1. Grammars
1

2. Grammars
Grammars express languages
Example: the English language
sentence noun _ phrase predicate
noun _ phrase article noun
predicate verb 2

3. article a
article the
noun boy
noun dog
verb runs
verb walks 3

4. A derivation of “the boy walks”:
sentence noun _ phrase predicate
noun _ phrase verb
article noun verb
the noun verb
the boy verb
the boy walks
4

5. A derivation of “a dog runs”:
sentence noun _ phrase predicate
noun _ phrase verb
article noun verb
a noun verb
a dog verb
a dog runs
5

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”, 6

7. Notation
noun boy
noun dog
Variable Terminal
Production
or
rule
Non-terminal
7

8. Another Example
Grammar: S aSb
S
Derivation of sentence
ab :
S aSb ab
S aSb S
8

9. S aSb
Grammar:S
Derivation of sentence
aabb :
S aSb aaSbb aabb
S aSb S 9

10. S aSb aaSbb aaaSbbb aaabbb
Other derivations:
S aSb aaSbb aaaSbbb
aaaaSbbbb aaaabbbb
10

11. 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” }
11

12. Language of the grammar
S aSb
S
n n
L {a b : n 0}
12

13. More Notation
G V , T , P, S
Grammar
V : A finiteSet of variables
T : A finiteSet of terminal symbols
S : Start variable
P: Set of Production rules
13
V and T are assumed to be disjoint

14. Example
G S aSb
Grammar : S
G V , T , P, S
V {S } T {a, b}
P {S aSb, S
14
}

15. More Notation
Sentential Form:
A sentence that contains
variables and terminals
Example:
S aSb aaSbb aaaSbbb aaabbb
Sentential Forms sentence
15

16. *
S aaabbb
We write:
Instead of:
S aSb aaSbb aaaSbbb aaabbb
16

17. *
w1 wn
In general we write:
w1 w2 w3  wn
If:
17

18. *
w w
By default:
18

19. Example
Grammar Derivations
S aSb *
S
S
*
S ab
*
S aabb
*
S aaabbb
19

20. Example
Grammar Derivations
S aSb
S aaSbb
S
aaSbb aaaaaSbbbbb
20

21. Another Grammar Example
G S Ab
Grammar : A aAb
A
Derivations:
S Ab aAbb abb
S Ab aAbb aaAbbb aabbb
21

22. Example
Grammar Derivations
S aSb
S aaSbb
S
aaSbb aaaaaSbbbbb
22

23. More Derivations
S Ab aAbb aaAbbb aaaAbbbb
aaaaAbbbbb aaaabbbbb
S aaaabbbbb
S aaaaaabbbbbbb
n n
S a b b 23

24. Language of a Grammar
For a grammar G
with start variable S :
L(G ) {w : S w}
String of terminals
24

25. Example
G S Ab
For grammar : A aAb
A
n n
L(G ) {a b b : n 0}
n n
Since: S a b b
25

26. A Convenient Notation
A aAb
A aAb |
A
article a
article a | the
article the
26

27. Linear Grammars
27

28. Linear Grammars
Grammars with
at most one variable at the right side
of a production
S aSb S Ab
Examples: S A aAb
A
28

29. A Non-Linear Grammar
Grammar G : S SS
S
Why S SS
S aSb
What will happen if
S bSa
S S instead
L(G) {w : na ( w) nb ( w)}
Number of a in string w 29

30. Another Linear Grammar
G S A
Grammar : A aB |
B Ab
n n
L(G ) {a b : n 0}
30

31. Right-Linear Grammars
All productions have form: A xB
A x
string of
Example: terminals
S abS
S a
31

32. Left-Linear Grammars
A Bx
All productions have form: or
A x
string of
S Aab terminals
Example:
A Aab | B
B a 32

33. Regular Grammars
33

34. 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 34

35. Observation
Regular grammars generate regular
languages G2
G1
Examples: S Aab
S abS A Aab | B
S a B a
L(G1) (ab) * a L(G2 ) aab(ab) *
35

36. Regular Grammars
Generate
Regular Languages
36

37. Theorem
Languages
Regular
Generated by
Languages
Regular Grammars
37

38. Theorem - Part 1
Languages
Regular
Generated by
Languages
Regular Grammars
Any regular grammar generates
a regular language
38

39. Theorem - Part 2
Languages
Regular
Generated by
Languages
Regular Grammars
Any regular language is generated
by a regular grammar
39

40. Proof – Part 1
Languages
Regular
Generated by
Languages
Regular Grammars
The language L (G ) generated by
any regular grammar G is regular
40

41. The case of Right-Linear
Grammars
Let G be a right-linear grammar
We will prove: L(G ) is regular
L( M ) L(G )
Proof idea: We will construct NFA M
41
with L(M)=L(G)

42. Grammar G is right-linear
Example:
S aA | B
A aa B
B b B|a
42

43. Construct NFA M such that
every state is a grammar variable:
A special
S VF
final state
B
S aA | B
A aa B
B b B|a 43

44. Add edges for each production:
a A
S VF
B
S aA
44

45. a A
S VF
B
S aA | B
45

46. A
a a
S a VF
B
S aA | B
A aa B
46

47. A
a a
S a VF
B
S aA | B
b
A aa B
B bB 47

48. A
a a
S a VF
a
B
S aA | B
b
A aa B
B bB | a 48

49. A
a a
S a VF
a
B
b
S aA aaaB aaabB aaaba
49

50. NFA M Grammar
A G
a S aA | B
a
A aa B
S a B bB | a
VF
a
B
L( M ) L(G )
b
aaab * a b * a
50

51. In General
A right-linear grammar G
has variables: V0 ,V1,V2 ,
and productions: Vi a1a2 amV j
or
Vi a1a2 am
51

52. We construct the NFA M such that:
each variable Vicorresponds to a
node: V1 V3
V0
VF
V2 special
V4
final state
52

53. Vi a1a2 amV j
For each production:
we add transitions and intermediate
nodes
Vi a1 a2 ………
am Vj
53

54. For each production:
Vi a1a2 am
we add transitions and intermediate
nodes
Vi a1 a2 ………
am
VF
54

55. 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 ) 55

56. 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 )
56

57. Since G is left-linear grammar
the productions look like:
A Ba1a2 ak
A a1a2 ak
57

58. Left G
A Ba1a2 ak
linear
Construct right-linear grammar G
A Bv
Right
G A ak a2a1B
linear
R
A v B 58

59. Construct right-linear grammar G
Left
G A a1a2 ak
linear
A v
Right
G A ak a2a1
linear
R
A v 59

60. R
L(G ) L(G )
It is easy to see that:
G
Since is right-linear, we have:
R L(G )
L(G ) L(G )
Regular Regular Regular
Language Language Language
60

61. Proof - Part 2
Languages
Regular
Generated by
Languages
Regular Grammars
Any regular language L is generated
by some regular grammar G
61

62. 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 )
62

63. Since L is regular
there is an NFA M such that L L(M )
b
Example:
M a
a
q0 q1 q2
b
L ab * ab(b * ab) * q3
L L(M ) 63

64. Convert M to a right-linear grammar
b
M a
a
q0 q1 q2
b
q0 aq1
q3
64

65. b
M a
a
q0 q1 q2
q0 aq1 b
q1 bq1 q3
q1 aq2
65

66. b
M a
a
q0 q1 q2
q0 aq1
q1 bq1 b
q1 aq2 q3
q2 bq3
66

67. 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 67

68. In General
a
For any transition: q p
Add production: q ap
variable terminal variable
68

69. For any final state: qf
Add production: qf
69

70. Since G is right-linear grammar
G is also a regular grammar
with L(G ) L( M ) L
70

71. For any regular language one can
construct left linear as well as right
linear grammar.
71

72. A
a a
a VF
S a F → Ba
B B → Bb here F is
start symbol
S aA | B B → Aaa
b B→S
A aa B A → Sa
B bB | a S→ this is new rule
72
as S is now a final state

73. A → aB D → Cb
B → bB D→B
C→B a
B → aC
B →B b Now D is the
C→b D start variable
D→ B B → Aa
A→
D→
73

74. Language generated by both
aaab* a +b*a
74

75. 75

76. 76

77. 77

78. Context-Free Languages
78

79. Context-Free Languages
n n R
{a b } {ww }
Regular Languages
79

80. Context-Free Languages
Context-Free Pushdown
Grammars Automata
stack
automaton
80

81. Context-Free Grammars
81

82. Example
A context-free grammar G: S aSb
S
A derivation:
S aSb aaSbb aabb
82

83. S aSb
S
L(G ) n n
{a b : n 0}
(((( ))))
83

84. A context-free grammar G: S aSa
S bSb
S
Another derivation:
S aSa abSba abaSaba abaaba
84

85. S aSa
S bSb
S
L(G ) R
{ww : w {a, b}*}
85

86. Example
A context-free grammar G: S aSb
S SS
S
A derivation:
S SS aSbS abS ab
86

87. A context-free grammar G: S aSb
S SS
S
A derivation:
S SS aSbS abS abaSb abab
87

88. S aSb
S SS
S
L(G ) {w : na ( w) nb ( w),
and na (v) nb (v)
in any prefix v}
() ((( ))) (( )) 88

89. Definition: Context-Free
Grammars
Grammar
G (V , T , P, S )
Variables Terminal Start
symbols variable
Productions of the form:
A x
x is string of variables and terminals 89

90. Definition: Context-Free
Languages
A language L is context-free
if and only if
there is a C.F.grammar Gwith
L L(G )
90

91. Derivation Order
1. S AB 2. A aaA 4. B Bb
3. A 5. B
Leftmost derivation:
1 2 3 4 5
S AB aaAB aaB aaBb aab
Rightmost derivation:
1 4 5 2 3
S AB ABb Ab aaAb aab
91

92. S aAB
A bBb
B A|
Leftmost derivation:
S aAB abBbB abAbB abbBbbB
abbbbB abbbb
Rightmost derivation:
S aAB aA abBb abAb
abbBbb abbbb 92

93. Derivation Trees
93

94. Def:
G =(V,T,P,S)
An ordered tree is a derivation tree for G iff
1 The root is labeled S
2 every leaf has a label from T { }
3 Every interior vertex has a label from V
4 If a vertex has label A(variable) and its children
are labeled from (L to R)
a1,a2,…an then P must contain a production
A a1 a2 …an
5 A leaf labeled has no sibling.
94

95. Yield:
The string of terminals obtained by
reading the leaves of the tree from left
to right omitting any ’s encountered
is called yield of the tree.
95

96. Partial derivation tree
A tree that has properties 3,4,5 but 1
need not
And 2 is replaced by
V T { }
96

97. S AB A aaA | B Bb |
S AB
S
A B
97

98. S AB A aaA | B Bb |
S AB aaAB
S
A B
a a A
98

99. S AB A aaA | B Bb |
S AB aaAB aaABb
S
A B
a a A B b
99

100. S AB A aaA | B Bb |
S AB aaAB aaABb aaBb
S
A B
a a A B b
100

101. S AB A aaA | B Bb |
S AB aaAB aaABb aaBb aab
Derivation Tree S
A B
a a A B b
101

102. S AB A aaA | B Bb |
S AB aaAB aaABb aaBb aab
Derivation Tree S
A B
yield
a a A B b aa b
aab
102

103. Partial Derivation Trees
S AB A aaA | B Bb |
S AB
Partial derivation tree S
A B
103

104. S AB aaAB
Partial derivation tree S
A B
a a A
104

105. sentential
S AB aaAB
form
Partial derivation tree S
A B
yield
a a A
aaAB
105

106. Sometimes, derivation order doesn’t matter
Leftmost:
S AB aaAB aaB aaBb aab
Rightmost:
S AB ABb Ab aaAb aab
S
Same derivation tree
A B
a a A B b
106

107. Ambiguity
107

108. E E E | E E | (E) | a
a a a
E E E E a E a E E
a a E a a*a
E E
leftmost derivation
a E E
a a 108

109. E E E | E E | (E) | a
a a a
E E E E E E a E E E
a a E a a a
E E
leftmost derivation
E E a
a 109 a

110. E E E | E E | (E) | a
a a a
Two derivation trees
E E
E E E E
a E E E E a
a a a 110 a

111. The grammarE E E | E E | (E) | a
is ambiguous:
string a a a has two derivation trees
E E
E E E E
a E E E E a
a a a a
111

112. The grammarE E E | E E | (E) | a
is ambiguous:
string a a a has two leftmost derivations
E E E a E a E E
a a E a a*a
E E E E E E a E E
a a E a a a 112

113. Definition:
A context-free grammar G is ambiguous
if some string w L(G ) has:
two or more derivation trees
113

114. In other words:
A context-free grammar G is ambiguous
if some string w L(G ) has:
two or more leftmost derivations
(or rightmost)
114

115. Why do we care about ambiguity?
a a a
take a 2
E E
E E E E
a E E E E a
a a a 115 a

116. 2 2 2
E E
E E E E
2 E E E E 2
2 2 2 116 2

117. 2 2 2 6 2 2 2 8
6 8
E E
2 4 4 2
E E E E
2 2 2 2
2 E E E E 2
2 2 2 117 2

118. Correct result: 2 2 2 6
6
E
2 4
E E
2 2
2 E E
2 2 118

119. • Ambiguity is bad for programming languages
• We want to remove ambiguity
119

120. We fix the ambiguous grammar:
E E E | E E | (E) | a
New non-ambiguous grammar: E E T
E T
T T F
T F
F (E)
120
F a

121. E E T T T F T a T a T F
a F F a a F a a a
E a a a
E E T
E T
E T
T T F T T F
T F
F F a
F (E)
F a a a
121

122. Unique derivation tree
E a a a
E T
T T F
F F a
a a
122

123. The grammar G: E E T
E T
T T F
T F
F (E)
F a
is non-ambiguous:
Every string w L (G ) has
a unique derivation tree 123

124. Inherent Ambiguity
Some context free languages
have only ambiguous grammars
Example: n n m n m m
L {a b c } {a b c }
S S1 | S2 S1 S1c | A S2 aS2 | B
A aAb | B bBc |
124

125. n n n
The string a b c
has two derivation trees
S S
S1 S2
S1 c a S2
125

126. Compiler
Lexical
parser
analyzer
input output
machine
program
code
126

127. A parser knows the grammar
of the programming language
127

128. The parser finds the derivation
of a particular input
derivation
Parser
input E => E + E
E -> E + E
=> E + E * E
10 + 2 * 5 |E*E
=> 10 + E*E
| INT
=> 10 + 2 * E
=> 10 + 2 * 5
128

129. derivation tree
derivation
E
E => E + E E + E
=> E + E * E
=> 10 + E*E 10
E * E
=> 10 + 2 * E
=> 10 + 2 * 5 2 5
129