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February 12th, 2015, 01:01 AM   #1
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Defining a new symmetry group - n dimensional permutation invariance of lambda

Symmetry group - Wikipedia, the free encyclopedia

Edit Distance and Pascals Simplex both do this in any number of dimensions, are both examples of algorithms which use this symmetry group.

Edit distance - Wikipedia, the free encyclopedia
Given any 2 text, or more generally sequences of bits (or if you must think in text, "0" and "1" are on the keyboard even if they cost 8-16 bits in memory that way)... Edit Distance finds the smallest set of changes to 1 of them which results in the other, allowing the remove of any letter/digit (like push backspace or delete) or adding a new letter/digit (0 or 1). Removing or adding has a cost, which reduces the value of the "energy function" of Edit Distance. At each square in a 2d grid, when filled in Breadth First by Manhatten Distance (down movements + right movements), so you get a triangle increasing by 1 square thick on its (perpendicular to the common kind of) diagonal for all those squares simultaneously. Each square calculated next depends only on the square to its left and the square above it. That is the symmetry group I am defining. It gets the same result regardless of if its parameters are leftSquare then upSquare or upSquare then leftSquare. Any permutation of those gets the same lambda function calculated and stored in the next square downright of 2 other squares.

Pascal's simplex - Wikipedia, the free encyclopedia
The simple common case of 2 dimensions is Pascal's triangle - Wikipedia, the free encyclopedia and has the same flow of updating things in a 2d grid as the Edit Distance algorithm.

Clique (graph theory) - Wikipedia, the free encyclopedia and NP-complete - Wikipedia, the free encyclopedia
Now we come to the subject of a million dollar millenium prize. Hopfield/boltzmann energy function is, intuitively to those who can hold it in their mind, evidence P not equal NP because its not hillclimbable and gets stuck in local minimum. The clique math problem is also in this symmetry group, in as many dimensions as n nodes. Each lambda function is updated based on the n cells behind it in n perpendicular directions. For example, if n is 2, its the same directions of cells as in Pascals Triangle and Edit Distance. The symmetry group doesnt say what lambda function is run on those n other lambda functions (by currying n times); It only says that it returns the same lambda when given the same n parameters in any order (n factorial number of possible permutations), but only for any n lambda functions that it could have generated from an earlier n dimensional recursion.

n dimensional recursion, the unique property of this symmetry group. In Edit Distance and Pascals Triangle, n is 2 because each cell's value is defined by the values in 2 cells (up and left) before it. Pascals Simplex, Edit Distance, and Clique all expand to any number of dimensions the same way. P equals NP if any lambda function of those as fast as P describes, given each possible order of dimensions, choose what cell to calculate next using this symmetry group. Each next cell's value depends on at first fewer cells which are in a clique or not in a clique but being explored to determine which are in some other set of nodes which are a bigger clique. Whatever its path of exploration and logic that guides the order of cells to explore, it will find the clique after all 2^n cells are calculated, the last being the corner of the 2x2x2x2x2... n dimensions hypercube of 2 cells on each side. The question of P equals NP is, can it be found in every possible case faster than 2^n, faster than exponential, something more like n^5 or n^3000?

Matrix multiplication - Wikipedia, the free encyclopedia
To most experts surprise, what had been done in a triple loop (3 dimensions of counting) for many years was found to take less than size^3 time, and repeatedly faster algorithms where found and today its around size^2.3 instead of 3. This is another sample of, at least something related to, the symmetry group in how it appears to depend on each cell depending on adjacent cells in 3 dimensions. I'm not sure what lambda function does that, but the shape of the problem appears that way.

Less directly related to... RSA (cryptosystem) - Wikipedia, the free encyclopedia and Chinese remainder theorem - Wikipedia, the free encyclopedia and Modular arithmetic - Wikipedia, the free encyclopedia RSA uses a symmetry in integers x^y and y^x circling around the multiply of 2 other primes. The church encoding of modular arithmetic is the normal church encoding for arithmetic where parameter function returns next higher integer or 0 if equal to the multiply of those 2 primes, and the other parameter is any integer in that range. Those 2 parameters are normally 0 and +1 function to start, and then they become other numbers as arithmetic is done relative to the current sum/multiply/etc, and those do not ring. The strength of RSA could therefore be proven, whatever it may be, by a brute force search of such lambda functions on all RSA keys up to a small integer whose kolmogorov complexity exceeds the smallest definition of NPComplete or any chosen measure of difficulty to prove it harder than. The symmetry of it is, like x+y=y+x, "x^y and y^x circling around the multiply of 2 other primes", and its not directly related to the symmetry group defined above, but it appears to be somewhere branched from a few levels deeper recursion of it. The symmetry group is on the level of adding multidimensional array indexs. The next level is multiplying indexs. Then RSA is the level of exponent indexs.

Less directly related to... Octave - Wikipedia, the free encyclopedia describes each adjacent 12 keys on a musical keyboard, each 12th next key 2 times the frequency. The related duplication in sine waves of any frequencies in "cooley tukey fast fourier transform" (FFT) Cooley–Tukey FFT algorithm - Wikipedia, the free encyclopedia is optimized from observations^2 to observations*logBase2(observations), both of which by todays standards ignore how many digits are in the observed numbers (like 2 64 bit floating points in a complex number), but the point about FFT is it folds a symmetry onto itself repeatedly.
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