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In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement $\overline{\mathcal{X}}$ is in the complexity class NP. Instances of decision problems in co-NP are sometimes called "counterexamples". In simple terms, co-NP is the class of problems for which efficiently verifiable proofs of "no" instances exist. Equivalently, co-NP is the set of decision problems where the "no" instances can be solved in polynomial time by a theoretical non-deterministic Turing machine.

An example of an NP-complete problem is the subset sum problem: given a finite set of integers, is there a non-empty subset that sums to zero? To give a proof of a "yes" instance, one must specify a non-empty subset that does sum to zero. The complementary problem is in co-NP and asks: "given a finite set of integers, does every non-empty subset have a non-zero sum?".