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A Dichotomy Theorem for Ordinal Ranks in MSO

Authors Damian Niwiński , Paweł Parys , Michał Skrzypczak

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Author Details

Damian Niwiński
  • Institute of Informatics, University of Warsaw, Poland
Paweł Parys
  • Institute of Informatics, University of Warsaw, Poland
Michał Skrzypczak
  • Institute of Informatics, University of Warsaw, Poland

Funding

All authors supported by the National Science Centre, Poland (grant no. 2021/41/B/ST6/03914).

Acknowledgements

The authors would like to thank Marek Czarnecki for preliminary discussions on related subjects.

Cite As Get BibTex

Damian Niwiński, Paweł Parys, and Michał Skrzypczak. A Dichotomy Theorem for Ordinal Ranks in MSO. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 69:1-69:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.69

Abstract

We focus on formulae ∃X.φ(Y, X) of monadic second-order logic over the full binary tree, such that the witness X is a well-founded set. The ordinal rank rank(X) < ω₁ of such a set X measures its depth and branching structure. We search for the least upper bound for these ranks, and discover the following dichotomy depending on the formula φ. Let η_φ be the minimal ordinal such that, whenever an instance Y satisfies the formula, there is a witness X with rank(X) ≤ η_φ. Then η_φ is either strictly smaller than ω² or it reaches the maximal possible value ω₁. Moreover, it is decidable which of the cases holds. The result has potential for applications in a variety of ordinal-related problems, in particular it entails a result about the closure ordinal of a fixed-point formula.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Tree languages
Keywords
  • dichotomy result
  • limit ordinal
  • countable ordinals
  • nondeterministic tree automata

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