handouts/Advanced/Lambda Calculus/parts/01 combinators.tex

\section{Combinators}

\definition{}
A \textit{free variable} in a $\lm$-expression is a variable that isn't bound to any input. \par
For example, $b$ is a free variable in $\lm a. b$. The same is true of $\star$ in any of the previous pages.

A \textit{combinator} is a function with no free variables.



\definition{The Kestrel}

Notable combinators are often named after birds.\hspace{-0.5ex}\footnotemark{} We've already met a few: \par
The \textit{Idiot}, $I = \lm a.a$ \par
The \textit{Mockingbird}, $M = \lm f.ff$ \par
The \textit{Cardinal}, $C = \lm fgx.(~ f(g(x)) ~)$

\footnotetext{Raymond Smullyan's \textit{To Mock a Mockingbird} is responsible for this.}

\vspace{2ex}

Another notable combinator is $K$, the \textit{Kestrel}:
$$
	K = \lm ab . a
$$



\problem{}
What does the Kestrel do? Explain in plain English. \par
\hint{What is $(K~\heartsuit~\star)$?}

\vspace{2cm}

\problem{}
Reduce $(K~I)$ to derive the \textit{Kite}. How does the Kite compare to the Kestrel? \par
We'll call the Kite KI.

\begin{solution}
	$\text{KI} = \lm ab . b$.
\end{solution}

\vfill
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