"Proof of Main Theorem: We now prove the Main Theorem. Let B be the set of programs that on input 0 either halt before repeating a state or fall off the tape before repeating a state." Q3. This method will not work assuming we have a Two way infinite tape Universal Turing Machine. Thus, by definition of halting problem one should be able to comment on all UTM models. Is there any general method using which we can comment on the halting Probability of a Random <Program, Input> Pair ?

"Lemma 1.2 The collection of programs having no transition reaching the halt state has asymptotic probability 1/e2, which is about 13.5%." Q1. Is the model proposed in the proof a Universal Turing Machine i.e. we consider all the programs that are possible (with both repeating and non-repeating states) and all possible inputs ? Q2. A randomly chosen program might have a transition to the halt state, but as in original Query we have a random <Program, Input> Pair. But the current input might not lead to a halt state. Thus, input also effects the probability.?

Thanks a lot for your reply. But I have few doubts/issues regarding the explanation(paper) and it would be great if you could clarify.