Lambda edits

Advanced/Lambda Calculus/parts/04 recursion.tex

@@ -35,21 +35,24 @@ Write an expression that resolves to itself. \par
 \vspace{1ex}
 
 This expression is often called $\Omega$, after the last letter of the Greek alphabet. \par
-$\Omega$ useless on its own, but it gives us a starting point for recursion.
+$\Omega$ useless on its own, but it gives us a starting point for recursion. \par
 
 \begin{solution}
 	$\Omega = M~M = (\lm x . xx) (\lm x . xx)$
 
-	\vspace{1ex}
-
+	\vspace{1mm}
+	
 	An uninspired mathematician might call the Mockingbird $\omega$, \say{little omega}.
 \end{solution}
 
+
 \vfill
+\pagebreak
+
+
 
 \definition{}
-This is the \textit{Y-combinator}, easily the most famous $\lm$ expression. \par
-You may notice that it's just $\Omega$, put to work.
+This is the \textit{Y-combinator}. You may notice that it's just $\Omega$ put to work.
 $$
 	Y = \lm f . (\lm x . f(x~x))(\lm x . f(x~x))
 $$
@@ -58,5 +61,27 @@ $$
 What does this thing do? \par
 Evaluate $Y f$.
 
+
 \vfill
-\pagebreak
+
+\definition{}
+We say $x$ is a \textit{fixed point} of a function $f$ if $f(x) = x$.
+
+\problem{}
+Show that $Y F$ is a fixed point of $F$.
+
+\vfill
+
+\problem{}
+Let $\theta = (\lm xy . y(xxy))$ and $\Theta = \theta \theta$. \par
+Let $N = \Theta F$ for an arbitrary lambda expression $F$. \par
+
+Show that $F N = N$.
+
+\vfill
+
+\problem{Bonus}
+Find a fixed-point combinator that isn't $Y$ or $\Theta$.
+
+\vfill
+\pagebreak