I’m confused about the notion of (semi-)decidable sets in the context of transition systems.
Suppose we have some transition relation $\rightarrow$ over some infinite set of states $\Sigma$. Let’s say the set $\Sigma_T \subset \Sigma$ is the set of terminal states (also infinite). Given a starting state $\sigma$ we can now define a function $T(\sigma)$ by $T(\sigma) = \{ \sigma’ \in \Sigma_T \mid \sigma \rightarrow^{*} \sigma’ \}$, where $\rightarrow^{*}$ is the transitive closure of $\rightarrow$. Thus, $T(\sigma)$ is the set of terminal states that we can end up in if we start in state $\sigma$.
My question is: what can we say about the decidability of the set $T(\sigma)$ for a given $\sigma$ and how does this relate to its (in)finiteness? More precisely:
The relation $\rightarrow$ may be such that $T(\sigma)$ is infinite. Suppose it is. Does this alone imply that $\sigma’ \in T(\sigma)$ is undecidable/semi-decidable?
Suppose for a given state $\sigma$ there is only one execution path $\sigma \rightarrow \sigma_1 \rightarrow \sigma_2 \rightarrow \ldots$ that never reaches a terminal state. Thus an attempt at computing the terminal states starting with $\sigma$ will never terminate. Does this imply that $T(\sigma)$ is undecidable, even though it is not an infinite set?