42 lines
1.1 KiB
TeX
Executable File
42 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 $(\lm a.a)~b$.
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\definition{Combinators}
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A \textit{combinator} is a lambda expression with no free variables.
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\vspace{1mm}
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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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The \textit{Kestrel}, $K = \lm ab . a$
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\problem{}
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If we give the Kestrel two arguments, it does something interesting: \par
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It selects the first and rejects the second. \par
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Convince yourself of this fact by evaluating $(K~\heartsuit~\star)$.
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\vfill
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\problem{}<kitedef>
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Modify the Kestrel so that it selects its \textbf{second} argument and rejects the first. \par
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\begin{solution}
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$\lm ab . b$.
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\end{solution}
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\vfill
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\problem{}
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We'll call the combinator from \ref{kitedef} the \textit{Kite}, $KI$. \par
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Show that we can also obtain the kite by evaluating $(K~I)$.
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\vfill
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\pagebreak |