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Is there a reduction from ATM to ATM-complement?

I have been thinking about it too much and couldn't find the answer.

I know that reduction from ATM-complement to ATM is not possible becouse if it was, ATM would not be in RE. But how can I proove/profe the other way around?

Thank you very much :)

Dvir Samuel
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    I'm voting to close this question as off-topic because this is a purely theoretical question and is therefore a better fit at cs.stackexchange.com. – templatetypedef Jan 26 '16 at 19:57

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There is no mapping reduction from (ATM)c to ATM. To see this, note that ATM is Turing-recognizable, so if (ATM)c ≤m ATM, we'd have that (ATM)c would be Turing-recognizable. But that's impossible, since we know that (ATM)c isn't Turing-recognizable, since if it were, ATM would be decidable (since any language that's Turing-recognizable and co-Turing recognizable is decidable).

However, there is a Turing reduction from (ATM)c to ATM. Just invoke the subroutine for ATM and return the opposite result.

templatetypedef
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  • I am so sorry, but by mistake I asked the opposite quetion.. I ment if there is a reduction from ATM to ATM-CO – Dvir Samuel Jan 26 '16 at 20:06
  • @DvirSamuel Turns out it has the same answer. :-) Can you prove that if A <=M B, then A^c <=M B^c? – templatetypedef Jan 26 '16 at 21:04
  • of course.. But pay attantion that ATM-Co is not really all the "rest of the world" that doesn't belong to ATM... What I mean is: if you take your world as N, and A={1,2,3} so A-co is all the numbers except 1, and 3.. But here with ATM its'nt the same.. – Dvir Samuel Jan 26 '16 at 21:06
  • Can you define co-ATM for me? I've always seen it as the set complement of ATM. – templatetypedef Jan 26 '16 at 22:38
  • ATM is the language that contains all (p,w) when p is a program/turing machine and p accept w and ATM-co is the language that contains all (q,x) when q is a program/turing machine and q doesn't accept x. – Dvir Samuel Jan 27 '16 at 07:50
  • Maybe I am getting confused with something... I am really thankful for your help :) – Dvir Samuel Jan 27 '16 at 07:50