Rice's theorem for non-semantic properties

Question

Rice's theorem tell us that the only semantic properties of Turing Machines (i.e. the properties of the function computed by the machine) that we can decide are the two trivial properties (i.e. always true and always false).

But there are other properties of Turing Machines that are not decidable. For example, the property that there is an unreachable state in a given Turing machine is undecidable$^{\dagger}$.

Is there a similar theorem to Rice's theorem that categorizes the decidability of similar properties? I don't have a precise definition. Any known theorem that covers the example I have given would be interesting for me.

$^\dagger$ it is easy to prove that this set is undecidable using Kleene's Recursion/Fixed Point theorems.

Answer 1

A general theorem that partially covers the example given is that any $\Sigma^0_1$-hard property of the machine will be undecidable. The halting problem is $m$-reducible to the state-reachability problem, so that shows the state reducibility problem is $\Sigma^0_1$-hard.

However, this is not an "if and only if" theorem like Rice's theorem. If every $\Sigma^0_1$ property of the index of the Turing machine counts as a property of the machine, then there is not going to be a nice characterization, because there is not any nice characterization of which r.e. sets are decidable in terms of the index of the r.e. set.