# Proving undecidability of a language with mapping reductions

I'm referring to questions like this one: Mapping reduction to show NeverHalt is undecidable

I understand with Turing reductions, you have to use oracle calls of the unknown language you're trying to prove is undecidable to solve a known undecidable language.

However, with mapping reductions, am I right in assuming these calls aren't needed? In addition, in the link provided, the solution pseudocode says

For input x:
Simulate M for input w
if it accepts, loop
if it rejects accept x


How can you say "if it accepts"? How can you determine this, what if it loops forever and this is never found out? Why can you make such statements with a mapping reduction but not with Turing reductions? Could I make a statement like "if M halts on w, do ...". I mentioned this to my teaching assistant and he said you can't make any statements like these unless you're accessing an oracle and doing a during reduction, but I see loads of examples which seem to show otherwise. Hopefully this makes sense

Usually, when the context of the statement "if $$M$$ accepts $$x$$" is in a loop simulating the execution of $$M$$ on $$x$$, we just mean to say that if the simulation stopped and $$M$$ accepted/rejected then do something, otherwise just continue simulating (the more precise way of writing this would be to say "if $$M$$ accepted after $$t$$ steps then...").
Your pseudocode doesn't do anything magical, it simply checks at a specific point of the simulation whether or not $$M$$ already accepted/rejected.
• Since the halting problem is not decidable, there isn't an always halting procedure that would allow you to check whether or not $M$ halts on $x$ for arbitrary $M,x$. If you had a halting oracle then this would have been easy, simply query it on $(M,x)$. Apr 29 at 9:06
• Both are ok in the context I described, you can always simulate $M$ and at any given point check if it already accepted/halted. The problem arises if you use this as a command which supposedly always returns an answer. Apr 29 at 9:09