Question

Design a DFA with 2 states that accepts the language of all binary numbers that are divisible by 3. Demonstrate it with a two-state DFA and a proof that the accepted language is precisely binary strings representing numbers divisible by 3. Otherwise, prove that such a two-state DFA is impossible.

Answer 1

Let us find the Minimal DFA that accepts the language of all binary numbers that are divisible by 3.

• State q₀ reached if reminder is 0. State q₀ is the final state(accepting state) because if remainder is 0 it is divisible by 3. It is also an initial state.
• As the divisor is 3 the number of remainders are {0,1,2} = 3. So take 3 states.
• q₀ is the state which accepts if the remainder is 0
• q₁ is the state which accepts if the remainder is 1
• q₂ is the state which accepts if the remainder is 2

• Initially the transition is on the state q0. δ:(q₀, 0)→q_0.   On seeing the input the transition is on to the same state q0.
• To process the number 1 then the binary string '1' there should be a transition rule δ:(q₀, 1)→q₁

• To process the number 2 then the binary representation is '10', end-state should be q₂, and to process '10', we just need to add one more transition rule δ:(q₁, 0)→q₂

• To process the number 3 then the binary it is '11', end-state is q₀, and we need to add a transition rule δ:(q₁, 1)→ q₀ because the remainder is 0 in where state q0 is already defined.

• Decimal	Binary	Remainder	End State
  0	0	0	q0
  1	1	1	q1
  2	10	2	q2
  3	11	0	q0
  4	100	1	q1
  5	101	2	q2
  6	110	0	q0
  7	111	1	q1

• Similarly we can construct the minimal dfa that accepts strings that are divisible by 3.
• Here q0 is final state and initial state.

This is the final and minimal DFA that accepts the laguage of all binary numbers that are divisible by 3.

with this minimal DFA we can not remove any state to form 2 state dfa that accepts the language of all binary numbers that are divisible by 3.

So we can conclude that a two state DFA is not possible such that it accepts the language of all binary numbers that are divisible by 3.

In general for accepting al language of all binary numbers that are divisible by "n" requires "n" - State DFA.