Problem $1(2$ points$)$: Draw syntactic trees for the following two sentences and apply the Hobbs tree$-$search algorithm to it for the pronoun $$his$,$ showing each step in the search:

Alan Smithee, a prolific movie director, has over $50$ movies to his name, although he is considered one of the worst directors of all time. For over $50$ years, he has delivered substandard films to the big screen.

Problem $2$: $(3$ points$)$ For the sentences in problem $1,$ use the syntactic tree to now apply Hobbs tree search algorithm to it for all the pronouns in the sentences. Remember to show each step in the search.

Problem $3$: $(2$ points$).$ Hobbs $(1977)$ cities the following examples $($a$)-($e$)$ from his corpus as being problematic for his tree$-$search algorithm. In each case, identify the correct referent of the underlined pronoun and the one that the algorithm will identify incorrectly. Discuss any factors that come into pay in determining the correct referent in each case, and what types of information might be necessary to account for them.

$($a$)$ The positions of pillars in one hall were marked by river boulders and a shaped convex cushion of bronze that had served as their footings.

$($b$)$ They were at once assigned an important place among the scanty remains which record the physical developments of the human race from the time of its first appearance in Asia.

$($c$)$ Sites at which the coarse grey pottery of the Shang period has been discovered do not extend far beyond the southernmost reach of the Yellow river, or westward beyond its junction with the Wei.

$($d$)$ The thin, hard, black$-$burnished pottery, made in shapes of angular profile, which archaeologists consider as the clearest hallmark of the Lung Shan culture, developed in the east. The site from which it takes its name is in Sahntung. It is traced to the north$-$east as far as Liao$-$ning province.

Problem $4(3$ points$)$: Segment the following passage into reasonable nuclei and satellites according to RST as described in section $21\mathrm{.}2\mathrm{.}1.$ Then mark all the RST relations you can find.

It was New York City$$s first new subway station in a quarter century, and official and riders marked the opening of the Hudson Yards station in September with much fanfare. But less than six months later, riders began to notice water leaking from above. At a meeting last week, board members demanded to know how the Metropolitan Transportation Authority could have spent $$2\mathrm{.}4$ billion on a project that leaks. In fact, leaks had plagued the station on the Far West Side of Manhattan for years while it was under construction. As the transit agency investigates what exactly went wrong, documents from a continuing legal dispute among the site$$s contractors reveal early concerns about how the waterproofing system was built and the type of concrete that was used.