Solution

The correct answer is : option 3

Key Points

Comprehension Recap: The DFA must recognize strings over Σ = {a, b} that satisfy the following:

• They may start and end with any combination of 'a' and 'b'.
• The string must contain the substring 'abb' anywhere within it.

Goal: We want a DFA that detects the presence of 'abb' anywhere, and allows characters before and after it.

Basic Structure of DFA to Accept 'abb':

• q1: Start state, accepts any number of a or b.
• q2: Transition on a from q1 (start of "abb").
• q3: Transition on b from q2 (second symbol).
• q4: Transition on b from q3 – string "abb" confirmed. This is a final state.
• q4 loops on both a and b to accept trailing characters.

Option-Wise Analysis:

✅ Option 3 (Correct):

• q1 loops on a and b – allows any prefix.
• q1 --a--> q2
• q2 --b--> q3
• q3 --b--> q4 (Final State)
• q4 loops on a and b – accepts suffix after "abb".

✅ This matches the behavior expected: accepts all strings containing "abb" regardless of their prefix or suffix.

 Option 1:

• Correct prefix handling (q1 loops on a, b).
• Transitions toward detecting "abb" are correct.
• However, q4 lacks a loop on a or b.

❌ This DFA accepts only strings ending at 'abb', and fails if more characters follow.

Option 2:

• q1 does NOT loop on a or b.
• This means only strings starting with a are accepted.
• Violates the rule that string can start with any symbol.

❌ Fails to meet the start flexibility requirement.

Option 4:

• Transitions are not consistent with detecting "abb".
• q3 returns back to q1 on b, preventing reaching a final state on "abb".

❌ Cannot reach accepting state on any valid 'abb' sequence.

Final Answer: Option 3

This DFA fulfills the language requirements: accepts any string containing 'abb', with any combination of 'a' and 'b' before or after it.