earley-parsing.pdf

7/28/2019 Earley-Parsing.pdf

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11-711 Algorithms for NLP

The Earley Parsing Algorithm

Reading:

Jay Earley,

An Efficient Context-Free Parsing Algorithm

Comm. of the ACM vol. 13 (2), pp. 94102


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General Principles:

A clever hybrid Bottom-Upand Top-Downapproach

Bottom-Upparsing completely guided by Top-Downpredictions

Maintains sets of dotted grammar rules that:

Reflect what the parser has seen so far

Explicitly predict the rules and constituents that will combine

into a complete parse

Similar to Chart Parsing - partial analyses can be shared

Time Complexity

3

, but better on particular sub-classes

Developed prior to Chart Parsing, first efficient parsing

algorithm for general context-free grammars.

1 11-711 Algorithms for NLP


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The Earley Parsing Method

Main Data Structure: The state(or item)

A state is a dotted rule and starting position:

1

The algorithm maintains sets of states, one set for each

position in the input string (starting from 0)

We denote the set for position

by



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Three Main Operations:

Predictor: If state

1

!

then for every

rule of the form # 1 # $ , add to the state

# 1 # $

Completer: If state

1

!

then for every state

in

% of form &

1

$

'

, add to

the state

&

1 $

'

Scanner: If state

1 )

!

and the next

input word is0

1

12

)

, then add to

1

1 the state

1 )



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The Earley Recognition Algorithm

Simplified version with no lookaheads and for grammars

without epsilon-rules

Assumes input is string of grammar terminal symbols

We extend the grammar with a new rule

3

$

The algorithm sequentially constructs the sets

for

05

5

6

1

We initialize the set 0 with 02

7

3

$

08



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The Earley Recognition Algorithm

The Main Algorithm: parsing input0

2

0 1 09

1. 02

7

3

$

08

2. For 05

5

do:

Process each itemA

!

in order by applying to it the single

applicable operation among:

(a) Predictor (adds new items to )

(b) Completer (adds new items to

)

(c) Scanner (adds new items to

1

1

3. If

1

12B

, Rejectthe input

4. If

2

and

9

1

127

3

$

08

then Acceptthe input



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Earley Recognition - Example

The Grammar:

1

CD

E

D

2

CD

)FG )H

3

CD

)FG

4

CD

)H

5

E

D

)I 0 E

D

6

E

D

P

CD

The original input: 0

2 The large can can hold the water

POS assigned input: 0

2 art adj n aux v art n

Parser input: 0

2 art adj n aux v art n $



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The input: 0


0: 3

$ 0

CD

E

D

0

CD

)FG )H

0

CD

)FG

0

CD

)H

0

1:

CD

)FG

)H

0

CD

)FG

0



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The input: 0


1: CD )FG )H 0

CD

)FG

0

2:

CD

)FG )H

0



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The input: 0


2: CD )FG )H 0

3:

CD

)FG )H

0



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The input: 0


3: CD )FG )H 0

CD

E

D

0

E

D

)I 0 E

D

3

E

D

P

CD

3

4: E

D

)I 0

E

D

3



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The input: 0


4: ED )I 0 ED 3

E

D

)I 0 E

D

4

E

D

P

CD

4

5:

E

D

P

C D

4



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The input: 0


5: ED P C D 4

CD

)FG )H

5

CD

)FG

5

CD

)H

5

6: CD

)FG

)H

5

CD

)FG

5



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The input: 0


6: CD )FG )H 5

CD

)FG

5

7:

CD

)FG

5



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The input: 0


7: CD )FG 5

E

D

P

CD

4

E

D

)I 0 E

D

3

CD

E

D

0

3

$

0

8:

3

$

0



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Time Complexity of Earley Algorithm

Algorithm iterates for each word of input (i.e.

iterations)

How many items can be created and processed in

?

Each item in

has the form

1

,0

5

5

Thus

items

The Scannerand Predictoroperations on an item each requireconstant time

The Completeroperation on an item adds items of form

&

1 $

'

to

, with 05

'

5

, so it may require up

to

time for each processed item

Time required for each iteration (

) is thus

2

Time bound on entire algorithm is therefore

3



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Time Complexity of Earley Algorithm

Special Cases:

Completeris the operation that may require

2

time in

iteration

For unambiguous grammars, Earley shows that the completer

operation will require at most

time

Thus time complexity for unambiguous grammars is

2

For some grammars, the number of items in each

is

bounded by a constant

These are called bounded-stategrammars and include even

some ambiguious grammars.

For bounded-state grammars, the time complexity of the

algorithm is linear -



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Parsing with an Earley Parser

As usual, we need to keep back-pointers to the constituents

that we combine together when we complete a rule

Each item must be extended to have the form

1

G 1

, where the

G

are pointers to the

already found RHS sub-constituents

At the end - reconstruct parse from the back-pointers

To maintain efficiency - we must do ambiguity packing



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Earley Parsing - Example

The input: 0




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The input: 0


0: 3

$ 0

CD

E

D

0

CD

)FG )H

0

CD

)FG

0

CD

)H

0

1:

CD

)FG 1 )H

0

1)FG

CD

)FG 1

0



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The input: 0


1: CD )FG 1

)H

0

CD

)FG 1

0

2:

CD

)FG 1 )H

2

0

2)H



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The input: 0


2: CD )FG 1 )H 2

0

3:

CD 4 )FG 1 )H

2 3

0

3

4 C D )FG 1 )H

2 3



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The input: 0


3: CD 4 )FG 1 )H 2 3 0

CD 4 ED

0

E

D

)I 0 E

D

3

E

D

P

CD

3

4:

E

D

)I 0

5

E

D

3

5)I 0



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The input: 0


4: E

D

)I 0 5

E

D

3

E

D

)I 0 E

D

4

E

D

P

CD

4

5:

E

D

P 6 CD

4

6P



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The input: 0


5: E

D

P 6

CD

4

CD

)FG )H

5

CD

)FG

5

CD

)H

5

6:

CD

)FG

7

)H

5

7)FG

CD

)FG 7

5



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The input: 0


6: CD

)FG 7

)H

5

CD

)FG 7

5

7:

CD 9 )FG 7 8

5

8

9C D

)FG 7 8



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The input: 0


7: CD 9 )FG 7 8

5

E

D 10 P 6 CD 9

4

10E

D

P 6 CD 9

E

D 11 )I 0 5 ED 10

3

11E

D

)I 0 5 ED 10

12 CD 4 ED 11

0

12

CD 4 ED 11

3

$

0

8:

3

$

0


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