87 lines
2.1 KiB
TeX
Executable File
87 lines
2.1 KiB
TeX
Executable File
\documentclass[../main.tex]{subfiles}
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\begin{document}
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\section{Challenges}
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Do \ref{Yfac} first, then finish the rest in any order. \\
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Have fun!
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\problem{}<Yfac>
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Design a recursive factorial function using $Y$. \\
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\vfill
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\problem{}
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Design a non-recursive factorial function. \\
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\note{This one is a lot easier than \ref{Yfac}, but I don't think it will help you solve it.}
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\problem{}
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Using pairs, make a \say{list} data structure. Define a GET function, so that $\text{GET}~L~n$ reduces to the nth item in the list. $\text{GET}~L~0$ should give the first item in the list, and $\text{GET}~L~1$, the \textit{second}. \\
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Lists have a defined length, so you should be able to tell when you're on the last element.
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\problem{}
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Write POW $a$ $b$, which raises $b$ to the $a$th power.
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\problem{}
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Write a MOD $a$ $b$ function that reduces to the remainder of $a \div b$.
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\begin{solution}
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\textbf{Factorial with recursion:}
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\vspace{3ex}
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$\text{FAC} = \lm yn.[Z~n][1][\text{MULT}~n~(y~(\text{D}~n))]$
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\linehack{}
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\textbf{Factorial without recursion:}
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\vspace{3ex}
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$\text{FAC}_0 = \lm p .
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\Bigl\langle~~
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\Bigl[D~(p~t)\Bigr]
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~,~
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\Bigl[\text{MULT}~(p~T)~(p~F)\Bigr]
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~~\Bigr\rangle
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$
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\vspace{2ex}
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$
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\text{FAC} = \lm n .
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\bigl( n~\text{FAC}_0~\langle n, 1 \rangle \bigr)
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$
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\linehack{}
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\textbf{Lists:}
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\vspace{3ex}
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One possible implementation is
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$\Bigl\langle
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\langle \text{is last} ~,~ \text{item} \rangle
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~,~
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\text{next}...
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\Bigr\rangle$, where:
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\vspace{1ex}
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\say{is last} is a boolean, true iff this is the last item in the list. \\
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\say{item} is the thing you're storing \\
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\say{next...} is another one of these list fragments. \\
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It doesn't matter what \say{next} is in the last list fragment. A dedicated \say{is last} slot allows us to store ANY function in this list.
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\vspace{1ex}
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Here, $\text{GET} = \lm nL.[(n~L~F)~T~F$] \\
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This will break if $n$ is out of range.
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\end{solution}
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\vfill
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\problem{Bonus}
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Play with \textit{Lamb}, an automatic lambda expression evaluator. \\
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\url{https://git.betalupi.com/Mark/lamb}
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\end{document} |