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I know that if there is a Turing Reduction from $A$ to $B$, say $A \le_T B$, and $B \in R$ then $A \in R$.

I also know that Turing Reduction is for Decision, and not Recognition.

Is it possible to design a new type of reduction, such that if $B \in RE$ then $A \in RE$ ?

Thank you!

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  • $\begingroup$ To my knowledge, many-one reduction does that. $\endgroup$
    – Nathaniel
    Commented Feb 13, 2023 at 9:33
  • $\begingroup$ Also RE contains languages, i.e., decision problems. $\endgroup$
    – Steven
    Commented Feb 13, 2023 at 10:42
  • $\begingroup$ Can you further explain please? :) $\endgroup$
    – Geo
    Commented Feb 13, 2023 at 14:54

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Actually Mapping reduction is the reduction that can do that.

Mapping reduction is defined as A is many reducible to B if there is some computable((recursive))function say f such that For all x, x belongs to A if and only if f(x) belongs to B. It can be proved that in many one reducability if B is r.e, then so is A.

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