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Say screw it and publish polynomial article

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Danila Fedorin 2023-05-22 21:42:32 -07:00
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content/blog/search_polynomials.md
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--- ---
title: "Search as a Polynomial" title: "Search as a Polynomial"
date: 2022-10-22T14:51:15-07:00 date: 2023-05-22T21:39:00-07:00
draft: true
tags: ["Mathematics"] tags: ["Mathematics"]
discussionRooms: ["!qLoehEvJNRndNrdlyU:matrix.org"] discussionRooms: ["!qLoehEvJNRndNrdlyU:matrix.org"]
--- ---
@ -486,3 +485,21 @@ two trips of equal druation but different length (two-hour trip from A to B
followed by a two-hour trip from B to C, or one-hour trip from A to C followed followed by a two-hour trip from B to C, or one-hour trip from A to C followed
by a three-hour trip from B to C). The first trip wins out, since it requires by a three-hour trip from B to C). The first trip wins out, since it requires
only \\(3.0\\) units of distance. only \\(3.0\\) units of distance.
### Anything but Routes
So far, all I've done can be reduced to variations on a theme: keeping track
of some aspects of a trip between cities, using polynomials for structure.
However, that's just the beginning. This sort of trick can be be made even
more powerful by further relaxing the notion of a "polynomial". By doing so,
we can make our polynomials represent arbitrary _effects_ (in the computer
science sense -- things like errors, logging to a console, storing and
accessing information from a database). Relying for just a little longer on
our example of journeys between cities, we might be able to represent trips
with random variation (traffic can be unpredicatable!), or maybe cities where
you will get stuck. But the point isn't routes: the same approach can be
used to represent traversing a binary tree, performing Prolog-like proof
search, or evaluating a non-deterministic program. The sky's the limit!
Unfortunately, doing so would require even more background and buildup, for
which I just don't have space for in this article. I'll save these
things for next time, though -- stay tuned!