[Fordim Hedgethistle]

Point of Order: Frodo never actually forgives Gollum he says that he pities him. These are two different things -- one can feel pity for the criminal whom is being sentenced to death but feel that the sentence is just: "You deserve to die for your crime, but I feel sorry for you insofar as you made bad choices in your life which have led to this".

In the conversation with Gandalf in "Shadow of the Past" Frodo says that Gollum "deserves death" and Gandalf agrees; he goes on to argue that Bilbo was right not to kill Gollum because "he has some part yet to play ere the end" and because the right to end life belongs properly to a higher power. No talk anywhere here of forgiveness or redemption.

So while I agree wholeheartedly that the Quest has succeeded thanks to Frodo's actions with Gollum, I think it's pretty clear that Frodo has decided that he doesn't have the right to kill Gollum, even though he may "deserve death" and not because he's forgiven him. That having been said, it's hard to see the drama enacted at the Crack of Doom as being about forgiveness; rather, it's about Pity.