How many minimum number of states will be there in the DFA accepting all strings (over the alphabet {a,b}) that do not contain two consecutive a’s?a)2b)3c)4d)5Correct answer is option 'B'. Can you explain this answer?

Question

How many minimum number of states will be there in the DFA accepting all strings (over the alphabet {a,b}) that do not contain two consecutive a&rsquo;s?a)2b)3c)4d)5Correct answer is option 'B'. Can you explain this answer?

Answer

To construct a DFA that accepts all strings over the alphabet {a, b} that do not contain two consecutive a's, we can start with the initial state and transition to a new state for each symbol in the alphabet.

Let's create a table to represent the transition function of the DFA:

| State | a | b |
|-------|---|---|
| q0 | q1 | q0 |
| q1 | q2 | q0 |
| q2 | q1 | q3 |
| q3 | q3 | q3 |

In this table, each row represents a state of the DFA, and each column represents a symbol in the alphabet. The entry in the table represents the next state that the DFA transitions to when it is in a certain state and reads a certain symbol.

The initial state is q0, and it transitions to q1 when it reads an 'a' and stays in q0 when it reads a 'b'. From q1, it transitions to q2 when it reads an 'a' and stays in q0 when it reads a 'b'. From q2, it transitions to q1 when it reads an 'a' and transitions to q3 when it reads a 'b'. Finally, from q3, it stays in q3 regardless of the symbol it reads.

The accepting state of the DFA is q0, as it represents the state where there are no consecutive 'a's.

Therefore, the minimum number of states in this DFA is 4.

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