Ordinal Measures of the Set of Finite Multisets

Author Isa Vialard

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Author Details

Isa Vialard
  • Université Paris-Saclay, CNRS, ENS Paris-Saclay, Laboratoire Méthodes Formelles, 91190, Gif-sur-Yvette, France

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Isa Vialard. Ordinal Measures of the Set of Finite Multisets. In 48th International Symposium on Mathematical Foundations of Computer Science (MFCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 272, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)


Well-partial orders, and the ordinal invariants used to measure them, are relevant in set theory, program verification, proof theory and many other areas of computer science and mathematics. In this article we focus on a common data structure in programming, finite multisets of some well partial order. There are two natural orders one can define on the set of finite multisets of a partial order: the multiset embedding and the multiset ordering. Though the maximal order type of these orders is already known, other ordinal invariants remain mostly unknown. Our main contributions are expressions to compute compositionally the width of the multiset embedding and the height of the multiset ordering. Furthermore, we provide a new ordinal invariant useful for characterizing the width of the multiset ordering.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity theory and logic
  • Theory of computation → Logic and verification
  • Theory of computation → Proof theory
  • Theory of computation → Program reasoning
  • Well-partial order
  • finite multisets
  • termination
  • program verification


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