# What would be the philosophical implications of a solution to the P versus NP problem?

## What would happen if the P vs NP problem was solved?

If P equals NP, every NP problem would contain a hidden shortcut, allowing computers to quickly find perfect solutions to them. But if P does not equal NP, then no such shortcuts exist, and computers’ problem-solving powers will remain fundamentally and permanently limited.

## Can P vs NP be solved?

Although one-way functions have never been formally proven to exist, most mathematicians believe that they do, and a proof of their existence would be a much stronger statement than P ≠ NP. Thus it is unlikely that natural proofs alone can resolve P = NP.

## What is the implication of finding a proof that P NP?

A proof would involve finding a polynomial time algorithm for an NP-complete problem. And when you find one polynomial algorithm, you can use it to solve all other NP-complete problems by reducing the problems to a common form. This means that a proof for P=NP and algorithms that use it will appear at the same time.

## What is the P versus NP problem what happens if P NP How would this affect computing problems as we currently know them?

Roughly speaking, P is a set of relatively easy problems, and NP is a set that includes what seem to be very, very hard problems, so P = NP would imply that the apparently hard problems actually have relatively easy solutions.

## Can NP problems be solved in polynomial time?

NP stands for Non-deterministic Polynomial time. This means that the problem can be solved in Polynomial time using a Non-deterministic Turing machine (like a regular Turing machine but also including a non-deterministic “choice” function).

## Can NP-hard problems be solved in polynomial time?

Consequences. If P ≠ NP, then NP-hard problems could not be solved in polynomial time. Some NP-hard optimization problems can be polynomial-time approximated up to some constant approximation ratio (in particular, those in APX) or even up to any approximation ratio (those in PTAS or FPTAS).

## What is P vs NP problem Justify your answer include definitions of P and NP class?

P versus NP

It is not known whether P = NP. However, many problems are known in NP with the property that if they belong to P, then it can be proved that P = NP. If P ≠ NP, there are problems in NP that are neither in P nor in NP-Complete. The problem belongs to class P if it’s easy to find a solution for the problem.

## What is the class of decision problems that can be solved by non-deterministic polynomial algorithm?

4. _________ is the class of decision problems that can be solved by non-deterministic polynomial algorithms. Explanation: NP problems are called as non-deterministic polynomial problems. They are a class of decision problems that can be solved using NP algorithms.

## Can you explain P NP NP completeness & NP-hard briefly?

Problems of NP can be verified by a Turing machine in polynomial time.
Types of Complexity Classes | P, NP, CoNP, NP hard and NP complete.

Complexity Class Characteristic feature
NP-hard All NP-hard problems are not in NP and it takes a long time to check them.
NP-complete A problem that is NP and NP-hard is NP-complete.

## What is P problem what is NP problem and NP complete problem?

P is a set of problems that can be solved by a deterministic Turing machine in Polynomial time. NP is set of decision problems that can be solved by a Non-deterministic Turing Machine in Polynomial time.

## What is the relationship between the classes P and NP explain?

NP is set of problems that can be solved by a Non-deterministic Turing Machine in Polynomial time. P is subset of NP (any problem that can be solved by deterministic machine in polynomial time can also be solved by non-deterministic machine in polynomial time) but P≠NP.

## What is the relationship between P NP and NP-complete problems?

Difference between NP-Hard and NP-Complete:

NP-hard NP-Complete
NP-Hard problems(say X) can be solved if and only if there is a NP-Complete problem(say Y) that can be reducible into X in polynomial time. NP-Complete problems can be solved by a non-deterministic Algorithm/Turing Machine in polynomial time.

## What is the relationship among P NP NP-complete and NP-hard problems?

All other problems in class NP can be reduced to problem p in polynomial time. NP-hard problems are partly similar but more difficult problems than NP complete problems. They don’t themselves belong to class NP (or if they do, nobody has invented it, yet), but all problems in class NP can be reduced to them.