How to detect infinite loop exist in linear bounded automata (LBA)?
The machine $M$ is deterministic. This means that, if $M$ is in a certain configuration $c$, then there is a single fixed configuration $c'$ (determined by the rules of $M$) which the execution of one step will lead it to. If $M$ ever reaches the configuration $c$ again, then the configuration $c'$ will follow no matter what. Hence, if the computation of $M$ causes it to be assume configurations $c_1, \dots, c_n$ and $c_n = c_1$, then $M$ will repeat the loop $c_1, \ldots, c_n$ indefinitely.