Solution

The correct answer is :option 3

Key Points

Comprehension Summary: The DFA should accept all strings over Σ = {a, b} that:

• Start and end with any combination of 'a' and 'b'
• Must contain the substring 'abb'

Concept:

• The DFA must identify the occurrence of the substring abb in the input string.
• After recognizing abb, it can accept any combination of 'a' and 'b' till the end.

Option 1:

• Only has 4 states and doesn’t seem to process the abb sequence correctly.
• No transitions explicitly validating abb.
• ❌ Incorrect DFA

Option 2:

• Starts with 'b' transitions from q1 and q2, which doesn’t help track abb.
• States don't represent memory of 'a' followed by two 'b's.
• ❌ Incorrect DFA

Option 3 (✅ Correct Answer):

• This DFA uses a sequence of 6 states to track each character of the required substring:
• q1 → q2 → q3 → q4 → q5 → q6 represents matching a → b → b
• Final state q6 ensures that the substring abb occurred.
• Transitions allow acceptance of any characters before and after the match.
• ✅ Correct DFA — accepts all strings containing 'abb'

Option 4:

• Structure is similar to option 3 but ends prematurely at state 6.
• There’s ambiguity on whether abb is fully validated with proper state transitions.
• ❌ Incomplete DFA

Final Answer: Option 3