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Cat Herding Game Played on Infinite Trees

Authors Rylo Ashmore , Sophie Pinchinat

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LIPIcs.FSTTCS.2025.10.pdf
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  • 15 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Higher order logic
  • Theory of computation → Verification by model checking
  • Theory of computation → Automata over infinite objects
  • Theory of computation → Tree languages
Keywords
  • Pursuit-evasion games
  • Cat Herding
  • Cat number
  • Infinite trees
  • Monadic Second Order Logic
  • Ordinals

Metrics

Abstract

The game of Cat Herding is played on a graph between two players, the cat and the herder. The game setup consists of the cat choosing a starting vertex for their cat token. Then, both players alternate turns, beginning with the herder: they delete (any) one edge, called a cut, and the cat moves along a path to a new vertex. While this game has been studied on finite graph arenas regarding how optimally herder wins, we shift our attention to an infinite version of the game where the cat may now survive indefinitely. We show that cat winning positions in an infinite tree can be characterized by a second-order monadic statement, also amounting to having a complete infinite binary tree minor, or having uncountably many distinct rays. We take advantage of the logical characterization of cat winning positions to generalize a measure known as the cat number, to ordinals.

Cite As Get BibTex

Rylo Ashmore and Sophie Pinchinat. Cat Herding Game Played on Infinite Trees. In 45th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 360, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.FSTTCS.2025.10

Author Details

Rylo Ashmore
  • Memorial University of Newfoundland, St. John’s, Canada
Sophie Pinchinat
  • IRISA/University of Rennes, France

Funding

  • Ashmore, Rylo: Rennes Métrople, Incoming International Mobility Grant for doctoral researchers.

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