Is this sentence grammatically incorrect?
The cat's—who was still sleeping—bowl is filled with milk.
If not, what does it imply?
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According to McCawley's discussion of appositive relative clauses in TSPE, an appositive relative clause is not part of the NP referred to by its relative pronoun, but it has to be next to it. In your example, "who is sleeping" is an appositive clause whose relative pronoun "who" refers to "the cat". Yet the appositive clause is not next to "the cat", and that's the problem:
[NP [NP The cat]'s—[S who was still sleeping] —bowl ] is filled with milk.
To fix that difficulty, we'd have to move the appositive clause to the other side of the "'s" possessive ending:
[NP [NP The cat]—[S who was still sleeping]'s —bowl ] is filled with milk.
Unfortunately, this introduces a new difficulty. Now, the "'s" is no longer suffixed to the NP it goes with, "the cat", because in McCawley's treatment, "the cat, who was still sleeping" is not a NP. It's not even a constituent.
So this example substantiates McCawley's theory. There is no grammatical way to associate an appositive relative clause with the NP of a "'s" possessive phrase.