1
Chart Parsing
Allen
’
s Chapter 3
J & M
’
s Chapter 10
2
Chart Parsing
General Principles:
A
Bottom

Up
parsing method
–
Construct a parse starting from the input symbols
–
Build constituents from sub

constituents
–
When all constituents on the RHS of a rule are matched, create a
constituent for the LHS of the rule
The
Chart
allows storing partial analyses, so that they can be shared.
Data structures used by the algorithm:
–
The Key:
the current constituent we are attempting to “match”
–
An Active Arc:
a grammar rule that has a partially matched RHS
–
The Agenda:
Keeps track of newly found unprocessed constituents
–
The Chart:
Records processed constituents (non

terminals) that span
substrings of the input
3
Chart Parsing
Steps in the Process:
Input is processed left

to

right, one word at a time
1. Find all POS of the current word (terminal

level)
2. Initialize Agenda with all POS of the word
3. Pick a Key from the Agenda
4. Add all grammar rules that start with the Key as active
arcs
5. Extend any existing active arcs with the Key
6. Add LHS constituents of newly completed rules to the
Agenda
7. Add the Key to the Chart
8. If Agenda not empty
–
go to (3), else go to (1)
4
The Chart Parsing Algorithm
Extending Active Arcs with a Key:

Each
Active Arc
has the form:
<p
i
>
[A
X
1
…
C…X
m
] <p
j
>

A Key constituent has the form: <p
j
>C<p
k
>

When processing the Key <p
1
>C<p
2
>, we search the active
arc list for an arc <p
0
>[A
X
1
…
C…X
m
]<p
1
>, and
then create a new active arc
<p
0
>[A
X
1
…C
… X
m
]<p
2
>

If the new active arc is a completed rule:
<p
0
>[A
X
1
…C
]<p
2
>, then we add <p
0
>A<p
2
> to the
Agenda

After “using” the key to extend all relevant arcs, it is entered
into the Chart
5
The Chart Parsing Algorithm
The Main Algorithm: parsing input x = x
1
…x
n
1
. i =
0
2
. If Agenda is empty and i < n then set i = i +
1
, find all
POS of x
i
and add them as constituents <p
i
>C<p
i+
1
> to
the Agenda
3
. Pick a Key constituent <p
j
>C< p
k
> from the Agenda
4
. For each grammar rule of form A
CX
1
…X
m
, add
<p
j
>[A
CX
1
…X
m
]<p
j
> to the list of active arcs
5
. Use Key to extend all relevant active arcs
6
. Add LHS of any completed active arcs into the Agenda
7
. Insert the Key into the Chart
8
. If Key is <
1
>S<n> then Accept the input Else goto (
2
)
6
Chart Parsing

Example
The Grammar:
(
1
) S
NP VP
(
2
) NP
ART ADJ N
(
3
) NP
ART N
(
4
) NP
ADJ N
(
5
) VP
AUX VP
(
6
) VP
V NP
7
Chart Parsing

Example
The input:
“x = The large can can hold the water”
POS of Input Words:
–
the:
ART
–
large:
ADJ
–
can:
N, AUX, V
–
hold:
N, V
–
water:
N, V
8
The large
can can hold the water
9
The large
can
can hold the water
10
The large
can can hold
the water
11
The large
can can hold the water
12
The final Chart
13
Time Complexity of Chart Parsing
•
Algorithm iterates for each word of input (i.e. n iterations)
•
How many keys can be created in a single iteration i?
–
Each key in Agenda has the form <p
j
>C< p
i+
1
>,
1
j
i
–
Thus O(n) keys
•
Each key <p
j
>C<p
i+
1
> can extend active arcs of the form
•
<p
k
>[A
X
1
…
C…X
m
]<p
j
>, for
1
k
j
•
Each key can thus extend at most O(n) active arcs
•
Time required for each iteration is thus O(n
2
)
•
Time bound on entire algorithm is therefore O(n
3
)
14
Improved Chart Parser
Increasing the Efficiency of the Algorithm
•
Observation: The algorithm creates constituents that
cannot “fit” into a global parse structure. Example:
<
2
>[NP
ADJ N]<
3
>
•
These can be eliminated using
predictions
•
Using Top

down predictions:
–
For each non

terminal A, define First(A) = the set of all leftmost
non

terminals derivable from A
–
Compute in advance First(A) for all non

terminals A
–
When processing the key <p
j
>C<p
i+
1
> we look for grammar rules
that form an active arc <p
j
>[A
C X
2
…X
m
]<p
j
>
–
Add the active arc
only
if A is
predicted
: there already exists an
active arc <p
k
>[B
X
1
…
D…X
m
]<p
j
> such that A
First(D)
–
For p
k
=
1
(arcs of the form <
1
>[A
C X
2
…X
m
]<
1
>), add the
active arc
only
if A
First(S)
15
Improved Chart Parser

Example
16
The large
can can hold
the water
17
The large
can can hold the water
18
19
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