The Myhill-Nerode theorem is an important characterization of regular languages, and it also has many practical implications.

One consequence of the theorem is an algorithm for minimizing DFAs which is a vital step in automata theory

Theorem:

The MyhillNerode Theorem states that for a language L such that L C Σ*, the following statements hold good :-

There is a DFA that accepts L(L is regular)

There is a right invariant equivalence relation ~ of finite index such L is a union of some of the equivalence classes of ~.

~L is of finite index.

**Example:**

**Step 1:** Consider every final-nonfinal state pair and tick it working only on the lower triangular part of the table

**Step 2:** Consider all the un-ticked areas of step1

For an input(either a or b) for each un-ticked state, see the intermediate state For the area (r,t):

(r,a) => {r} and (t,a) =>s

So, here the intermediate state is ‘s’

Now check if {r,s} is ticked in step1.

If yes, tick {r,t} as well.

Similarly, {q,u} and {r,q} are also ticked

**Step3:** Continue step2 until all states have been processed. Once no more can be ticked, algorithm terminates.

Hence, here {s,u} is also ticked.

Final table now becomes

**Step 4:** Check the spaces which are still un-ticked and such states can be merged together.

In the final minimized DFA, q-s are the new states and p-t are the new states