\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
