Assignment 2
1. Write down the definition of Regular Language with suitable example.
2. Determine the following claims are true or false for all regular expressions 𝑟1 and 𝑟2. The symbol
≡ stands for equivalence.
a. (𝑟1
∗)∗≡ 𝑟1
∗
b. (𝑟1
∗𝑟2
∗)∗≡ (𝑟1+ 𝑟2)∗
c. (𝑟1
∗𝑟2
∗)∗≡(𝑟1𝑟2)∗
3. Give the regular expression of following languages with Σ = {𝑎, 𝑏}:
a. 𝐿1= {𝑎𝑚𝑏𝑛|𝑚 ≤ 4 𝑎𝑛𝑑 𝑛 ≥ 1}
b. 𝐿2= {𝑎𝑚𝑏𝑛|(𝑚 + 𝑛) 𝑖𝑠 𝑜𝑑𝑑}
c. 𝐿3= {𝑎𝑚𝑏𝑛|𝑚 ≥ 3 𝑎𝑛𝑑 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛}
d. 𝐿1
e. 𝐿2
f. 𝐿3
g. 𝐿4= {𝑣𝑤𝑣|𝑣, 𝑤𝜖{𝑎 , 𝑏}∗ 𝑎𝑛𝑑 |𝑣|≤ 2}
h. 𝐿5= {𝑤|𝑤𝜖{𝑎, 𝑏 }∗ 𝑎𝑛𝑑 |𝑤|𝑚𝑜𝑑3 = 0}
i. 𝐿5
4. Give the regular expression of the following languages with Σ = {𝑎, 𝑏, 𝑐}:
a. All strings having exactly two a’s.
b. All strings having at least two a’s.
c. All strings having at most two a’s.
d. All strings having at least one occurrence of each alphabet.
e. All strings in which all runs of a’s lengths that are multiples of three.
5. Give the regular expression of following languages with Σ = {0, 1}:
a. All strings ending with 11.
b. All strings not ending with 11.
c. All strings, in which first two and last two symbols are same and the length of string is
more than 3.
d. All strings with even number of 1’s.
e. All strings having two times occurrence of 00. 000 is also counted as two time
occurrence of 00.
6. Construct the Generalized Transition Graph (GTG) and find the regular expression of the
following languages:
a. 𝐿 = {𝑤𝜖{0,1}∗|𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓(0)= 2𝑛 + 1 𝑎𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓(1)= 2𝑚 + 1 𝑤ℎ𝑒𝑟𝑒 𝑛, 𝑚 ≥
0}
b. 𝐿 = {𝑤𝜖{0,1}∗|𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓(0)= 2𝑛 𝑎𝑛𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓(1)= 2𝑚 𝑤ℎ𝑒𝑟𝑒 𝑛, 𝑚 ≥ 0}
7. Proof that if 𝐿 is a regular language then there exist a regular expression 𝑟 such that 𝐿 = 𝐿(𝑟).
8. Construct the NFA that accepts the following languages on alphabets Σ = {𝑎, 𝑏 }:
a. 𝐿(𝑎∗+ 𝑎∗(𝑎 + 𝑏)𝑐∗)
b. 𝐿(𝑎𝑏∗𝑎𝑎 +𝑏𝑏𝑎∗𝑎𝑏)
c. 𝐿((𝑎𝑏𝑎𝑏)∗+ (𝑎𝑎𝑎∗+ 𝑏)∗)
