# Is the language of turing machines, which return epsilon on its own encoding, decidable?

Is the language $$\{ \langle M\rangle | f(\langle M\rangle)=\epsilon\}$$ decidable? $$f()$$ means, that the turing machine returns $$\epsilon$$ on its own encoding and $$\langle M\rangle$$ stands for the encoding (a "string-version" of the TM).

I have no idea how I should start. How can I decide, if I use a reduction, a sub program technique or the diagonalizability? I would really appreciate if you could help me, thanks! :)

• Hint: Given a Turing Machine $M$ you can always define a turning Machine $M'$ that ignores its input, simulates $M$ on the empty input, and when $M$ halts (possibly never), returns $\epsilon$. – Steven Nov 13 '19 at 23:21
• "$f()$ means, that the turing machine..." What Turing machine? Do you mean $M(\langle M\rangle)$? – xskxzr Nov 14 '19 at 8:19