations that are finite unions of deterministic top-down tree-automatic relations, then the problem is decidable for synchronous transducers, which produce exactly one output symbol in each step (but can be non-linear).
Keywords
  • tree transducers
  • tree automatic relation
  • uniformization

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References

  1. J. Richard Büchi and Lawrence H. Landweber. Solving sequential conditions by finite-state strategies. Transactions of the American Mathematical Society, 138:295-311, 1969. URL: http://dx.doi.org/10.1090/S0002-9947-1969-0280205-0.
  2. Aranud Carayol, Christof Löding, Damian Niwiński, and Igor Walukiewicz. Choice functions and well-orderings over the infinite binary tree. Central European Journal of Mathematics, 8(4):662-682, 2010. Google Scholar
  3. Alonzo Church. Logic, arithmetic and automata. In Proceedings of the International Congress of Mathematicians, pages 23-35, 1962. Google Scholar
  4. Thomas Colcombet and Christof Löding. Transforming structures by set interpretations. Logical Methods in Computer Science, 3(2):1-36, 2007. URL: http://dx.doi.org/10.2168/LMCS-3(2:4)2007.
  5. Thomas Colcombet and Christof Löding. The non-deterministic Mostowski hierarchy and distance-parity automata. In Proceedings of the 35th International Colloquium on Automata, Languages and Programming, ICALP 2008, volume 5126 of Lecture Notes in Computer Science, pages 398-409. Springer, 2008. Google Scholar
  6. Thomas Colcombet and Christof Löding. The non-deterministic Mostowski hierarchy and distance-parity automata. In International Colloquium on Automata, Languages and Programming, ICALP 2016, 2016. to appear, full version on URL: http://arxiv.org/abs/1602.08565.
  7. Hu. Comon, M. Dauchet, R. Gilleron, C. Löding, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree automata techniques and applications, 2007. Release October, 12th 2007. URL: http://www.grappa.univ-lille3.fr/tata.
  8. C. C. Elgot and J. E. Mezei. On relations defined by generalized finite automata. IBM J. Res. Dev., 9(1):47-68, 1965. Google Scholar
  9. Ferenc Gécseg and Magnus Steinby. Tree Automata. Akadémiai Kiadó, Budapest, 1984. Google Scholar
  10. Yuri Gurevich and Saharon Shelah. Rabin’s uniformization problem. J. Symb. Log., 48(4):1105-1119, 1983. Google Scholar
  11. Michael Holtmann, Łukasz Kaiser, and Wolfgang Thomas. Degrees of lookahead in regular infinite games. In Foundations of Software Science and Computational Structures, volume 6014 of Lecture Notes in Computer Science, pages 252-266. Springer, 2010. URL: http://dx.doi.org/10.1007/978-3-642-12032-9_18.
  12. Frederick A. Hosch and Lawrence H. Landweber. Finite delay solutions for sequential conditions. In ICALP, pages 45-60, 1972. Google Scholar
  13. Kojiro Kobayashi. Classification of formal languages by functional binary transductions. Information and Control, 15(1):95-109, 1969. Google Scholar
  14. Dietrich Kuske and Thomas Weidner. Size and computation of injective tree automatic presentations. In Mathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings, volume 6907 of Lecture Notes in Computer Science, pages 424-435. Springer, 2011. Google Scholar
  15. C. Löding. Logic and automata over infinite trees, 2009. Habilitation Thesis, RWTH Aachen, Germany. Google Scholar
  16. Christof Löding and Sarah Winter. Synthesis of deterministic top-down tree transducers from automatic tree relations. In Proceedings Fifth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2014, Verona, Italy, September 10-12, 2014., volume 161 of EPTCS, pages 88-101, 2014. URL: http://dx.doi.org/10.4204/EPTCS.161.
  17. T. Milo, D. Suciu, and V. Vianu. Typechecking for xml transformers. J. Comput. Syst. Sci., 66(1):66-97, 2003. URL: http://dx.doi.org/10.1016/S0022-0000(02)00030-2.
  18. Yiannis Nichola Moschovakis. Descriptive Set Theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, Amsterdam, New York, Oxford, 1980. Google Scholar
  19. Dirk Siefkes. The recursive sets in certain monadic second order fragments of arithmetic. Arch. für mat. Logik und Grundlagenforschung, 17:71-80, 1975. Google Scholar
  20. Wolfgang Thomas. Automata on infinite objects. In Handbook of Theoretical Computer Science, volume B: Formal Models and Semantics, pages 133-192. Elsevier Science Publishers, Amsterdam, 1990. Google Scholar
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