fall 2005costas busch - rpi1 context-free languages

Fall 2005 Costas Busch - RPI 1

Context-Free Languages


Regular Languages

}0:{ nba nn }{ Rww

**ba *)( ba


Regular Languages

}{ nnba }{ Rww




PushdownAutomata

Context-FreeGrammars

stack

automaton


Context-Free Grammars


GrammarsGrammars express languages

Example: the English language

verbpredicate

nounarticlephrasenoun

predicatephrasenounsentence

_

_


sleepsverb

runsverb

dognoun

catnoun

thearticle

aarticle


A derivation of “the dog walks”:

sleepsdogthe

verbdogthe

verbnounthe

verbnounarticle

verbphrasenoun


_

_


A derivation of “a cat runs”:

runscata

verbcata

verbnouna

verbnounarticle

verbphrasenoun


_

_


Language of the grammar:

L = { “a cat runs”, “a cat sleeps”, “the cat runs”, “the cat sleeps”, “a dog runs”, “a dog sleeps”, “the dog runs”, “the dog sleeps” }


Notation

dognoun

catnoun

Variable Terminal

Production Rules


Another ExampleGrammar:

Derivation of sentence :

S

aSbS

abaSbS

ab

aSbS S


Grammar:

Derivation of sentence :

aabbaaSbbaSbS

aSbS S

aabb

S

aSbS


aaabbbaaaSbbbaaSbbaSbS

aaaabbbbaaaaSbbbb

aaaSbbbaaSbbaSbS

Other derivations:

S

aSbSGrammar:


Language of the grammar

S

aSbS

}0:{ nbaL nn


More Notation

Grammar PSTVG ,,,

:V

:T

:S

:P

Set of variables

Set of terminal symbols

Start variable

Set of Production rules


Example

Grammar :

S

aSbSG

PSTVG ,,,

}{SV },{ baT

},{ SaSbSP


More NotationSentential Form: A sentence that contains variables and terminals

Example:


Sentential Forms sentence


We write:

Instead of:

aaabbbS*



nww*

1

nwwww 321

In zero or more steps

In general we write:

If:

ww*Trivially:


S

aSbS

aaabbbS

abS

S

*

*

*

Example Grammar Possible Derivations

baaaaaSbbbbaaSbb

aaSbbS


Another Grammar ExampleGrammar :

A

aAbA

AbS

Derivations:

aabbbaaAbbbaAbbAbS

abbaAbbAbS

bAbS

⇒⇒⇒⇒

⇒⇒⇒

⇒⇒

G


More Derivations

aaaabbbbbaaaaAbbbbb

aaaAbbbbaaAbbbaAbbAbS

bbaS

bbbaaaaaabbbbS

aaaabbbbbS

nn

A

aAbA

AbSGrammar


Language of a Grammar

For a grammar with start variable :

GS

}:{)( wSwGL

String of terminals


ExampleFor grammar :

A

aAbA

AbS

}0:{)( nbbaGL nn

Since: bbaS nn

G


A Convenient Notation

A

aAbA|aAbA

thearticle

aarticle

theaarticle |


Example

Context-free grammar : |aSbS

aabbaaSbbaSbS

G

Possible Derivations:


)(GL

(((( ))))

}0:{ nba nn

Describes parentheses:)b (, a


||bsbaSaS

abbaabSbaaSaS

Context-free grammar :G

Derivations:

Example

abaabaabaSabaabSbaaSaS

)(GL }*},{:{ bawwwR

Palindromes of even length


|| SSaSbS

ababSaSbSSSS

A context-free grammar :G

Derivations:

Example

abababaSbabSaSbSSSS

}prefixanyin

)()( and

),()(:{

v

vnvn

wnwnw

ba

ba

)(GL

() ((( ))) (( ))

Describes matched parentheses: )b (, a


Definition: Context-Free Grammars

Grammar

Productions of the form:

xAString of variables and terminals

),,,( PSTVG

Variables Terminalsymbols

Startvariable

Variable


*},:{)(*

TwwSwGL

),,,( PSTVG

The language of the grammar is:


Definition: Context-Free Languages

A language is context-free

if and only if

there is a context-free grammar with

L

G)(GLL


Derivation Orderand

Derivation Trees


Derivation Order ABS .1

A

aaAA

.3

.2

B

BbB

.5

.4

aabaaBbaaBaaABABS54321

Leftmost derivation:

aabaaAbAbABbABS32541

Rightmost derivation:


|AB

bBbA

aABS

Leftmost derivation:

abbbbabbbbB

abbBbbBabAbBabBbBaABS

Rightmost derivation:

abbbbabbBbb

abAbabBbaAaABS


ABS

ABS |aaAA |BbB

S

BA

Derivation Trees


ABS |aaAA |BbB

aaABABS

a a A

S

BA


ABS |aaAA |BbB

aaABbaaABABS S

BA

a a A B b


ABS |aaAA |BbB

aaBbaaABbaaABABS S

BA

a a A B b


ABS |aaAA |BbB

aabaaBbaaABbaaABABS S

BA

a a A B b

Derivation Tree


aabaaBbaaABbaaABABS

yield

aab

baa

S

BA

a a A B b

Derivation Tree

ABS |aaAA |BbB


Partial Derivation Trees

ABS

S

BA

Partial derivation tree

ABS |aaAA |BbB


aaABABS

S

BA

a a A



aaABABS

S

BA

a a A


sententialform

yield

aaAB


aabaaBbaaBaaABABS

aabaaAbAbABbABS S

BA

a a A B b

Same derivation tree

Sometimes, derivation order doesn’t matter

Leftmost:

Rightmost:


Ambiguity


aEEEEEE |)(|| aaa

E

EE

EE

a

a a

aaaEaa

EEaEaEEE

*

leftmost derivation


aEEEEEE |)(|| aaa

E

EE

a a

EE a

aaaEaa

EEaEEEEEE

leftmost derivation


aEEEEEE |)(|| aaa

E

EE

a a

EE a

E

EE

EE

a

a a

Two derivation trees


The grammar aEEEEEE |)(|| is ambiguous:

E

EE

a a

EE a

E

EE

EE

a

a a

string aaa has two derivation trees


string aaa has two leftmost derivations

aaaEaa

EEaEEEEEE

aaaEaa

EEaEaEEE

*

The grammar aEEEEEE |)(|| is ambiguous:


Definition:

A context-free grammar is ambiguous

if some string has:

two different derivation trees OR two leftmost derivations

G

)(GLw


Why do we care about ambiguity?

E

EE

a a

EE a

E

EE

EE

a

a a

aaa

take 2a


E

EE

EE

E

EE

EE

222

2

2 2 2 2

2


E

EE

EE

E

EE

EE

6222

2

2 2 2 2

2

8222

4

2 2

2

6

2 2

24

8

Good derivation Tree

Bad derivation Tree


• We want to remove ambiguity

• Ambiguity is bad for programming languages


We fix the ambiguous grammar:

aEEEEEE |)(||

New unambiguous grammar:

aEF

FFTT

TTEE

|)(

|

|


aaaFaaFFa

FTaTaTFTTTEE

E

E T

T F

F

a

T

F

a

a

aaa

aEF

FFTT

TTEE

|)(

|

|

Unique derivation tree


Another Ambiguous Grammar

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


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


Inherent Ambiguity

Some context free languageshave only ambiguous grammars

Example: }{}{ mmnmnn cbacbaL

||11

aAbA

AcSS

||22

bBcB

BaSS

21 | SSS

0, mn


The string nnn cbahas always two derivation trees(for any grammar)

S

1S

S

2S

1S c 2Sa

For example

fall 2005costas busch - rpi1 context-free languages

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