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	\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

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