I am looking at this task:
Draw the diagram of a two tape Non deterministic Turing Machine M that decides the language
𝐿 = { 𝑤 ∈ Σ* | 𝑤 = 𝑢*𝑢*, 𝑢 ∈ Σ* }
Which are the steps to construct the NDTM (linguistically), I believe I could draw the diagram, but I couldn't come out with an answer.
By u*u*u (viewed in the edit history), I presume what you intend is the language of all words of the form u^3 (u repeated three times) where u is any string over the alphabet.
Our NDTM needs to accept strings in the language in at least one way, and it must never accept anything not in the language. In particular, the key is that an NDTM can reject strings in the language, as long as some path through the NDTM does accept every string in the language.
Given that, our first step can be do guess about the length of u. The NDTM can mark three tape symbols (say, by writing versions of the symbols that are underlined) by nondeterministically transitioning from state q0 to q1 then q2 at arbitrary points while scanning right. Then, we can reset the tape head and use a deterministic TM to answer the question: did the split we guessed in the first step result in a string of the form u^3?
This is deterministic since we know the delineation of parts. We can check the first two parts (say, by bouncing back ad forth and marking symbols we've already processed), and then the second two parts (using the same technique, but applied to the 2nd and 3rd parts).
We have reduced the problem to that of checking whether a string is of the form w|w where we know the split. This deterministic TM is easier to come up with. When we put it after the NDTM that guesses about how to split up the initial input, we get a NDTM that can (and for exactly one guess, does) accept any string of the form u^3, but cannot possibly accept anything else. This is what we were after and we are done.