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2023-10-04 09:42:09 -07:00
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14
Advanced/Lambda Calculus/parts/04 recursion.tex
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@ -12,7 +12,7 @@ We cannot re-create this in lambda notation. Functions in lambda calculus are \t
\vspace{1ex}
As an example, consider the statement $A = \lm a. A~a$ \\
As an example, consider the statement $A = \lm a. A~a$ \par
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}
@ -20,7 +20,7 @@ This means \say{write $(\lm a.A~a)$ whenever you see $A$.} However, $A$ is \text
\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. \\
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}
@ -28,12 +28,12 @@ This means \say{write $(\lm a.A~a)$ whenever you see $A$.} However, $A$ is \text
\end{instructornote}
\problem{}
Write an expression that resolves to itself. \\
Write an expression that resolves to itself. \par
\note{Your answer should be short and sweet.}
\vspace{1ex}
This expression is often called $\Omega$, after the last letter of the Greek alphabet. \\
This expression is often called $\Omega$, after the last letter of the Greek alphabet. \par
$\Omega$ useless on its own, but gives us a starting point for recursion.
\begin{solution}
@ -41,20 +41,20 @@ $\Omega$ useless on its own, but gives us a starting point for recursion.
\vspace{1ex}
An uninspired mathematician might call the Mockingbird $\omega$, \say{little omega}. \\
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. \\
This is the \textit{Y-combinator}, easily the most famous $\lm$ expression. \par
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? \\
What does this thing do? \par
Evaluate $Y f$.
\vfill