String and Regular Expression

CS 3133 Foundations of Computer Science
C term 2014

Solutions of the Sample Problems for the Midterm
Exam
1. Give a regular expression that represents the set of strings over Σ =
{a, b} that contain the substring ab and the substring ba.
Solution:
a+ b+ a(a ∪ b)∗ ∪ b+ a+ b(a ∪ b)∗
(20 points)
2. Consider the following grammar G:
S → SAB|λ
A → aA|a
B → bB|λ
(a) Give a leftmost derivation of abbaab.
(b) Build the derivation tree for the derivation in part (1).
(c) What is L(G)?
Solution:

1

(a) The following is a leftmost derivation of abbaab:
S

⇒ SAB
⇒ SABAB
⇒ ABAB
⇒ aBAB
⇒ abBAB
⇒ abbBAB
⇒ abbAB
⇒ abbaAB
⇒ abbaaB
⇒ abbaabB
⇒ abbaab

(b)
S

A

S

S

A

B a a

B

b

A

b

B

B a b

B

(c)
L(G) = a(a ∪ b)∗ ∪ λ
(20 points)
3. Construct a regular grammar over the alphabet Σ = {a, b, c, d} whose language is the set of strings that contain exactly two b-s.
Solution:
The following is a regular grammar over {a, b, c, d} whose language is the set of strings containing exactly two b-s:
S → aS | cS | dS | bB
B → aB | cB | dB | bC
C → aC | cC | dC | λ
2

(20 points)
4. Consider the following grammar G:
S → aSA|λ
A → bA|λ
(a) Give a regular expression for L(G).
(b) Is G ambiguous? Explain your answer.
Solution:
(a) The following is a regular expression for L(G): a+ b∗ ∪ λ
(b) Yes the grammar is ambiguous. Here are two different leftmost derviations for the string aabb:
S

and

⇒ aSA
⇒ aaSAA
⇒ aaAA
⇒ aabAA
⇒ aabbAA
⇒ aabbA
⇒ aabb

S

⇒ aSA
⇒ aaSAA
⇒ aaAA
⇒ aaA
⇒ aabA
⇒ aabbA
⇒ aabb

(20 points)
5. Design a DFA that accepts the language consisting of the set of those strings over {a, b, c} in which the number of a’s plus the number of b’s plus twice the number of c’s is divisible by six.
3

Solution:
The state diagram of a DFA is

a,b q 1

q
2

a,b

a,b

c c c

c

c

q0

q3 c a,b q 5

a,b

(20 points)

4

q

a,b