I know how to determine whether a language is not regular using Myhill-Nerode theorem.

Consider the language L = {0ⁿ1ⁿ | n >= 0}. We can use the Myhill-Nerode theorem to show that this language is not regular.

Step 1: Define the relation L on the language L as follows:

For any two strings x and y in *, x L y if and only if, for any string z in *, either both xz and yz are in L, or neither of them are in L.

In this case, if x and y have a different number of zeros:

x = 0^m and y = 0ⁿ where m n

Then we can choose z = 1ⁿ.

This will result in xz not being in L and yz to be in L.

Therefore, x and y are not equivalent under the relation L.

Step 2: Construct an infinite set of pairwise distinguishable strings {w1, w2, w3, ...} such that for any i and j, if i j, then wi and wj are not equivalent under the relation L.

In this case, we can choose wi = 0ⁱ and wj = 0^j, where i j. These strings are pairwise distinguishable because we can choose z = 1^j, which will result in wiz not being in L and wjz being in L.

Step 3: Since we have shown that there are an infinite number of pairwise distinguishable strings, there must be an infinite number of equivalence classes of strings under L, which implies that the language L is not regular.

But I really don't understand how to use Myhill Nerode theorem to show that a language is regular. I know the fact that if there are an infinite number of equivalence classes, you can conclude that the language L is not regular. Language is regular if and only if there are a finite number of equivalence classes of strings under a certain relation. But I really don't understand how this works out. Explain me how to determine whether a language is regular using Myhill-Nerode theorem with examples so that I can easily understand.