Recognisability Equals Definability for Finitely Representable Matroids of Bounded Path-Width - Campbell, Rutger; Guillon, Bruno; Kante, Mamadou Moustapha; Kim, Eun Jung; Oum, Sang-il (2025 40th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), 2025)
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.