Context-Free Languageshs/teach/13w/intro/pdf/08syn.pdf · CIS, LMU, 2013-11-25 Slides based on RPI...

Context-Free Languages Hinrich Schütze

CIS, LMU, 2013-11-25 Slides based on RPI CSCI 2400

Thanks to Costas Busch

Take-away Definition context-free grammar Definition context-free language Derivation, sentential form, sentence Derivation trees Ambiguity Context-free grammars for natural language

2

Terminology For our purposes in this class: Context free grammar = Constituency grammar = Phrase structure grammar

3

4

Grammars Grammars express languages Example: the English language

verbpredicate

nounarticlephrasenoun

predicatephrasenounsentence

→

→

→

_

_

5

walksverb

runsverb

dognoun

catnoun

thearticle

aarticle

→

→

→

→

→

→

6

A derivation of “the dog walks”:

walksdogtheverbdogthe

verbnounthe

verbnounarticle

verbphrasenoun


⇒

⇒

⇒

⇒

⇒

⇒

__

7

A derivation of “a cat runs”:

runscataverbcata

verbnouna

verbnounarticle

verbphrasenoun


⇒

⇒

⇒

⇒

⇒

⇒

__

8

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

dognoun

catnoun

→

→

Variable Terminal

Production Rules

10

Another Example Grammar: Derivation of sentence :

λ→

→

SaSbS

abaSbS ⇒⇒

ab

aSbS→ λ→S

11

Language?

12

aabbaaSbbaSbS ⇒⇒⇒

aSbS→ λ→S

aabb

λ→

→

SaSbSGrammar:

Derivation of sentence :

13

Other derivations:

aaabbbaaaSbbbaaSbbaSbS ⇒⇒⇒⇒

aaaabbbbaaaaSbbbbaaaSbbbaaSbbaSbS

⇒⇒

⇒⇒⇒

14

Language of the grammar

λ→

→

SaSbS

}0:{ ≥= nbaL nn

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More Notation Grammar ( )PSTVG ,,,=

:V

:T

:S

:P

Set of variables

Set of terminal symbols

Start variable

Set of Production rules

16

Example Grammar :

λ→

→

SaSbSG

( )PSTVG ,,,=

}{SV = },{ baT =

},{ λ→→= SaSbSP

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More Notation Sentential Form: A sentence that contains variables and terminals Example: aaabbbaaaSbbbaaSbbaSbS ⇒⇒⇒⇒

Sentential Forms sentence

18

We write: Instead of:

aaabbbS*⇒


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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

λ→

→

SaSbS

aaabbbS

aabbS

abS

S

*

*

*

*

⇒

⇒

⇒

⇒λ

Grammar Derivations

22

baaaaaSbbbbaaSbb

aaSbbS

∗

∗

⇒

⇒λ→

→

SaSbS

Grammar

Example

Derivations

23

Another Grammar Example Grammar :

λ→

→

→

AaAbAAbSG

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Language?

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Grammar :

λ→

→

→

AaAbAAbS

Derivations:

€

S→ Ab→ bS→ Ab→ aAbb→ abbS→ Ab→ aAbb→ aaAbbb→ aabbb

G

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More Derivations

aaaabbbbbaaaaAbbbbbaaaAbbbbaaAbbbaAbbAbS

⇒⇒

⇒⇒⇒⇒

bbaS

bbbaaaaaabbbbS

aaaabbbbbS

nn∗

∗

∗

⇒

⇒

⇒

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Language of a Grammar For a grammar with start variable :

GS

}:{)( wSwGL∗⇒=

String of terminals

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Example For grammar :

λ→

→

→

AaAbAAbS

}0:{)( ≥= nbbaGL nn

Since: bbaS nn∗⇒

G

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A Convenient Notation

λ→

→

AaAbA

λ|aAbA→

thearticleaarticle

→

→ theaarticle |→

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Revisit first grammar A context-free grammar :

λ→

→

SaSbS

aabbaaSbbaSbS ⇒⇒⇒

G

A derivation:

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A context-free grammar :

λ→

→

SaSbS


G

Another derivation:

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λ→

→

SaSbS

=)(GL

(((( ))))

}0:{ ≥nba nn

Describes parentheses:

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λ→

→

→

SbSbSaSaSA context-free grammar : G

Example

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Language?

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λ→

→

→

SbSbSaSaS

abaabaabaSabaabSbaaSaS ⇒⇒⇒⇒

A context-free grammar : G

Another derivation:

36

λ→

→

→

SbSbSaSaS

=)(GL }*},{:{ bawwwR ∈

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λ→

→

→

SSSSaSbSA context-free grammar : G

Example

38

Language?

39

λ→

→

→

SSSSaSbS

abababaSbabSaSbSSSS ⇒⇒⇒⇒⇒

A context-free grammar : G

Two derivations:

ababSaSbSSSS ⇒⇒⇒⇒

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λ→

→

→

SSSSaSbS

}prefixanyin )()( and

),()(:{

vvnvn

wnwnw

ba

ba≥

==)(GL

Interpretation?

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λ→

→

→

SSSSaSbS

}prefixanyin )()( and

),()(:{

vvnvn

wnwnw

ba

ba≥

=

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

=)(GL

Describes matched parentheses:

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Definition: Context-Free Grammars

Grammar

Productions of the form: xA→String of variables and terminals

),,,( PSTVG =

Variables Terminal symbols

Start variable

Variable

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*},:{)(*

TwwSwGL ∈⇒=

),,,( PSTVG =

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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

λ→

→

AaaAA

.3

.2λ→

→

BBbB

.5

.4

aabaaBbaaBaaABABS54321⇒⇒⇒⇒⇒

Leftmost derivation:

aabaaAbAbABbABS32541⇒⇒⇒⇒⇒

Rightmost derivation:

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Language?

47

λ|ABbBbAaABS

→

→

→

Leftmost derivation:

abbbbabbbbBabbBbbBabAbBabBbBaABS

⇒⇒

⇒⇒⇒⇒

Rightmost derivation:

abbbbabbBbbabAbabBbaAaABS

⇒⇒

⇒⇒⇒⇒

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Language?

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Derivation Trees

50

ABS⇒

ABS→ λ|aaAA→ λ|BbB→

S

BA

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aaABABS ⇒⇒

a a A

S

BA

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aaABbaaABABS ⇒⇒⇒S

BA

a a A B b

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aaBbaaABbaaABABS ⇒⇒⇒⇒S

BA

a a A B b

λ

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aabaaBbaaABbaaABABS ⇒⇒⇒⇒⇒S

BA

a a A B b

λ λ

Derivation Tree

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aabaaBbaaABbaaABABS ⇒⇒⇒⇒⇒

yield

aabbaa

=

λλ

S

BA

a a A B b

λ λ

Derivation Tree


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Partial Derivation Trees

ABS⇒

S

BA

Partial derivation tree


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aaABABS ⇒⇒

S

BA

a a A


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aaABABS ⇒⇒

S

BA

a a A


sentential form

yield aaAB

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aabaaBbaaBaaABABS ⇒⇒⇒⇒⇒

aabaaAbAbABbABS ⇒⇒⇒⇒⇒S

BA

a a A B b

λ λ

Same derivation tree

Sometimes, derivation order doesn’t matter Leftmost:

Rightmost:

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Ambiguity

61

aEEEEEE |)(|| ∗+→

aaa ∗+

E

EE

EE

+

a

a a

∗

aaaEaaEEaEaEEE*+⇒∗+⇒

∗+⇒+⇒+⇒

leftmost derivation

62


aaa ∗+

E

EE

+

a a

∗

EE a

aaaEaaEEaEEEEEE

∗+⇒∗+⇒

∗+⇒∗+⇒∗⇒

leftmost derivation

63


aaa ∗+

E

EE

+

a a

∗

EE a

E

EE

EE

+

a

a a

∗

Two derivation trees

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The grammar aEEEEEE |)(|| ∗+→

is ambiguous:

E

EE

+

a a

∗

EE a

E

EE

EE

+

a

a a

∗

string aaa ∗+ has two derivation trees

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string aaa ∗+ has two leftmost derivations

aaaEaaEEaEEEEEE

∗+⇒∗+⇒

∗+⇒∗+⇒∗⇒

aaaEaaEEaEaEEE*+⇒∗+⇒

∗+⇒+⇒+⇒

The grammar aEEEEEE |)(|| ∗+→

is ambiguous:

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Definition:

A context-free grammar is ambiguous if some string has: two or more derivation trees

G

)(GLw∈

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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)

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Why do we care about ambiguity?

E

EE

+

a a

∗

EE a

E

EE

EE

+

a

a a

∗

aaa ∗+

take 2=a

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E

EE

+

∗

EE

E

EE

EE

+

∗

222 ∗+

2

2 2 2 2

2

70

E

EE

+

∗

EE

E

EE

EE

+

∗

6222 =∗+

2

2 2 2 2

2

8222 =∗+

4

2 2

2

6

2 2

24

8

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E

EE

EE

+

∗

6222 =∗+

2

2 2

4

2 2

2

6

Correct result:

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•  We want to remove ambiguity

•  Ambiguity is bad for programming languages

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We fix the ambiguous grammar:


New non-ambiguous grammar:

aFEFFTFTT

TETEE

→

→

→

∗→

→

+→

)(

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aFEFFTFTT

TETEE

→

→

→

∗→

→

+→

)(

aaaFaaFFaFTaTaTFTTTEE

∗+⇒∗+⇒∗+⇒

∗+⇒+⇒+⇒+⇒+⇒

E

E T+

T ∗ F

F

a

T

F

a

a

aaa ∗+

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E

E T+

T ∗ F

F

a

T

F

a

a

aaa ∗+

Unique derivation tree

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The grammar :

aFEFFTFTT

TETEE

→

→

→

∗→

→

+→

)(

is non-ambiguous:

Every string has a unique derivation tree

G

)(GLw∈

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Another Ambiguous Grammar

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

Ambiguity?

78

If expr1 then if expr2 then stmt1 else stmt2

IF_STMT

expr1 then

else if expr2 then

STMT

stmt1

if

IF_STMT

expr1 then else

if expr2 then

STMT stmt2 if

stmt1

stmt2

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Inherent Ambiguity Some context free languages have only ambiguous grammars

Example: }{}{ mmnmnn cbacbaL ∪=

λ||11

aAbAAcSS

→

→

λ||22

bBcBBaSS

→

→21 | SSS→

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The string nnn cba

has two derivation trees

S

1S

S

2S

1S c 2Sa

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Ambiguity in natural language?

Take-away Definition context-free grammar Definition context-free language Derivation, sentential form, sentence Derivation trees Ambiguity Context-free grammars for natural language

82

Context-Free Languageshs/teach/13w/intro/pdf/08syn.pdf · CIS, LMU, 2013-11-25 Slides based on RPI...

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