We prove that a connected graph of bounded degree with only nitely many orbits has a decidable MSO-theory if and only if it is context-free. This implies that a group is context-free if and only if its Cayley-graph has a decidable MSO-theory. On the other hand, the rst-order theory of the Cayley-graph of a group is decidable if and only if the group has a decidable word problem. For Cayley-graphs of monoids we prove the following closure properties. The class of monoids whose Cayley-graphs have decidable MSO-theories is closed under free products. The class of monoids whose Cayley-graphs have decidable rstorder theories is closed under general graph products. For the latter result on rst-order theories we introduce a new unfolding construction, the factorized unfolding, that generalizes the tree-like structures considered by Walukiewicz. We show and use that it preserves the decidability of the rst-order theory.

The model theory of finite structures is intimately connected to various fields