CD for this type of descriptions along the lines of Section 4 is not difficult once relational descriptions are integrated with a standard GRE algorithm (Krahmer and Theune 2002, Section 8.6.2): Suppose an initial description is generated describing the set of all those dogs that are in sheds over a given size (say, size 5); if this description happens to distinguish an individual dog then this legitimizes the use of the noun phrase the dog in the large shed.
note that this is felicitous even if the shed is not the largest one in the domain, as is true for d 2 in the following situation (contains-a=b means that a is contained by b): the dog in the large shed denotes 'the dog such that there is no other shed that is equally large or larger and that contains a dog'.