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Questions about decision problems that can be solved on nondeterministic Turing machines in time polynomial in the length of the input.

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How does a co-NP problem differ from an NP (its complement) one?

I have quite a hard time understanding co-NP problems. If we can reduce every problem to decision problem. NP problems should accept YES instances -> instances where the answer is yes. … For co-NP problem, they should accept NO instance -> answer to them is NO. But given this, how would we formulate co-NP problem to Traveling Salesman problem I mentioned earlier? …
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