Proving that DFA equivalence is decidable

Question

The following question is taken from Sipser:

Prove that $EQ_{\mathsf{DFA}}$ is decidable by testing the two DFAs on all strings up to a certain size. Calculate a size that works.

Here is the answer:

For any DFAs $A$ and $B$, $L(A) = L(B)$ if and only if $A$ and $B$ accept the same strings up to length $mn$, where $m$ and $n$ are the numbers of states of $A$ and $B$, respectively. Obviously, if $L(A) = L(B)$, $A$ and $B$ will accept the same strings. If $L(A) \neq L(B)$, we need to show that the languages differ on some string $s$ of length at most $mn.$ Let $t$ be a shortest string on which the languages differ. Let $l$ be the length of $t$. If $l \leq mn$ we are done. If not, consider the sequence of states $q_0,q_1,\dots,q_l$ that $A$ enters on input $t$, and the sequences of states $r_0,r_1,\dots,r_l$ that $B$ enters on input $t$. Because $A$ has $m$ states and $B$ has $n$ states, only $mn$ distinct pairs $(q,r)$ exist, where $q$ is a state of $A$ and $r$ is a state of $B$. By the pigeon hole principle, two pairs of states $(q_i,r_i)$ and $(q_j,r_j)$ must be identical. If we remove the portion of $t$ from $i$ to $j-1$, we obtain a shorter string on which $A$ and $B$ behave as they would on $t$. Hence we have found a shorter string on which the two languages differ, even though $t$ was supposed to be shortest. Hence $t$ must be no longer than $mn$.

Unfortunately, I do not understand the sequence of steps in the answer. Could anyone explain the answer for me in a simpler way?

Answer 1

Here is a slightly different proof, along similar lines.

Lemma 1. Let $A$ be a DFA having $n$ states, and let $B$ be a DFA having $m$ states. There is a DFA $C$ having $nm$ states such that $L(C) = [L(A) \setminus L(B)] \cup [L(B) \setminus L(A)]$.

The proof uses the product construction (exercise).

Lemma 2. Let $C$ be a DFA having $p$ states. If $L(C) \neq \emptyset$ then $L(C)$ contains some word of length at most $p-1$.

Proof sketch. The language $L(C)$ is non-empty if some final state $q_f$ is reachable from the initial state $q_0$. The shortest path from $q_0$ to $q_f$ has length at most $p-1$, and this corresponds to a word of length at most $p-1$ in $L(C)$.

Theorem. Let $A$ be a DFA having $n$ states, and let $B$ be a DFA having $m$ states. Then $L(A) = L(B)$ iff for all words $w$ of length at most $nm-1$, either $w \in L(A),L(B)$ or $w \notin L(A),L(B)$.

Proof. If $L(A) = L(B)$ then the condition clearly holds. If $L(A) \neq L(B)$ then $\Delta := [L(A) \setminus L(B)] \cup [L(B) \setminus L(A)] \neq \emptyset$. According to Lemma 1, there is a DFA having $nm$ states whose language is $\Delta \neq \emptyset$. According to Lemma 2 (with $p = nm$), there exists a word $w \in \Delta$ of length at most $nm-1$. This is a word for which it doesn't hold that either $w \in L(A),L(B)$ or $w \notin L(A),L(B)$.