Question: Suppose you have a finite state machine that accepts only strings with an equal number of zeros and ones. Show that you can then construct a finite state machine that accepts only strings of the form 0^n 1^n, i.e., an arbitrary finite number of zeros followed by the same number of ones.
My attempt at a response:
A Finite State Machine that accepts strings with an equal number of 0’s and 1’s (e.g. n zeros and n one's, where n is some arbitrary finite number), must have a counter that keeps track of the total number of 0's and 1's. This is impossible, since by definition a FSM has no counter or memory. But suppose this were possible. Then it’d be possible to construct another FSM that keeps track of the count of consecutive 0's in the prefix of a given string, and ensure that number equals the count of consecutive 1's in the suffix of that string.
I feel like there is more to the answer than this, but I don't really know what else can be said. I know that no FSM can be constructed to accept only strings in the format 0^n 1^n where n could be any arbitrary nonnegative integer, but that it is possible to do for any fixed, specific value of n. For example, it's possible to build FSM to accept only strings that contain three 0's, followed by three 1's. But I don't know whether this detail is relevant at all. Thanks for any help.