1. A decision problem is a computational problem that can be posed as a yes-no question on an infinite set of inputs. It typically involves determining whether a given input satisfies a specific property or condition. Examples include deciding if a number is prime or if one number evenly divides another.
2. For a decision problem to be decidable, there must exist an algorithm that can solve it in a finite number of steps for any given input. This means there is a systematic procedure that will always terminate and provide a definitive yes or no answer. Undecidable problems, on the other hand, have no such algorithm that can solve them for all possible inputs.
3. The class P (Polynomial time) consists of decision problems that can be solved by a deterministic Turing machine in polynomial time relative to the input size. This means that for an input of size n, the algorithm's running time is bounded by a polynomial function of n. Problems in P are considered efficiently solvable and include tasks like sorting, searching in a sorted list, and determining if a number is prime. The class NP (Nondeterministic Polynomial time) contains decision problems whose solutions can be verified in polynomial time by a deterministic Turing machine. NP problems have the characteristic that if a solution is proposed, it can be quickly checked for correctness. Examples include the Boolean satisfiability problem and the traveling salesman problem.
4. The "P versus NP" question asks whether every problem whose solution can be quickly verified (NP) can also be quickly solved (P). In other words, it seeks to determine if P = NP. This problem is considered one of the most important open questions in computer science and mathematics. If P were equal to NP, it would imply that problems currently believed to be computationally difficult (like many optimization problems) could actually be solved efficiently.
5. Resolving the P versus NP question would likely result in significant financial and professional rewards. The Clay Mathematics Institute has offered a $1 million prize for its solution as part of the Millennium Prize Problems. However, the true value of solving this problem extends far beyond this monetary reward. The solver would gain immense prestige in the academic community, likely securing top positions at leading institutions and research labs. Furthermore, if the solution shows that P = NP, it could lead to revolutionary algorithms with enormous commercial value, potentially resulting in substantial wealth through patents and applications in various industries.
