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CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 10 Mälardalen University 2012. Content The Pumping Lemma for CFL Applications of the Pumping Lemma for CFL Example of Midterm Exam 2 (CFL). The Pumping Lemma for Context-Free Languages.

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slide1

CDT314

FABER

Formal Languages, Automata and Models of Computation

Lecture 10

Mälardalen University

2012

slide2

ContentThe Pumping Lemma for CFLApplications of the Pumping Lemma for CFLExample of Midterm Exam 2 (CFL)

the pumping lemma for context free languages

The Pumping LemmaforContext-Free Languages

Based on C Busch, RPI, Models of Computation

slide4

Take an infinite context-free language.

It generates an infinite number of different strings:

Example:

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string

Derivation tree

slide7

string

Derivation tree

repeated

slide14

Therefore, the string

is also generated by the grammar

slide15

We know:

We also know the following string is generated:

slide16

We know:

Therefore, the following string is also generated:

slide17

We know:

Therefore, the following string is also generated:

slide18

We know:

Therefore, the following string is also generated:

slide19

Therefore, knowing that

is generated by grammar

We also know that

is generated by

slide20

We are given an infinite

context-free grammar .

In general

Assume has no unit-productions

and no -productions.

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Take a string

with length bigger than

(Number of productions) x

(Largest right side of a production)

>

Consequence:

Some variable must be repeated

in the derivation of .

slide22

string

Last repeated variable

repeated

stringsof terminals

slide23

Possible

derivations

slide24

We know:

Following string is also generated:

slide25

We know:

This string is also generated:

The original

slide26

We know:

This string is also generated:

slide27

We know:

This string is also generated:

slide28

We know:

This string is also generated:

slide29

Therefore, any string of the form

is generated by the grammar

slide30

Therefore

knowing that

we also know that

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(Number of productions) x

(Largest right side of a production)

Observation:

Since is the last repeated variable

A string has length bigger than

>

slide32

Observation

Since there are no unit or productions

slide33

For infinite context-free language

there exists an integer such that

for any string

we can write

with lengths

and

The Pumping Lemma for CFL

slide35

Unrestricted grammarlanguages

Non-regular languages

Context-Free Languages

Regular Languages

slide36

Example

Theorem

The language

is not context free.

Proof

Use the Pumping Lemma

for context-free languages.

slide37

Assume thecontrary, that

is context-free.

Since is context-free and infinite

we can apply the pumping lemma.

slide38

Pumping Lemma gives a number

such that:

for any string with length

We can choose e.g.

slide39

We can write:

with lengths and

slide42

Case 1:

is within

slide43

Case 1:

and consist from only

slide44

Case 1:

Repeating and

slide45

Case 1:

From Pumping Lemma:

slide46

Case 1:

From Pumping Lemma:

However:

Contradiction!

slide47

Case 2:

is within

slide49

Case 3:

is within

slide51

Case 4:

overlaps and

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Case 4:

contains only

Possibility 1:

contains only

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Case 4:

Possibility 1:

contains only

contains only

slide54

Case 4:

From Pumping Lemma:

slide55

Case 4:

From Pumping Lemma:

However:

Contradiction!

slide56

Case 4:

Possibility 2:

contains and

contains only

slide57

Case 4:

Possibility 2:

contains and

contains only

slide58

Case 4:

From Pumping Lemma:

slide59

Case 4:

From Pumping Lemma:

However:

Contradiction!

slide60

Case 4:

Possibility 3:

contains only

contains and

slide61

Case 4: Possibility 3: contains only

contains and

Similar analysis with Possibility 2

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Case 5:

overlaps and

slide64

(Since , string cannot

overlap , and at the same time)

There are no other cases to consider.

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In all cases we obtain a contradiction.

Therefore:

The original assumption that

is context-free must be wrong.

Conclusion:

is not context-free.

END OF PROOF

what is the difference between context free languages and regular languages

......

......

What is the difference between Context Free Languages and Regular Languages?

In regular languages a single symbol/substring in the string w can be “pumped”.

the difference between context free languages and regular languages
The difference between Context Free Languages and Regular Languages

In CFL’s multiple symbols/substrings in the string w can be “pumped”.

Consider the language {an bn | n > 0}

No single symbol can be pumped and allow us to stay in the language.

However, there do exist pairs of symbols which can be pumped resulting in strings which stay in the language.

Thus a CFL pumping lemma applies.

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stringsof terminals

String

Last repeated variable

repeated

pumping conditions for rl and cfl
A language L satisfies the RLpumping condition if:

there exists an integer m > 0 such that

for all strings w in L of length at least m

there exist strings x, y, zsuch that

w = xyzand

|xy| ≤ mand

|y| ≥ 1and

for all i ≥ 0, xyizis in L

|xy| ≤ m is in the beginning of the w and can be pumped within m symbols.

A language L satisfies theCFLpumping condition if:

there exists an integer m > 0such that

for all strings w in L of length at least m

there exist strings u, v, x, y, z such that

w = uvxyzand

|vxy| ≤ mand

|vy| ≥ 1and

for all i ≥ 0, uvixyiz is in L

|vxy| ≤ mand u comes first in the w and it can be arbitrarily long.

Pumping Conditions for RL and CFL
pumping lemma

CFL’s

“Pumping Languages”

All languages over {a,b}

Pumping Lemma

All CFL’s satisfy the CFL pumping condition

But some languages that satisfy CFL pumping condition are not CFL!

implications
Implications

CFL’s

“Pumping Languages”

All languages over {a,b}

We can use the pumping lemma to prove a language Lis not a CFL.

Show that L does not satisfy the CFL pumping condition.

We cannot use the pumping lemma to prove a language is CFL.

Showing L satisfies the pumping condition does not guarantee that L is context-free.

pumping lemma1

Pumping Lemma

What does it mean?

pumping condition
Pumping Condition
  • A language L satisfies the CFL pumping condition if:
    • there exists an integer m > 0such that
    • for all strings w in L of length at least m
    • there exist strings u, v, x, y, zsuch that
      • w = uvxyzand
      • |vxy| ≤ m and
      • |vy| ≥ 1 and

Then for all i ≥ 0, uviwyizis in L

v and y can be pumped
v and y can be pumped

1) w in L2) w = uvxyz3) for all i ≥ 0, uvixyiz is in L

  • Let w = abcdefgbe in L
  • Then there exist substrings v and y in w such that v and y can be repeated (pumped) and the resulting string is still in L:

uvixyiz is in L for all i ≥ 0

slide77
For example w =abcdefg

v = cdand y = f

uv0xy0z = uxz =abegis in L

uv1xy1z = uvxyz = abcdefgis in L

uv2xy2z = uvvxyyz = abcdcdeffgis in L

uv3xy3z = uvvvxyyyz = abcdcdcdefffg is in L

what the other parts mean
What the other parts mean

A language L satisfies the CFL pumping condition if:

there exists an integer m > 0 such that

for all strings w in L of length at least m

w must be in L

what the other parts mean1
What the other parts mean

There exist strings u, v, x, y, z such that

  • w = uvxyz and
  • |vxy| ≤ m and

v and y are contained within m characters of w

Note: these are NOT necessarily the first m characters of w

|vy| ≥ 1

(v and y cannot both be , one of them might be , but not both)

  • For all i ≥ 0, uvixyiz is in L
how we use the pumping lemma
How we use the Pumping Lemma
  • We choose a specific language L.
  • We show that L does not satisfy the pumping condition.
  • We conclude that L is not context-free.
showing that l does not pump
A language L satisfies the CFL pumping condition if:

there exists an integer m > 0 such that

for all strings w in L of length at least m

there exist strings u, v, x, y, z such that

w = uvxyz and

|vxy| ≤ m and

|vy| ≥ 1 and

for all i ≥ 0, uvixyiz is in L

A language L does not satisfy the CFL pumping condition if:

for all integers m of sufficient size

there exists a string w in L of length at least m such that

for all strings u, v, x, y, z where

w = uvxyz and

|vxy| ≤ m and

|vy| ≥ 1

there exists a i ≥ 0 such that uvixyiz is not in L

Showing that L “does not pump”
slide83

Unrestricted grammarlanguages

Non-regular languages

Context-Free Languages

Regular Languages

slide85

For infinite context-free language

there exists an integer such that

for any string

we can write

with lengths

then

The Pumping Lemma for CFL

and

slide86

Example

Theorem

The language

is not context free.

Proof

Use the Pumping Lemma

for context-free languages.

slide87

Assume the contrary - that

is context-free.

Since is context-free and infinite

we can apply the pumping lemma.

slide88

Pumping Lemma gives a number

such that we can

pick any string of

with length at least

So we pick:

slide89

and

with lengths

We can write:

Pumping Lemma says:

for all

slide91

Case 1:

is within the first

slide92

Case 1:

is within the first

slide93

Case 1:

is within the first

slide94

Case 1:

is within the first

However, from Pumping Lemma:

Contradiction!

slide95

Case 2:

is in the first

is in the first

slide96

Case 2:

is in the first

is in the first

slide97

Case 2:

is in the first

is in the first

slide98

Case 2:

is in the first

is in the first

However, from Pumping Lemma:

Contradiction!

slide99

Case 3:

overlaps the first

is in the first

slide100

Case 3:

overlaps the first

is in the first

slide101

Case 3:

overlaps the first

is in the first

slide102

Case 3:

overlaps the first

is in the first

However, from Pumping Lemma:

Contradiction!

slide103

Case 4:

in the first

overlaps the first

Analysis is similar to case 3

slide104

or

or

Other cases:

is within

Analysis is similar to case 1:

slide105

More cases:

overlaps

or

Analysis is similar to cases 2,3,4:

slide106

There are no other cases to consider.

Since , it is impossible for

to overlap:

neither

nor

nor

slide107

is not context-free.

In all cases we obtain a contradiction.

Therefore:

The original assumption that

is context-free must be wrong.

Conclusion:

END OF PROOF

slide108

Unrestricted grammarlanguages

Non-regular languages

Context-Free Languages

Regular Languages

slide109

Example

Theorem

The language

is not context free.

Proof

Use the Pumping Lemma

for context-free languages.

slide110

Since is context-free and infinite

we can apply the pumping lemma.

Assume to the contrary that

is context-free.

slide111

Pumping Lemma gives a number

such that we can:

pick any string of with length at least

so we pick:

slide112

We can write:

with lengths

and

Pumping Lemma says:

for all

slide113

We examine all the possible locations

of the string in

There is only one case to consider.

slide118

Since

for

we have:

slide121

We obtained a contradiction

Therefore:

The original assumption that

is context-free must be wrong

Conclusion:

is not context-free

END OF PROOF

slide122

Unrestrictedgrammarlanguages

Context-free languages

Regular Languages

slide123

Example

Theorem

The language

is not context free

Proof

Use the Pumping Lemma

for Context-free languages

slide124

Since is context-free and infinite

we can apply the pumping lemma.

Assume to the contrary that

is context-free.

slide125

Pumping Lemma gives a number

such that we can:

pick any string of with length at least

so we choose:

slide126

We can write:

with lengths and

Pumping Lemma says:

for all

slide130

and

The most complicated sub-case:

slide131

and

The most complicated sub-case:

slide132

and

The most complicated sub-case:

slide136

When we examine the rest of the cases

we also obtain a contradiction.

slide137

is not context-free.

Conclusion:

In all cases we obtain a contradiction.

Therefore:

The original assumption that

is context-free must be wrong.

END OF PROOF

check your knowledge before the midterm 2 selected examples of context free language problems

Check your knowledge before the Midterm 2!Selected ExamplesofContext Free Language Problems

problem 1 find a cfg for the following language
Problem 1. Find a CFG for the following language

Solution 1.

Let G be the grammar with productions:

Claim: L(G) = L

find a cfg for the following language
Find a CFG for the following language:

Proof:

Consider the following derivation:

(the first * applies S  aSc n times, the second * to B  bBc m times)

Since all words in L(G) must follow this pattern in their derivations, it is clear that L(G)  L

find a cfg for the following language1
Find a CFG for the following language

Considerw  L, w = anbmc(n + m)for some n, m  0

The derivation

S * anScn anBcn * anbmBcmcn anbmc(n + m)

clearly produces w for any n, m.

 L  L(G)

 L  L(G)

G is a CFG for L

END OF PROOF

find a pda and cfg for the following language

Problem 2.

Find a PDA and CFG for the following language

Solution 2.

Is the automaton deterministic? Yes.

It acts in a unique way in each state, no l-transitions.

slide146

Problem 4.

Prove that the language L is context-free

Consider the following two languages:

L1 ={w : w is made from a’s and b’s

and the length of w is a multiple of 10}

L2 = {an bn: n  0}

Solution 4.

slide147
L1 ={w : w is made from a’s and b’s and the length of w is a multiple of ten}

L2 = {an bn: n  0}

LetL1cdenote the complement of L1.

We have that: L = L1c  L2.

L1is a regular language, since we can easily build a finite automaton with 10 states that accepts any string in this language.

L1cis regular too, since regular languages are closed under complement.

slide148
The language L2is context-free.

The grammar is: S  aSb | 

Therefore, the language L = L1c  L2is also context-free,

since context-free languages are closed under regular intersection (Regular Closure).

END O PROOF

slide153

CFG, direct construction

Strings start and finish with different symbols

Strings contain at least one more a than b

(we must have AA here as only one A just balances b)

slide154

Further Reading

Famous Pushdown Automata Examples

http://www.liacs.nl/~hoogeboo/praatjes/tarragona/schoolpda-VIII.pdf

Computational and evolutionary aspects of language

http://www.nature.com/nature/journal/v417/n6889/full/nature00771.html

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