8/9/2019 CYK Lecture

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6.3

A

MEMBERSHIP ALGORITHM

FOR

CONTEXT-FREE GRi\MMARS

171

A IVmmbershfip

Algorithm

for Conte

xt-Fr

ee

Grammars*

In Section 5.2, we claim, without any

elaboration, that

membership and

parsing

algorithms

for context-free

grammars

exist

that

require approxi

mately u

3

steps to

parse

a

string

w. We

are

now in a position to justify

this

claim.

The

algorithm we will descr ibe here is called the CYK algorithm,

after

its originators J. Cocke, D. H. Younger,

and

T. Kasami. The algo

rithm works only if

the

grammar is-in Chomsky no rmal form

and

succeeds

by

breaking one problem into

a sequence of smaller ones in

the

following

way. Assume that we have a grammar

G

=

F,

T, S, P)

in Chomsky normal

form and a

string

We define

substrings

Wi j

=

ai'

a j

and

subsets of V

Clearly,

W

E

L

(G) i f and

only if

S

E FIn'

To

compute Vij

observe that

A

E

Vii

i f

and

only if

G

contains a pro

duction

A 4 ai .

Therefore,

Vi i

can be computed for all 1 S i ::; n by

inspection of

w

and

the

productions

of the grammar. To continue, notice

that

for

j

>

i,

A

derives

Wij

i

and

only

i f

there

is a

production

A

4

BG,

with

B ik

and

G

WkH j

for

some

k

with i

S k k

< j . In other

words,

ij = U {A:

A 4

BG, with

B

E

V ik,GE

Vk+l,j}

{G.t l

kE {i , i+I, ... j - I }

An inspection of

the

indices in (6.8) shows that it

can

be used to

compute

all

the Vij if we proceed in

the

sequence

1. Compute

V

 

, V

22

,

... , V

n n

>

2. Compute V

12

,

11

3

,

..

 

Vn-I,n,

and

so on.

8/9/2019 CYK Lecture

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Determine whether the string

w = ruibbb

is in the language generated by

the grammar

S > AB,

A

> BBla,

B

>

AB/b.

First

note that

Wll =

a,

so Vl l is the

set

of all variables that immedi-

ately derive a, that is, V

u

= {A}.

Since W22

=

U, we also have

V

22

= {A}

and, similarly,

V

 

= {it} ,V

22

= {A} , V33 = {B} ,V44 = {B}

1

V55 = {B} .

Now we use 6.8) to get

V

I 2

=

{A: A > BC,B

E

Vil C

E

V

n}

Since V

 

= {A}

and

V

22

=

{A},

the set consists of

all

variables that occur

on

the

left side of a

production

whose right

side

ill AA. Since

there

are

none, Vi is empty Next,

V

23

= {A: A > BC,B E V

2

Z C

E V

33

}

,

so the required right side is AB, and we have

V

Z3

=

{S,B}.

A straightfor-

ward

argument

along these lines

then

gives

VIZ =   23 = {S,B}, V

34

= {A},

45

= {A},

Vi

=

{S,B} ,V

24

=

{A}, 35

= {S,B},

V

14

= {A}, V

25

= {S,B},

Vis =

{S,B} ,

so that w

E

L G).

The

CYK algorithm, as

described

here,

determines

membership for any

lcnguage

generated

by grammar in Chomsky normal form. With some

additions to keep track of how

th

elements of are derived it can be

converted into a parsing method. To see

that the

CYK

membership algo-

rithm requires 0 n

3

 

steps, notice

that

exactly n n

+

1)

/2

sets of ii have

to be computed. Each involves the evaluation of

at

most n

terms

in 6.8),

30

the

claimed result follows.