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March 7th, 2013, 08:36 PM   #1
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Lambda Tensor - a minimalist general computing math operator

LambdaTensors are the analog expansion of discrete Lambda Calculus.

Lambda Tensors are proven Turing Complete because Lambda Calculus is a subset at the tensor level of vectors and a subset of that where all vectors have position 1.0 in any dimension and position 0 in all other dimensions. Every dimension is a Lambda Tensor.

To call a LambdaTensor on another LambdaTensor, you call all its parts on all parts of the parameter. When both are Lambdas (vectors with position in only 1 dimension), everything the Lambdas do is multiplied by 1. When its more complex tensors, it does weighted sums (at vector level) of what each dimension of LambdaTensor returns or affine transforms (at the next level, vectors of vectors, also known as networks or 2d matrix), and so on.

What would you use a LambdaTensor for? We use them all the time.

At cardinality 0, the first LambdaTensor is nothing, void, null, no degrees of freedom, unity, all possibilities cancelling out to zero.

At cardinality 1, we have normal numbers, 1 amount that applies to all things equally.

At cardinality 2, we have vectors, a different number for different objects, like prices for each item in a store or a direction or position in 3d space.

At cardinality 3, we have vector fields. A picture is a vector field mapping x and y position to red, green, and blue, 2 input dimensions and 3 output.

I need this new kind of math I call LambdaTensors because people think in too many dimensions, which we call words, but we don't understand how to think in less dimensions so we can't explain what words mean in terms of math or anything simpler. Only complexity is in most people's minds. I need LambdaTensor to build a new kind of search engine that we play with as a flowing dream-like space on the Internet, not searching words or files, instead searching for simple shapes and colors relative to eachother, like parts of pictures or blobs of color in higher dimensions to represent more complex ideas.

Feature Vectors in AI theory are LambdaTensors that look at another LambdaTensor (parameter) and return a LambdaTensor level 1 (a scalar number, the "hot and cold" game, more or less like the feature being measured or hill climbed in an exploration of the high dimensional space.

Think of particles as bell curves that overlap eachother in some places and extend out to infinity in all directions but with quickly lowering density as you get farther away. Their amount of overlap between each pairs of particles would be at the network tensor level. We could evolve LambdaTensors to describe various properties of groups of particles, parts of pictures, or any object in the world or part of math, and use those Feature Vectors to define a high dimensional space where all these things can be organized and searched using the Vector Space Model which is at the core of most search engine theory and simply means that in a space of descriptions of kinds of things, you search for a query vector in that space and get search results of whichever vectors are near. Google, for example, does this probably in a variety of ways, to turn the text you type into a high dimensional vector and give you back the closest website links in that space. Each website, picture, and other thing on the Internet has some position in the high dimensional space.

Its very simple. We've been using LambdaTensors without knowing it. We don't call it that, but if we start thinking in terms of analog computing, the hard emotionless brittle cold calculating ways of normal computers can instead flow with subtleties we all find so intuitive that we rarely talk about or try to understand these parts of our minds.

Words are very high dimensional, but we should learn to walk before we run, to define what is the simplest possible model of an idea and build an Internet where addresses are ideas instead of text, like a Wikipedia where we edit the shapes and patterns of ideas to define them as a network of other ideas, visually on screen like high dimensional playdough.

There is no supercomputing grid, no amount of raw power or brute force strategies, which can compete with the accurate use of recursive math to do deep learning the right way, even on a slow computer. Our brains run at most 200 cycles per second. When we react to something in half a second, like steering a car away from someone jumping in front (as Google must program driverless cars to do), every calculation in our head to make that work is at most a path of 100 neurons, many in parallel but none longer than 100. If we find it hard to program for decentralized networks like grids of cell processors, its because of the obsession with large sequential calculations, while peer to peer (no central parts) is the future, not because you can download copyrighted things for free, instead because peer to peer is how the universe works, how cell processors work, what makes Wikipedia converge toward democracy of all Human knowledge, and its how the next generation of Internet will have to work to scale up. We have to remove the bottlenecks in many parts of our world. LambdaTensors are a universal math operator, a way to think about the world in a unified way, a minimalist foundation we can build on, an open ended space of possibilities.

This is what LambdaTensors can do. The next step is to find a practically efficient way to calculate them.
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March 7th, 2013, 09:59 PM   #2
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Re: Lambda Tensor - a minimalist general computing math oper

I then searched for lambdatensor and found that a Lisp software exists of that name, which only has 4000 search results on Google which means its importance, if the software actually implements the full power of the unification of lambda and tensor, has been hidden in such technical jargon and optimization algorithms that its not been seen in our everyday lives and its potential as a new form of flowing visual communication. I take the fact that their LambdaTensor software is for calculating quantum gravity as confirmation that my research leading to this has not been completely insane, has some important reality to it, and that its going to work when I build the shared Internet space where we can play with it, instead of letting it go to waste only used for the most technical things. Lambda Tensor is the only thing it could be accurately called because that's what it does in math words, and as it is fundamental to all possible theories of computing, physics, and AI theory, it is Obvious (a legal word) and Unavoidable in any serious pursuit of the simplest model of everything, therefore not patentable (we always need a legal defense when proceeding on important paths others may want to take from the world). We should look into that software, but what I'm planning is optimized for a different purpose, a massively multiplayer paint program as a shared high dimensional space on the Internet, and my version of Lambda Tensors (excluding versions before 1.2.2 as they only lead to this unification path) is called Physicsmata (a public domain open source software). You don't really understand anything until you derive it, or to summarize Socrates, I know that I know nothing (Lambda Tensor cardinality 0) and from that I start to know one thing. Quantum Gravity is simpler than Newtonian physics for the same reason 2+2=4 is simpler than 2+2=5. One is true. The other is not and leads to deep exploration into never ending contradictions. The universe instead quines around in the space of all possibilities leaving no loose ends.
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March 8th, 2013, 07:57 PM   #3
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Re: Lambda Tensor - a minimalist general computing math oper

You use the word "dimension" here in a number of different ways, some ways incompatible with the others. That makes it very difficult to understand what you are saying.
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