in theoretical computer science I learned for every recursive enumerable language there would be an enumerator and a grammar. So since word problem and halting problem are recursively enumerable, I was wondering what kind of grammar and enumerator could this be. And let the Wordproblem be $ L = \{ \langle M, w \rangle | M \space is \space TM \space and \space w \in L(M) \} $ and Halting Problem $ L = \{ \langle M, w \rangle | M \space is \space TM \space and \space M \space halts \space on \space w\} $
Ok for word problem: since there exists a sequence of $M_i$ I would start with $M_1$ and find all words for this TM and give them out. So if I have any TM is there a possibility to give all words out which are accepted by this TM? I probably would have to give all $w_i$ to it and compute the first i words for i steps, then i+1 words for i+1 steps and so on for a sequence of computable words $w_1, w_2,.. \in \Sigma^* $ Or maybe something like DFS on all configurations. This really sounds like that only for one TM this could go on forever. So I would need to start the second TM for the same period of time after a while... Seems as if something similiar could work for Halting Problem. Do you have any more refined thoughts on this one?
Greets,
Felix