Question

(4) (a) Construct a DFA which recognizes the set of words in which the number of a 's and the number of b 's are congruent modulo 3. (You can do this with 9 states by keeping separate mod3 counts of both the number of a 's and the number of b 's. But you can use many fewer states, which will greatly simplify the next step, if you instead keep track of the value mod 3 of the difference between the number of a 's and the number of b 's.) (b) Convert this DFA into an equivalent regular expression.

Answer 1

To construct a DFA that recognizes words with the number of 'a's and 'b's congruent modulo 3, use a DFA with 3 states representing the congruence values. To convert DFA to a regular expression, use state elimination method or derive from the state transition table.

To construct a DFA that recognizes the set of words in which the number of 'a's and the number of 'b's are congruent modulo 3, we can keep track of the value modulo 3 of the difference between the number of 'a's and the number of 'b's. We can represent this using a DFA with 3 states: State 0 for when the difference is congruent to 0 modulo 3, State 1 for when the difference is congruent to 1 modulo 3, and State 2 for when the difference is congruent to 2 modulo 3.

1. Start with the initial state, State 0.

2. Transition to State 1 when seeing an 'a' and stay in State 0 when seeing a 'b'.

3. Transition to State 2 when seeing a 'b' and stay in State 0 when seeing an 'a'.

4. Transition to State 0 when seeing an 'a' or a 'b' in State 1 or State 2.

5. Repeat Steps 2-4 for all three states to cover all possible transitions.

The resulting DFA will have 3 states and accept words where the number of 'a's and 'b's are congruent modulo 3.

To convert this DFA into an equivalent regular expression, we can use the state elimination method or use the state transition table to derive the regular expression.