1. 3.3 Grammars
HW on p.97 : 1~13, 17, 26(p.111)
2008 Fall 1

2. Recall Notation
 Grammar G V ,T , S , P p.21
V : Set of variables
T : Set of terminal symbols
S : Start variable
P: Set of Production rules
2008 Fall 2

3. Example
S aSb
 Grammar G: S
G V ,T , S , P
V {S } T {a, b}
P {S aSb, S }
2008 Fall 3

4. More Notation on Derivations
 Sentential Form:
A sentence that contains variables
 Example:
A derivation from the starting variable
S aSb aaSbb aaaSbbb aaabbb
Sentential Forms sentence
2008 Fall 4

5. *
 We write: S aaabbb
Instead of:
S aSb aaSbb aaaSbbb aaabbb
2008 Fall 5

6. Linear Grammars
2008 Fall 6

7. Linear Grammars
 Grammars with productions of the form
X Y with X V, Y V T*, and
Y has at most one variable from V
i.e. at most one variable at the right side of a production
Examples:
S Ab
S aSb |
A aAb |
2008 Fall 7

8. A Non-Linear Grammar
2 variables on the right
 Grammar G : S SS of the production
S
S aSb
S bSa
L(G) {w : na ( w) nb ( w)}
Number of a in string w
2008 Fall 8

9. Right-Linear Grammars
 All productions A xB
have form:
or
Example:
A x
S abS
x : string of terminals
S a including
2008 Fall 9

10. Left-Linear Grammars
All productions have form: A Bx
or
Example:
A x
S Aab
A Aab | B
x : string of terminals
B a including
2008 Fall 10

11. Regular Grammars
2008 Fall 11

12. Regular Grammars
 A regular grammar is any
right-linear or left-linear grammar
 Examples:
G1 G2
S abS S Aab
S a A Aab | B
B a
2008 Fall 12

13. Observation-1
 Regular grammars generate regular languages
 Examples: G2
G1 S Aab
A Aab | B
S abS
B a
S a
2008 Fall 13

14. Observation-2
S abS
G1
S a
Consider string w = abababa:
S abS ababS abababS abababa
a The derivation S * abababa
S b
corresponds to a walk in a FA with label
a
abababa from the state S to a final state.
2008 Fall 14

15. Regular Grammars
Generate
Regular Languages
2008 Fall 15

16. Theorem
Languages
Regular
Generated by
Languages
Regular Grammars
2008 Fall 16

17. Theorem - Part 1
Languages
Regular
Generated by
Languages
Regular Grammars
Any regular grammar generates
a regular language
G: regular grammar, show L(G) is regular
2008 Fall 17

18. Theorem - Part 2
Languages
Regular
Generated by
Languages
Regular Grammars
Any regular language is generated
by some regular grammar
L is regular, show there is a regular
grammar G such that L= L(G).
2008 Fall 18

19. Proof – Part 1
Languages
Regular
Generated by
Languages
Regular Grammars
The language L (G ) is regular if G is
any regular grammar
2008 Fall 19

20. The case of Right-Linear Grammars
 Let G be a right-linear grammar
We will prove: L(G ) is regular
Proof idea:
We will construct NFA M
with L( M ) L(G )
2008 Fall 20

21.  Grammar G is right-linear
Example: S aA | B
A aa B
B b B|a
Note that
V= {S, A, B} in the given regular grammar
2008 Fall 21

22.  Construct NFA M such that
every state is a grammar variable:
initial state
A
special
S VF
final state
B
S aA | B
A aa B
B b B|a
2008 Fall 22

23.  Add edges for each production:
a A
S VF
B
S aA
2008 Fall 23

24. a A
S VF
B
S aA | B
2008 Fall 24

25. A
a a
S a VF
B
S aA | B
A aa B
2008 Fall 25

26. A
a a
S a VF
B
S aA | B
b
A aa B
B
2008 Fall
bB 26

27. A
a a
S a VF
a
S aA | B B
A aa B b
B bB | a
2008 Fall 27

28. NFA M Grammar G
A
a S aA | B
a
A aa B
S a B bB | a
VF
a
B L( M ) L(G )
b L(( aaa )b * a )
2008 Fall 28

29. In General
 A right-linear grammar G
has variables:V0 ,V1,V2 ,
and productions:
Vi a1a2 ...amV j
or
Vi a1a2 ...am
2008 Fall 29

30.  We construct an NFA M such that:
each variableVi corresponds to a node:
V1 V3
V0
VF
V2 special
V4
final state
2008 Fall 30

31.  For each production: Vi a1a2 ...amV j
we add transitions and intermediate nodes from
Vi to Vj with the label a1a2…am
Vi a1 a2 ………
am V
j
2008 Fall 31

32.  For each production: Vi a1a2 ... am
 we add transitions and intermediate nodes from
Vi to VF with the label a1a2…am
a1 a2 ………
am
Vi VF
What if Vi  ?
2008 Fall 32

33.  Resulting NFA M looks like this:
a9
a2 a4
a1 V1 V3
a3 a5
V0
a3 a4
VF
a8 a9
V2 a5
V4
It holds that: L(G ) L( M )
2008 Fall 33

34. The case of Left-Linear Grammars
 Let G be a left-linear grammar
 We will prove: L(G ) is regular
Proof idea:
We will construct a right-linear
grammar G with
R
L(G ) L(G )
2008 Fall 34

35. 
Since G is left-linear grammar
the productions look like:
A Ba1a2 ... ak
A a1a2 ... ak
2008 Fall 35

36.  Construct right-linear grammar G from G by
G:X Y G' : X YR
( Bv )R = vR BR = vR B
Left linear G Right linear G
A → Bv A R
v B
A Ba1a2 ... ak A ak ... a2 a1 B
A v R
A v
A a1a2 ... ak A ak ... a2 a1
2008 Fall 36

37.  It is easy to see that: R
L(G ) L(G ) Hw#8 p.97
 Since G is right-linear, we have:
R L(G )
L(G ) L(G )
Regular Regular Regular
Language Language Language
2008 Fall 37

38. Proof - Part 2
Languages
Regular
Generated by
Languages
Regular Grammars
Any regular language L is generated
by some regular grammar G
2008 Fall 38

39. Any regular language L is generated
by some regular grammar G
Proof idea:
Let M be the NFA with L L(M ).
Construct from M a regular grammar G
such that L ( M ) L (G )
2008 Fall 39

40.  Since L is regular
there is an NFA M such that L L(M )
Example: L L(ab * ab(b * ab)*)
b
M a
a
q0 q1 q2
L L(M ) b
q3
2008 Fall 40

41.  Convert M to a right-linear grammar
b
M
a a
q0 q1 q2
b
q0 aq1
q3
2008 Fall 41

42. b
M a
a
q0 q1 q2
q0 aq1 b
q3
q1 bq1
q1 aq2
2008 Fall 42

43. b
M a
a
q0 q1 q2
q0 aq1
b
q1 bq1
q1 aq2 q3
q2 bq3
2008 Fall 43

44. L(G ) L( M ) L
G b
q0 aq1 M a
a
q0 q1 q2
q1 bq1
q1 aq2 b
q2 bq3 q3
q3 q1
q3 Since q3 is a final state
2008 Fall 44

45. In General, Converting given FA into a
regular grammar:
a
For any transition: q p
Add production: q ap
variable terminal variable
2008 Fall 45

46. For any final state: qf
Add production: qf
2008 Fall 46

47.  Since G is right-linear grammar
G is also a regular grammar
with L(G ) L( M ) L
2008 Fall 47

48. Construct a left-linear grammar
from FA: hw# 9 p.97
 If L={strings with abb as prefix}
From FA of the reverse of L, LR, we can
easily obtain a right linear grammar for LR.
Finally, convert this grammar into a left-
linear one for L.
2008 Fall 48

49. Example
 Sometimes we can construct a regular grammar
by inspecting the corresponding regular
expression.
Example: Find both a left-linear and right-
linear grammar for L(aab*a)?
L(aab*a)* ?
L((ab+a)b*a) ?
L(aa*(ab+a)*)?
2008 Fall 49

50. Example
 Consider language L = { anbm : n+m is even}
Find its right-, left-linear grammar by
(1) inspection from reg. exp. (2) FA.
We have so far introduce three ways to represent a
regular language: Finite State Machine, Regular
Expression, Regular Grammar.
2008 Fall 50