Find the regular expression for the language accepted by the automata given below:

Correct option is A

• The initial state is also an accepting state (double circle)

• There are three states connected using transitions labeled a and b
• The automaton contains cycles, meaning:
• Repetition of patterns is allowed
• Movement pattern:
• ‘a’ generally moves forward
• ‘b’ helps in transitions and returning paths

Hence, the language consists of:

• ​$$\varepsilon$$​​
  
• Repeated structured sequences of a’s and b’s

• Cycles exist → repetition needed
• Start state is accepting → $$\varepsilon$$​ must be included

• $$aa^* →$$​ at least one $$‘a’ → a^+$$​
• (a + b) → either ‘a’ or ‘b’
• ab → fixed suffix

• Zero or more repetitions of the block
• Hence:
• $$\varepsilon$$​ is included
• Repetition supported

Step 5: Consistency with Automaton

• They do not have Kleene star at the outermost level, hence:
• Do not generate ε
• Impose minimum length constraint
• Therefore, they cannot represent the given automaton

• If start state is accepting, always check:

​$$\varepsilon \in L$$​​

• Only expressions with:
• Kleene star (*)
• or explicit $$\varepsilon$$​
  can satisfy this

• $$\varepsilon$$​ is included
• Cyclic behavior is captured