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Questions about mathematical devices that read an input stream symbol by symbol and use a state transition map to produce an output stream, maybe using secondary storage.
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Example of two undecidable languages that cannot be mapping reduced to each other
A really simple example is $A_\mathrm{TM}$ (i.e. the language of $\langle M, w \rangle$ where TM $M$ accepts $w$) and its complement $\overline{A_\mathrm{TM}}$.
$A_\mathrm{TM}$ is recognizable but not …