Possessive misbehaving?

[vcvpaiva]

If I look at the representation of "My cat barked." it gives me "barked:(subj:My cat ) have:(subj:My , dobj:cat )" ok, I don't like the use of "have" would prefer possessive, (as 'have' will have many other uses) but whether using it or not, I'd expect that the subject of the possessive is the "cat", not the pronoun? how can the "cat" be a direct object? I do see that it would be the object of the sentence "I have a cat", but then I really think that one does not need to create this new proposition, that simply having the NP=my cat is a possessive would be enough.

[vcvpaiva]

also I'd expect "The picture of Anna arrived." and "Anna's picture arrived." to have the same representation and they do not.

The picture of Anna arrived = arrived:(subj:The picture of Anna ) Anna's picture arrived =arrived:(subj:Anna 's picture ) have:(subj:Anna 's , dobj:picture )