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:PROPERTIES:
:ID: 8f265f93-e5fd-4150-a845-a60ab7063164
:END:
#+title: recursion
#+author: Preston Pan
#+html_head: <link rel="stylesheet" type="text/css" href="../style.css" />
#+html_head: <script src="https://polyfill.io/v3/polyfill.min.js?features=es6"></script>
#+html_head: <script id="MathJax-script" async src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js"></script>
#+startup: latexpreview
#+OPTIONS: broken-links:t
#+description: A description of recursive hierarchies in everything.
* Recursion is Recursion
Exactly as I say in the title.
** but what is recursion?
Recursion.
** No, seriously, what is it?
Self reference.
** haha, very clever, it's not like that joke has been made 10 million times
Yeah, but I think it's a good introduction to the subject. You can think of recursion
as [[id:42dbae12-827c-43c4-8dfc-a2cb1e835efa][self-assembly]] and it has deep connections to topics such as [[id:b005fb71-2a16-40f9-9bb6-29138f4719a2][emergence]]. I will first
describe it in a mathematics context, and then a programming context.
For demonstration purposes, I will use my own programming language, [[https://ret2pop.nullring.xyz/blog/stem.html][Stem]] (warning: link takes you outside of mindmap).
Again, stem is a prerequisite as it is the standard programming language in the mindmap.
* [[id:a6bc601a-7910-44bb-afd5-dffa5bc869b1][Mathematics]] Describes Recursion
For this example, I will be using the [[id:aed6b5dc-c2ec-4e8c-b793-538cd5d6e355][factorial]]. One might define it like so:
\begin{align*}
f: \mathbb{N}\rightarrow\mathbb{N}\ s.t. \\
f(0) = 1 \\
f(n) = nf(n - 1)
\end{align*}
in other words, we want a [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] defined over [[id:2d6fb5ac-a273-4b33-949c-37380d03c076][natural numbers]] that is one when the input is zero,
and otherwise multiplies the input with a copy of itself, only the input is one less. Let's try evaluating
this [[id:b1f9aa55-5f1e-4865-8118-43e5e5dc7752][function]] at $x = 3$.
\begin{align*}
f(3) = 3f(3 - 1) = 3f(2) \\
f(2) = 2f(1) \\
f(1) = 1f(0) \\
f(0) = 1
\end{align*}
once we substitute $f(0) = 1$ in, you will see it all collapses.
\begin{align*}
f(0) = 1 \\
f(1) = 1f(0) = 1 \times 1 = 1 \\
f(2) = 2f(1) = 2 \times 1 = 2 \\
f(3) = 3f(2) = 3 \times 2 = 6
\end{align*}
and so the result is multiplying $3 \times 2 \times 1 \times 1 = 6$. If you observe what we did, you'll see that we started
by trying to replace unknown variables by trying to evaluate $f(x)$ one number down, and eventually we reach
a "base case" -- zero. As soon as the "base case" occurs, we then "go back up" by replacing all the unknown
values with known ones -- and that's how we evaluate recursive functions.
* Programming Describe Recursion
In stem, a factorial implementation might look like this:
#+begin_src stem :exports both
factorial [ dup 0 <= [ 1 + ] [ dup 1 - factorial * ] if ] def
5 factorial .
#+end_src
#+RESULTS:
: 120
and in stem, we can print out the stack every step of the way with the builtin word ~?~:
#+begin_src stem :exports both
factorial-debug [ dup 0 <= [ 1 + ] [ ? "\n" . dup 1 - factorial-debug ? "\n" . * ] if ] def
5 factorial-debug .
#+end_src
#+RESULTS:
#+begin_example
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#+end_example
as you can see, the stack is slowly built up to have all of the numbers needed, and then when we reach the basecase (the base case being the condition
that doesn't cause recursion in the if statement), in which case we "go back up" by multiplying and going back up the stack. This procedure of using a stack
is present in all programming languages, although in stem the operations are transparent as the stack is accessible by regular program users. In short, we keep
on going down and down until we hit the bottom, base case, in which case we have all the pieces we need in order to go back up again, where the stack stores
the information from most recent tasks to be done and we work back up in order to do the less recent tasks.
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