# Myhill-Nerode Theorem

This section will explore the Myhill-Nerode Theorem due to John Myhill and Anil Nerode who proved this result in 1958 (link to paper). It provides a necessary and sufficient condition for a language \(L\) to be regular. As we will see, it can be used to prove that a language is regular or not regular which none of the techniques weâ€™ve seen so far allow us to do. Furthermore, the M-N theorem can be used to prove the uniqueness of minimal DFAs. That is, we can use the M-N theorem to prove that the DFA \(M'\) that results from the minimization algorithm is the unique minimal DFA that accepts the language \(L(M')\). (This was something I claimed in a previous section without proof.)

The videos in this section might â€śfeelâ€ť somewhat different. This is because I prepared these for a guest lecture that I delivered at McGill University for the Theory of Computation class during the Fall 2022 semester. The class was taught by the exceptional Prakash Panangaden to whom I will be forever grateful for having given me this opportunity. You can get the notes for these videos here.

### Mathematical Preliminaries

In this first video, I review some important notions needed to understand the M-N theorem and I also introduce 2 important equivalence classes that will be needed to state and prove the theorem. Important note: In this set of videos, I use \(\epsilon\) to mean the empty string, but this is equivalent to using \(\lambda\).

### The Myhill-Nerode Theorem

Now that Iâ€™ve presented the required definitions, I present the Myhill-Nerode Theorem and explain what it is saying.

### Proving the Myhill-Nerode Theorem

The M-N theorem claims that three things (statements \((1)\), \((2)\) and \((3)\)) are equivalent. To prove this, I go in a cyclic fashion from \((1) \implies (2)\) to \((2) \implies (3)\) to \((3) \implies (1)\).

#### $$(1) \implies (2)$$

#### $$(2) \implies (3)$$

#### $$(3) \implies (1)$$

And weâ€™re done! This proves the M-N theorem, now how can we actually use it?

### Using the Myhill-Nerode Theorem to show that a language is not regular

### Proving uniqueness of minimal DFAs

Using the M-N theorem to prove non-regularity when the pumping lemma canâ€™t is a pretty interesting result. However, even more interesting is what M-N tells us about minimal DFAs. When you run the minimization algorithm (presented previously), is it possible to generate multiple different minimal DFAs? That is, if you run the minimization algorithm on all the DFAs that accept some regular language \(L\), will you **always** get the same minimal DFA? Surprisingly (or not), the answer is â€śYes!â€ť. We can use the statement in M-N to prove this indirectly. We do this by showing that for any regular language \(L\) the DFA we constructed in the proof from \((3) \implies (1)\) (i.e. the DFA with state size equal to the index of \(\equiv_L\)) is the unique minimal DFA that accepts \(L\). Therefore, if we use the minimization algorithm (whose correctness we can prove), it will land on this same minimal DFA.

### Exercises

**Exercise.**Show that you cannot use the pumping lemma to prove that the language \( \{a^ib^jc^k : \text{if } i = 1 \text{ then } j = k\} \) is not regular.

**Exercise.**Use M-N to show that \(\{a^n b^{n^2} : n \geq 0 \}\) is not regular.

**Exercise.**Use M-N to show that \(\{w \in \{0,1\}^* : |w| \mod 3 = 0\}\) is regular. (Hint: \(\equiv_L\) should have index 3.)

**Exercise.**Design an algorithm that checks where two NFAs \(N_1, N_2\) accept the same language. Design another algorithm that checks whether two regular expressions \(R_1, R_2\) denote the same language. In each case, you can use algorithms already discussed as building blocks.