Determining whether the language of a DFA is closed under reversal

The question and its answer is given below:

Let $S = \{ \langle M \rangle \mid \text{$M$is a$\textsf{DFA}$that accepts$w^{\mathcal{R}}$whenever it accepts$w$}\}$. Show that $S$ is decidable.

For any language $A$, let $A^{\mathcal{R}} = \{w^{\mathcal{R}} \mid w \in A\}$. If $\langle M \rangle \in S$, then $L(M) = L(M)^{\mathcal{R}}$. The following $\textsf{TM}$ $T$ decides $S$.

$T =$ “On input $\langle M \rangle$, where $M$ is a $\textsf{DFA}$:

1. Construct $\textsf{DFA}$ $N$ that recognizes $L(M)^{\mathcal{R}}$.
2. Run $\textsf{TM}$ $F$ from Theorem 4.5 on $\langle M,N \rangle$, where $F$ is the Turing machine deciding $EQ_{\textsf{DFA}}$.
3. If $F$ accepts, accept. If $F$ rejects, reject.”

But I do not understand in the first line of the answer, in the second statement why does it say that $L(M) = L(M)^{\mathcal{R}}$, could anyone explain this for me please?

• What does the definition of $S$ tell you about $M$? What does it tell you about $L(M)$? – D.W. Jun 1 '18 at 6:39
• OK ... is $L(M^R) = L(M)^R$? @D.W. – Intuition Jun 1 '18 at 6:55
• Not at all, if you mean $\langle M\rangle^R$. That reverses the description of $M$, so likely wouldn't even be the description of any DFA. – Rick Decker Jun 1 '18 at 14:11

For a language $L$, we define the reverse language $L^{\mathcal{R}}$ by $$L^{\mathcal{R}} = \{ x^{\mathcal{R}} : x \in L \}.$$ In words, $L^{\mathcal{R}}$ is obtained from $L$ by reversing all words.
It follows that $L = L^{\mathcal{R}}$ if and only if, for each word $x$, we have $x \in L \Leftrightarrow x^{\mathcal{R}} \in L$.