We conjecture that the class of frame matroids can be axiomatised in monadic second-order logic. We have succeeded in axiomatising the class of bicircular matroids. This implies that bicircularity can be tested in polynomial time when the input is restricted to finite-field representable matroids of bounded branch-width. Furthermore, a class of bicircular matroids with bounded branch-width has a decidable theory, so there is a finite procedure to test whether or not any given monadic second-order sentence is a theorem for the class.
This is joint work with Daryl Funk and Mike Newman.