A Characterisation of Pi^0_2 Regular Tree Languages

Authors Filippo Cavallari, Henryk Michalewski, Michal Skrzypczak

Thumbnail PDF


  • Filesize: 0.55 MB
  • 14 pages

Document Identifiers

Author Details

Filippo Cavallari
Henryk Michalewski
Michal Skrzypczak

Cite AsGet BibTex

Filippo Cavallari, Henryk Michalewski, and Michal Skrzypczak. A Characterisation of Pi^0_2 Regular Tree Languages. In 42nd International Symposium on Mathematical Foundations of Computer Science (MFCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 83, pp. 56:1-56:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


We show an algorithm that for a given regular tree language L decides if L is in Pi^0_2, that is if L belongs to the second level of Borel Hierarchy. Moreover, if L is in Pi^0_2, then we construct a weak alternating automaton of index (0, 2) which recognises L. We also prove that for a given language L, L is recognisable by a weak alternating (1, 3)-automaton if and only if it is recognisable by a weak non-deterministic (1, 3)-automaton.
  • infinite trees
  • Rabin-Mostowski hierarchy
  • regular languages


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


  1. Mikołaj Bojańczyk and Thomas Place. Regular languages of infinite trees that are Boolean combinations of open sets. In ICALP, pages 104-115, 2012. Google Scholar
  2. Julius Richard Büchi and Lawrence H. Landweber. Solving sequential conditions by finite-state strategies. Transactions of the American Mathematical Society, 138:295-311, 1969. Google Scholar
  3. Thomas Colcombet. Fonctions régulières de coût. Habilitation thesis, Université Paris Diderot - Paris 7, 2013. Google Scholar
  4. Thomas Colcombet, Denis Kuperberg, Christof Löding, and Michael Vanden Boom. Deciding the weak definability of Büchi definable tree languages. In CSL, pages 215-230, 2013. Google Scholar
  5. Thomas Colcombet and Christof Löding. The non-deterministic Mostowski hierarchy and distance-parity automata. In ICALP (2), pages 398-409, 2008. Google Scholar
  6. Jacques Duparc and Filip Murlak. On the topological complexity of weakly recognizable tree languages. Fundamentals of computation theory, 2007. Google Scholar
  7. Alessandro Facchini and Henryk Michalewski. Deciding the Borel complexity of regular tree languages. In CiE 2014, pages 163-172, 2014. Google Scholar
  8. Alessandro Facchini, Filip Murlak, and Michał Skrzypczak. Index problems for game automata. ACM Trans. Comput. Log., 17(4):24:1-24:38, 2016. Google Scholar
  9. Tomasz Gogacz, Henryk Michalewski, Matteo Mio, and Michał Skrzypczak. Measure properties of game tree languages. In MFCS, pages 303-314, 2014. Google Scholar
  10. Erich Grädel, Wolfgang Thomas, and Thomas Wilke, editors. Automata, Logics, and Infinite Games: A Guide to Current Research, volume 2500 of Lecture Notes in Computer Science. Springer, 2002. Google Scholar
  11. Alexander Kechris. Classical descriptive set theory. Springer-Verlag, New York, 1995. Google Scholar
  12. Denis Kuperberg and Michael Vanden Boom. Quasi-weak cost automata: A new variant of weakness. In FSTTCS, volume 13 of LIPIcs, pages 66-77, 2011. Google Scholar
  13. Orna Kupferman and Moshe Y. Vardi. The weakness of self-complementation. In STACS, pages 455-466, 1999. Google Scholar
  14. Christof Löding. Logic and automata over infinite trees. Habilitation thesis, RWTH Aachen, Germany, 2009. Google Scholar
  15. Satoru Miyano and Takeshi Hayashi. Alternating finite automata on omega-words. Theor. Comput. Sci., 32:321-330, 1984. Google Scholar
  16. Filip Murlak. The Wadge hierarchy of deterministic tree languages. Logical Methods in Computer Science, 4(4), 2008. Google Scholar
  17. Damian Niwiński and Igor Walukiewicz. A gap property of deterministic tree languages. Theor. Comput. Sci., 1(303):215-231, 2003. Google Scholar
  18. Damian Niwiński and Igor Walukiewicz. Deciding nondeterministic hierarchy of deterministic tree automata. Electr. Notes Theor. Comput. Sci., 123:195-208, 2005. Google Scholar
  19. Dominique Perrin and Jean-Éric Pin. Infinite Words: Automata, Semigroups, Logic and Games. Elsevier, 2004. Google Scholar
  20. Michael Oser Rabin. Decidability of second-order theories and automata on infinite trees. Trans. of the American Math. Soc., 141:1-35, 1969. Google Scholar
  21. Michael Oser Rabin and Dana Scott. Finite automata and their decision problems. IBM Journal of Research and Development, 3(2):114-125, April 1959. Google Scholar
  22. Michał Skrzypczak. Descriptive Set Theoretic Methods in Automata Theory - Decidability and Topological Complexity, volume 9802 of Lecture Notes in Computer Science. Springer, 2016. Google Scholar
  23. Michał Skrzypczak and Igor Walukiewicz. Deciding the topological complexity of Büchi languages. In ICALP (2), pages 99:1-99:13, 2016. Google Scholar
  24. Jerzy Skurczyński. The Borel hierarchy is infinite in the class of regular sets of trees. Theoretical Computer Science, 112(2):413-418, 1993. Google Scholar
  25. Wolfgang Thomas. Languages, automata, and logic. In Handbook of Formal Languages, pages 389-455. Springer, 1996. Google Scholar
  26. Wolfgang Thomas and Helmut Lescow. Logical specifications of infinite computations. In REX School/Symposium, pages 583-621, 1993. Google Scholar
  27. Boris A. Trakhtenbrot. Finite automata and the monadic predicate calculus. Siberian Mathematical Journal, 3(1):103-131, 1962. Google Scholar