Overview: Our daily experience is that it is harder to solve problems than it is to check the correctness of solutions to these problems. Is this experience merely a coincidence or does it represent a fundamental fact of life (or a property of the world)? This is the essence of the P versus NP Question, where P represents search problems that are efficiently solvable and NP represents search problems for which solutions can be efficiently checked.

Another natural question captured by the P versus NP Question is whether proving theorems is harder that verifying the validity of these proofs. In other words, the question is whether deciding membership in a set is harder than being convinced of this membership by an adequate proof. In this case, P represents decision problems that are efficiently solvable, whereas NP represents sets that have efficiently verifiable proofs of membership.

These two formulations of the P versus NP Question are indeed equivalent, and the common belief is that P is different from NP. That is, we believe that solving search problems is harder than checking the correctness of solutions for them and that finding proofs is harder than verifying their validity.

Organization. The two formulations of the P versus NP Question are rigorously presented and discussed in Sections 2.2 and 2.3, respectively. The equivalence of these formulations is shown in Section 2.4, and the common belief that P is different from NP is further discussed in Section 2.7. We start by discussing the notion of efficient computation (see Section 2.1).