Ideas create reality.

Realizations What makes the P.I.P. theorem so interesting to explore even further has something to do with what I have realized throughout 7+ years. Here are my realizations so far: I was essentially working on AI even before I knew such thing existed P.I.P. as a generalization of Taylor's expansion RS-ALC as a generalization of the P.I.P. theorem Similarities with Category Theory Before going any further, let me first tell you that I am no expert in any of those theories/fields although I can spot multiple similarities between PIP and those. 1. Working on a part of something that I didn't even know about P.I.P. is all about the prediction of the next observation (obtained by a polynomial function) by somehow processing the history of all observations, and it can be essentially formulated as a lossless compression algorithm since its outputs never contradict the previous observations. It can also be viewed as a compression technique since you only need to keep the frontier o...

Background. When I was high schooling at TDV-BTL (Türkiye Diyanet Vakfı - Bakü Türk Lisesi), I happened to be thinking about functions and their properties during the 10/11$^\text{th}$ grade(s). I was mainly thinking about what happens when we change the formula of a function a little bit; do its outputs completely change by large offset or is it predictable what the function looked like before changing its formula syntactically just by looking at its new inputs and outputs? One day something interesting occurred to me as I was playing with arbitrary linear and quadratic functions - when I fixed the inputs and outputs on a table vertically, so that the (euclidean) difference between each successive input was the same, the consequent differences in the Y column did not change after some iteration, no matter how many rows(i.e., data points or x and y pairs) I added to the table as long as I respected the same-successive-difference rule on X column. For linear functions, I just observe...