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For an assignment I have to proof that for two given decidable languages A,B, A\B is decidable too.

My idea is as follows: If B is empty or doesnt have elements in common with A, then A\B is decidable because A\B=A.

If B is not empty and intersects with A, A\B is just a smaller subset of A and therefore decidable.

I have a feeling that this is either the wrong idea or not formalized enough. I would really appreciate any hints regarding that.


EDIT: A subset of a decidable language is NOT always decidable too, this was a misconception.

Thanks in advance

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  • $\begingroup$ Why do you think a subset of a decidable language is still decidable? $\endgroup$ – xskxzr Nov 10 '18 at 11:01
  • $\begingroup$ This is obviously a misconception, thanks for pointing that out. That leaves me with no useful approach to this assignment though. Can you point me in the right direction? $\endgroup$ – Daniel Siegel Nov 10 '18 at 11:07
  • $\begingroup$ You can try to make use of the fact that $B$ is decidable. $\endgroup$ – xskxzr Nov 10 '18 at 11:26
  • $\begingroup$ I can't think of any way to utilize that. Can you elaborate? $\endgroup$ – Daniel Siegel Nov 10 '18 at 11:33
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Let $M_A$ and $M_B$ be deciders for $A$ and $B$ respectively. You can construct a TM $M$,

$M$ = "On input x,

$\qquad$ 1. Simulate $M_A$ on x.

$\qquad$ 2. Reject if $M_A$ rejects. Else simulate $M_B$ on x.

$\qquad$ 3. Accept if $M_B$ rejects. Else reject."

$Correctness$:

$\qquad$ $M$ always halts as $M_A$ and $M_B$ always halts. $M$ accepts if $M_A$ accepts and $M_B$ rejects i.e, $x \in A$ and $x \notin B$. $M$ rejects otherwise.

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