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A notion of branch-width, which generalizes the one known for graphs, can bedefined for matroids. We first give a proof of the polynomial timemodel-checking of monadic second-order formulas on representable matroids ofbounded branch-width, by reduction to monadic second-order formulas on trees.This proof is much simpler than the one previously known. We also provide alink between our logical approach and a grammar that allows to build matroidsof bounded branch-width. Finally, we introduce a new class of non-necessarilyrepresentable matroids, described by a grammar and on which monadicsecond-order formulas can be checked in linear time.

Monadic second-order model-checking on decomposable matroids