P = Np Problem Puns

So I recently learned about the method for proving a problem is NP-compete, namely

1. Convert problem from optimization problem to "equivalent" decision problem
2. Difficulty upper-bound: Show the problem is in NP
3. Difficulty lower-bound: Show it's in NP-Hard (i.e. HC∝TSP) by creating an algorithm that does the following:
   1. Reduce a known NP-Complete problem instance to an instance of the problem in question.
   2. Runs in polynomial time

My question is about the lower-bound part. If we don't currently know whether P = NP = NP-Complete, why is it even possible to show that NP-hard is the lower bound for the difficulty of the problem?

If P = NP, wouldn't the logic here break?

Both would be the empty set

Here's my understanding of the subset sum problem: You have a set of n integers, and you'd like to know if there is a subset of those integers that sums to some number k.

Specifically, if there were an O(n) algorithm for any given value of k, would that be enough to show that P = NP, even if it was not polynomial time with respect to k? I assume no, but I'm just trying to understand more about how a specific problem like this can be used to prove or disprove P = NP.

(apologies if this question doesn't even make sense - it's entirely possible that I'm just fundamentally misunderstanding something)

I'm about halfway through my quite-large phase 4 factory, which I shamelessly planned out using Satisfactory Tools (amazing tool). During this process, my co-op buddy and I have gotten into some intense debates about the use of production calculators, which has lead me down the thought rabbit hole about the solvability of the problems that production calculators solve.

P vs NP basically looks at whether or not a given problem can be solved efficiently or quickly. Without alternate recipes, I am fairly certain that Satisfactory production calculations are more-or-less a "P" problem, or one that can be solved quickly.

However, once you add in alternate recipes, outputs of existing factories, resource limits, weighted resource value, and different definitions of "best", I suspect that production calculations become an "NP" problem. Not only are NP problems tedious to solve by hand, they are impossible to solve in a way that is analytically efficient. The only way to solve these problems is to try different permutations and then compare the results.

Do we have any math nerds here who can definitively confirm or deny this hypothesis?

Title says it all really,

Potentially a very dumb question, but are there any problems where finding a valid solution can be done in Polynomial time but checking a solution cannot be done in Polynomial time?

A local minimum is where our best theories are close, but if we would back up and try it from a different perspective we could get closer. Unfortunately most scientists hate backing up and starting anything over.

Scientific method:

• See something strange

• Think of possible causes and call each a hypothesis

• Figure out what would happen if each hypothesis is true or false

• Do experiments to see which of those expected effects happpen, and start to have more confidence in hypothesis which implies those observations.

Scientific method is correct for hill climbable puzzles, but NPComplete puzzles are only hill climbable if P equals NP. So far nobody can solve the harder of NPComplete problems efficiently because they are so tangled. They wrap you around and around in many dimensions until you meet yourself in so many permutations that you have no idea what to do. This is where we get confused what algorithms to write, and where the scientific method falls into local minimums.

Here's a real world example of NPComplete problem: If you had a text file listing all pairs of facebook friends, find the biggest clique, a group of people who all friend eachother. Facebook's algorithms may get close, and they are good at recommending who you may know to send friend requests to, and they may not care about that last bit of accuracy, but its a severe flaw in the scientific method to only be able to look into specific groups within facebook instead of somehow calculating it all at once NPComplete style.

Thats why I only use scientific method for simple problems but must use NPComplete theory for anything advanced. Maybe theres a way to write a second level scientific method thats compatible with NPComplete?

So to prove that P = NP, we need an algorithm/method/way to solve an NP problem, such as 3SAT, in polynomial time or less, for all possible instances of the problem, with no regard to the number of clauses, or the number of variables, or the application of the instance in real life.

Now resolution, as I understand it is, if a clause is of the format: P(t),x and another is Q(t),-x, then the resolution of these two clauses is P(t),Q(t). and the instance is UNSAT, if there exists a clause where P and Q are empty, and is SAT if all possible P's and Q's are generated and none of them satisfy the condition that both are empty in the same clause, i.e. the empty clause.

So why/how/where is this wrong? and Pardon me for any mistakes I made.

Thank you

What to do now? Say for example that I even managed to apply it to the 3SAT problem, so that it could essentially be solved in O(n), aka in polynomial time, what is the next step?

I understand that logically it would be to write a paper and publish it, but here comes several of my problems, where they mainly start off this that I to some extent just don't want it completely out there, just for anyone and everyone to be able to see the proof, since if I understand correctly, this may turn this discovery from a helpful one to a hurtful one.

I am sure you of all people on Reddit would know both sides of the coin, and that we wouldn't be able to reap the benefits of this discovery without unfortunate malicious side products, which is why I am not completely certain that just posting it online would be the best move.

Nevertheless, when I thought about it a bit more I discovered that, don't all discoveries have that same property? That they could all be maliciously exploited? Sure they may not all have the same magnitude, but they still do have it, and what was the result? We still published them, and I believe that its because the benefits out weigh the negatives, and that we only think about it with P vs NP, because the negatives are actually well known, more prone to usage, related to all of us, and will affect almost everyone.

I guess one could say that its simply a major trolley problem, where our issue lies in that both tracks have so many people we can't even see which has more so that we could even take that into consideration. Its not like the numbers make it easier, but if we were to go with the same logic as that, we choose the most beneficial route in saving the most number of people it would help then. You could also say that we shouldn't publish at all, i.e. stop the train, but wouldn't that just delay the train? The train is inevitable if it is the case that P=NP, its just a matter of time before someone could prove it, in result sooner or later we will face the same issue once again.

So kind of like a TLDR summary, if it was that I have a constructive proof that P=NP, applied to the 3SAT problem, how could I publish it in a way that we could benefit the most from it and decrease the number of ways it could maliciously be used?

Having read the post about "Specializing in Problems we Don't Understand", now I want a good grad level introduction to this topic.

My basic level: I know calc through PDEs, Decent Linear Alg (del, double del, etc). Stats is fine (I TAed a Stats course for more than a year -- maybe I graded your work?). I have taken all my grad school maths through a reputable university Econ department so far. They were hard, but I just never came across P / NP. So, what's a great book I can spend a month or so on and walk away understanding it pretty well?

N = 1, duh

> As far as the entropy problem in economics. As I see it the same principle applies to P vs NP. if you take complexity far enough, then P=NP. So that's what we try to do. If you look at N as limited by real-world human needs, then it's within the realm of human possibility. As an example, the traveling salesman problem seems to be NP, but CSW solved it by creating a financial incentive using R-Puzzles. Things like cracking codes aren't really NP problems because there isn't a moral human need to do so. There is no shortcut to helping people with their human needs. The job of engineers is to solve real-world problems, not fantasy. Nothing in science has ever been created that hasn't been observed already in nature. So for all intents and purposes, N has no purpose being defined as anything that cannot be already observed in nature. Bitcoin as a global financially based supercomputer can solve real-world NP problems that single individuals have great difficulty doing so. CSW discovered that a financial system solves human problems. Adam Smith can suck it.

I'm having trouble understanding what it actually means for a P problem to be reduced to NP-Complete, but I have seen from other online answers that it is possible. So my question is, is it possible and if so, what does it mean?

It p = np was proven true, how would this actually help us determine the algorithms to solve non-polynomial time problems in polynomial time? Just because it’s proven that they could be reduced, that doesn’t necessarily mean we would be any closer to figuring out how to actually reduce them. Are we assuming that the proof for p = np would give us some grand insight that would show us how to reduce the problems?

This would be the holy grail for a security researcher, but, surprisingly, when me and my pals came up with the hypothetical, we couldn't find a way to actually make decent cash with it. Surely there must be some huge systems relying on the premise that cryptography actually works who could be exploited for cash?

A zero-day exploit broker offers to buy this exploit from you for $5 million, but you believe that you could make more by keeping it to yourself.

Can you do that without going to prison?

Hashing isn't NP-hard : Reversing hashes and finding collisions in Sha-2 is O(1). Bitcoin also hashes the public keys, so unless I'm mistaken, Bitcoin is off the table!

If P = 0 or N = 1

So to prove that P = NP, we need an algorithm/method/way to solve an NP problem, such as 3SAT, in polynomial time or less, for all possible instances of the problem, with no regard to the number of clauses, or the number of variables, or the application of the instance in real life.

Now resolution, as I understand it is, if a clause is of the format: P(t),x and another is Q(t),-x, then the resolution of these two clauses is P(t),Q(t). and the instance is UNSAT, if there exists a clause where P and Q are empty, and is SAT if all possible P's and Q's are generated and none of them satisfy the condition that both are empty in the same clause, i.e. the empty clause.

So why/how/where is this wrong? and Pardon me for any mistakes I made.

Thank you

P = NP

[divide both sides by P]

N = 1

Where do I apply to collect my US$1,000,000 Millenium Prize?

:Theorem:

Consider the set of sets P such that P is the empty set ∅

Then the intersection of P is NP:

P = ∅ ⟹ ⋂P = NP

where NP is the universe

_________________________

Intersection of Empty Class, in the same form:

Let NP be a basic universe

Let ∅ denote the empty class

Then the intersection of ∅ is NP:

⋂∅ = NP

_________________________

:Proof:

Let P = ∅

Then from the definition:

⋂P = {x : ∀X ∈ P : x ∈ X}

Consider any x ∈ NP

Then as P = ∅ , it follows that:

∀X ∈ P : x ∈ X

from the definition of vacuous truth

∀x : P(x) ⟹ NP(x)

It follows directly that:

⋂P = {x : x ∈ NP}

That is:

⋂P = NP

Well, since P are problems that are relatively simple to solve, they are actually "No Problem" to solve, i.e. P = No Problem = NP. Hence, P = NP. I really don't get what gets everyone so worked up about this.

:Theorem:

Consider the set of sets P such that P is the empty set ∅

Then the intersection of P is NP:

P = ∅ ⟹ ⋂P = NP

where NP is the universe

_________________________

Intersection of Empty Class, in the same form:

Let NP be a basic universe

Let ∅ denote the empty class

Then the intersection of ∅ is NP:

⋂∅ = NP

_________________________

:Proof:

Let P = ∅

Then from the definition:

⋂P = {x : ∀X ∈ P : x ∈ X}

Consider any x ∈ NP

Then as P = ∅ , it follows that:

∀X ∈ P : x ∈ X

from the definition of vacuous truth

∀x : P(x) ⟹ NP(x)

It follows directly that:

⋂P = {x : x ∈ NP}

That is:

⋂P = NP

What to do now? Say for example that I even managed to apply it to the 3SAT problem, so that it could essentially be solved in O(n), aka in polynomial time, what is the next step?

I understand that logically it would be to write a paper and publish it, but here comes several of my problems, where they mainly start off this that I to some extent just don't want it completely out there, just for anyone and everyone to be able to see the proof, since if I understand correctly, this may turn this discovery from a helpful one to a hurtful one.

I am sure you of all people on Reddit would know both sides of the coin, and that we wouldn't be able to reap the benefits of this discovery without unfortunate malicious side products, which is why I am not completely certain that just posting it online would be the best move.

Nevertheless, when I thought about it a bit more I discovered that, don't all discoveries have that same property? That they could all be maliciously exploited? Sure they may not all have the same magnitude, but they still do have it, and what was the result? We still published them, and I believe that its because the benefits out weigh the negatives, and that we only think about it with P vs NP, because the negatives are actually well known, more prone to usage, related to all of us, and will affect almost everyone.

I guess one could say that its simply a major trolley problem, where our issue lies in that both tracks have so many people we can't even see which has more so that we could even take that into consideration. Its not like the numbers make it easier, but if we were to go with the same logic as that, we choose the most beneficial route in saving the most number of people it would help then. You could also say that we shouldn't publish at all, i.e. stop the train, but wouldn't that just delay the train? The train is inevitable if it is the case that P=NP, its just a matter of time before someone could prove it, in result sooner or later we will face the same issue once again.

So kind of like a TLDR summary, if it was that I have a constructive proof that P=NP, applied to the 3SAT problem, how could I publish it in a way that we could benefit the most from it and decrease the number of ways it could maliciously be used?