If the solution to a problem is easy to check for correctness, must the problem be easy to solve?
For instance, it is easy to check if a filled sudoku grid is a valid solution. Must it therefore be easy to solve sudokus?
Most people would probably intuitively answer "no", and most computer scientists agree, but this has still not been proven, so we actually don't know.
Well, there's counterfactual examples of this, so it must not be true.
In pretty much every single relationship worldwide, one person can very easily determine if the recommendation from the other for where to eat or what to watch is correct or not.
And yet successfully figuring out where to eat or what to watch is nigh impossible.
I think there's a fighter further* problem, it may be true and we just don't know the easy way to do it.
That’s actually the simplest and clearest description of the P/NP problem I’ve ever read.
I think my favorite troll statement to a mathematician/comp scientist is:
"Actually, P > NP - there exist problems where it's harder to verify a solution than to arrive at one"
Isn't it proof enough? Using the Sudoku example: there are certainly different levels of difficulties, depending on how many numbers are set in the beginning and other parameters. Checking if the solved answer is correct, is always the same "difficulty" - thus there is no correlation between the difficulty of the puzzle at the beginning and checking the Correctness. Some people might not be able to solve it, but they certainly can check if the solution is right
Isn’t it proof enough?
Unfortunately no. The question is a simplification of the P versus NP problem.
The problem lies in having to prove that no method exists that is easy. How do you prove that no matter what method you use to solve the sudoku, it can never be done easily? You'll need to somehow prove that no such method exists, but that is rather hard. In principle, it could be that there is some undiscovered easy way to solve sudokus that we don't know about yet.
I'm using sudokus as an example here, but it could be a generic problem. There's also a certain formalism about what "easy" means but I won't get into it further, it is a rather complicated area.
Interestingly, it involves formal languages a lot, which is funny as you wouldn't think computer science and linguistics have a lot in common, but they do in a lot of ways actually.
You can solve any sudoku easily by trying every possible combination and seeing if they are correct. It'll take a long time, but it's fairly easy.
Well it just so happens that the definition of "easy" in the actual problem is essentially "fast". So under that definition, checking every single possible solution is not an "easy" method.
What if the sudoku is 1 milllion lines by 1 million lines? How about a trillion by a trillion? The answer is still easy to check, but it takes exponentially longer to solve the board as the board gets larger. That's the jist of the problem: Is there a universal solution to a problem like this that can solve any size sudoku before the heat death of the universe?
For the purposes of OPs problem (P v NP), it considers not particular solutions, but general algorithmic approaches. Thus, we consider things as either Hard (exponential time, by size of input), or Easy (only polynomial time, by size of input).
A number of important problems fall into this general class of Hard problems: Sudoku, Traveling Salesman, Bin Packing, etc. These all have initial setups where solving them takes exponential time.
On the other hand, as an example of an easy problem, consider sorting a list of numbers. It's really easy to determine if a lost is sorted, and it's always relatively fast/easy to sort the list, no matter what setup it had initially.
Most people would probably intuitively answer "no", and most computer scientists agree, but this has still not been proven, so we actually don't know.
I disagree, I think most computer scientists believe that P != NP, at least when it comes to classical computers. If we believed that P = NP, then why would we bother with encryption?
EDIT: nvm, I misread it.
I think you've misunderstood 😅. Answering "no" to that question corresponds to P != NP (there are problems that are easy to verify but not easy to solve), while "yes" means P = NP (if a solution is easy to check, the problem must be easy to solve). So I am saying most people and most scientists believe P != NP exactly as you say.
If the solution to a problem is easy to check for correctness, must the problem be easy to solve?
For instance, it is easy to check if a filled sudoku grid is a valid solution. Must it therefore be easy to solve sudokus?
Most people would probably intuitively answer "no", and most computer scientists agree, but this has still not been proven, so we actually don't know.
Well, there's counterfactual examples of this, so it must not be true.
In pretty much every single relationship worldwide, one person can very easily determine if the recommendation from the other for where to eat or what to watch is correct or not.
And yet successfully figuring out where to eat or what to watch is nigh impossible.
I think there's a
fighterfurther* problem, it may be true and we just don't know the easy way to do it.That’s actually the simplest and clearest description of the P/NP problem I’ve ever read.
I think my favorite troll statement to a mathematician/comp scientist is:
"Actually, P > NP - there exist problems where it's harder to verify a solution than to arrive at one"
Isn't it proof enough? Using the Sudoku example: there are certainly different levels of difficulties, depending on how many numbers are set in the beginning and other parameters. Checking if the solved answer is correct, is always the same "difficulty" - thus there is no correlation between the difficulty of the puzzle at the beginning and checking the Correctness. Some people might not be able to solve it, but they certainly can check if the solution is right
Unfortunately no. The question is a simplification of the P versus NP problem.
The problem lies in having to prove that no method exists that is easy. How do you prove that no matter what method you use to solve the sudoku, it can never be done easily? You'll need to somehow prove that no such method exists, but that is rather hard. In principle, it could be that there is some undiscovered easy way to solve sudokus that we don't know about yet.
I'm using sudokus as an example here, but it could be a generic problem. There's also a certain formalism about what "easy" means but I won't get into it further, it is a rather complicated area.
Interestingly, it involves formal languages a lot, which is funny as you wouldn't think computer science and linguistics have a lot in common, but they do in a lot of ways actually.
You can solve any sudoku easily by trying every possible combination and seeing if they are correct. It'll take a long time, but it's fairly easy.
Well it just so happens that the definition of "easy" in the actual problem is essentially "fast". So under that definition, checking every single possible solution is not an "easy" method.
What if the sudoku is 1 milllion lines by 1 million lines? How about a trillion by a trillion? The answer is still easy to check, but it takes exponentially longer to solve the board as the board gets larger. That's the jist of the problem: Is there a universal solution to a problem like this that can solve any size sudoku before the heat death of the universe?
For the purposes of OPs problem (P v NP), it considers not particular solutions, but general algorithmic approaches. Thus, we consider things as either Hard (exponential time, by size of input), or Easy (only polynomial time, by size of input).
A number of important problems fall into this general class of Hard problems: Sudoku, Traveling Salesman, Bin Packing, etc. These all have initial setups where solving them takes exponential time.
On the other hand, as an example of an easy problem, consider sorting a list of numbers. It's really easy to determine if a lost is sorted, and it's always relatively fast/easy to sort the list, no matter what setup it had initially.
I disagree, I think most computer scientists believe that P != NP, at least when it comes to classical computers. If we believed that P = NP, then why would we bother with encryption?
EDIT: nvm, I misread it.
I think you've misunderstood 😅. Answering "no" to that question corresponds to P != NP (there are problems that are easy to verify but not easy to solve), while "yes" means P = NP (if a solution is easy to check, the problem must be easy to solve). So I am saying most people and most scientists believe P != NP exactly as you say.
Reading comprehension is hard my bad.
Edit: wait no, it's "easy" I'm just dumb.