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For my classes in theoretical computer science the following proof must be shown to be wrong. However, this is the first time I am attempting myself at this topic, so I would be thankful for some help:

Lemma:

If $L \in NP $, then $ \bar{L} \in NP $

Proof:

  • Let N be a non-deterministic Turing machine, which accepts L in polynomial time.

  • Swap all final states of $N$ with non-final states and vice-versa.

  • This new Turing machine $N'$ does now accept $\bar{L}$ in polynomial time.

Therefore $\bar{L} \in NP$

I have doubts that simply swapping non-final and final states within $N$ will produce a $N'$ with the proposed properties, but how can I show this?

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nitowa
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1 Answers1

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You just need to show that swapping final and non-final states doesn't complement the language accepted by an NTM, e.g., by showing that, for at least one NTM $N$ (which you can construct to have whatever properties you need), there is a string that is accepted by both $N$ and $N'$, or rejected by both.

David Richerby
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  • How should I go about starting an argument like this? I neither have a precise definition of $N$ nor $L$, and since $N$ is nondeterminisitic I find it hard to understand how something like this would be stated. – nitowa Apr 17 '16 at 14:30
  • It's enough to do it for a particular $L$ and a particular $N$. – Yuval Filmus Apr 17 '16 at 18:47