On winning strategies in Ehrenfeucht-Fraïssé games

We present a powerful and versatile new sufficient condition for the second player (the "duplicator") to have a winning strategy in an Ehrenfeucht-Fraïssé game on graphs. We accomplish two things with this technique. First, we give a simpler and much easier-to-understand proof of Ajtai and Fagin's result that reachability in directed finite graphs is not in monadic NP. (Monadic NP, otherwise known as monadic ∑₁¹, corresponds to existential second-order logic with the restriction that the second-order quantifiers range only over sets, and not over relations of higher arity, such as binary relations.) Second, we show that this result holds in the presence of a larger class of built-in relations than was known before.