Trees, grids, and MSO decidability: From graphs to matroids
Theoretical Computer Science
Monadic second order (MSO) logic has proved to be a useful tool in many areas of application, reaching from decidability and complexity to picture processing, correctness of programs and parallel processes. To characterize the structural borderline between decidability and undecidability is a classical research problem here. This problem is related to questions in computational complexity, especially to the model checking problem, for which many tools developed in the area of decidability have
... roved to be useful. For more than two decades it was conjectured in [D. Seese, The structure of the models of decidable monadic theories of graphs, Ann. Pure Appl. Logic 53 (1991) ] that decidability of monadic theories of countable structures implies that the theory can be reduced via interpretability to a theory of trees. It is one of the main goals of this article to prove a variant of this conjecture for matroids representable over a finite field. (Matroids can be viewed as a wide generalization of graphs, and they seem to capture some second order properties in a more suitable way than graphs themselves, cf. the recent development in matroid structure theory [J.F. Geelen, A.H.M. Gerards, G.P. Whittle, Branchwidth and well-quasi-ordering in matroids and graphs, J. Combin. Theory Ser. B 84 (2002) 270-290; J.F. Geelen, A.H.M. Gerards, N. Robertson, G.P. Whittle, Excluding a planar graph from a GF(q)-representable matroid, manuscript, 2003 ].) More exactly we prove, for every finite field F, that any class of F-representable matroids with a decidable MSO theory must have uniformly bounded branch-width. Moreover, we show that bounding the branch-width of all matroids in general is not sufficient to obtain a decidable MSO theory. Our paper gives a (rather detailed) introduction to these different subjects, and shows that a blend of ideas and methods from logic together with structural matroid theory can lead to new tools and algorithms, and can shed light on some old open problems.