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CS 301 - Lecture 16 Pumping Lemma for

Context Free Grammars Fall 2008

Review •  Languages and Grammars

–  Alphabets, strings, languages •  Regular Languages

–  Deterministic Finite and Nondeterministic Automata –  Equivalence of NFA and DFA and Minimizing a DFA –  Regular Expressions –  Regular Grammars –  Properties of Regular Languages –  Languages that are not regular and the pumping lemma

•  Context Free Languages –  Context Free Grammars –  Derivations: leftmost, rightmost and derivation trees –  Parsing and ambiguity –  Simplifications and Normal Forms –  Nondeterministic Pushdown Automata –  Pushdown Automata and Context Free Grammars –  Deterministic Pushdown Automata

•  Today: –  Pumping Lemma for context free grammars

The Pumping Lemma for

Context-Free Languages

Take an infinite context-free language

Example:

bBSbBaBbAABS

→

→

→

→

Generates an infinite number of different strings

2

abbabbbbabbabbBbabbaBbBbabbABbabbSb

abbBaBbBABS

⇒⇒

⇒⇒⇒⇒

⇒⇒⇒

bBSbBaBbAABS

→

→

→

→

A derivation:

In a derivation of a long string, variables are repeated

S

A B

bBa

b

bS

A B

bBa

b

b

Derivation tree string abbabbbb

S

A B

bBa

b

bS

A B

bBa

b

b

repeated

Derivation tree string abbabbbb B

bS

A B

bBa

b

b

aBbbbaBbBbABbSbB⇒⇒

⇒⇒⇒

bB⇒aBbbbB*⇒

3

B

bS

A B

bBa b

Repeated Part

aBbbbB*⇒

B

bS

A B

ba bB

bS

A B

bBa b

aaBbbbbbbaBbbbB**⇒⇒

Another possible derivation from

aBbbbB*⇒

B

B

bS

A B

ba bB

bS

A B

bBa b

22** )()()()( bbbBabbbBaB ⇒⇒

aBbbbB*⇒

22* )()( bbbbaabbS⇒

)()()( 22 GLbbbbaabb ∈

aBbbbB*⇒ bB⇒abbBbS

*⇒

4

S

A B

bBa

baBbbbB*⇒

bB⇒

abbBbS*⇒

abbbbabbBbS ⇒⇒*

b

00 )()( bbbbaabb=

00* )()( bbbbaabbS⇒

)()()( 00 GLbbbbaabb ∈

aBbbbB*⇒ bB⇒abbBbS

*⇒

3333* )()()()( bbbbaabbbbbBaabbS ⇒⇒

S

A B

bBa

b

bS

A B

ba bB

bS

A B

bBa b

bS

A B

bBa b

b

aBbbbB*⇒

bB⇒

abbBbS*⇒

33* )()( bbbbaabbS⇒

)()()( 33 GLbbbbaabb ∈

aBbbbB*⇒ bB⇒abbBbS

*⇒

5

ii bbbbaabbS )()(*⇒

)()()( GLbbbbaabb ii ∈

aBbbbB*⇒ bB⇒abbBbS

*⇒

In General:

0≥i

Consider now an infinite context-free language

G

Take so that I has no unit-productions no -productions

Gλ

L

Let be the grammar of }{λ−L

(Number of productions) x (Largest right side of a production)

= Let

bBSbBaBbAABS

→

→

→

→Example :

131=+= pm

G

p

Let 1+= pm

1234 =×=p

Take a string with length

)(GLw∈mw ≥||

We will show: in the derivation of a variable of is repeated

wG

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wvvv k ⇒⇒⇒⇒ 21

wS*⇒

1vS =

wvvv k ⇒⇒⇒⇒ 21

fvv ii +< + |||| 1 maximum right hand side of any production

fkw ⋅

Number of productions in grammar

>k

Some production must be repeated

wvvv k ⇒⇒⇒⇒ 21

2

2

1

rBrArS

→

→

→

Repeated variable

wAaaAaav ⇒⇒⇒⇒⇒⇒ 43211

7

)(GLw∈ mw ≥||

wAaaAaaS ⇒⇒⇒⇒⇒⇒ 4321

Some variable is repeated

Derivation of string wS

A

A

x

Last repeated variable u z

v yuvxyzw =:,,,, zyxvu

repeated

Strings of terminals

Derivation tree of string w

S

A

A

x

u z

v y

uAzS∗⇒

vAyA∗⇒

xA∗⇒

Possible derivations:

uAzS∗⇒ vAyA

∗⇒ xA

∗⇒

We know:

This string is also generated:

uxzuAzS*⇒⇒

∗

zxyuv 00

8

This string is also generated:

uvxyzuvAyzuAzS**⇒⇒⇒

∗

The original zxyuvw 11=

uAzS∗⇒ vAyA

∗⇒ xA

∗⇒

We know:

This string is also generated:

uvvxyyzuvvAyyzuvAyzuAzS***⇒⇒⇒⇒

∗

zxyuv 22

uAzS∗⇒ vAyA

∗⇒ xA

∗⇒

We know:

This string is also generated:

uvvvxyyyzuvvvAyyyz

uvvAyyzuvAyzuAzS**

**

⇒⇒

⇒⇒⇒⇒∗

zxyuv 33

uAzS∗⇒ vAyA

∗⇒ xA

∗⇒

We know:

This string is also generated:

yyyzvxyuvvvyyyzvAyuvvv

uvvvAyyyzuvvAyyzuvAyzuAzS

…

⇒

⇒⇒

⇒⇒

⇒⇒⇒⇒

zxyuv ii

uAzS∗⇒ vAyA

∗⇒ xA

∗⇒

We know:

*

***

*****

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Therefore, any string of the form

zxyuv ii

is generated by the grammar G

0≥i

knowing that )(GLuvxyz ∈

we also know that )(GLzxyuv ii ∈

Therefore,

Lzxyuv ii ∈

}{)( λ−= LGL

S

A

A

x

u z

v y

mvxy ≤||Observation: Since is the last repeated variable A

S

A

A

x

u z

v y

1|| ≥vyObservation: Since there are no unit or -productions λ

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The Pumping Lemma:

there exists an integer such that m

for any string mwLw ≥∈ || ,

we can write

For infinite context-free language L

uvxyzw =

with lengths 1||and || ≥≤ vymvxy

and it must be:

0 allfor , ≥∈ iLzxyuv ii

What’s Next •  Read

–  Linz Chapter 1,2.1, 2.2, 2.3, (skip 2.4), 3, 4, 5, 6.1, 6.2, (skip 6.3), 7.1, 7.2, 7.3, (skip 7.4), and 8.1

–  JFLAP Chapter 1, 2.1, (skip 2.2), 3, 4, 5, 6, 7 •  Next Lecture Topics

–  More Pumping Lemma for Context Free Grammars •  Quiz 3 in Recitation on Wednesday 11/12

–  Covers Linz 7.1, 7.2, 7.3, (skip 7.4), 8, and JFLAP 5,6,7 –  Closed book, but you may bring one sheet of 8.5 x 11 inch paper with any notes you

like. –  Quiz will take the full hour

•  Homework –  No homework today –  New Homework Available Friday Morning –  New Homework Due Next Thursday