Question

1. Let L be the language over {a, b, c} accepting all strings so that: 

1. No b's occur before the first c. 

2. No a's occur after the first c. 

3. The last symbol of the string is b. 

4. Each b that is not the last symbol is immediately followed by at least two d's. 

Choose any constructive method you wish, and demonstrate that L is regular. You do not need an inductive proof, but you should explain how your construction accounts for each rule.

Answer 1

Here is a DFA which accepts the given language. All missing transitions go to the reject state.

a b C C C 4

In the given DFA, the state 1 takes any number of 'a's before seeing a 'c'. Therefore it ensures that there are no 'a's after the first 'c'. Similarly, there will be no 'b' before the first 'c' as there is no allowed transition for that. After the first 'c' is seen, it moves to state 2.

At state 2, it allows any number of 'c's, and then if it sees a 'b', it moves to final state 3. Therefore, if 'b' is the last letter, the word will be accepted. Otherwise, if 'b' is not the last letter, then it goes to state 4 which comes back to state 2, therefore at least two 'c's have been seen after the 'b'.

This covers all the rules as required, proving that the language is regular.

Comment in case of any doubts.