For example, many reductions from $A_{TM}$ will often have the lines:Simulate M on w:    if M accepts        ...    elseBut: a reduction is a computable function, right? So why are we allowed to say if M accepts?Also, I can take the same logic and reduce $A_{TM}$ to a deciable language:Let L be the empty language. Clearly L is decidable.Given $(M,w)$: Output output:      If M accepts w            output = '0'      else            output = $\emptyset$Obviously I'm wrong here. But why?Thanks in advance.
Why are mapping reductions from incomputable Languages OK?
automata;reductions
There are two options:Either the pseudocode means run $A_{TM}$ on the pair $M,w$ when it says Simulate $M$ on $w$;or it really means to simulate $M$ on $w$, and if it ever stops then do something; there is no else clause in that case.