Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic second-order sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain "built-in" relations, such as the successor relation). The proof makes use of Ehrenfeucht-Fraisse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic second-order sentence.

Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures.In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy.As applications of this technique it is shown that • — Graph Connectivity is not expressible in existential monadic second-order (...) logic , even in the presence of a built-in linear order,• — Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n0, and• — the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation. (shrink)