On monadic NP vs. monadic co-NP

It is proved that connectivity of finite graphs is not in monadic NP, even in the presence of arbitrary built-in relations of moderate degree (that is, degree (log n) /sup o(1)/). This results in a strong separation between monadic NP and monadic co-NP. The proof uses a combination of three techniques: (1) a technique of W. Hanf (1965) for showing that two (infinite) structures agree on all first-order sentences, under certain conditions; (2) a recent approach to second-order Ehrenfeucht-Fraisse games by M. Ajtai and R. Fagin (1990); and (3) playing Ehrenfeucht-Fraisse games over random structures. Regarding (1), a version of Hanf´s result that is better suited for use as a tool in inexpressibility proofs for classes of finite structures is given. The power of these techniques is further demonstrated by using the first two techniques to give a very simple proof of the separation of monadic NP from monadic co-NP without the presence of built-in relations.