So, I was reading this pdf on complexity theory. On page 18 pf pdf (Page 12 of book) The Immerman-Szelepcsenyi Theorem is mentioned with proof. The following lines are from the book :
The idea is to cycle through all possible configurations $\sigma$ of $M$, for each one checking whether it is a final configuration that is reachable from the initial configuration by a computation of $M$. In case no such accepting configuration is found we know $x\notin A$ so we let $\overline M$ accept $x$.
My question here that the machine $M$ may not halt for a string $x\notin A$. Then the logic of "cycle through all possible configurations" won't make any sense. What am I missing here?
On a side note is this book too high level for me? (I am graduate in Computer Science) Getting stuck at page 12 is not a good sign right?