Added Crypto & Lambda handouts

Advanced/Lambda Calculus/parts/04 recursion.tex

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+\documentclass[../main.tex]{subfiles}
+
+\begin{document}
+
+	\section{Recursion}
+
+	Say we want a function that computes the factorial of a positive integer. Here's one way we could define it:
+	$$
+		x! = \begin{cases}
+			x \times (x-1)! & x \neq 0 \\
+			1 & x = 0
+		\end{cases}
+	$$
+
+	We cannot re-create this in lambda notation. Functions in lambda calculus are \textit{anonymous}, which means we can't call them before they've been fully defined.
+
+	\vspace{1ex}
+
+	As an example, consider the statement $A = \lm a. A~a$ \\
+	This means \say{write $(\lm a.A~a)$ whenever you see $A$.} However, $A$ is \textit{inside} what we're rewriting. We'd fall into infinite recursion before even starting our $\beta$-reduction!
+
+	\begin{instructornote}
+		We're talking about recursion, and \textit{computability} isn't far away. At one point or another, it may be good to give the class a precise definition of \say{computable by lambda calculus:}
+
+		\vspace{4ex}
+
+		Say we have a device that reduces a $\lm$ expression to $\beta$-normal form. We give it an expression, and the machine simplifies it as much as it can and spits out the result. \\
+
+		\vspace{1ex}
+
+		An algorithm is \say{computable by lambda calculus} if we can encode its input in an expression that resolves to the algorithm's output.
+	\end{instructornote}
+
+	\problem{}
+	Write an expression that resolves to itself. \\
+	\note{Your answer should be short and sweet.}
+
+	\vspace{1ex}
+
+	This expression is often called $\Omega$, after the last letter of the Greek alphabet. \\
+	$\Omega$ useless on its own, but gives us a starting point for recursion.
+
+	\begin{solution}
+		$\Omega = M~M = (\lm x . xx) (\lm x . xx)$
+
+		\vspace{1ex}
+
+		An uninspired mathematician might call the Mockingbird $\omega$, \say{little omega}. \\
+	\end{solution}
+
+	\vfill
+
+	\definition{}
+	This is the \textit{Y-combinator}, easily the most famous $\lm$ expression. \\
+	You may notice that it's just $\Omega$, put to work.
+	$$
+		Y = \lm f . (\lm x . f(x~x))(\lm x . f(x~x))
+	$$
+
+	\problem{}
+	What does this thing do? \\
+	Evaluate $Y f$.
+
+	\vfill
+	\pagebreak
+
+\end{document}