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A Pumping Lemma forand Closure Proper ties ofConte xt-Free Langua ges

Martin Franzle

Informatics and Mathematical Modelling

The Technical University of Denmark

Context-free languages III – p.1/20

What you’ ll learn

1. A pumping lemma for CFLs:

� The lemma� Its use in proving languages non-CFL

� Insight in the limits of CFLs

2. Closure proper ties of CFLs:

� The operation of substitution — generalizinghomomorphisms

� Closure under union, concatenation, Kleene closure,homomorphism� Non-closure under intersection, complement, difference� Closure under intersection with a regular language� Closure under inverse homomorphism


A pumping lemma for CFLs


Rationale

� As CFGs build strings of arbitrary length – if the languagedefined is not finite — from a finite set of rules, it is reasonableto expect that all sufficiently long words of the language havesome kind of repeatable pattern.

� Obviously, this pattern can be more complex than that occurringin regular languages (why?)

Program:

1. Identify the kind of pattern;

2. derive a method for proving languages non-contextfree.


Idea

Any parse-tree deeper than the number of variables in the CFG musthave some variable occur at least twice on its deepest path:

S

V

V

S

V

V

S

V

V

This provides “pluggable” subtrees!


Pumping it

Parse tree

can be shrunk

u v w x y

S

V

V

y

w

u

S

V

Caution: To really change yield of the parse tree, we have tomake sure that or .


Pumping it

Parse tree can be shrunk

u v w x y

S

V

V

y

w

u

S

V

Caution: To really change yield of the parse tree, we have tomake sure that or .


Pumping it

Parse tree can be shrunk

u v w x y

S

V

V

y

w

u

S

V

Caution: To really change yield of the parse tree, we have tomake sure that � �� or � �� .


Pumping it

Parse tree

can be expanded

u v w x y

S

V

V

u v

v w x

x y

S

V

V

V

Again, change of yield depends on .


Pumping it

Parse tree can be expanded � � �

u v w x y

S

V

V

u v

v w x

x y

S

V

V

V

Again, change of yield depends on .


Pumping it

Parse tree can be expanded � � �

u v w x y

S

V

V

u v

v w x

x y

S

V

V

V

Again, change of yield depends on � � �� .Context-free languages III – p.7/20

Ensuring � or � �

can be done by using Chomsky-normal-form (CNF) grammars:

� they don’t contain nullable symbols,

� the production used for the upper

in the parse tree must be ofthe form � �� as it did not directly derive a terminal

the lower

must be derived from either

�� or

�

� in the first case, � �� as

� is non-nullable,� in the second case, � �� as

� is non-nullable.


The pumping lemma for CFLs

Thm: For any CFL

�

there is constant � �

IN such that any string� � �

with

� � �� can be split into � � � �� with1.

� � � � �� ,

i.e. we find the pumpable portion in a “short” substring,

2. � � �� ,

i.e. at least one of and is non-trivial,

3.

��

IN � � ! � � ! � � �

,

i.e. and can be simultaneously “pumped”.


The pumping lemma for CFLs

Thm: For any CFL

�

there is constant � �

IN such that any string� � �

with

� � �� can be split into � � � �� with1.

� � � � �� ,i.e. we find the pumpable portion in a “short” substring,

2. � � �� ,i.e. at least one of � and � is non-trivial,

3.

��

IN � � ! � � ! � � �

,i.e. � and � can be simultaneously “pumped”.


An auxiliar y lemma on CNF grammar s

Lem: If

" � # %$ &$ ' $ ( )

is a CNF grammar and � � &*

is the yield of aparse tree of

"

of depth � then

� � �� +,.- �

.

Prf: By complete induction on :

Induction hypothesis: For all it is true that the yield of any parsetree of depth satisfies , if is terminal.

Induction step: Parse tree depth . We deduce from the inductionhypothesis that for all terminal yields of parse trees of depth .

Case 1: . There is no parse tree of depth having a terminal yield.Thus, the conjecture is trivially true.Case 2: . As is of CNF and is terminal, can only be derivedby a production with . Hence, .

Case 3: . As tree depth , the root of the parse tree uses aproduction, which is of the form because is of CNF. As theoverall tree depth is , the subtrees rooted at and have depth ,s.t. the induction hypothesis applies to them.Thus, holds for the yields of these subtrees. Now,

such that .



Lem: If

" � # %$ &$ ' $ ( )



"

of depth � then

� � �� +,.- �

.

Prf: By complete induction on /:

Induction hypothesis: For all it is true that the yield of any parsetree of depth satisfies , if is terminal.




such that .



Lem: If

" � # %$ &$ ' $ ( )



"

of depth � then

� � �� +,.- �

.

Prf: By complete induction on /:Induction hypothesis: For all / 0 1

it is true that the yield 2 of any parsetree of depth / satisfies

3 2 34 5687 9

, if 2 is terminal.




such that .



Lem: If

" � # %$ &$ ' $ ( )



"

of depth � then

� � �� +,.- �

.



3 2 34 5687 9

, if 2 is terminal.

Induction step: Parse tree depth /: 1

. We deduce from the inductionhypothesis that

3 2 34 5 ;7 9

for all terminal yields of parse trees of depth

1

.



such that .



Lem: If

" � # %$ &$ ' $ ( )



"

of depth � then

� � �� +,.- �

.



3 2 34 5687 9

, if 2 is terminal.



3 2 34 5 ;7 9


1

.�

Case 1:

1: <

. There is no parse tree of depth

<

having a terminal yield.Thus, the conjecture is trivially true.

Case 2: . As is of CNF and is terminal, can only be derivedby a production with . Hence, .


such that .



Lem: If

" � # %$ &$ ' $ ( )



"

of depth � then

� � �� +,.- �

.



3 2 34 5687 9

, if 2 is terminal.



3 2 34 5 ;7 9


1

.�

Case 1:

1: <


<

having a terminal yield.Thus, the conjecture is trivially true.�

Case 2:

1: =

. As

>

is of CNF and 2 is terminal, 2 can only be derivedby a production

?A@ B with B C D. Hence,

3 2 3 : = : 5 E : 5 ;7 9

.


such that .



Lem: If

" � # %$ &$ ' $ ( )



"

of depth � then

� � �� +,.- �

.



3 2 34 5687 9

, if 2 is terminal.



3 2 34 5 ;7 9


1

.�

Case 1:

1: <


<

having a terminal yield.Thus, the conjecture is trivially true.�

Case 2:

1: =

. As

>

is of CNF and 2 is terminal, 2 can only be derivedby a production

?A@ B with B C D. Hence,

3 2 3 : = : 5 E : 5 ;7 9

.�

Case 3:

1GF =

. As tree depth1F =

, the root of the parse tree uses aproduction, which is of the form

?A@ H 9 HJI because

>

is of CNF. As theoverall tree depth is

1, the subtrees rooted at

H 9 and

HI have depth 0 1

,s.t. the induction hypothesis applies to them.Thus,

3 2 9 34 5 ;7 ILK 3 2 I 3

holds for the yields 28M of these subtrees. Now,2 : 2 9 2 I such that

3 2 3 : 3 2 9 38N 3 2 I 34 5 ;7 I N 5 ;7 I : 5 ;7 9

.Context-free languages III – p.10/20

Pumping lemma as a proof schemeGiven a language

�

deemed to be non-CFL, proceed as follows:

1. Take arbitrary � �

IN,

(Selection of is not under your control — you have to accept any .)

2. provide a construction of � depending on �,

(You select .)

3. arbitrarily break � into � �� subject to the constraints(a)

� � � � �� ,(b) � � �� ,

(Selection of is not under your control — you have to accept anysplit that satisfies the two constraints.)

4. pick

� �

IN depending on � $ � $ � $ � $ � and � such that� � ! � � ! � ��

.

(You select .)

This constitutes a proof of

�

being not context-free.Context-free languages III – p.11/20

Pumping lemma as a proof schemeGiven a language

�

deemed to be non-CFL, proceed as follows:

1. Take arbitrary � �

IN,(Selection of / is not under your control — you have to accept any /.)

2. provide a construction of � depending on �,(You select O.)

3. arbitrarily break � into � �� subject to the constraints(a)

� � � � �� ,(b) � � �� ,(Selection of PRQ S Q 2 Q T Q U is not under your control — you have to accept anysplit that satisfies the two constraints.)

4. pick

� �

IN depending on � $ � $ � $ � $ � and � such that� � ! � � ! � ��

.(You select

V

.)

This constitutes a proof of

�

being not context-free.Context-free languages III – p.11/20

Closure proper ties of CFLs


Substitutions

Idea: Generalize the notion of homomorphism by substituting a fullCFL (instead of just a word) for each terminal symbol.

Def: Given a set

&

of terminals, a substitution for

&is a mappingWX & �

CFL.

Given � � &*

and a substitution W on

&,

W # � ) def� Y � � � � � � � �[Z]\ Z � � ! � W # � ! ) for all

� � � � �^ $

i.e. W # � )

is the concatenation of the languages W # � � ) $ W # � � ) $ � � � .Given

�_ &*

and a substitution W on

&

,W # � ) def�

\ ` aW # � ) �


Closure under substitution

Thm: If

�

is a CFL over alphabet

&

and W is a substitution for&

thenW # � )

is a CFL.

Prf: Essentially, we take a CFG

>

for

b

and replace each terminal B by the startsymbols of a CFG for c d B e .

Let and be a CFG for for each. Then generates , where

is the disjoint union of and all ’s,

,

consists of1. the productions from , but with each terminal being replaced

by (actually a homomorphism on the productions),

2. .


Closure under substitution

Thm: If

�

is a CFL over alphabet

&

and W is a substitution for&

thenW # � )

is a CFL.

Prf: Essentially, we take a CFG

>

for

b

and replace each terminal B by the startsymbols of a CFG for c d B e .Let

> : d HQ DQ fQ ? e

and

>hg : d Hg Q Dg Q fg Q ?g e

be a CFG for c d B e for eachB C D

. Then

> i : d H i Q D i Q f i Q ? e

generates c d b e, where� H i

is the disjoint union of

H

and all

Hg ’s,� D i : jg k l Dg ,� f i

consists of1. the productions from

f

, but with each terminal B C D

being replacedby

?g (actually a homomorphism on the productions),

2.

jg k l fg .


Implications of substitution closureCor: The CFLs are closed under� union,� concatenation,� Kleene closure (

*

) and positive closure (

m

),� homomorphism.

Prf: For the first three items, we state regular (and thus context-free) languagesplus a substitution that maps them into the desired language:1. ,2. ,3. and , respectively.

Applying the substitution and to these RLs, we obtain

1. ,2. ,3. and .Homomorphism, finally, is a special case of substitution.Hence, substitution closure implies all these closure properties.



*


m

),� homomorphism.

Prf: For the first three items, we state regular (and thus context-free) languagesplus a substitution that maps them into the desired language:1. BN n

,2. B n

,3. B o and B B o , respectively.

Applying the substitution c d B e : band c d n e : p

to these RLs, we obtain

1.

bq p

,2.

b p

,3.

b o

and

b r

.

Homomorphism, finally, is a special case of substitution.Hence, substitution closure implies all these closure properties.



*


m

),� homomorphism.

Prf: For the first three items, we state regular (and thus context-free) languagesplus a substitution that maps them into the desired language:1. BN n

,2. B n

,3. B o and B B o , respectively.

Applying the substitution c d B e : band c d n e : p

to these RLs, we obtain

1.

bq p

,2.

b p

,3.

b o

and

b r

.Homomorphism, finally, is a special case of substitution.Hence, substitution closure implies all these closure properties.


Non-c losure under inter section

Thm: The CFLs are not closed under intersection.

N.B.: This is in contrast to the RLs, which are closed underintersection.

Prf: By contraposition:p : s B6 n 6]t u 3 / Q v C

IN

w

and

x : s B6 n ut u 3 / Q v CIN

w

are CFLs. If theCFLs were closed under intersection then

s B6 n 6t 6 3 / C

IN

w: py x

werea CFL, which it is not.


Implications of non-c losure under inter section

Cor: The CFLs are not closed under� complement,� difference.

Prf: By contraposition:

If the CFLs were closed under complement, then they were also closed

under intersection, as

py x : p q x

.

If the CFLs were closed under difference, then they were also closed undercomplement, as

D o

is a CFL and

b : D o z b.


Closure under reversal

Thm: The CFLs are closed under reversal of words.

Prf: Take a CFG for the language and reverse the bodies of all productions.


Closure under inter section with RL

Thm: The CFLs are closed under intersection with a regularlanguage.I.e.,

�{ |

is a CFL if

�

is a CFL and

|

is an RL.

Prf: Let

f : d}R~ Q � Q �Q �~ Q �~ Q � E Q �~ e

be a PDA acceptingb

by final state and let� : d} g Q � Q �J� Q � � Q �� e

be a DFA for

�

.The PDA

} i : d} ~ � } g Q � Q �Q � Q d �~ Q � � e Q � E Q �~ � �g ewith

� d d � Q � e Q B Q � e def: d d � i Q � i e Q � e��d � i Q � e C �~ d � Q B Q � e

� � i : �� d � Q B e

then accepts

by �

by final state.

Cor: If is a CFL and is an RL, then is a CFL.

Prf: . Thus, is a CFL because of closure of the CFLs underintersection with RLs and because of closure of the RLs under complement.


Closure under inter section with RL

Thm: The CFLs are closed under intersection with a regularlanguage.I.e.,

�{ |

is a CFL if

�

is a CFL and

|

is an RL.

Prf: Let

f : d}R~ Q � Q �Q �~ Q �~ Q � E Q �~ e

be a PDA acceptingb

by final state and let� : d} g Q � Q �J� Q � � Q �� e

be a DFA for

�

.The PDA

} i : d} ~ � } g Q � Q �Q � Q d �~ Q � � e Q � E Q �~ � �g ewith

� d d � Q � e Q B Q � e def: d d � i Q � i e Q � e��d � i Q � e C �~ d � Q B Q � e

� � i : �� d � Q B e

then accepts

by �

by final state.

Cor: If

�

is a CFL and

|is an RL, then

� � |

is a CFL.

Prf:

b z � : by �

. Thus,b z �

is a CFL because of closure of the CFLs underintersection with RLs and because of closure of the RLs under complement.


Closure under inverse homomorphismThm: The CFLs are closed under reverse homomorphism.

Prf: We add a length v B T s 3� d B e 3 3 B C D w

buffer to a PDA for the CFL:

Accept/reject

Bufferh(a)a h

statePDA

Stack

�

Buffer is part of finite state set,�

whenever it is empty, it can befilled with

� dnext input

e

, therebykeeping the stack and theoriginal PDA’s state,�

when buffer is not empty, theoriginal PDA’s non- � moves canbe performed on the frontmostelement of the buffer, therebyremoving that element from thebuffer,�

the original PDA’s � moves canalways be performed, leaving thebuffer intact.


Download - A Pumping Lemma for and Closure Properties of Context ... · A Pumping Lemma for and Closure Properties of Context-FreeLanguages Martin Franzle¨ Informatics and Mathematical Modelling

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