CKY Assignment

Below we give the CKY chart for a parse of the 10-word sentence:

The agency sees use of the codes as a way.

The assignment is to explain some features of the chart resulting from use of the CKY algorithm to parse this sentence.

We first give the grammar the parser uses (in Chomsky Normal Form), as well as the lexicon.

Grammar

pp -> p np
vpg -> vbg np
X2 -> X1 pp
vp -> vbz np
vp -> X1 pp
vp -> X2 pp
ap -> rb a
s -> np vp
np -> dt nbar
np -> X3 np
np -> ap nbar
np -> nbar pp
np -> n n
np -> vbg np
X3 -> np cc
X1 -> vbz np
nbar -> ap nbar
nbar -> nbar pp
nbar -> n n
nbar -> vbg np

Lexicon

a: dt
and: cc
use: n np nbar
codes: n np nbar
sees: vbz
of: p
agency: n np nbar
rapidly: rb
handling: vbg
labor: n np nbar
volume: n np nbar
as: p
costs: n np nbar
controlling: vbg
way: n np nbar
growing: a ap
mail: n np nbar
the: dt
widespread: a ap

Part A

Your task concerns only the entries in the last column of the chart, which includes all entries ending at index 10. In "chart parsing" speak. These entries are referred to as edges.

Basically there are two possible kinds of explanation for an edge appearing in the chart.

1. It is a lexical edge for category cat for the word spanning from 9 to 10 (way), and the lexicon has an entry for way with that category. For example there is an entry associating way with the category n in the lexicon above, and there is an entry for an n in the (9,10) cell of the chart below.

   You can report your discovery of a lexical edge in a very simple format known as a lexical record. Just report the word, the span, and the category:

   Lexical edge
   way (9,10) n
   

2. It is a rule edge for category cat for the span from i to j, and there is rule for building cat using cat1 and cat2 in the grammar, and some k (between i and j) such that there is an edge from i to k of category cat1 and edge from k to j of category cat2 in the chart. So for example, there is an entry for np in the (8,10) cell of the chart below, and this is explained by the fact that there is a rule in the grammar.

   np -> dt nbar
   

   and there is a dt in the (8,9) cell of the chart and an nbar in the (9,10) cell.

   You can report your discovery of a rule edge in a simple format called a dtr record. Just give cat, the span, cat1 and cat2, and the value of k. So for our example, this is:

   np (8,10)
   (det,9,nbar)
   

   Your task is to find all the lexical records and daughter recods for the last column in the chart (that is, for all the edges whose span ends at 10).

Part B

Finally, this sentence has 5 parses in this grammar. This is because certain edges contributing to the s edge in the (0,10) cell can be built in more than one way, given the grammar. These are called ambiguous edges. Find these ambiguous edges. They may not be in the last column, but they are in the chart, Give the lexical records or daughter records for these edges. You do not have to find all the ambiguous edges, just the ones contributing to the final 5 parses. Finally you do not have to find all the edges contributing to the final 5 parses, just the ambiguous ones.

When you have found the ambiguous edges, you are ready to draw the five parse trees that this chart contains. Each parse tree starts with the edge S(0,10). Each will continue with the dtr record or records you found for that S. Each daughter record will lead you to two other edges that are daughters for that S. Ambiguous edges (with more than one daughter record) will generate more than one parse tree. But just because an edge is not ambiguous does not mean it has only one parse tree. It may be ambiguous not because it has more than one daughter record, but because one or more of its daughters does. You should be able to build five parse trees

	1 the	2 agency	3 sees	4 use	5 of	6 the	7 codes	8 as	9 a	10 way
0	dt	np		s			s			s
1		np nbar n		s			s			s
2			vbz	vp X1			vp X2 X1			vp X2 X1
3				np nbar n			np nbar			np nbar
4					p		pp			pp
5						dt	np			np
6							np nbar n			np nbar
7								p		pp
8									dt	np
9										np nbar n