My wife keeps bugging me to enclose our 10'x26' concrete patio i poured about 10 years ago. I do not think we can do this, however. The concrete is a 4" (at least) slab on a bed of limestone. Since the roof would not be supported on something that is below the frost line, I'm afraid it might heave up in the winter (central PA). It would be difficult to dig 30-36" down just outside the patio, too, since there is only about a foot on one side between the edge of the patio and our 8' shed. So, can I build on the patio, or will this not be a good idea?
Attaching a non-frost protected structure directly to a frost protected structure is rarely a good idea, for the frost heave concerns you mentioned. As well it's rarely allowed by code.
You could either drill and pour a few "sono-tubes" below the frost line and anchor posts to them, leaving some room for movement, or build on the patio and have NO direct connection to the house. The latter is far more complicated than it sounds, and in being involved in hundreds of home additions, I've only come across one legitimate one.