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A Complexity Approach to Tree Algebras: the Bounded Case

Authors Thomas Colcombet , Arthur Jaquard



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LIPIcs.ICALP.2021.127.pdf
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Author Details

Thomas Colcombet
  • Université de Paris, CNRS, IRIF, F-75006, Paris, France
Arthur Jaquard
  • Université de Paris, CNRS, IRIF, F-75006, Paris, France

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Thomas Colcombet and Arthur Jaquard. A Complexity Approach to Tree Algebras: the Bounded Case. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 127:1-127:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ICALP.2021.127

Abstract

In this paper, we initiate a study of the expressive power of tree algebras, and more generally infinitely sorted algebras, based on their asymptotic complexity. We provide a characterization of the expressiveness of tree algebras of bounded complexity. Tree algebras in many of their forms, such as clones, hyperclones, operads, etc, as well as other kind of algebras, are infinitely sorted: the carrier is a multi sorted set indexed by a parameter that can be interpreted as the number of variables or hole types. Finite such algebras - meaning when all sorts are finite - can be classified depending on the asymptotic size of the carrier sets as a function of the parameter, that we call the complexity of the algebra. This naturally defines the notions of algebras of bounded, linear, polynomial, exponential or doubly exponential complexity... We initiate in this work a program of analysis of the complexity of infinitely sorted algebras. Our main result precisely characterizes the tree algebras of bounded complexity based on the languages that they recognize as Boolean closures of simple languages. Along the way, we prove that such algebras that are syntactic (minimal for a language) are exactly those in which, as soon as there are sufficiently many variables, the elements are invariant under permutation of the variables.

Subject Classification

ACM Subject Classification
  • Theory of computation → Tree languages
  • Theory of computation → Regular languages
Keywords
  • Tree algebra
  • infinite tree
  • language theory

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References

  1. Achim Blumensath. Recognisability for algebras of infinite trees. Theor. Comput. Sci., 412(29):3463-3486, 2011. URL: https://doi.org/10.1016/j.tcs.2011.02.037.
  2. Achim Blumensath. Regular tree algebras. Logical Methods in Computer Science, 16, 2020.
  3. Mikolaj Bojanczyk and Bartek Klin. A non-regular language of infinite trees that is recognizable by a sort-wise finite algebra. Log. Methods Comput. Sci., 15(4), 2019. URL: https://lmcs.episciences.org/5927, URL: https://doi.org/10.23638/LMCS-15(4:11)2019.
  4. Mikolaj Bojanczyk and Michal Pilipczuk. Definability equals recognizability for graphs of bounded treewidth. In Martin Grohe, Eric Koskinen, and Natarajan Shankar, editors, Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science, LICS'16, New York, NY, USA, July 5-8, 2016, pages 407-416. ACM, 2016. URL: https://doi.org/10.1145/2933575.2934508.
  5. Mikolaj Bojanczyk and Luc Segoufin. Tree languages defined in first-order logic with one quantifier alternation. Log. Methods Comput. Sci., 6(4), 2010. URL: https://doi.org/10.2168/LMCS-6(4:1)2010.
  6. Mikolaj Bojanczyk, Luc Segoufin, and Howard Straubing. Piecewise testable tree languages. Log. Methods Comput. Sci., 8(3), 2012. URL: https://doi.org/10.2168/LMCS-8(3:26)2012.
  7. Mikolaj Bojanczyk and Igor Walukiewicz. Forest algebras. In Jörg Flum, Erich Grädel, and Thomas Wilke, editors, Logic and Automata: History and Perspectives [in Honor of Wolfgang Thomas], volume 2 of Texts in Logic and Games, pages 107-132. Amsterdam University Press, 2008.
  8. Bruno Courcelle and Joost Engelfriet. Graph structure and monadic second-order logic: a language-theoretic approach, volume 138. Cambridge University Press, 2012.
  9. Zoltán Ésik and Pascal Weil. Algebraic recognizability of regular tree languages. Theor. Comput. Sci., 340(1):291-321, 2005. URL: https://doi.org/10.1016/j.tcs.2005.03.038.
  10. Zoltán Ésik and Pascal Weil. Algebraic characterization of logically defined tree languages. Int. J. Algebra Comput., 20(2):195-239, 2010. URL: https://doi.org/10.1142/S0218196710005595.
  11. Marcel-Paul Schützenberger. On finite monoids having only trivial subgroups. Information and Control, 8:190-194, 1965.
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