Solution

The correct answer is : option 1

Key Points

Problem Statement:

We are given four DFAs and must identify the minimum state DFA that accepts the language L, where:

• Σ = {a, b}
• L = Set of all strings where "abb" is a substring

This means the DFA should:

• Accept any string that contains "abb" anywhere in the string
• Reject strings that do not contain the exact substring "abb"

Designing the DFA for 'abb' recognition:

The DFA must track three progress levels to reach the substring "abb":

1. q0: Initial state, no match yet
2. q1: After reading 'a'
3. q2: After reading 'ab'
4. q3: After reading 'abb' → Final State

After reaching the final state, any number of further characters should be accepted (loop at final state).

Option 1 (Correct):

Let's trace the transitions:

• q1 (Start) — a → q2
• q2 — b → q3
• q3 — b → q4 (final state starts)
• q4 → loops on a
• q5 is a trap or accepting completion state

✔ Accepts 'abb' as required and uses minimum 5 states to precisely track substring occurrence. This matches the minimum-state DFA for 'abb'.

❌ Option 2:

• Only 4 states used
• Transitions are not correctly capturing the 'abb' sequence
• No backtracking after false match paths

 Invalid – misses important transition sequences for 'abb'

❌ Option 3:

• Correct transitions but uses 6 states
• Matches 'abb' and has extra states for suffix handling

⚠ Valid DFA but not minimum

❌ Option 4:

• Similar issue as Option 2
• Does not correctly identify and confirm the full sequence 'abb'

 Final Answer: Correct Option: Option 1

This DFA efficiently tracks the substring "abb" with minimal 5 states and is functionally correct and minimal for the language.