I'm reading right now Computers and Intractability by Garey and Johnson. I've found the definition of PTIME complexity class a little strange. This is how it looks:
Let $\Sigma = \{ 0,1\}$. By $\Sigma^{*}$ we denote the set of all finite strings created from symbols from $\Sigma$. If a computation on a Deterministic Turing Machine with input $x \in \Sigma^{*}$ ends in $q_{y}$ (yes-state), it means that TM accepts this input. Language $L_{M}$ recognised by the machine $M$ is given by:
$L_{M} = \{ x \in \Sigma^{*} :$ $M$ accepts $x \}$.
Time complexity function is given by:
$T_{M} = max\{ m : $ there is an $x \in \Sigma^{*}$, with $|x| = n$, such that the computation of $M$ on input $x$ takes time $m$$\}$.
We call program $M$ a polynomial time DTM program if there exists a polynomial $p$ such that, for all $n \in \mathbb{Z}^{+}$, $T_{M}(n) \leq p(n)$.
Then they give such $P$ class definition:
$P = \{ L :$ there is a polynomial time DTM program $M$ for which $L=L_{M} \}$
Does this mean that their P complexity class contains only languages that are composed of yes-instances, i.e. languages recognised by $M$, consisted of $x$ strings such that $M$ always halts in accepting state for every of them as an input? What about input that also causes halting, but in rejecting state $q_{n}$? Shouldn't these be also in PTIME?
Do I understand this correctly? If yes, why they decided to define P class in such way?
P.S. Maybe that matters, I have the first edition of the book (1979).