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November 19th, 2013, 06:34 PM   #1
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Degree of freedom in signal processing (gravity) NP Complete

Clique Cover is NP Complete, interchangible with all other NP Complete problems which are a core mystery of computing and less directly related to physics theory as they think quantum math should have its own complexity classes because of some kind of magic "qu" in the qubits which as I see it disappears when you use ordinary bits to define the complex numbers, pascals triangle, and other math related to how you get hyperspheres and hyperbells from coin flips but that's all they can really do without putting information in. Wave particle duality is the only thing the set of all random bits can do.

n nodes
Does it have a clique cover of c cliques (which are incompatible with the other cliques by at least 1 node)?

Exactly when it fits on the surface of a c dimensional hypersphere, and not a c-1 dimensional hypersphere, of radius 1, and with distance constraint held between each pair of nodes lacking an edge (cant have both at once, goes in only 1 clique and pushes all the others away), held to a distance of sqrt(2) which is simply between the middle of the hypercube faces where we put the nodes (the diagonal between 2 poles if you think of c dimension axises as poles).

Imagine a sphere with a triangle of 3 nodes. If all 3 are a clique, they can move around their whole 2-lorentz(epsilon) dimensional surface, but if theres an edge between 2 of them then the other can go to one or the other but they will force any number of other nodes away if any breaks the distance constraint, nomatter how long the chain of constraints is. If I'm near you and you're near a bad node, I'm near the bad node and you either stay with me without it or go with it, depending on which clique (in the clique cover) you can fit in, and usually theres alot of combinations that can fit, but this is an exact statement... clique cover of c on c dimensional hypersphere (or c-1-lorentz(epsilon) if you only count the surface not the flat space its radius is in).

About "2-lorentz(epsilon)", which you don't need to know but its an interesting related thing.
(or we can say 2 dimensional if you don't care about that missing qubit between hyperbell and alternating hyperspheres, but that's another subject, multiply height of 2 bell curves and you get a circle of constant density at each radius, divide by that circle and you only have half a bell curve left (is radius negative or positive?), so we can either say negative radius like a lightcone or some kind of parity or dimensional recursion bit... needs research.)

We might find useful in researching ways to crunch the nodes into the smaller dimensions, than number of nodes, hypersphere surface, (after the universe ends in heat death unless P is practically close to NP or equal, which I've not given up on) but to do it faster...
aod(eachVector) setEqualTo aod(eachVector)/sumForAllVector(aod(thatVector)*dotProductBetweenS elfAndThatVector^2) where all vector are length 1 from hypersphere center to point on its surface. If you run this approximately log number of times it converges to the integer number of dimensions the hypersphere is embedded in, the total aods for all vectors sum (Amount Of Dimension), exact if maximally automorphic spread of vectors (like carbon in diamond is 4 way in 3d or like an equilateral triangle spread around a circle, and including every sine wave or discrete spread of its peaks/valleys as in what fourier measures).

Hyperbell and hypersphere are interchangible. Every ring of complex numbers, as transformed by fourier (especially fractional fourier which is continuous, unitary, and interval divisible by pi, and as I see it, its just asking to be pigeonholed by a binary search cutting between its hypersphere dimensions as given and a duplicate to store the output in then fast fourier in on it recursively if you can keep the network of data points balanced and working both directions while you slide, but that's for some other time).

The universe is an infinite dimensional hypersphere surface, or hyperbell is equal. Those madscientist or philosopher enough to take it seriously will find it very useful that every point in the universe has exactly the same number of automorphisms as every other and in exactly the same combinations, or in other words, the universe is nothing at all until you observe it as your inverse (equal and opposite force in every way imaginable).

Gravity is simply what happens when information exists, like 2 bits get entangled to be not equal (xor) or any combination of many of them, and the wavefunction of the universe is profoundly changed from complete balance and cancelling out (hypersphere surface) to there being more possible states where you fall a little in that direction. Instead of an infinite line (bellcurve view instead of hypersphere) you get an orbit of finite length, or at least an infinite number of reals minus those few bits in length which you lost by observing a specific information, which you now fall toward. You dont need gravitons for gravity. Gravity is any unbalance from all possibilities together cancelling out, any observation of information, which transforms your view of reality from an infinite dimensional and very boring (if you could be aware of the boredom in such a nonexistence) hypersphere surface to potentially any system of waves as we see around us.

That hypersphere of 2 dimensions per frequency which fourier's input and output signal fits on spread evenly, is the universe and the midpoint of gravity which you should inverse from if we can ever find a way to represent it in specific logic.

Degree of freedom in signal processing (gravity) NP Complete because clique cover of c cliques fits on the surface, with distance constraints between pairs lacking an edge, of (c-1)-sphere (c flat dimensions) and not the next lower dimension (ways to define fractional dimensions are a subject to get into later).

I'm considering using this in my open source AI system which will be on the Internet as soon as possible, for things like to represent thoughts as a growing network of permutations which partially overlap eachother and compare as analogies, among the 1024 something-like-a-qubit which are each a dimension of hypersphere and pixel on screen displaying position as brightness (so all possible things that could be on screen (TODO 3 times more for color) are some unit vector on that hypersphere. I would encode patterns as a bunch of cliques size 300 or so, and it has to be lightning fast for the graphics. I need to get thousands of near maxcliques per second to cluster ideas in a way thats relevant to what people paint on screen with the mouse and it dreams back a prediction/response to fit into your ideas and what its recently thought about.. in progress. You could use boltzmann machines for near clique finding but its energy function is a sum of nands of the lacking edges, not the same thing you get with distance constraints and hypersphere degrees of freedom... the clique would randomly move while others are held in place.

Imagine what we could do with that dimension lacking by holding radius constant. We have c cliques in c points, but a surface of c-1 (also minus lorentz(epsilon) unless you believe you can crush a flat paper onto a sphere without losing a little wrinkle piece of dimension, so Poincare is not about the 3-sphere, its about slightly less dimensions than that)... Anyways, this is where the real potential for P equals NP may be, that it fits in less dimensions than cliques sitting together at those same points on hypersphere surface, blocking eachother but allowing within each clique to be at the same position. I will not pay c dimensions of calculation for what fits in c-1, and if it fits in c-1 whats to stop us from taking our c-1 dimensional structure, whatever form we find works for that potential reduction, and asking what is the clique cover of it? You can't have a bigger clique cover than dimensions. But thats very speculative for now. We just know what complexity class these things are in.
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November 21st, 2013, 05:43 AM   #2
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Re: Degree of freedom in signal processing (gravity) NP Comp

Actually, what I should have said is Minimum Clique Cover, a set of cliques covering all nodes once each which has the least cliques of all ways to do that, because nomatter what you do there will always be at least that many things that have to be perpendicular to eachother. The useful thing about a node-dimensional hypersphere surface is if you have each clique moving around randomly in its own lower dimensional hypersphere (of as many dimensions as that clique size) then for each pair of hyperspheres we have 2 continuous surfaces whose every point is the same distance to every point on the other hypersphere. If you move randomly in both, you have not gotten closer or farther to any part of any other hypersphere. Even if you travel an infinite distance of all paths in each hypersphere individually, you have travelled no distance relative to the other hyperspheres. The distance to them is always sqrt(2) a quarter turn. But where are these hyperspheres is the question... They should be whichever pairs of nodes have a random spread of dot product (blurring around sqrt(2) distance said the flat way) because... Theres no randomness anywhere else, all gone from holding those lacking an edge as perpendicular. If we knew how to exactly hold them perpendicular.
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