1. Parsing with Context Free Grammars Reading: Chap 13, Jurafsky & Martin
This slide set was adapted from J. Martin, U. Colorado
Instructor: Paul Tarau, based on Rada Mihalcea’s original slides

2. Parsing
Parsing with CFGs refers to the task of assigning correct trees to input strings
Correct here means a tree that covers all and only the elements of the input and has an S at the top
It doesn’t actually mean that the system can select the correct tree from among the possible trees
As with everything of interest, parsing involves a search that involves the making of choices

3. Some assumptions.. Assume… These are all (quite) feasible
You have all the words already in some buffer
The input is/isn’t pos tagged
All the words are known
These are all (quite) feasible
State-of-the art in POS tagging?
“all words are known” ?

4. Top-Down Parsing
Since we are trying to find trees rooted with an S (Sentences) start with the rules that give us an S.
Then work your way down from there to the words.

5. Top Down Space

6. Bottom-Up Parsing
Of course, we also want trees that cover the input words. So start with trees that link up with the words in the right way.
Then work your way up from there.

7. Bottom-Up Space

8. Top-Down VS. Bottom-Up Top-down Bottom-up
Only searches for trees that can be answers
But suggests trees that are not consistent with the words
Guarantees that tree starts with S as root
Does not guarantee that tree will match input words
Bottom-up
Only forms trees consistent with the words
Suggest trees that make no sense globally
Guarantees that tree matches input words
Does not guarantee that parse tree will lead to S as a root
Combine the advantages of the two by doing a search constrained from both sides (top and bottom)

9. Top-Down, Depth-First, Left-to-Right Search + Bottom-up Filtering

10. Example (cont’d)

11. Example (cont’d)
flight
flight

12. Example (cont’d)
flight
flight

13. Possible Problem: Left-Recursion
What happens in the following situation
S -> NP VP
S -> Aux NP VP
NP -> NP PP
NP -> Det Nominal …
With the sentence starting with
Did the flight…

14. Solution: Rule Ordering
S -> Aux NP VP
S -> NP VP
NP -> Det Nominal
NP -> NP PP
The key for the NP is that you want the recursive option after any base case.

15. Avoiding Repeated Work
Parsing is hard, and slow. It’s wasteful to redo stuff over and over and over.
Consider an attempt to top-down parse the following as an NP
A flight from Indianapolis to Houston on TWA

16. flight

17. flight
flight

20. Dynamic Programming
We need a method that fills a table with partial results that
Does not do (avoidable) repeated work
Does not fall prey to left-recursion
Solves an exponential problem in (approximately) polynomial time

21. Earley Parsing Fills a table in a single sweep over the input words
Table is length N+1; N is number of words
Table entries represent
Completed constituents and their locations
In-progress constituents
Predicted constituents

22. States
The table-entries are called states and are represented with dotted-rules.
S -> · VP A VP is predicted
NP -> Det · Nominal An NP is in progress
VP -> V NP · A VP has been found

23. States/Locations
It would be nice to know where these things are in the input so…
S -> · VP [0,0] Predictor
A VP is predicted at the start of the sentence
NP -> Det · Nominal [1,2] Scanner
An NP is in progress; the Det goes from 1 to 2
VP -> V NP · [0,3] Completer
A VP has been found starting at 0 and ending at 3

24. Graphically

25. Earley
As with most dynamic programming approaches, the answer is found by looking in the table in the right place.
In this case, there should be an S state in the final column that spans from 0 to n+1 and is complete.
If that’s the case you’re done.
S -> α · [0,n+1]
So sweep through the table from 0 to n+1…
Predictor: New predicted states are created by states in current chart
Scanner: New incomplete states are created by advancing existing states as new constituents are discovered
Completer: New complete states are created in the same way.

26. Earley More specifically… 1. Predict all the states you can upfront
2. Read a word
Extend states based on matches
Add new predictions
Go to 2
3. Look at N+1 to see if you have a winner

27. Earley and Left Recursion
So Earley solves the left-recursion problem without having to alter the grammar or artificially limit the search
Never place a state into the chart that’s already there
Copy states before advancing them
S -> NP VP
NP -> NP PP
The first rule predicts
S -> · NP VP [0,0] that adds
NP -> · NP PP [0,0]
stops there since adding any subsequent prediction would be fruitless
When a state gets advanced make a copy and leave the original alone
Say we have NP -> · NP PP [0,0]
We find an NP from 0 to 2 so we create NP -> NP · PP [0,2]
But we leave the original state as is

28. Example Book that flight
We should find… an S from 0 to 3 that is a completed state…

29. Example (cont’d)

30. Example (cont’d)

31. Example (cont’d)