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