Knowledge Representation

When using Semantic MediaWiki to represent knowledge, it would be nice to learn about Category Theory, and use its super simple linguistic construct, directed arrow, to represent data. Arrow as a symmetry breaking structure, provides universal expressiveness to encode any information, therefore, is being used to represent knowledge. To learn more about how Category Theory can help represent knowledge, please watch the following videos.

Category Theory for Beginners: Everyday Language

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Category Theory for Beginners: Machine Executable Programs

The following segment of video (just click on the image, and it will jump to 32'10", where the segment talks about computer programs represented by ologies.) shows how to use Category Theory to represent computable programs. This following video segment defines a kind of recursive algorithm using commuting diagrams. The original explanation text of this video segment is shown below: (text extract from Richard Southwell's Youtube Channel)

In the last scene I use recursion to represent a function h which takes an input (n,x) and outputs (0,f^n (x)) where f^n (x) is the element of S obtained by taking x and applying f to it n times. This can be seen as follows. We call the bottom left inclusion function in0. We call the right inclusion function in1. We call the top inclusion function in2.  When n is greater than 0 we can think of (n,x) as belonging to the top left type. Since (h after in2)=k, we have (h after in2)((n,x))= h((n,x))= (k((n,x)) = h((n-1,f(x)) in this case. Also, since (h after in0)=in1 we have (h after in0 ((n,x))=h((0,x))=in1((0,x))=(0,x). So if n is greater than zero than h((n,x))= h((n-1,f(x)). Also h((0,x))=(0,x).  So when we start with a positive n, we have that h keeps applying f to the right entry, while reducing the left entry by one. This continues until the left entry is zero, when it outputs the result. 

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Category Theory for Beginners: Knowledge Representation

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