This slide contains, 1) Some terminologies like yields, derives, word, derivation 2) Leftmost and Rightmost derivation 3) Ambiguity checking 4) Parse tree generation and ambiguity checking

1. Theory of Computation
CSE 2233
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

2. CFG – Context-free Grammar
2
A Context-free Grammar is a 4 tuple (V, 𝛴, R, S), where
▸ V is a finite set called the variables
▸ Σ is a finite set, disjoint from V, called the terminals
▸ R is a finite set of rules, with each rule being a variable and a string of variables and terminals
▸ S ϵ V is the start variable
Example:
Grammar, G1 = ( { S }, { a, b }, R, S )
V = { S }
Σ = { a, b }
R is,
S → aSb | SS | ε
Example:
Grammar, G2 = ( V, Σ, R, <EXPR> )
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R is,
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

3. CFL – Context-free Language
3
Context-free Language:
– The collection of languages generated by some context-free grammars are called Context-free Languages.
– They include all the regular languages and many additional(recursive structure) languages.
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

4. CFG >> Terminologies – yields, derives, derivation
4
▸ If u, v, w are strings of variables and terminals, and A → w is a rule of the grammar, we say that
uAv ⟹ uwv i.e. uAv yields uwv
▸ We say u derives v, written as
- if u = v or
- if a sequence u1, u2, u3, … … … , uk exists for k >= 0 and u ⟹ u1 ⟹ u2 ⟹ u3 ⟹ … … ⟹ uk ⟹ v
▸ A word is a string of terminals and the language of the grammar is { w ∈ 𝛴* | S =*> w }
▸ A derivation of a word w from a CFG G = ( V, 𝛴, R, S ) is a sequence of strings over V U 𝛴 ,
S = s0 => s1 => s2 => … … … => sk = w
where
- s0 = S is the start variable of G
- For each 1 <= i <= k,
si is obtained by activating a single production rule of G on one of the variables of si-1
vu
*

Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

5. CFG >> Derivation
5
Derivation – you can apply rules to any variables, no order is maintained. [single rules at a time]
For example, a + a x a
<EXPR>
⇒ <EXPR> + <TERM>
⇒ <TERM> + <TERM>
⇒ <TERM> + <TERM> x <FACTOR>
⇒ <TERM> + <TERM> x a
⇒ <FACTOR>+ <TERM> x a
⇒ a + <TERM> x a
⇒ a + <FACTOR> x a
⇒ a + a x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}

6. CFG >> Leftmost Derivation
6
Leftmost Derivation – A derivation of a string w in a grammar G is a leftmost derivation
if at every step the leftmost remaining variable is the one replaced.
For example, a + a x a
<EXPR>
⇒ <EXPR> + <TERM>
⇒ <TERM> + <TERM>
⇒ <FACTOR> + <TERM>
⇒ a + <TERM>
⇒ a + <TERM> x <FACTOR>
⇒ a + <FACTOR> x <FACTOR>
⇒ a + a x <FACTOR>
⇒ a + a x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}

7. CFG >> Rightmost Derivation
7
Rightmost Derivation – A derivation of a string w in a grammar G is a rightmost derivation
if at every step the rightmost remaining variable is the one replaced.
For example, a + a x a
<EXPR>
⇒ <EXPR> + <TERM>
⇒ <EXPR> + <TERM> x <FACTOR>
⇒ <EXPR> + <TERM> x a
⇒ <EXPR> + <FACTOR> x a
⇒ <EXPR> + a X a
⇒ <TERM> + a X a
⇒ <FACTOR> + a X a
⇒ a + a X a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}

8. CFG >> Leftmost vs Rightmost Derivation
8
G = ( V, Σ, R, S )
Where,
V = { S }
Σ = { (, ) }
R={
S → SS | ( S ) | 𝜀
}
Leftmost derivation of “( ) ( )”: S ⇒ SS ⇒ ( S ) S ⇒ ( ) S ⇒ ( ) ( S ) ⇒ ( ) ( )
Rightmost derivation of “( ) ( )”: S => SS => S ( S ) => S ( ) => ( S ) ( ) => ( ) ( )
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

9. CFG >> Parse Tree
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Root – must be labeled by the start variable
Leaves – labeled by a terminal or 𝜀
Interior nodes – labeled by a variable
Yield – the concatenation of the labels of the leaves in left to right order
For example, S → SS | ( S ) | ( )
yields ( ( ) ) ( )
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

10. CFG >> Example 1
10
▸ Let, CFG G = ( V, Σ, R, <EXPR> )
where V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}
▸ Derive the following strings
- a + a x a
- ( a + a ) x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

11. CFG >> “a+ a x a” Leftmost Derivation
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<EXPR>
⇒ <EXPR> + <TERM>
⇒ <TERM> + <TERM>
⇒ <FACTOR> + <TERM>
⇒ a + <TERM>
⇒ a + <TERM> x <FACTOR>
⇒ a + <FACTOR> x <FACTOR>
⇒ a + a x <FACTOR>
⇒ a + a x a
So, <EXPR> derives a + a x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
Practice the rightmost derivation.
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}

12. CFG >> “a+ a x a” Parse Tree
12
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}
Is there any leftmost/rightmost version of parse tree??

13. CFG >> “( a+ a ) x a” Leftmost Derivation
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<EXPR>
⇒ <TERM>
⇒ <TERM> x <FACTOR>
⇒ <FACTOR> x <FACTOR>
⇒ ( <EXPR> ) x <FACTOR>
⇒ ( <EXPR> + <TERM> ) x <FACTOR>
⇒ ( <TERM> + <TERM> ) x <FACTOR>
⇒ ( <FACTOR> + <TERM> ) x <FACTOR>
⇒ ( a + <TERM> ) x <FACTOR>
⇒ ( a + <FACTOR> ) x <FACTOR>
⇒ ( a + a ) x <FACTOR>
⇒ ( a + a ) x a
So, <EXPR> derives ( a + a ) x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}

14. CFG >> “( a+ a ) x a” Parse Tree
14
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a
}

15. CFG >> Practice 1
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▸ Let, CFG G = ( V, Σ, R, <EXPR> )
where V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, b, +, x, (, ) }
R={
<EXPR> → <EXPR> + <TERM> | <TERM>
<TERM> → <TERM> x <FACTOR> | <FACTOR>
<FACTOR> → ( <EXPR> ) | a | b
}
▸ Derive the following strings:
- ( a + b ) x a
- a + b x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

16. CFG >> Practice 2
16
▸ Let us consider a CFG with the following production rules,
<SENTENCE> → <NOUN-PHRASE> <VERB-PHRASE>
<NOUN-PHRASE> → <CMPLX-NOUN> | <CMPLX-NOUN> <PREP-PHRASE>
<VERB-PHRASE> → <CMPLX-VERB> | <CMPLX-VERB> <PREP-PHRASE>
<PREP-PHRASE> → <PREP> <CMPLX-NOUN>
<CMPLX-NOUN> → <ARTICLE> <NOUN>
<CMPLX-VERB> → <VERB> | <VERB> <NOUN-PHRASE>
<ARTICLE> → a | the
<NOUN> → boy | girl | flower
<VERB> → touches | likes | sees
<PREP> → with
▸ Now derive the following strings:
- a boy sees
- The boy sees a flower
- a girl with a flower likes the boy
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

17. CFG >> Ambiguity
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▸ Grammar G is ambiguous if it generates some string ambiguously.
▸ A string w is derived ambiguously in context-free grammar G if it has two or more different leftmost derivations.
▸ A Context Free Grammar G = ( V, 𝛴, R, S ) is ambiguous if there is some string w ∈ 𝛴* such that there are two distinct parse
trees T1 and T2 having S at the root and having yield w.
[ Note: Two parse trees are equal if they are equal as trees and if all productions corresponding to inner nodes are also equal ]
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

18. CFG >> Ambiguity – Example 1
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CFG G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR> }
Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <EXPR> | <EXPR> X <EXPR> | ( <EXPR> ) | a
}
Now derive the string : a + a x a
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

19. CFG >> Ambiguity – a + a x a Derivation
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CFG G = ( V, Σ, R, <EXPR> ) where, V = { <EXPR> } and Σ = { a, +, x, (, ) }
R={
<EXPR> → <EXPR> + <EXPR> | <EXPR> X <EXPR> | ( <EXPR> ) | a
}
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
Parse Tree 1 Parse Tree 2

20. CFG >> Ambiguity – Practice 1
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CFG G = ( V, Σ, R, <EXPR> )
Where,
V = { <EXPR>, <TERM>, <FACTOR> }
Σ = { a, b, +, x }
R={
<EXPR> → <EXPR> + <EXPR> | <EXPR> X <EXPR> | a | b
}
Does the grammar generate the string “a + b x a” ambiguously???
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

21. CFG >> Ambiguity – Practice 2
21
CFG G = ( V, Σ, R, S )
Where,
V = { S }
Σ = { (, ) }
R={
S → SS | ( S ) | ( )
}
Does the grammar generate the string “( )( )( )” ambiguously???
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

22. CFG >> Ambiguity – Practice 3
22
CFG G = ( V, Σ, R, S )
Where,
V = { S, A, B, C, D }
Σ = { 0, 1, 2 }
R={
S → AB | CD
A → 0A1 | 01
B → 2B | 2
C → 0C | 0
D → 1D2 | 12
}
Does the grammar generate the string “012” or, “001122” ambiguously???
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU

23. a)
S → AB | C
A → aAb | ab
B → cBd | cd
C → aCd | aD
D → bDc | bc
b)
S → ABA
A → Aa | eps
B → bB | eps
CFG >> Ambiguity – Practice 4
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Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
▪ Can the string ‘aabbccdd’ be derived from the above CFG? If yes, show the
leftmost and rightmost derivation.
▪ Is the leftmost derivation unique? If not, show the other leftmost derivation.
▪ Is the grammar ambiguous? Justify your answer with a suitable example.

24. a)
E → E + T | T
T → T x F | F
F → (E) | a
b)
S → S + S | S * S | A | B
A → aA | 1
B → bB | 2
CFG >> Ambiguity – Practice 5
24
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
Now give parse trees and elft-most derivations for each of the following:
▪ a + a
▪ a x a + a
▪ ((a))
▪ Show a leftmost derivation of the string: aa1 + bb2 * a1
▪ Show whether the string, bbb2 + aa1 + b2, makes the grammar ambiguous

25. a)
E → E + E | E – E | ( E ) | V
V → x | y | z | A
A → A * A | A % A | C
C → 0 | 1
b)
E → E + E | E * E | ( E ) | N
N → 0N | 1N | 0 | 1
CFG >> Ambiguity – Practice 6
25
Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
▪ Find out if the following grammar is ambiguous or not, using the string
x+(0*1%0).
▪ If the grammar is ambiguous, show two different parse trees for this string.
▪ Show a leftmost derivation and rightmost derivation for the string:
( 1 + 10 ) * 1

26. a)
E → E + E | E * E | ( E ) | a | b | c
b)
S → WX | Y
W → aWb | ab
X → cXd | cd
Y → aYd | aZc
Z → aZb | bb
CFG >> Ambiguity – Practice 7
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Mohammad Imam Hossain | Lecturer, Dept. of CSE | UIU
▪ Show the rightmost derivation and the leftmost derivation for the input string
a + ( b + c ) * c
▪ Determine whether the following CFG is ambiguous or not with the help of
following input string ‘aabbcd’

27. 27
References:
Chapter 2(section 2.1), Introduction to the Theory of Computation, 3rd Edition by Michael Sipser
THANKS!
Any questions?
You can find me at imam@cse.uiu.ac.bd