$M=(Q,\sum,\Gamma, \delta, q_{0}, q_{accept}, q_{reject})$ is a TM with one tape. let $c_{1}, c_{2}$ be two configurations of $M$.
A configuration is defined like this:
$uqv$ where $(q\in Q; u,v\in \Gamma^{*} )$
we say that $c_{1}$ is reachable from $c_{2}$ in $M$, if when $M$ is in configuration $c_{1}$ it can reach the configuration $c_{2}$ in a finite number of steps that is greater than 0.
Is the following language TR (Turing Recognizable), is it Decidable?
$L=\{\langle M, c \rangle \mid M\text{ is a TM; configuration }c\text{ is reachable from configuration }c\text{ in }M\}$?
I know that the answer to this is no, $L$ is not Decidable. I just fail to see the logic behind that.
See I thought $L$ is Decidable because of the following justification:
$L$ is decided by the structure of $M$ it's self with no regards to whatever language $M$ accepts so it is intuitive.
Let's run $M$ non diterministically on any letter of it's alphabet until we get to cofiguration $d\ne c$ then again to reach another config. and so on until we get to configuration $c$ it's self after a threshold number of steps, let's say $N^{q}$ number of steps, where $N$ is the number of configurations in $M$ and $q$ is the number of letters in alphabet.
Kindly explain why language $L$ is TR and not Decidable.
Thank you in advance.