[thelopa]

Some models of the Turing machine have an infinite tape in only one direction. Some also don’t require every state to have an exhaustive set of transitions. Those models generally consider going off the end of the tape or failing to find a transition to be a halting state.

[tromp]

The paper makes explicit how this is dealt with:

> For the purposes of defining the halting problem H, one should specify whether it officially counts as halting or not if the head should happen to fall off the left edge of the tape. Although the truth of the main theorem will not depend on these details, provided we adopt a uniform answer, let us be definite and regard such computations as having not officially halted, as the halt state was not reached.

[l33t233372]

It’s unclear to me why restricting ourselves to that model is useful here

[knome]

Changing the model wouldn't change anything. A turing machine is equivalent to any computational machine you want to put in there. It doesn't matter what the machine is. The turing machine is the goto because it was first used for them.

Any other machine that can simulate a turing machine can simulate failing in those same ways. You can't escape the problems of turing completeness without losing it.

[fooker]

It’s the simplest model which can simulate all useful models of computation we have so far.

So, if you want to prove something about the limits of computation, a Turing machine is usually the right choice.

[mgraczyk]

It is true that some models define halting that way, but in this paper they explicitly do not.