Tree Automata and Pigeonhole Classes of Matroids: I

Abstract Hliněný’s Theorem shows that any sentence in the monadic second-order logic of matroids can be tested polynomial time, when input is limited to a class $${\mathbb {F}}$$ F -representable with bounded branch-width (where finite field). If each matroid decomposed by subcubic tree such way only amount information flows across displayed separations, then has decomposition-width. We introduce pigeonhole property for classes matroids: if every subclass also decomposition-width, pigeonhole. An efficiently stronger property, involving an efficiently-computable equivalence relation on subsets ground set. show extends class. In sequel paper, we use these ideas extend fundamental transversal matroids, lattice path bicircular and $$H$$ H -gain-graphic where H group. give characterisation families hypergraphs described via automata: family defined automaton it Furthermore, logic, decidable theory.