1

Grammars

2

GrammarsGrammars express languages

Example: the English language

verbpredicate

nounarticlephrasenoun

predicatephrasenounsentence

_

_

3

walksverb

runsverb

dognoun

boynoun

thearticle

aarticle

4

A derivation of “the boy walks”:

walksboythe

verbboythe

verbnounthe

verbnounarticle

verbphrasenoun

predicatephrasenounsentence

_

_

5

A derivation of “a dog runs”:

runsdoga

verbdoga

verbnouna

verbnounarticle

verbphrasenoun

predicatephrasenounsentence

_

_

6

Language of the grammar:

L = { “a boy runs”, “a boy walks”, “the boy runs”, “the boy walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }

7

Notation

dognoun

boynoun

Variable orNon-terminal

TerminalProductionrule

8

Another ExampleGrammar:

Derivation of sentence :

S

aSbS

abaSbS

ab

aSbS S

9

Derivation of sentence :aabb

aabbaaSbbaSbS

aSbS S

10

Other derivations:

aaabbbaaaSbbbaaSbbaSbS

aaaabbbbaaaaSbbbb

aaaSbbbaaSbbaSbS

11

Language of the grammar

S

aSbS

0: nbaL nn

12

More Notation

Grammar PSTVG ,,,

V

T

S

P

Set of variables

Set of terminal symbols

Start variable

Set of Production rules

13

ExampleGrammar :

S

aSbSG

PSTVG ,,,

}{SV

},{ baT

},{ SaSbSP

14

More NotationSentential Form: A sentence that contains Variables and terminals

Example:

aaabbbaaaSbbbaaSbbaSbS

Sentential Forms sentence

15

We write:

Instead of:

aaabbbS*

aaabbbaaaSbbbaaSbbaSbS

16

In general we write:

If:

nww*

1

nwwww 321

17

By default:

ww*

18

Examples

S

aSbS

aaabbbS

aabbS

abS

S

*

*

*

*

Grammar:

baaaaaSbbbbaaSbb

aaSbbS

19

Another ExampleGrammar G:

A

aAbA

AbS

Derivations:

aabbbaaAbbbaAbbS

abbaAbbAbS

bAbS

20

aaaabbbbbaaaaAbbbbb

aaaAbbbbaaAbbbaAbbAbS

bbaS

bbbaaaaaabbbbS

aaaabbbbbS

nn

21

Language of a Grammar

For a grammar with start variable

GS

}:{)( wSwGL

22

ExampleFor grammar G:

A

aAbA

AbS

}0:{)( nbbaGL nn

Since: bbaS nn

23

A Convenient Notation

A

aAbA|aAbA

thearticle

aarticle

theaarticle |

24

Linear Grammars

25

A Linear GrammarGrammars with at most one variable on the right sideof a production

Examples:

A

aAbA

AbS

S

aSbS

26

A Non-Linear Grammar

bSaS

aSbS

S

SSS

Grammar :G

)}()(:{)( wnwnwGL ba

27

Another Linear GrammarGrammar

AbB

aBA

AS

|

}0:{)( nbaGL nn

:G

28

Right-Linear GrammarsAll productions have the form:

Example:

xBA

xAor

aS

abSS

29

Left-Linear GrammarsAll productions have form:

Example:

BxA

xA

or

aB

BAabA

AabS

|

30

Regular Grammars

31

Regular GrammarsDefinition: A regular grammar is any right-linear or left-linear grammar

Examples:

aS

abSS

aB

BAabA

AabS

|

32

ObservationRegular grammars generate regular languages

Examples:

aS

abSS

Grammar G:

aabGL *)()(

Grammar G:

aB

BAabA

AabS

|

*)()( abaabGL

33

Theorem

A Language is regular if and only ifthere is a grammar such that

L

G )(GLL