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	\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.

	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: \\
	The \textit{Idiot}, $I = \lm a.a$ \\
	The \textit{Mockingbird}, $M = \lm f.ff$ \\
	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. \\
	\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? \\
	We'll call the Kite KI.

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

	\vfill
	\pagebreak


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