transferring unrecognizability

Mapping reductions carry more than decidability; they also carry the weaker notion of recognizability, the property of a machine that says yes on every string in the language but may simply loop forever on strings outside it. Recognizability rides the same one-way pipe: if A <=m B and B is recognizable, then A is recognizable too. Read by contraposition, if A is unrecognizable, then B is unrecognizable. The translator preserves the yes-answers exactly, which is all a recognizer ever commits to.

Here is the mechanism. A recognizer for B is a machine R that halts and accepts exactly when its input is in B, and may loop on inputs not in B. If A <=m B by computable f, build a recognizer for A: on input w, compute f(w) and run R on it. Since w is in A if and only if f(w) is in B, this machine accepts exactly the strings of A, looping when it should, so A is recognizable. Turn it around: if A is known unrecognizable, B must be unrecognizable, or else the construction would recognize the unrecognizable A. This is the standard way to certify a language as genuinely beyond even recognition, not merely undecidable.

A subtle but vital point about symmetry. Because a mapping reduction sends yes to yes and no to no, it simultaneously gives (complement of A) <=m (complement of B). So A <=m B also transfers co-recognizability. This two-sided symmetry is exactly why mapping reductions, and not the stronger Turing reductions, are the right tool for separating recognizable from unrecognizable. The classic payoff: A_TM is recognizable but its complement is not, and the complement of A_TM is the canonical example used to seed unrecognizability across other languages by exactly this transfer.