Among what has been known since the works by Sag and Williams in the mid 1970s is that VPE (VP ellipsis) in English does not allow readings such as indicated in (2) for (1).

(1)　　John fed his cat; Bill did too.
(2)　a.　John fed John's cat; Bill fed Mike's cat.
　　b.　John fed Mike's cat; Bill fed Bill's cat.
　　c.　John fed Mike's cat; Bill fed John's cat.
　　d.　John fed Bill's cat; John fed John's cat.

If his in (1) is used to refer to John, as indicated in the interpretation in the first conjunct in (2a), the second conjunct in (1) can only mean (3a) or (3b).

(3)　a.　Bill fed John's cat.
　　b.　Bill fed Bill's cat.

If his in (1) is used to refer to someone other than John, as indicated in the first conjunct of (2b), (2c), and (2d), that person must be the person whose cat Bill fed. Hence the unavailability of the readings in (2b-d) for (1).
　　In Hoji 1998, it is pointed out that such a restriction on the interpretive possibilities is missing in the case of the Null Object Construction in Japanese (NOC), and that is taken as evidence that NOC, at least, need not be an instance of surface anaphora. Given the hypothesis that the restriction in question is due to the syntactic (i.e., structural) parallelism that must hold between the ellipsis site of a certain type and 'its antecedent' (and this type of ellipsis is called surface anaphora), and given the hypothesis that do the same thing, do that, and do it in English are not of this type of linguistic object, it follows that the restriction noted above is not observed with these expressions. In my "Surface and Deep .." paper, this point is not made. Neither was it made in my Sytax+ presentation at USC on 10/29/3003.
The paradigms below indicates that the expectation just noted is in fact fulfilled.

(4)　　John fed his cat with chopsticks; Bill did the same thing.

It seems that, given an appropriate context, (4) is compatible with any of the readings in (2). As in the case of NOC (see Hoji 1998: 3.3), (4) also seems compatible with (5).

(5)　　John fed John's cat with chopsticks; Bill fed a cat with chopsticks.

The felicitous use of do the same thing might tend to require some additional materials such as with chopsticks. But that does not seem to be a requirement imposed by grammar since we can think of an appropriate context in which such additional materials are not needed to get the reading in question.