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For people who like Maths ( P versus NP problem )

Zyphullen

The Millennium Prize Problems P versus NP problem - I've been searching and thinking about this for quite some time, maybe pushing on over a year, and just felt like giving my unfiltered views on it. So P = NP is a very interesting math puzzle that's trying to see if making computational searches can be reduced down to less steps. If P is equal to NP, that means it would be possible to hack anything per se, due to encryption relying heavily on this not being true, e.g., P not equal to NP. (But no-one has been able to fully prove this to be impossible.) But my logic is buzzing me and telling me the opposite, that P can become equal to NP. Reason for my thinking, and taking an abnormal approach: it doesn't matter for internet security – because even if you say could hack into anything knowing this method in the internet's current form, it won't matter. I have a solution to this problem – taking from cloud-based gaming tech, e.g., (you can play high-end games on a 2000 desktop – as long as you have good internet). Since cloud gaming streams the game to you and it's not running on your desktop, you're just watching it, but it also allows inputs to be sent to the server only when allowed. Now, if we used this tech for security on every site with sensitive data, it would be impossible to hack or retrieve the data. Why? Because let's say you were seeing someone else's desktop from a webcam only* – could you on your keyboard access that desktop's data and copy and paste it over? Not what's showed on the screen, what's stored in the hardware behind what you're viewing. So you wouldn't be able to touch the backend code or even the frontend code; only thing you could do at most is shut down the stream. But with Twitch and YouTube live over the years, the tech has gotten better where a stream can't easily be shut down. Yes, it would cost more data streaming every website, but internet speed has increased massively recently with major breakthroughs in fiber options without needing to change the physical form.

So back to why I think P = NP: maths sometimes is very very limiting, set rules you must follow, and I feel sometimes it lacks outside-the-box thinking. Since the real world physics where everything is moving in the whole universe, it's not static. Bit of backstory leading to this thought: I was working on trying to solve the 3x3 magic square of squares some months ago, starting from very basic random.range in all 9 cells numbers to find this while trying to understand the puzzle and get my head around the rules. Back then when I started, I didn't even know what a squared number was, or better said the term. So I did end up thinking a match 8 after some optimizing of said random script to be less random, but not all 9 numbers were perfect square numbers, which breaks the puzzle. And a 8 match with none squared numbers was found during the pyramids times, built into a rock table in museums. But the point of the magic square of squares is it is possible to have a match 8 where all 9 numbers in the 3x3 grid are distinct squared numbers, e.g., 11, 22, 33, 44 etc. So once I got my head around that, I completely rebuilt the whole script – took like a month to get my head around going from optimized random numbers brute force checks – skipping ahead 2 months, my scripts became so optimized for efficiency and set accuracy I could check in some patterns from 1 million to 86 billion numbers check per min, going from 1 counter++ next repeat loop from at start struggling to check the first 1000 numbers while running the script all night to taking 7 full days none stop leaving my desktop running algorithms all night long to find an already single known pattern on a set TargetSum to checking every number between 1 - 1 billion over 12 hours. I optimized the speed of the script many many times, having sayings like "I NEED MORE POWER" since I was limited to my desktop, I kept finding faster and faster ways to speed up the brute force search, stored all my data in Excel Google Doc with 50 different tabs for every known pattern I found myself to help find patterns to the patterns I found that are none other known examples on the internet other than the ones I posted about.

But my point of the above journey (was not done alone, I had an old friend who helped me understand the maths and teach me his A-level maths; he had to dumb a lot of things down for me since I have no education and took me forever to get my head around how I'm going to make a script to brute force check. But a lot mathematicians don't just scripts, and ofc before computers they couldn't, so I do have a massive advantage using technology to try to solve problems. But my habit is always using brute force to try to learn things; I struggle to learn from studying, kinda why no education. I can only do what I want to do per se, which is bad very bad in the real world, kinda why I'm poor and unemployable! But that's why I'm posting in an autistic forum anyways.

Back onto P = NP is from the none stop finding ways to improve the speed at which a script doing the same thing runs! Why not for anything else, like let's say I had a data center the size of the universe, well my script but run infinitely faster. This is what I mean by maths can be limiting in the real world: you can use tech/build tech to find things faster and faster infinitely, but it's having that idea how can I do it. There is no single algorithm or pattern that covers every possibly; everything needs its own unique approach to keep speeding it up. Anything and everything can be improved no matter how little or how much, but sometimes it would take so much effort to improve you might as well work on the smaller easier things at first till that difficult one becomes easy in itself. Like now we have cars, see them everyday but do we really understand what we have? 200 years ago horse and cart and train it took days to weeks to months to years travel from place to place of which we can do anywhere on earth in less than a day! But not all publicly might point being P != NP through logic of the real world maybe on paper with limitations, but let's say I tied you up and say type on the keyboard to solve a problem... well possible but would take quick a bit more time. So it's less about the steps it takes and more about Time itself. Even if it's more steps, say 1 billion steps but another method takes 10 steps but them 10 steps takes 2 hours but them 1 billion steps can be done in 1 nanosecond it doesn't matter p = NP! We're looking at it the wrong way.

Anyways what do you think?

Also nothing is impossible and there is a chance i have miss-understood some parts of the problems at it's core, but knowing that will also help me understand it on a deep level with more knowledge overtime I'm getting there slowly i think, please correct me! it's part of my learning process.

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  • Zyphullen said:
    But my logic is buzzing me and telling me the opposite, that P can become equal to NP.

    It seems not too unexpected to me that P is different to NP, which I translate concretely as "It is easier to check that a Sudoku was correctly solved than to find the solution." 

    • In that particular case, we look at dimensionality. Finding a solution requires more work of checking neighbours etc. Checking seems more straightforward, there is less to consider. This would make P != NP intuitive. 
    • But one could ask, what if there is a rule? I think it is possible but it is easier to think of a counter proof. What if some processes can not be improved further? (this goes below.)

    Another intuition for me is related to predictability. I take predictability as a similar case to solving the puzzle, because it requires searching possibilities. But your question here may be: is there a formula, so we don't need to search? So if you accept my framing that P=NP requires us to find a "trick" to calculate fast, then below there is a sort of contradiction.

    We could look at computational irreducibility:

    Computational irreducibility is the concept that certain computational processes cannot be simplified, meaning the only way to determine their outcomes is to perform each step of the computation.

    For example, if you run a cellular automata (like the Game Of Life by Conway) there are configurations for which we can not predict the configuration at a subsequent step, unless we run it. In this case, we can't even check fast. But my point here is that limitations are there.

    This could mean that there are cases for which there is no shortcut (no symmetry nor clever technique we can exploit); it does not mean that P=NP is false by default, but it makes it possible to understand that some processes do not have wild reductions to a very simple case.

    I'm curious to read your thoughts.

  • in reply to Deleted User

    Just recently, I've been spending every night solving one Killer Sudoku puzzle before bed—really cool you mentioned that! So, I did have a thought while doing the puzzle a couple of nights ago: I could take a screenshot of the book with the empty grid and paste it into Grok/AI to see if it could solve it (which it massively failed to do!). But there's no reason it couldn't be trained like a chess engine to solve them, or there might already be an auto-solver made. But then the reverse could happen instead: NP ≠ P because now NP takes longer than P steps! In reality, in the existence of time itself, everything in this universe is moving nonstop—time never stands still—while math alone is static (then we can run loops).

    But let's say, in fact, the NP at the core still took many more steps; in reality, to the human perspective, it took longer to check if the solution was right than for the program to run and solve it, so if we take Time* into the equation and weight the both between steps taken and time it takes to complete each step then if it takes me as a human longer to check the solution then solving it then NP is therefore faster that maybe also disproves my thinking of P = NP, but I'm thinking along the lines dose this problem really matter? i don't believe there is one-in-all equation to solve any problem quickly every problem will have it's own solutions to working faster in less time/more steps or more time / less steps.

    I think there have been too many rules implemented into this thought process that reinforce the difficulty, but we have climbed the wrong branch of thinking. since if we always purely look at steps and steps alone for real world then NP ≠ P, but the point polynomial time vs Nondeterministic Polynomial time - we take this into the real world and it changes

    since like humans running speed vs car speed - the car takes more steps to build ( extracting materials from the ground, power the machines to build it the humans who need to put it together ect ect ) but then can go faster and reusable ( unlike a lot of rockets ), so back to the computer science of it - lets say i have a CPU the Size of the whole universe ignore physics limitations then i try to brute force hack anything NP might take more steps but it's just cracked it in 1 second! so moving away from the whole problem all together and reaching in what were the problems* and befits* be of said P = NP and fix them problems.

    problem is i reversed my reversely of my own logic several times there.. hmmm maybe I'm taking this to far into real world scenarios but then again at the end of the day it is the real world since a data center physically hard drive holding said data from being stolen still exists. or if you want to solve something faster don't just look at the steps alone time it more important.

    maybe I'm waffling to much... and went off track myself.. hmm time to sit and think more!

    Edit: To summarize: There's no single equation that fits all solutions with a small number of steps, but crank it to near-infinite steps/compute (e.g., evolving AI → AGI → ASI), it's not even AGI yet and it can speed up everything making it feel like less steps then me manually coding - make a script to search for x,y,z - speeding it up even more, who cares if it takes more steps if in real life i get the solution faster.

Reply
  • in reply to Deleted User

    Just recently, I've been spending every night solving one Killer Sudoku puzzle before bed—really cool you mentioned that! So, I did have a thought while doing the puzzle a couple of nights ago: I could take a screenshot of the book with the empty grid and paste it into Grok/AI to see if it could solve it (which it massively failed to do!). But there's no reason it couldn't be trained like a chess engine to solve them, or there might already be an auto-solver made. But then the reverse could happen instead: NP ≠ P because now NP takes longer than P steps! In reality, in the existence of time itself, everything in this universe is moving nonstop—time never stands still—while math alone is static (then we can run loops).

    But let's say, in fact, the NP at the core still took many more steps; in reality, to the human perspective, it took longer to check if the solution was right than for the program to run and solve it, so if we take Time* into the equation and weight the both between steps taken and time it takes to complete each step then if it takes me as a human longer to check the solution then solving it then NP is therefore faster that maybe also disproves my thinking of P = NP, but I'm thinking along the lines dose this problem really matter? i don't believe there is one-in-all equation to solve any problem quickly every problem will have it's own solutions to working faster in less time/more steps or more time / less steps.

    I think there have been too many rules implemented into this thought process that reinforce the difficulty, but we have climbed the wrong branch of thinking. since if we always purely look at steps and steps alone for real world then NP ≠ P, but the point polynomial time vs Nondeterministic Polynomial time - we take this into the real world and it changes

    since like humans running speed vs car speed - the car takes more steps to build ( extracting materials from the ground, power the machines to build it the humans who need to put it together ect ect ) but then can go faster and reusable ( unlike a lot of rockets ), so back to the computer science of it - lets say i have a CPU the Size of the whole universe ignore physics limitations then i try to brute force hack anything NP might take more steps but it's just cracked it in 1 second! so moving away from the whole problem all together and reaching in what were the problems* and befits* be of said P = NP and fix them problems.

    problem is i reversed my reversely of my own logic several times there.. hmmm maybe I'm taking this to far into real world scenarios but then again at the end of the day it is the real world since a data center physically hard drive holding said data from being stolen still exists. or if you want to solve something faster don't just look at the steps alone time it more important.

    maybe I'm waffling to much... and went off track myself.. hmm time to sit and think more!

    Edit: To summarize: There's no single equation that fits all solutions with a small number of steps, but crank it to near-infinite steps/compute (e.g., evolving AI → AGI → ASI), it's not even AGI yet and it can speed up everything making it feel like less steps then me manually coding - make a script to search for x,y,z - speeding it up even more, who cares if it takes more steps if in real life i get the solution faster.

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