Question

For the language L = w over  = {a, b}, where #(a) and #(b) represent the number of a’s and that of b’s in w, respectively, Construct a minimal DFA M that accepts L. Draw its transition diagram.

Answer 1

To construct a minimal DFA for the given language, create three states representing the possible remainders when the difference between the number of 'a's and 'b's is divided by 3 and define the transitions based on the input characters.

Step-by-step explanation:

To construct a minimal DFA M that accepts the language L = w , we can start by understanding the properties of the language. The condition (#(a) - #(b)) mod 3 = 0 means that the difference between the number of 'a's and the number of 'b's in w should be divisible by 3.

One way to construct a minimal DFA is to have three states representing the possible remainders when the difference between the number of 'a's and the number of 'b's is divided by 3. These states can be labeled as state 0, state 1, and state 2. We can then define the transitions based on the input characters 'a' and 'b'.

For example, if we are in state 0 and we see an 'a', we stay in state 0 because the difference between the number of 'a's and 'b's remains divisible by 3. If we are in state 0 and we see a 'b', we transition to state 1 because the difference between the number of 'a's and 'b's increases by 1, which is congruent to 1 modulo 3. In this manner, we can construct a minimal DFA with three states and transitions labeled with 'a' and 'b'.