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I would like to define a Turing machine as follows.

Let Z be an alphabet, and let L be the set of all sentences of this alphabet. For instance, Z={0,1}, then L is the set of all binary sentences.

A Turing machine is a function which takes a (binary) input and produces a (binary) output:

$$ TM: L \to L $$

However, it is also possible that it doesn't halt. In this case, it is not from $L\to L $.

Initially, I was thinking of mapping $TM: L \to L \cup \nexists$. And claim if it doesn't halt, it maps to $\nexists$.

BUT, here are my concerns:

  1. If TM never halts because it keeps printing symbols forever, then it does map to a sentence of L. It is just that the sentence is infinite.

  2. If TM never halts because it gets stuck somewhere (after n symbols are printed), then it did produce a sentence of L, its just that it is stuck.

So I all cases it maps from L to L.

So is the right answer simply to claim from L to L (where sentences can contain infinitely many symbol) and just claim TM is either computable or non computable?

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This is not the definition of a Turing machine. "Turing machine" is a term of art with a standard, accepted definition. The right answer is "none of the above" - the premise is faulty, as Turing machine already has a definition that is different from what you've written.

You seem to be attempting to come up with a way to represent the externally visible behavior of a Turing machine. And you have correctly identified reasons why your approach does not work. (It's not correct that in case 1 it does map to a sentence of $L$, because an infinite sequence is not in $L$; $L$ consists of only finite sequences.) No, your final question is not the correct approach either. A TM cannot be "computable" or "not computable"; computability is a property of a language, not a Turing machine.

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