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I was trying to solve a problem with a mapping reduction from $A_{TM}$ to $INFINITE_{TM}$, and came across a solution that was 100% identical to another solution I saw for $A_{TM} \leq_M ALL_{TM}$. This is the reduction:


$M_f$: Given an input of $\langle M, w\rangle$, where $M$ is a $TM$ and $w$ is a word:

Define a TM $M_1$:

Given the input $x$ do:

  1. Run $M$ on $w$ and return its result
  2. Return $\langle M_1\rangle$

This is correct for $ALL_{TM}$ because if $M$ accepts $w$, $L(M_1)=\Sigma^*$, whereas if it rejects, $L(M_1)=\emptyset$

However, it is also given as a solution for $A_{TM} \leq_M INFINITE_{TM}$, where

$INFINITE_{TM}=\{\langle {M \rangle} |$ $M$ is a $TM$, $L(M)$ is an infinite language $\}$, for example here: http://www.sfu.ca/~kabanets/308/lectures/lec7.pdf

I don't fully understand why theirs, or other similar explanations, are correct. What about infinite languages that aren't $\Sigma^*$? What about finite languages that aren't $\emptyset$?

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The m-reducibility definition says

Language $A$ is m-reducible to $B$ (denoted $A\leq_mB$) if there is a computable function $f:\Sigma^* \rightarrow \Sigma^*$ such that for every $w\in \Sigma^*$, $$w\in A \iff f(w)\in B$$

The point you should pay attention to is that there is no need to talk about every $x\in B$. If you can map all elements of $A$ to one special element of $B$ say $x_0$ and all strings that don't belong to $A$ to string which doesn't belong to $B$ say $y_0$ then you are done!

Maybe if you think of $A\leq_m B$ as language $B$ is harder than $A$ then you feel better about the reduction. This means if you can decide $B$ then you also can also decide $A$.

So by the proof, you have provided we map every $\langle M,w\rangle \in A_{TM}$ to $\Sigma^*$ and every $w' \notin A_{TM}$ to $\emptyset$. Thus $INFINITE_{TM}$ is harder than $A_{TM}$. If you could decide just one elements of $INIFINITE_{TM}$ which is $\langle M \rangle$ that $L(M)=\Sigma^*$ then you could decide $A_{TM}$.

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