What is desicion problem?

he decision problem in computer science refers to a fundamental question related to the solvability of computational tasks or problems. It is a formal way to ask whether there exists an algorithm or a procedure that can determine whether a given input satisfies a particular property or condition. The decision problem is typically framed as a yes-or-no question. Given an input, the goal is to decide whether the answer is "yes" (the input satisfies the specified property) or "no" (the input does not satisfy the specified property).

What does it mean to desicion probem to be decidable?

In computer science and the theory of computation, a decision problem is said to be "decidable" if there exists an algorithm (a computational procedure) that can correctly determine the answer for all possible inputs to the problem in a finite amount of time. In other words, if a decision problem is decidable, it means that there is a systematic way to solve it, and the algorithm will always produce the correct result within a reasonable amount of time for any input.

What is the class P and what is the class NP?

In theoretical computer science, the classes P and NP are two important complexity classes that classify decision problems based on their computational complexity and the efficiency of algorithms that can solve them. These classes play a significant role in the study of algorithmic efficiency and are central to discussions about the tractability of problems.

Class P (Polynomial Time):

• The class P consists of decision problems for which there exists a deterministic Turing machine that can solve the problem in polynomial time with respect to the size of the input.
• In simpler terms, if a problem is in P, it means that there exists an efficient algorithm that can solve it, and the running time of this algorithm grows at most polynomially as the input size increases.
• Problems in P are considered tractable or efficiently solvable. Common examples include sorting, searching, and various mathematical computations.

Class NP (Nondeterministic Polynomial Time):

• The class NP consists of decision problems for which, given a proposed solution, you can verify in polynomial time whether the solution is correct.
• In other words, if a problem is in NP, it means that if someone claims to have a solution, you can efficiently check if that claim is valid.
• It's important to note that NP does not necessarily imply that finding a solution is also efficient; it only concerns the verification of solutions.
• Many important problems in computer science, such as the Traveling Salesman Problem and the Knapsack Problem, are believed to be NP-complete, meaning they are among the most challenging problems in NP.

What is intiutive meaning of the "P versus NP" question?

The "P versus NP" question, in simple terms, asks whether every problem that is easy to verify (in NP) is also easy to solve (in P). In other words: Is every problem with a quickly checkable solution also quickly solvable from scratch? If the answer is "yes," then P equals NP, and it implies that efficient algorithms exist for a wide range of problems, including many currently believed to be difficult. This would have significant implications for fields like cryptography, optimization, and more, as it could potentially break the security of many encryption schemes and make complex optimization problems efficiently solvable.

If you solve the P versus NP question, how much richer will you be?

If I were to prove that P = NP it would disrupt entire field of computing creating many security challanges to the humanity. The will be high need to the development of cryptography as result new efficient algorthms that can solve high-level encryptions.