63 lines
1.9 KiB
TeX
Executable File
63 lines
1.9 KiB
TeX
Executable File
\section{Recursion}
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Say we want a function that computes the factorial of a positive integer. Here's one way we could define it:
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$$
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x! = \begin{cases}
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x \times (x-1)! & x \neq 0 \\
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1 & x = 0
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\end{cases}
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$$
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We cannot re-create this in lambda calculus, since we aren't given a way to recursively call functions.
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\vspace{2mm}
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One could think that $A = \lm a. A~a$ is a recursive function. In fact, it is not. \par
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Remember that such \say{definitions} aren't formal structures in lambda calculus. \par
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They're just shorthand that simplifies notation.
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\begin{instructornote}
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We're talking about recursion, and \textit{computability} isn't far away. At one point or another, it may be good to give the class a precise definition of \say{computable by lambda calculus:}
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\vspace{4ex}
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Say we have a device that reduces a $\lm$ expression to $\beta$-normal form. We give it an expression, and the machine simplifies it as much as it can and spits out the result.
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\vspace{1ex}
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An algorithm is \say{computable by lambda calculus} if we can encode its input in an expression that resolves to the algorithm's output.
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\end{instructornote}
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\problem{}
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Write an expression that resolves to itself. \par
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\note{Your answer should be quite short.}
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\vspace{1ex}
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This expression is often called $\Omega$, after the last letter of the Greek alphabet. \par
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$\Omega$ useless on its own, but it gives us a starting point for recursion.
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\begin{solution}
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$\Omega = M~M = (\lm x . xx) (\lm x . xx)$
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\vspace{1ex}
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An uninspired mathematician might call the Mockingbird $\omega$, \say{little omega}.
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\end{solution}
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\vfill
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\definition{}
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This is the \textit{Y-combinator}, easily the most famous $\lm$ expression. \par
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You may notice that it's just $\Omega$, put to work.
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$$
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Y = \lm f . (\lm x . f(x~x))(\lm x . f(x~x))
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$$
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\problem{}
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What does this thing do? \par
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Evaluate $Y f$.
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\vfill
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\pagebreak
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