# Purpose of Acceptance Problem

I am confused about the purpose/statement of the Acceptance problem:

$$A_{TM} =\{\langle M\rangle\,s |$$ Turing machine $$M$$ accepts $$s\}$$

It can be shown that $$A_{TM}$$ is uncomputable, so we know that, in general, one cannot compute whether an arbitrary Turing machine will accept some input.

Common sense tells me that for a total Turing machine $$T$$ (one that always halts on any input in finite time) we can easily check if $$T$$ accepts $$s$$ by simply running $$T$$. So $$A_{TM}$$ is only uncomputable if non-total Turing machines are allowed.

However, given that HALT is undecidable, what is the point of knowing that $$A_{TM}$$ is uncomputable? HALT is arguably a more relevant result, and $$A_{TM}$$ is anyway only uncomputable if we allow Turing machines that may not halt - and if we allow Turing machines that may not halt, then we cannot decide if they halt on input $$s$$.

Further, HALT and $$A_{TM}$$ can be many-one reduced to one another. So, what is the point of the Acceptance problem, given that HALT is among the most well-known decision problems?

• The acceptance problem is a variant of the halting problem. That's it. Apr 29, 2021 at 6:39
• @YuvalFilmus thank you for your comment. I was just wondering if I missed any deeper meaning/implications behind the acceptance problem. If there are none, then my question is answered Apr 29, 2021 at 6:51