linguistic:paraphrasing, stylistic:tone
Verifier-based definition

In order to explain the verifier-based definition of NP, consider the subset sum problem:Assume that we are given some integers, {−7, −3, −2, 5, 8}, and we wish to know whether some of these integers sum up to zero. Here the answer is "yes", since the integers {−3, −2, 5} corresponds to the sum (−3) + (−2) + 5 = 0.

To answer whether some of the integers add to zero we can create an algorithm that obtains all the possible subsets. As the number of integers that we feed into the algorithm becomes larger, both the number of subsets and the computation time grows exponentially.

But notice that if we are given a particular subset, we can efficiently verify whether the subset sum is zero, by summing the integers of the subset. If the sum is zero, that subset is a proof or witness for the answer is "yes". An algorithm that verifies whether a given subset has sum zero is a verifier. Clearly, summing the integers of a subset can be done in polynomial time, and the subset sum problem is therefore in NP.

The above example can be generalized for any decision problem. Given any instance I of problem Π {\displaystyle \Pi } and witness W, if there exists a verifier V so that given the ordered pair (I, W) as input, V returns "yes" in polynomial time if the witness proves that the answer is "yes" or "no" in polynomial time otherwise, then Π {\displaystyle \Pi } is in NP.

The "no"-answer version of this problem is stated as: "given a finite set of integers, does every non-empty subset have a nonzero sum?". The verifier-based definition of NP does not require an efficient verifier for the "no"-answers. The class of problems with such verifiers for the "no"-answers is called co-NP. In fact, it is an open question whether all problems in NP also have verifiers for the "no"-answers and thus are in co-NP.

In some literature the verifier is called the "certifier", and the witness the " certificate ".