Recognisability Equals Definability for Finitely Representable Matroids of Bounded Path-Width

Let {\mathbb{F}} be a finite field. We prove that there is an MSO-transduction which, given an {\mathbb{F}}-representable matroid of path-width k, produces a branch-decomposition of width at most f(k), for some function f. As a corollary, any recognizable property of {\mathbb{F}}-representable matro...

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Bibliographic Details
Published in:2025 40th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) pp. 678 - 690
Main Authors: Campbell, Rutger, Guillon, Bruno, Kante, Mamadou Moustapha, Kim, Eun Jung, Oum, Sang-il
Format: Conference Proceeding
Language:English
Published: IEEE 23.06.2025
Subjects:
branch-width
Computer science
decomposition-width
definability
Galois fields
Logic
matroid
monadic second-order logic
path-width
recognisability
transduction
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Summary:Let {\mathbb{F}} be a finite field. We prove that there is an MSO-transduction which, given an {\mathbb{F}}-representable matroid of path-width k, produces a branch-decomposition of width at most f(k), for some function f. As a corollary, any recognizable property of {\mathbb{F}}-representable matroids with bounded path-width is definable in MSO logic, and therefore recognizability is equivalent to MSO-definability on classes of {\mathbb{F}}-representable matroids of bounded path-width. This generalizes the result of Bojańczyk, Grohe and Pilipczuk [Logical Methods in Computer Science 17(1), 2021] which asserts the equivalence of the two notions on graphs of bounded linear clique-width.
DOI:10.1109/LICS65433.2025.00057