1

Simplifications of

Context-Free Grammars

2

A Substitution Rule

substitute B

bB

abbAB

abBcA

aaAA

aA

abbcA

ababbAcA

aaAA

aA

equivalentgrammar

3

In general:

321 ||| yyyB

xBzA

zxyzxyzxyA n||| 21

substitute B

equivalentgrammar

4

Useless Productions

aAA

AS

S

aSbS

5

aAA

AS

S

aSbS

aAaaaaAaAAS

never terminates...

6

aAA

AS

S

aSbS

Useless Production

7

bAB

A

aAA

AS

Another grammar:

8

bAB

A

aAA

AS

Not reachable from S

Useless Production

9

In general:

If wxAyS

Then A

is uselessOtherwise A

is useful

)(GLw

10

Removing Useless Productions

Example Grammar

aCbC

aaB

aA

CAaSS

||

11

First: find all variables that producestrings with only terminals

aCbC

aaB

aA

CAaSS

|| },{ BA

},,{ SBA

Round 1:

Round 2:

12

Keep only the variablesthat produce terminal symbols

aCbC

aaB

aA

CAaSS

||

},,{ SBA

aaB

aA

AaSS

|

13

Second: Find all variablesreachable from S

aaB

aA

AaSS

|

S A B

Dependency Graph

notreachable

14

Keep only the variablesreachable from S

aaB

aA

AaSS

|

aA

AaSS

|

Final Grammar

15

Nullable Variables

:production A

Nullable Variable: A

16

Removing Nullable Variables

Example Grammar:

M

aMbM

aMbS

Nullable variable

17

Substitute

M

M

aMbM

aMbS

abM

aMbM

abS

aMbS

Final Grammar

18

Unit-Productions

BAUnit Production:

19

Removing Unit Productions

Observation:

AA

Is removed immediately

20

Example Grammar:

bbB

AB

BA

aA

aAS

21

bbB

AB

BA

aA

aAS

substitute

BA

bbB

BAB

aA

aBaAS

|

|

22

bbB

BAB

aA

aBaAS

|

|

bbB

AB

aA

aBaAS

|

23

substitute

AB

bbB

aA

aAaBaAS

||

bbB

AB

aA

aBaAS

|

24

Remove repeated productions

bbB

aA

aBaAS

|

bbB

aA

aAaBaAS

||

Final grammar

25

Removing All

1. Remove Nullable Variables

2. Remove Unit-Productions

3. Remove Useless Variables

26

Normal Formsfor

Context-free Grammars

27

Chomsky Normal Form

All productions have form:

BCA

variable variable

aAOR

terminal

28

Examples:

bA

SAA

aS

ASS

ChomskyNormal Form

aaA

SAA

AASS

ASS

NOTChomskyNormal Form

29

Convertion to Chomsky Normal Form

Example:

AcB

aabA

ABaS

ChomskyNormal Form

NOT

30

AcB

aabA

ABaS

Introduce variables for terminals:

cT

bT

aT

ATB

TTTA

ABTS

c

b

a

c

baa

a

cba TTT ,,

31

Introduce intermediate variables:

cT

bT

aT

ATB

TTTA

ABTS

c

b

a

c

baa

a

cT

bT

aT

ATB

TTV

VTA

BTV

AVS

c

b

a

c

ba

a

a

2

2

1

1

21, VV

32

Final grammar inChomsky Normal Form:

cT

bT

aT

ATB

TTV

VTA

BTV

AVS

c

b

a

c

ba

a

a

2

2

1

1

AcB

aabA

ABaS

Initial grammar

33

From any context-free grammar Not in Chomsky Normal Form

We can obtain: An equivalent grammar in Chomsky Normal Form

In general:

34

The Procedure:

Removenullable variablesunit productions

35

For every symbol :

aT

a

Add new variable

In productions, replace with a aT

Add production aTa

36

Replace any production: nCCCA 21

with:

nnn CCV

VCV

VCA

12

221

11

New variables: 221 ,,, nVVV

37

Theorem:

For any context-free grammarthere is an equivalent grammarin Chomsky Normal Form

38

Greinbach Normal Form

All rules have form:

kVVVaA 21

symbol variables

0k

39

Examples:

bB

bbBaAA

cABS

||

GreinbachNormal Form

aaS

abSbS

GreinbachNormal Form

NOT

40

aaS

abSbS

Conversion to Greinbach Normal Form

bT

aT

aTS

STaTS

b

a

a

bb

GreinbachNormal Form

41

Theorem:

For any context-free grammar

there is an equivalent grammarin Greinbach Normal Form

42

An Applicationof

Chomsky Normal Forms

43

The CYK Membership Algorithm

Inputs: • Grammar in Chomsky Normal Form

G

• String w

Output:

find if )(GLw

44

Algorithm: Input Examples

Grammar :G

bB

ABB

aA

BBA

ABS

String : w aabbb

45

a a b b b

aa ab bb bb

aab abb bbb

aabb abbb

aabbb

aabbb

46

aA

aA

bB

bB

bB

aa ab bb bb

aab abb bbb

aabb abbb

aabbb

bB

ABB

aA

BBA

ABS

47

aA

aA

bB

bB

bB

aa abS,B

bbA

bbA

aab abb bbb

aabb abbb

aabbb

bB

ABB

aA

BBA

ABS

48

aA

aA

bB

bB

bB

aa abS,B

bbA

bbA

aabS,B

abbA

bbbS,B

aabbA

abbbS,B

aabbbS,B

bB

ABB

aA

BBA

ABS

49

Therefore: )(GLaabbb

Time Complexity:3||w

It can be easily converted to a parser