- #1
Dragonfall
- 1,030
- 4
I don't understand why if P vs NP can't be decided in ZFC, then there exist "near" polynomial time algorithms for NP. How would that possibly work?
P and NP are classifications used in computer science to categorize problems based on the time it takes for a computer to solve them. P stands for "polynomial time," meaning that the amount of time it takes to solve a problem grows at a reasonable rate as the size of the input increases. NP stands for "nondeterministic polynomial time," meaning that the problem can be verified in polynomial time, but the solution may not be found in polynomial time.
An undecidable problem is one that cannot be solved by a computer in any amount of time. This means that there is no algorithm that can be used to determine if a solution exists or not. In other words, there is no way to write a computer program that can always give a correct answer for every possible input.
If P vs NP is proved to be undecidable, it would have major implications for computer science and mathematics. It would mean that there are problems that cannot be efficiently solved, even with the most advanced technology. It would also challenge the widely believed idea that all problems can ultimately be solved with enough computing power.
Yes, there are several real-world applications of P vs NP being undecidable. One example is in cryptography, where the security of many encryption algorithms relies on the assumption that P and NP are not equal. If P vs NP is undecidable, it would mean that these algorithms may not be as secure as previously thought. Additionally, many optimization problems in fields such as logistics and scheduling would become much more difficult to solve if P vs NP is undecidable.
P vs NP is still an open problem in computer science and mathematics. While there have been attempts to prove its undecidability, no definitive answer has been reached. It remains one of the most famous unsolved problems in computer science, and its resolution would have a significant impact on the field.