Thinking about the halting problem for TM's, I came up with a statement that I can't prove or disprove easily and would want your suggestions.
Conjecture: Given a TM whose digraph has a single cycle , and given that it loops forever on a word $w_1 \in \Sigma^*$. Then the set of words on which it loops forever is a regular language.
Question: Is this true or false , and how so?
Note that, if the TM's source code were written out in (say) C, it would have only a single loop (for or while ). Edit: The digraph consists of states for its nodes; if there is a transition $\delta(q_1, s) = (q_2, t, L/R)$, then there is a directed edge from $q_1$ to $q_2$, marked with $s \rightarrow t, L/R$ in the digraph.