TY - GEN
T1 - The Uniformisation of Monadic Second-Order Logic over Countable Ordinals
AU - Colcombet, Thomas
AU - Rabinovich, Alexander
N1 - Publisher Copyright: © Thomas Colcombet and Alexander Rabinovich.
PY - 2026/7/29
Y1 - 2026/7/29
N2 - We study the uniformisation problem for monadic second-order logic (MSO) over countable ordinal chains. Given a formula defining a relation between subsets of the input structure, the question is whether there exists a formula that defines a function selecting, for every set in the domain of the relation, a unique set such that the pair belongs to the relation. It is known, due to Lifsches and Shelah [9], that MSO cannot, in general, be uniformised over the class of countable ordinals. We show that the maximal uniformisation degree is reached by extending the logic with a predicate that, given a set, selects (when possible) a cofinal subset of order type ω. Equivalently, every MSO formula can be uniformised over the class of countable ordinal chains using a formula in this extended logic.
AB - We study the uniformisation problem for monadic second-order logic (MSO) over countable ordinal chains. Given a formula defining a relation between subsets of the input structure, the question is whether there exists a formula that defines a function selecting, for every set in the domain of the relation, a unique set such that the pair belongs to the relation. It is known, due to Lifsches and Shelah [9], that MSO cannot, in general, be uniformised over the class of countable ordinals. We show that the maximal uniformisation degree is reached by extending the logic with a predicate that, given a set, selects (when possible) a cofinal subset of order type ω. Equivalently, every MSO formula can be uniformised over the class of countable ordinal chains using a formula in this extended logic.
KW - Monadic second-order logic
KW - chains
KW - composition method
KW - factorization trees
KW - ordinals
KW - uniformisation
UR - https://www.scopus.com/pages/publications/105045282092
U2 - 10.4230/LIPIcs.LICS.2026.30
DO - 10.4230/LIPIcs.LICS.2026.30
M3 - Conference contribution
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 41st Annual Symposium on Logic in Computer Science, LICS 2026
A2 - Claudia, Faggian
A2 - Joost-Pieter, Katoen
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 41st Annual Symposium on Logic in Computer Science, LICS 2026
Y2 - 20 July 2026 through 23 July 2026
ER -