Mark/handouts
1
1
Fork 0
Code Pull Requests 2 Actions Packages Releases Activity

Cleanup

Browse Source
This commit is contained in:
Mark
2023-02-05 21:02:05 -08:00
parent e3e09c9c95
commit 494e91dc98
7 changed files with 574 additions and 612 deletions
Download Patch File Download Diff File Expand all files Collapse all files

64
Advanced/Lambda Calculus/parts/01 combinators.tex
View File

@@ -1,47 +1,39 @@
\documentclass[../main.tex]{subfiles}
\section{Combinators}
\definition{}
A \textit{free variable} in a $\lm$-expression is a variable that isn't bound to any input. \\
For example, $b$ is a free variable in $\lm a. b$. The same is true of $\star$ in any of the previous pages.
\begin{document}
A \textit{combinator} is a function with no free variables.
\section{Combinators}
\definition{The Kestrel}
\definition{}
A \textit{free variable} in a $\lm$-expression is a variable that isn't bound to any input. \\
For example, $b$ is a free variable in $\lm a. b$. The same is true of $\star$ in any of the previous pages.
Notable combinators are often named after birds.\hspace{-0.5ex}\footnotemark{} We've already met a few: \\
The \textit{Idiot}, $I = \lm a.a$ \\
The \textit{Mockingbird}, $M = \lm f.ff$ \\
The \textit{Cardinal}, $C = \lm fgx.(~ f(g(x)) ~)$ \\
A \textit{combinator} is a function with no free variables.
\footnotetext{Raymond Smullyan's \textit{To Mock a Mockingbird} is responsible for this.}
\definition{The Kestrel}
\vspace{2ex}
Notable combinators are often named after birds.\hspace{-0.5ex}\footnotemark{} We've already met a few: \\
The \textit{Idiot}, $I = \lm a.a$ \\
The \textit{Mockingbird}, $M = \lm f.ff$ \\
The \textit{Cardinal}, $C = \lm fgx.(~ f(g(x)) ~)$ \\
Another notable combinator is $K$, the \textit{Kestrel}:
$$
K = \lm ab . a
$$
\problem{}
What does the Kestrel do? Explain in plain English. \\
\hint{What is $(K~\heartsuit~\star)$?}
\footnotetext{Raymond Smullyan's \textit{To Mock a Mockingbird} is responsible for this.}
\vspace{2cm}
\vspace{2ex}
\problem{}
Reduce $(K~I)$ to derive the \textit{Kite}. How does the Kite compare to the Kestrel? \\
We'll call the Kite KI.
Another notable combinator is $K$, the \textit{Kestrel}:
$$
K = \lm ab . a
$$
\problem{}
What does the Kestrel do? Explain in plain English. \\
\hint{What is $(K~\heartsuit~\star)$?}
\begin{solution}
$\text{KI} = \lm ab . b$. \\
\end{solution}
\vspace{2cm}
\problem{}
Reduce $(K~I)$ to derive the \textit{Kite}. How does the Kite compare to the Kestrel? \\
We'll call the Kite KI.
\begin{solution}
$\text{KI} = \lm ab . b$. \\
\end{solution}
\vfill
\pagebreak
\end{document}
\vfill
\pagebreak