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Function to Multiway system ?

Posted 1 year ago
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Hey everyone, So I try to make sense of :

Traditional Riemannian manifolds are full of structure that our hypergraphs do not have. Nevertheless, we are beginning to see that there are analogs of many ideas from geometry and calculus on manifolds that can be applied to our hypergraphs—at least in some appropriate limit as they become sufficiently large. To continue the analogy, consider trying to define a function on a hypergraph. For a scalar function, we might just assign a value to each node of the hypergraph. And if we want the function to be somehow smooth, we should make sure that nearby nodes are assigned similar values. But what about a vector function? An obvious approach is just to assign values to each directed edge of the hypergraph. And given this, we can find the component in a direction corresponding to a particular geodesic just by averaging over all edges of the hypergraph along that geodesic. (To recover results for continuous spaces, we must take all sorts of potentially intricate limits.) (At a slightly more formal mathematical level, to define vectors in our system, we need some analog of a tangent space. On manifolds, the tangent space at a point can be defined in terms of the equivalence class of geodesics passing through that point. In our systems, the obvious analog is to look at the edges around a point, which are exactly what any geodesic through that point must traverse.) 127 For a rank-p tensor function, we can assign values to p edges associated either with a single node, or with a neighborhood of nearby nodes. And, once again, we can compute “projections” of the tensor in particular “directions” by averaging values along p geodesics. The gradient of a scalar function ∇f at a particular point X can be defined by starting at X and seeing along what geodesic the (suitably averaged) values decrease fastest, and at what rate. The results of this can then be assigned to the edges along the geodesic so as to specify a vector function. The divergence of a vector function ∇.f can be defined by looking at a ball in the hypergraph, and asking for the total of the values of the function on all hyperedges in the ball. The analog of Gauss’s theorem then becomes a fairly straightforward “continuity equation” statement about sums of values on edges inside and at the surface of part of a hypergraph. Page 127 from : https://www.wolframphysics.org/technical-introduction/inc/Wolfram-ModelsForPhysics.pdf

How can one turn a function to rule and so into a multiway graph then causal graph. Is it a empiric process or can you take a give function and turn it into a rule or a given space/function and turn it into rule.

For example Mandelbrot set which turn out to be something like(if i remember well) {x,y} -->{z,z} {x,z} {y,z}

Or can I combine rule like the one which describe a straight line and replace it into the rule of Mandelbrot set. And so build from the already discovered rule a new system which contain those 2 rules...

Or can I take a function which describe an hypergraph like in https://arxiv.org/pdf/1901.00899.pdf (with combinatory) And turn it into a rule ?

Or can I turn a Turing machine into a set of rule ?

Visual summary: https://www.wolframphysics.org/visual-summary/img/visual-summary-darkmode-4k.jpg

3 Replies

Try to encode Goodstein's function into a rewriting rule of a hypergraph and you will know the main difficulty of the problem.

Posted 1 year ago

So if I try to find a rule which mimic the exponential function or any other function(in Euclidean space) it's impossible?

In a word, yes, because emergent properties are, by definition, not derivable by any means other than direct observation (which some might even argue is entirely subjective).

Put another way, the only way to detect any emergent mathematical pattern(s) of behavior from any complex system is to:

a) detect some isomorphic correspondence between what you want to find, and what is observed from the complex system as it evolves, then

b) calculate an error function capable of producing a metric for the difference between what you're looking for, and what is observed in the evolving complex system, then

c) if the error function is significant, tweak or replace the rules of the complex system, then go back to step (a).

If the above looks suspiciously like the scientific method, that's because it basically is. And just as ambiguous, as it turns out (if not more so), because there are (currently?) no guidelines, much less rules, for how to ""tweak" the rules of a complex system to better "fit" what you're looking for. Neither is there (currently?) any guidelines for how one might go about "detecting" emergent mathematical patterns in any given complex system (again, highly subjective, possibly very much so).

Alternatively, you might consider taking a hint from modern-day Cosmology / Fundamental Physics, and changing step (c) to:

c) if the error function is significant, announce to the world that you have discovered a new "particle" and call it "Dark ______" (fill in the blank), the properties of which are exactly defined by the aforementioned error function.

Regardless, implementation of the above could theoretically be accomplished by using the aforementioned "error function" as an inverse fitness function in a large-scale Evolutionary Computation.

Piece of cake, really, if you happen to have a couple of unused networked supercomputers lying around.. ;)

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