反转确定性有限自动机

Question

I just started learning about automata theory in my computability and complexity course. I was given an assignment that asks us to design DFA's based on certain conditions. However, one of my assignment questions is having us write a reverse DFA (assuming I understand the question correctly). My professor did not give us any examples relating to reversing a DFA. The question is as follows:

For any w ∈ {a, b}∗, we write w^r for the reverse of w. For example, if w = ababbb, then w^r = bbbaba.

Show that {wxw^R | x ∈ {a, b}∗, w ∈ {a, b}^2} is regular by designing a DFA that accepts it.

Any advice/help as to how to solve this would be greatly appreciated

Answer 1

I won't do your homework for you, but here's a basic start. Your machine accepts any string of four or more characters such that the first two characters are the reverse of the last two.

So your machine is going to go into any of four states based on whether the first two characters are 'aa', 'ab', 'ba', or 'bb.

From the 'aa' state, you now have to accept [aa]*aa. From the 'ab' state, you now have to accept [ab]*ba. And likewise for the other two.

You should be able to write a DFA that accepts these.