
Reducing Sequences In this article, we'll discuss the entropy reduction of a sequence by representing it as a binary tree which is multidimensional. In computer programming there are standard methods of representing the elements of a sequences with a binary tree whose elements are the nodes. The multidimensional tree [1] is the progressive expansion of sequences from a vector of elements, to a binary tree of nodes, and into forest of multiple overlapping of binary trees. In the context of building neural networks, a sequence is an ordered list of elements. It's a time ordered, Markov process which is embedded in a type of binary tree called a trie. You can visualize a trie as a two dimensional lattice or trellis. A three dimensional trie is an overlay of a pair of two dimensional tries. Propagating or traversing the 3D tree is equivalent to propagating each of the pairs of binary trees. The order of looping on 2D trees is 2 whereas the order on a 3D tree is 1. In practice, in computer programs we only propagate over 2D trees instead of multidimensional trees. The philosophy, or the vocabulary of the metaworld, for content and form are, respectively, parts and rules. The nodes at the vertices of trees are the parts of the forest, and the rules are the links in the tree of a rewriting system. [2] With respect to the programming language Lisp, the content and form are the variables and functions. The neural network is a dynamic system in which the content and form of the trees are represented by neurons and their connections. The computer simulation of this network is a recurrent temporally ordered Markov process. In using the Markov model for simulating neural nets you would, by trail and error, compute the probabilities for the state transitions. In the binary tree formulation you would use the structure or form of the connections between the nodes. By trail and error you would try to geometrically build the connections between the neurons. This is as simple as I can get it so far. If you have any ideas, whether from a philosophical, mathematical or programming view, and want to collaborate on this, please contact me at: glenn @ neuralmachines . com. [1] Multidimensional Trees, William Baldwin and George Strawn, 1991 [2] Pattern Recognition, Satosi Watanabe, 1985, p. 385

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