1. Chapter 9: Parsing with Context-Free Grammars
Heshaam Faili
University of Tehran

2. Context-Free Grammars
Context-Free Grammars are of the form:
A  , where  is a string of terminals and/or non-terminals
Note that the regular grammars are a proper subset of the context-free grammars.
This means that every regular grammar is context-free, but there are context-free grammars that aren’t regular
CFGs only specify what trees look like, not how they should be computationally derived  We need to parse the sentences

3. Parsing Intro Input: a string Output: a (single) parse tree Strategies
A useful step in the process of obtaining meaning
We can view the problem as searching through all possible parses (tree structures) to find the right one
Strategies
Top-Down (goal-directed) vs. Bottom-Up (data-directed)
Breadth-First vs. Depth-First
Adding Bottom-Up to Top-Down: Left-Corner Parsing
Example
Book that flight!

4. Grammar and Desired Tree

5. Top-Down Parsing
Expand rules, starting with S and working down to leaves
Replace the left-most non-terminal with each of its possible expansions.
The full search is on p. 361, Fig. 10.3
While we guarantee that any parse in progress will be S-rooted, it will expand non-terminals that can’t lead to the existing input
e.g., 5 of 6 trees in third ply = level of the search space
None of the trees take the properties of the lexical items into account until the last stage

6. Top-down (breadth-first) parsingNP VP S
Aux VP NP S VP S NP VP
Det
Nom S NP VP
PropN S NP VP
Det
Nom
Aux S NP VP
Aux
PropN S VP V NP S VP V

7. Expansion techniques Breadth-First Expansion (shown in figure)
All the nodes at each level are expanded once before going to the next (lower) level.
This is memory intensive when many grammar rules are involved
Depth-First (shown on p. 367, Fig. 10.7)
Expand a particular node at a level, only considering an alternate node at that level if the parser fails as a result of the earlier expansion
i.e., expand the tree all the way down until you can’t expand any more

8. Top-down (depth-first) parsingNP VP
Det
Nom
Does
FAIL S NP VP
Aux
Does S S NP VP S NP VP
Det
Nom
Does
Aux
this
Does this flight include a meal ?

9. Top-Down Depth-First Parsing
There are still some choices that have to be made:
1. Which leaf node should be expanded first?
Left-to-right strategy moves through the leaf nodes in a left-to-right fashion
2. Which rule should be applied first?
There are multiple NP rules; which should be used first?
Can just use the textual order of rules from the grammar
There may be reasons to take rules in a particular order (e.g., probabilities)

10. Parsing with an agenda Search states are kept in an agenda
Search states consist of partial trees and a pointer to the next input word in the sentence
Based on what we’ve seen before, apply the next item on the agenda to the current tree
Add new items to (the front of) the agenda, based on the rules in the grammar which can expand at the (leftmost) node
We maintain the depth-first strategy by adding new hypotheses (rules) to the front of the agenda
If we added them to the back, we would have a breadth-first strategy
See figure 10.6 pg. 366E

11. Bottom-Up Parsing
Bottom-Up Parsing is input-driven  start from the words and move up to form a tree
Here we match one or more nodes on the upper fringe of the parse tree against the right-hand side of a CFG rule, building the left-hand side as a parent node of those nodes.
We can also have breadth-first and depth-first approaches
The example on the next slide (p. 362, Fig. 10.4) moves in a breadth-first fashion
While any parse in progress will be tied to the input, many may not lead to an S!
e.g., left-most trees in plies 1-4 of Fig 10.4

12. Bottom-up parsing Book that flight Book that flight Noun Det Book that
Verb
Det
Noun
Book
that
flight
Noun
Det
NOM
Book
that
flight
Verb
Det
Noun
NOM
Book
that
flight
Noun
Det
NOM NP
Book
that
flight
Verb
Det
Noun
NOM NP
Book
that
flight
Verb
Det
Noun
NOM VP
Book
that
flight
Verb
Det
Noun
NOM NP VP
Book
that
flight
Verb
Det
Noun
NOM VP NP

13. Comparing Top-Down and Bottom-Up Parsing
While we guarantee that any parse in progress will be S-rooted, it will expand non-terminals that can’t lead to the existing input, e.g., first 4 trees in third ply.
Bottom-Up:
While any parse in progress will be tied to the input, many may not lead to an S, e.g., left-most trees in plies 1-4 of p. 362, Fig 10.4.
So, both pure top-down and pure bottom up approaches are highly inefficient.

14. Left-Corner Parsing Motivation:
Both pure top-down and bottom-up approaches are inefficient
The correct top-down parse has to be consistent with the left-most word of the input
Left-corner parsing: a way of using bottom-up constraints as part of a top-down strategy.
Left-corner rule: expand a node with a grammar rule only if the current input can serve as the left corner from this rule.
Left-corner from a rule: first word along the left edge of a derivation from the rule
Put the left-corners into a table, which can then guide parsing

15. Left-Corner Example S NP VP Left Corners S VP S Aux NP VP
NP Det Nominal | ProperNoun
Nominal  Noun Nominal | Noun
VP Verb | Verb NP
Noun  book | flight | meal | money
Verb  book | include | prefer
Aux  does
ProperNoun  Houston | TWA
Left Corners
S => NP …=> Det, ProperNoun
VP => Verb
Aux … => Aux
NP => Det, ProperNoun
Nominal => Noun

16. Other problems: Left-Recursion
Left-corner parsers still guided by top-down parsing
Consider rules like:
S  S and S
NP  NP PP
A top-down left-to-right depth-first parser could apply a rule to expand a node (e.g., S), and then apply that same rule again, and again, ad infinitum.
Left Recursion: A grammar is left-recursive if a non-terminal leads to a derivation that includes itself as its leftmost immediate or non-immediate child (i.e., along its leftmost branch).
PROBLEM: Top-Down parsers may not terminate on a left-recursive grammar

17. Other problems: Repeated Parsing of Subtrees
When parser backtracks to an alternative expansion of a non-terminal, it loses all parses of subconstituents that it built.
There is a good chance that it will rebuild the parses of some of those constituents again.
This can occur repeatedly.
a flight from Indianopolis to Houston on TWA
NP  Det Nom
Will build an NP for a flight, before failing when the parser realizes the input PPs aren’t covered
NP  NP PP
Will again build an NP for a flight, before failing when the parser realizes the two remaining PPs in the input aren’t covered

18. Other problems: Ambiguity
Repeated parsing of subtrees is even more of a problem for ambiguous sentences
PP attachment:
NP or VP: I shot an elephant in my pajamas.
NP bracketing: the meal [on flight 286] [from SF] [to Denver]
Coordination:
[old men] and women vs. old [men and women]
3 kinds of ambiguities: attachment, coordination, noun-phrase bracketing.
Parsers have to disambiguate between lots of valid parses or return all parses
Will repeat a lot of work parsing the commonalities of each ambiguity

19. Ambiguity

20. Addressing the problems: Chart Parsing
More or less a standard method for carrying out parsing; keeps tables of constituents that have been parsed earlier, so it doesn’t reduplicate the work.
Each possible sub-tree is represented exactly once.
This makes it a form of dynamic programming (which we saw with minimum edit distance and the Viterbi algorithm)
Combines bottom-up and top-down parsing
Rather simple and elegant in the way it works!

21. Earley Chart Parsing Representation
The parser uses a representation for parse state based on dotted rules. S  NP  VP
Dotted rules distinguish what has been seen so far from what has not been seen (i.e., the remainder).
The constituents seen so far are to the left of the dot in the rule, the remainder is to the right.
Parse information is stored in a chart, represented as a graph.
The nodes represent word positions.
The labels represent the portion (using the dot notation) of the grammar rule that spans that word position.
 In other words, at each position, there is a set of labels (each of which is a dotted rule, also called a state), indicating the partial parse tree produced until then.

22. Example: Chart for A Dog
Given a trivial grammar
NP  D N
D  a
N  dog
Here’s the chart for the complete parse of “a dog”
[0] D  a [1] (scan)
[1] N dog [2] (scan)
[ 0] NP  D N [0] (predict)
[0] NP  D  N [1] (complete)
[0] NP  D N  [2] (complete)

23. More Chart Parsing Terminology
A state is complete if it has a dot at the right-hand side of its rule. Otherwise, it is incomplete.
At each position, there is a list (actually, a queue) of states.
The parser moves through the N+1 sets of states in the chart left-to-right, processing the states in each set in order.
States will be stored in a FIFO (first-in first-out) queue at each start position
The processing applies one of three operators, each of which takes a state and produces new states added to the chart.
Scanner, Predictor, Completer
There is no backtracking.

24. Earley Parsing Algorithm
The parsing algorithm is just a few lines long, as can be seen on p. 381, Figure 10.16
In the top level loop, for each position, for each state, it calls the predictor, or else the scanner, or else the completer.
The algorithm never backtracks and never removes states, so we don’t redo any work
The goal is to have S  α• as the last chart entry, i.e. the dot has moved over the entire input to derive an S

25. The Earley Algorithm

26. The 3 Operators: Predictor, Scanner, Completer
Procedure PREDICTOR((AB, [i, j]))
For each (B) in grammar do
Enqueue((B  , [j, j]), chart[j])
End
Procedure SCANNER ((AB, [i, j]))
If B is a part-of-speech for word[j] then
Enqueue((B  word[j], [j, j+1]), chart[j+1])
Procedure COMPLETER((B, [j, k]))
For each (AB, [i, j]) in chart[j] do
Enqueue((A B, [i, k]), chart[k])

27. Prediction Procedure PREDICTOR((AB, [i, j]))
For each (B) in grammar do
Enqueue((B  , [j, j]), chart[j])
End
Predicting is the task of saying we kinds of input we expect to see
Add a rule to the chart saying that we have not seen , but when we do, it will form a B
The rule covers no input, so it goes from j to j
Such rules provide the top-down aspect of the algorithm

28. Scanning Procedure SCANNER ((AB, [i, j]))
If B is a part-of-speech for word[j] then
Enqueue((B  word[j], [j, j+1]), chart[j+1])
Scanning reads in lexical items
We add a dotted rule indicating that a word has been seen between j and j+1
This is then added to the following (j+1) chart
Such a completed dotted rule can be used to complete other dotted rules
These rules also show how the Earley parser has a bottom-up component

29. Completion Procedure COMPLETER((B, [j, k]))
For each (AB, [i, j]) in chart[j] do
Enqueue((A B, [i, k]), chart[k])
End
Completion combines two rules in order to move the dot, i.e., indicate that something has been seen
A rule covering B has been seen, so any rule A which refers to B in its RHS moves the dot
Instead of spanning from i to j, A now spans from i to k, which is where B ended
Once the dot is moved, the rule will not be created again

30. Example (Book that flight)

31. Example(Book that flight)

32. Example(Book that flight), cont

33. Example(Book that flight), cont

34. Example(Book that flight), cont

35. Earley parsing
The Earley algorithm is efficient, running in polynomial time.
Technically, however, it is a recognizer, not a parser
To make it a parser, each state needs to be augmented with a pointer to the states that its rule covers
For example, a VP would point to the state where its V was completed and the state where its NP was completed

36. Other Dynamic Programming methods
CYK (Cocke-Kasami-Younger) Parser
Using CNF grammar rules
Chart Parsing
Modified version of Earley parsing with dynamic ordering of states in the algorithm

37. CYK Parsing The DP method by using CNF grammar
ABC
Am
Any CFG can be converted to CNF,
So, don’t loss anything …
AB : unit productions (can be rewrited by A for any A)
Like other DP methods, a simple (n+1)*(n+1) matrix used to encode the structure of the sentence (n: sentence length)
Indexed is the gap between words
[0 Book 1 that 2 flight 3 ]
[i,j] : is a set of non-terminals that represent all the constituents that span positions i through j of the input

38. CYK Parsing, cont,d
Since our grammar is in CNF, the non-terminal entries in the table have exactly two daughters in the parse.
for each constituent represented by an entry [i, j] in the table there must be a position in the input, k, where it can be split into two parts such that i < k < j.
Given such a k, the first constituent [i,k] must lie to the left of entry [i, j] somewhere along row i, and the second entry [k, j] must lie beneath it, along column j

39. CYK Algorithm

41. CYK example (CNF Grammar)

42. CYK example (Book the flight through Houston)

43. CYK in practice Does not have major problem theoretically
The resulted parse tree are not consistent to syntacticians…(because of CNF formal)
Syntax to Semantic approach complicated …
Post-processing needed to return-back the result to more acceptable form

44. Chart Parser
In both the CKY and Earley algorithms, the order in which events occur (adding entries to the table, reading words, making predictions, etc.) is statically determined by the procedures that make up these algorithms.
Unfortunately, dynamically determining the order in which events occur based on the current information is often necessary
Chart Parsing facilitates just such dynamic determination of the order in which chart entries are processed.
Using Agenda

45. Chart Parser
fundamental rule: generalized the ideas in CYK and Earley:
if the chart contains two edges A → α • B β , [i, j] and B → γ •, [ j,k] then we should add the new edge A →α B • β [i,k] to the chart
Prediction can be top-down of botton-up

47. Prediction in Chart Parser

48. Inadequacies of parsing with plain CFGs
While the Earley algorithm works well for CFGs, we have to at some point question the validity of using plain CFGs
We’ll show this by looking at two phenomena (although, there are many more):
Subject-verb agreement
Subcategorization frames

49. Modeling Subject-Verb Agreement in CFGs
The flights leave vs. The flight leaves.
S  3sgNP 3sgVP
S  PluralNP PluralVP
3sgVP 3sgVerb # flies
| 3sgVerb NP # wants + a flight
| 3sgVerb NP PP # leaves + Boston + in the morning
| 3sgVerb PP # leaves + on Thursday
3sgNP Pronoun # I
| ProperNoun # Denver
| Det 3sgNominal # a + flight
3sgNominal  Noun 3sgNominal # morning + flight
| 3sgNoun #flight

50. Problems with Modeling Agreement in CFGs
You can see how messy this is, resulting in a massive increase in the size of the grammar.
Of course, once we add in determiner-noun agreement (e.g., “a flight” vs. “(the) flights”), it would get even larger.
Other languages which have gender agreement (e.g., French) will make it even worse.
Furthermore, we miss generalizations: all transitive verbs have an NP object, regardless of whether the verb is 3rd singular or not
We will need to go to feature-based grammars to address these problems.

51. Subcategorization Frames in CFGs
V1.  eat, sleep I want to eat
V2. NP prefer, find, leave Find [NP the flight from Pittsburgh to Boston]
V3. NP NP show, give, find Show [NP me] [NP the airlines with flights from Pittsburgh]
V4. PPfrom PPto fly, travel I would like to fly [PP from Boston] [PP to Philadelphia]
V5. NP PPwith help, load Can you help [NP me] [PP with a flight]
V6. VPto prefer, want, need I would prefer [VP to go by United Airlines]
V7. VPbare_stem can, would, might I can [VP go from Boston]
V8. V_S mean, imply Does this mean [S American has a hub in Boston]

52. CFG Grammar For Subcategorization
VP  V1
| V2 NP
| V3 NP NP
| V4 PPfrom PPto
| V5 NP PPwith
|V6 VPto
|V7 VPbare_stem
|V8 S
V1 eat | sleep,

53. Problem with Modeling Subcat in CFGs
Again, this results in an explosion in the number of rules, especially when a full set of subcategorization frames is included.
If we combine these rules with the agreement rules, it gets even worse
Also, nouns, adjectives, and prepositions can also subcategorize for complements.
And again, we have no way to state what’s in common about these rules
So, we turn to feature-based grammars