Chapter 4 PDA

7/28/2019 Chapter 4 PDA

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Chapter 4

Context-Free Grammars and

Languages

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2

Context-Free Languages


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Regular Languages

0:{ nbann

{ Rww

**ba *)( ba+


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Regular Languages

{ nnba { Rww



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PushdownAutomata

Context-Free

Grammars

stack

automaton


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Context-Free Grammars


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Grammars

Grammars express languages

Example: the English language

verbpredicate

nounarticlephrasenoun

predicatephrasenounsentence

_

_


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walksverb

runsverb

dognouncatnoun

thearticle

aarticle


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Example 1:

A derivation of the dog walks:

walksdogthe

verbdogthe

verbnounthe

verbnounarticle

verbphrasenoun


_

_


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Example 2:

A derivation of a cat runs:

runscata

verbcata

verbnouna

verbnounarticle

verbphrasenoun


_

_


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Language of the grammar:

L = { a cat runs,

a cat walks,

the cat runs,the cat walks,

a dog runs,

a dog walks,the dog runs,

the dog walks }


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Notation

dognouncatnoun

Variable Terminal

Production Rules


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Another Example

Grammar:

Derivation of sentence :

S

aSS

aaSbS

a

aSS S


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aabaaSbbaSbS

aSbS S

aab

S

aSSGrammar:

Derivation of sentence :


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Other derivations:

aaabbaaaSbbbaaSbbaSbS

aaaabbbbaaaaSbbbb

aaaSbbaaSbbaSbS


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Language of the grammar

S

aSS

0:{ = nbaL nn


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More Notation

Grammar ( )PSTVG ,,,=

:

:

:S

:P

Set of variables

Set of terminal symbols

Start variable

Set of Production rules


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Example

Grammar :

S

aSSG

( )PSTVG ,,,=

{S= ,{ ba=

,{ = SaSbSP


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More Notation

Sentential Form:

A sentence that containsvariables and terminals

Example:


Sentential Forms sentence


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We write:

Instead of:

aaabbS*



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In general we write:

If:

nww

*

1

nwwww

321


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By default: ww*


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Example

S

aSS

aaabbS

aabbS

abS

S

*

*

*

*

Grammar Derivations


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Another Grammar Example

Grammar :

A

aAbA

AbS

Derivations:

aabbaaAbbbaAbbAbS

abbaAbbAbSbAbS

G


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A Convenient Notation

A

aAA|aAbA

thearticle

aarticle

thaarticle |


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Definition: Context-Free Grammars

Grammar

Productions of the form:

AString of variables

and terminals

),,,( PSTVG =

Variables Terminalsymbols Startvariable

Variable


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Definition: Context-Free Languages

A language is context-free

if and only if

there is a context-free grammar

with

L

G)(GLL =


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Derivation Order

ABS.1

A

aaA

.3

.2

B

BB

.5

.4

aaaaBbaaBaaABABS54321

Leftmost derivation:

aaaaAbAbABbABS32541

Rightmost derivation:

ABS


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|AB

bBbA

aABS

Leftmost derivation:

abbbbabbbbBabbBbbBabAbBabBbBaABS

Rightmost derivation:

abbbbabbBbb

abAabBbaAaABS


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Derivation Trees


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ABS

ABS |aaAA |BbB

S

B


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ABS |aaAA |BbB

aaABABS S

B


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ABS |aaAA |BbB

aaABaaABABS S

B

B b


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ABS |aaAA |BbB

aaBaaABbaaABABS S

B

B b


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ABS |aaAA |BbB

aaaaBbaaABbaaABABS S

B

B b

Derivation Tree

| |


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aaaaBbaaABbaaABABS

yield

aabaa=

S

B

B b

Derivation Tree

ABS |aaAA |BbB

P i l D i i T


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Partial Derivation Trees

ABSS

B

Partial derivation tree

ABS |aaAA |BbB


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aaABABS

S

B


sentential


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aaABABS

S

B


sentential

form

yield

aaAB

Sometimes derivation order doesnt matter


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aaaaBbaaBaaABABS

aaaaAbAbABbABS S

B

B b

Same derivation tree

Sometimes, derivation order doesn t matter

Leftmost:

Rightmost:


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Ambiguity

aEEEEEE |)(|| +


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aEEEEEE |)(|| +

aaa +

E

EE

EE

+

aaaEaa

EEaEaEEE

*++

+++

leftmost derivation

aEEEEEE |)(|| +


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aEEEEEE |)(|| +

aaa +

E

EE

+

EE

aaaEaa

EEaEEEEEE

++

++

leftmost derivation

aEEEEEE |)(|| +


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aEEEEEE |)(|| +

aaa +

E

EE

+

EE

E

EE

EE

+

Two derivation trees

Th EEEEEE |)(||


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The grammar aEEEEEE |)(|| +

is ambiguous:

E

EE

+

a

EE

E

EE

EE

+

a

string aaa + has two derivation trees

Th EEEEEE |)(||


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string aaa + has two leftmost derivations

aaaEaa

EEaEEEEEE

++

++

aaaEaa

EEaEaEEE

*++

+++

The grammar aEEEEEE |)(|| +

is ambiguous:

D fi iti


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Definition:

A context-free grammar is ambiguous

if some string has:

two or more derivation trees

G

)(GLw

h d


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In other words:

A context-free grammar is ambiguous

if some string has:

two or more leftmost derivations

G

)(GLw

(or rightmost)

Wh d b t bi it ?


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Why do we care about ambiguity?

E

EE

+

EE

E

EE

EE

+

aaa +

take 2=a


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E

EE

+

EE

E

EE

EE

+

222+

2

2 2 2 2

2


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E

EE

+

EE

E

EE

EE

+

222 =+

2

2 2 2 2

2

222 =+

2 2

2

2 2

2


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E

EE

EE

+

222 =+

2 2

2 2

Correct result:


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We want to remove ambiguity

Ambiguity is bad for programming languages

We fix the ambiguous grammar:


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We fix the ambiguous grammar:

aEEEEEE |)(|| +

New non-ambiguous grammar:

aF

EF

F

FT

TE

TEE

+

)(

FTaTaTFTTTEE +++++


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aF

EF

F

FT

TE

TEE

+

)(

aaaFaaFFa +++

E

E +

F

FF

aaa+

U i d i ti t


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E

E +

F

FF

aaa+

Unique derivation tree

The grammar : TEE +G


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The grammar :

aF

EF

F

FT

TE

TEE

+

)(

is non-ambiguous:Every string has

a unique derivation tree

G

)(GLw

Another Ambiguous Grammar


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Another Ambiguous Grammar

IF_STMT if EXPR then STMT| if EXPR then STMT else STMT

If expr1 then if expr2 then stmt1 else stmt2


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If expr1 then if expr2 then stmt1 else stmt2

IF_STMT

expr1 then

elseif expr2 then

STMT

stmt1

if

IF_STMT

expr1 then else

if expr2 then

STMT stmt2if

stmt1

stmt2


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APPLICATIONS OFCONTEXT-FREE GRAMMARS

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CFGs and Programming Languages

Important uses of formal languages: to define precisely a programming language.

to construct an efficient translator for it.

RLs are used for simple patterns, while CFLs for morecomplicated aspects.

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S-grammars:

::= if

::= then ::= else

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Chapter 4 PDA