Ambiguous & Unambiguous Grammar

Topic: Ambiguous grammar & parse Tree
COURSE TITLE: THEORY OF COMPUTING
COURSE CODE: SE-234
1
Department Of Software Engineering

Ambiguous & Unambiguous Grammar
Depending on Number of Derivation Trees,
CFGs are sub-divided into 2 types:
 Ambiguous Grammars
 Unambiguous Grammars
1/3/2020Ambiguous Grammar & Parse Tree
2
The Context Free Grammars(CFGs) are
classified based on:
 Number of Derivation trees
 Number of strings

What is Ambiguous Grammar?
3
A grammar is said to be Ambiguous if
there exists two or more derivation &
parse tree for a string w.
Figure: 1
s a
s
s
s
a s
aa
a

What is Unambiguous Grammar?
4
A grammar can be unambiguous if the grammar does not contain more than one
leftmost derivation or more than one rightmost derivation or more than one parse
tree for the given input string.
b x
b
x
x
b

5
G = (V,T,P,S) is a CFG is said to be ambiguous if and only if
there exist a string in T* that has more than one parse tree.
where,
• V = finite set of variables.
• T = finite set of terminals.
• P = finite set of productions of the form, A → α, where A is
a variable and α ∈ (V ∪ T)*
• S = designated variable called the start symbol.
Definition of ambiguous & unambiguous
grammar:

6Example - 1
Check the grammar G with production rules −
X → X+X | X*X |X| a
is ambiguous or not?
Let’s find out the derivation tree for the string "a+a*a". It has two leftmost
derivations.
Derivation – 1
X → X*X
→ X+X*X [X→X+X]
→ a+X*X [X →X*X]
→ a+a*X [X→a]
→ a+a*a
Derivation – 2
X → X+X
→ a+X [X→a]
→ a+X*X [X →X*X]
→ a+a*X [X→a]
→ a+a*a

7
Figure: 3Figure: 2
Parse Tree - 1 Parse Tree - 2
Since there are two different parse trees for a single
string "a+a*a",
So the grammar G is ambiguous.
x +
x
x
x
a x
aa
*
x *
x
x
x
+x
a a
a

8Example - 2
The grammar G is with production rules-
S → T
T → 01
T → T T
From the above grammar, Check the ambiguity by the derivation &
parse tree for string “010101”.
Derivation – 1
S → T
→ TT [T→TT]
→ TTT [T→TT]
→ 01TT [T→01]
→ 0101T [T→01]
→ 010101 [T→01]
Derivation – 2
S → T
→ TT [T→TT]
→ TTT [T→TT]
→ TT01 [T→01]
→ T0101 [T→01]
→ 010101 [T→01]

9
Figure: 3Figure: 2
Parse Tree - 1 Parse Tree - 2
S
T
T T
T
T
0 1 0 1
T
0 1
S
T
T T
T
T
0 1 0 1
T
0 1
The grammar has two or more derivation & parse tree. So this is also an Ambiguous.

10
Example - 3
Given a context-free grammar G with start symbol S, terminal symbols T and
productions P, the language L(G) that G generates is defined to be the set of strings
of terminal symbols that can be obtained by derivation from S using the productions
P, i.e., the set {w ∈ T ∗ | S ⇒ w}.
T → R
T → aTc
R → ∈
R → RbR
Check that the grammar is Ambiguous or Unambiguous for the string ‘aabbbcc’ ?

11
T ⇒ aTc
⇒ aaTcc [T ⇒ aTc]
⇒ aaRcc [T ⇒ R]
⇒ aaRbRcc [R ⇒ RbR]
⇒ aaRbRbRcc [R ⇒ RbR]
⇒ aaRbRbRbRcc [R ⇒ RbR]
⇒ aabRbRbRcc [R ⇒ ∈]
⇒ aabbRbRcc [R ⇒ ∈]
⇒ aabbbRcc [R ⇒ ∈]
⇒ aabbbcc [R ⇒ ∈]
Leftmost Derivation of the string ‘aabbbcc’
T ⇒ aTc
⇒ aaTcc [T ⇒ aTc]
⇒ aaRcc [T ⇒ R]
⇒ aaRbRcc [R ⇒ RbR]
⇒ aaRbRbRcc [R ⇒ RbR]
⇒ aaRbRbRbRcc [R ⇒ RbR]
⇒ aaRbRbRbcc [R ⇒ ∈]
⇒ aaRbRbbcc [R ⇒ ∈]
⇒ aaRbbbcc [R ⇒ ∈]
⇒ aabbbcc [R ⇒ ∈]
Rightmost derivation of the string ‘aabbbcc’

12
b R
b
T
R R
R
∈
R
c
a
a T
T c
b R
∈
∈∈
b
R
b
T
R R
R
∈
R
c
a
a T
T c
∈
R
b
∈
∈
Parse Tree-1 Parse Tree-2
R
R

13
Example - 4
Consider VN = {S, E}, VT = {if, then, else} and a grammar G = (VT,VN, S, P) such that the
following grammar
S → if E then S | if E then S else S
Are productions of G. Then the string
W → if E then if E then S else S
Now check the ambiguity.
Derivation – 1
S → if E then S else S
→ if E then if E then S else S
Derivation – 2
S → if E then S
→ if E then if E then S else S

14S
Eif then S else S
if E Sthen
S
Eif then S
else Sif E Sthen
Parse Tree-1
Parse Tree-2
The string has two derivation trees. So it is ambiguous.

15
Consider the following grammar- S → aS | ∈
The language generated by this grammar-
L = { an , n>=0 } or a*
Example - 5
Let us consider a string w = “aaa”.
First Derivation-
S → aS
→ aaS [S → aS]
→ aaaS [S → aS]
→ aaa∈ [S → ∈ ]
→ aaa
Second Derivation-
S → aS
→ aaS [S → aS]
→ aaaS [S → aS]
→ aaa∈ [S → ∈]
→ aaa
Figure - 4
Parse Tree
S
Sa
a S
Sa
∈
 Here first derivation and second derivation are exactly same.
So, this is an example of an unambiguous grammar.

16Example - 6
The following Grammar generates prefix expressions with
operands x and y and binary operators +, -, and *:
E → +EE | *EE | -EE | x | y
Now find leftmost and rightmost derivations and derivation
tree for the string ‘+*-xyxy’.
+ E E
E
y
* E E
EE-
yx
x
Parse Tree:
E → +EE
→ +*EEE [E → *EE]
→+*-EEEE [E → -EE]
→+*-xEEE [E → x]
→+*-xyEE [E → y]
→+*-xyxE [E → x]
→+*-xyxy [E → y]
Left Derrivation:
In this grammar, there is no more derivation and parse tree.
So this is also unambiguous.

17Example - 7
Consider the given grammar
S → S+S | S-S | S*S | S/S | (S) | x | y | z
is the productions of G. Then for the string W → (x+y)*x-z*y/(x+x) ; check the ambiguity.
S → S*S
→ (S)*S
→ (S+S)*S
→ (x+y)*S
→ (x+y)*S-S
→ (x+y)*x-S
→ (x+y)*x-S*S
→ (x+y)*x-z*S
→ (x+y)*x-z*S/S
→ (x+y)*x-z*y/S
→ (x+y)*x-z*y/(S)
→ (x+y)*x-z*y/(S+S)
→ (x+y)*x-z*y/(x+x)
S → S/S
→ S/(S)
→ S/(S+S)
→ S/(x+x)
→ S*S/(x+x)
→ S*y/(x+x)
→ S-S*y/(x+x)
→ S-z*y/(x+x)
→ S*S-z*y/(x+x)
→ S*x-z*y/(x+x)
→ (S)*x-z*y/(x+x)
→ (S+S)*x-z*y/(x+x)
→ (x+y)*x-z*y/(x+x)
Derivation – 1 Derivation – 2
In this grammar, there are more than two derivation and parse tree. So this is an ambiguous.

18
Reference from:
 Basics of Compiler Design - by Torben Ægidius Mogensen
 Introduction to the Theory of Computation_3rd Edition – by Michael Sipser
 Introduction to Automata Theory, Language, and Computation – by John E. Hopcroft
 https://www.javatpoint.com/automata-ambiguity-in-grammar
 https://www.geeksforgeeks.org/ambiguous-grammar/

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