On the Expansion of Monadic Second-Order Logic with Cantor-Bendixson Rank and Order Type Predicates

Authors Thomas Colcombet , Alexander Rabinovich



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Thomas Colcombet
  • Université Paris Cité, CNRS, IRIF, France
Alexander Rabinovich
  • The Blavatnik School of Computer Science, Tel-Aviv University, Israel

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Thomas Colcombet and Alexander Rabinovich. On the Expansion of Monadic Second-Order Logic with Cantor-Bendixson Rank and Order Type Predicates. In 33rd EACSL Annual Conference on Computer Science Logic (CSL 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 326, pp. 11:1-11:26, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CSL.2025.11

Abstract

In this work, we consider two extensions of monadic second-order logic, and study in what cases the classical decidability results are preserved.
The first extension, MSO[CBrank_β], is MSO (over the signature of the binary tree) augmented with the extra ability to express that the subtree over a set X has Cantor-Bendixson rank β, for some fixed countable ordinal β. We show that this extension is decidable over the binary tree if and only if β is finite, which means that it is decidable if and only if it is equivalent in expressiveness to MSO.
The second extension, MSO[otp_α], is MSO (over the signature of order) augmented with the extra ability to express that the suborder induced by a set X has order type α for some fixed countable ordinal α. We show that this extension is decidable over countable ordinals if and only if α < ω^ω, which means that it is decidable if and only if it is equivalent in expressiveness to MSO.
The first result can be established as a consequence of the second. The second result relies on the undecidability results of the logic BMSO (itself relying on the undecidability of MSO+U) in the case of ω^β for β a limit ordinal, and on entirely new techniques when β is a successor ordinal. We also have some partial extensions of the second result to some uncountable cases.

Subject Classification

ACM Subject Classification
  • Theory of computation → Tree languages
  • Theory of computation → Logic and verification
Keywords
  • Logic
  • Algorithmic model theory
  • Monadic second-order logic
  • Ordinals
  • Binary tree

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