Assignment I: PCFGs
Cognitive Models of Language and Beyond
0440949 Andreas van Cranenburgh

1.
       S
       |
      VP        
     /  \
    /     \
   /        NP
  /        /  \
 |        /     \ 
 |        NP      \
 |       /   \      \
 |      |      \      \____
 |      NP      PP         PP
 |     /  \     /  \       / \
 |    |    |    |  NP     |  NP
 |    |    |    |   |     |  |
 V   DET   N    P   N     P  N  
List the sales of products in 2008

List the sales-of-products in 2008



       S
       |
      VP        
     /  \
    /    \
   /      NP
  /      /   \_____
 |      |          \
 |      NP         PP
 |     /  \       /  \
 |    |    |     /    NP__  
 |    |    |    /     |    \
 |    |    |    |     NP    PP
 |    |    |    |     |     | \
 |    |    |    |     |     |  NP
 |    |    |    |     |     |  |
 V   DET   N    P     N     P  N  
List the sales of products in 2008

List the sales of products-in-2008


           S
           | 
           VP   
    _____/ |  \______________  
   /       |                  \
  /        NP                PP
 /       /   \               / \
 |      |     \___          |  |
 |      NP        PP        |  |
 |     /  \      /  \       |  NP
 |    |    |    /    NP     |  |
 |    |    |    |     |     |  | 
 V   DET   N    P     N     P  N  
List the sales of products in 2008

List the-sales-of-products and do it in 2008

2a. The first tree is the preferred interpretation, where "products" and
"2008" both modifiy "sales".

b.

 S -> VP	1
 VP -> V NP	1
 NP -> N        2/5
 NP -> NP PP	2/5
 NP -> DET N	1/5
 PP -> P NP	1
 V -> List	1
 DET -> the	1
 N -> sales	1/3
 N -> products	1/3
 N -> 2008	1/3
 P -> of	1/2
 P -> in	1/2

3.
First tree:
 S -> VP   VP -> V NP   NP -> NP PP   NP -> DET N   PP -> P NP  NP -> N
   1          1            2/5           1/5           1           2/5

 PP -> P NP    NP -> N
    1             2/5

joint probability: 0.0128


Second tree:
 S -> VP   VP -> V NP   NP -> NP PP   NP -> DET N    PP -> P NP  
   1          1            2/5           1/5            1
  
NP -> NP PP     NP -> N   PP -> P NP     NP -> N
   2/5             2/5       1              2/5

joint probability: 0.00512


Third tree:
 S -> VP    VP -> V NP PP    NP -> NP PP   NP -> DET N   PP -> P NP   
   1           0                2/5           1/5           1

NP -> N      PP -> P NP     NP -> N
   2/5          1              2/5

joint probability: 0 (ungrammatical, due to PP attaching to VP)

4. Recently activated rules might temporarily receive higher probabilities,
taking probability mass from the rules that have not been recently activated.

Additionally, recently parsed chunks might be memorized temporarily, so that 
if they re-occur, they will receive consistent parses.
Eg., perhaps "sales of products" has been mentioned previously, if it is
mentioned again there is a high probability it should be parsed in the same
way, which could resolve potential ambiguities.

Instead of probabilities for single rules we could use probabilities for
rules given their parent constituent.