  We consider distributed model-checking of Monadic Second-Order logic (MSO) on
graphs which constitute the topology of communication networks. The graph is
thus both the structure being checked and the system on which the distributed
computation is performed. We prove that MSO can be distributively model-checked
with only a constant number of messages sent over each link for planar networks
with bounded diameter, as well as for networks with bounded degree and bounded
tree-length. The distributed algorithms rely on nontrivial transformations of
linear time sequential algorithms for tree decompositions of bounded tree-width
graphs.
