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Advanced/Lambda Calculus/parts/03 numbers.tex
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@ -1,10 +1,10 @@
\section{Numbers}
Since the only objects we have in $\lm$-calculus are functions, it's natural to think of quantities as \textit{adverbs} (once, twice, thrice,...) rather than \textit{nouns} (one, two, three ...) \\
Since the only objects we have in $\lm$-calculus are functions, it's natural to think of quantities as \textit{adverbs} (once, twice, thrice,...) rather than \textit{nouns} (one, two, three ...)
\vspace{1ex}
We'll start with zero. If our numbers are \textit{once,} \textit{twice,} and \textit{twice}, it may make sense to make zero \textit{don't}. \\
We'll start with zero. If our numbers are \textit{once,} \textit{twice,} and \textit{twice}, it may make sense to make zero \textit{don't}. \par
Here's our \textit{don't} function: given a function and an input, don't apply the function to the input.
$$
0 = \lm fa.a
@ -18,13 +18,13 @@ Write $1$, $2$, and $3$. We will call these \textit{Church numerals}.\hspace{-0.
\footnotetext{after Alonzo Church, the inventor of lambda calculus and these numerals. He was Alan Turing's thesis advisor.}
\begin{solution}
$1$ calls a function once on its argument: \\
$1$ calls a function once on its argument: \par
$1 = \lm fa . (f~a)$.
\vspace{1ex}
Naturally, \\
$2 = \lm fa . [~f~(f~a)~]$ \\
Naturally, \par
$2 = \lm fa . [~f~(f~a)~]$ \par
$3 = \lm fa . [~f~(f~(f~a))~]$
\vspace{1ex}
@ -49,7 +49,7 @@ What is $(4~I)~\star$?
\vfill
\problem{}
What is $(3~NOT~T)$? \\
What is $(3~NOT~T)$? \par
How about $(8~NOT~F)$?
\vfill
@ -57,13 +57,13 @@ How about $(8~NOT~F)$?
\pagebreak
\problem{}
This handout may remind you of Professor Oleg's handout on Peano's axioms. Good. \\
Recall the tools we used to build the natural numbers: \\
This handout may remind you of Professor Oleg's handout on Peano's axioms. Good. \par
Recall the tools we used to build the natural numbers: \par
We had a zero element and a \say{successor} operation so that $1 \coloneqq S(0)$, $2 \coloneqq S(1)$, and so on.
\vspace{1ex}
Create a successor operation for the Church numerals. \\
Create a successor operation for the Church numerals. \par
\hint{A good signature for this function is $\lm nfa$, or more clearly $\lm n.\lm fa$. Do you see why?}
\begin{solution}
@ -83,14 +83,14 @@ Verify that $S(0) = 1$ and $S(1) = 2$.
\pagebreak
Assume that only Church numerals will be passed to the functions in the following problems. \\
Assume that only Church numerals will be passed to the functions in the following problems. \par
We make no promises about their output if they're given anything else.
\problem{}
Define a function ADD that adds two Church numerals.
\begin{solution}
$\text{ADD} = \lm mn . (m~S~n) = \lm mn . (n~S~m)$ \\
$\text{ADD} = \lm mn . (m~S~n) = \lm mn . (n~S~m)$
\end{solution}
\begin{instructornote}
@ -111,8 +111,8 @@ Define a function ADD that adds two Church numerals.
\vfill
\problem{}
Adding is nice, but we can do better. \\
Design a function MULT that multiplies two numbers. \\
Adding is nice, but we can do better. \par
Design a function MULT that multiplies two numbers. \par
\hint{The easy solution uses ADD, the elegant one doesn't. Find both!}
\begin{solution}
@ -126,22 +126,22 @@ Design a function MULT that multiplies two numbers. \\
\problem{}
Define the functions $Z$ and $NZ$. $Z$ should reduce to $T$ if its input was zero, and $F$ if it wasn't. \\
Define the functions $Z$ and $NZ$. $Z$ should reduce to $T$ if its input was zero, and $F$ if it wasn't. \par
$NZ$ does the opposite. $Z$ and $NZ$ should look fairly similar.
\vspace{1ex}
\begin{solution}
$Z\phantom{N} = \lm n .[n~(\lm a.F)~T]$\\
$Z\phantom{N} = \lm n .[n~(\lm a.F)~T]$ \par
$NZ = \lm n .[n~(\lm a.T)~F]$
\end{solution}
\vfill
\problem{}
Data structures will be useful. Design an expression PAIR that constructs two-value tuples. \\
For example, say $A = \text{PAIR}~1~2$. Then, \\
$(A~T)$ should reduce to $1$ \\
Data structures will be useful. Design an expression PAIR that constructs two-value tuples. \par
For example, say $A = \text{PAIR}~1~2$. Then, \par
$(A~T)$ should reduce to $1$ \par
$(A~F)$ should reduce to $2$
\begin{solution}
@ -149,21 +149,21 @@ $(A~F)$ should reduce to $2$
\end{solution}
\vfill
From now on, I'll write (PAIR $A$ $B$) as $\langle A,B \rangle$. \\
From now on, I'll write (PAIR $A$ $B$) as $\langle A,B \rangle$. \par
Like currying, this is only notation. The underlying $\lm$ logic remains the same.
\pagebreak
\problem{}<shiftadd>
Write a function $H$, which we'll call \say{shift and add.} \\
It does exactly what it says on the tin: \\
Write a function $H$, which we'll call \say{shift and add.} \par
It does exactly what it says on the tin:
\vspace{1ex}
Given an input pair, it should shift its right argument left, then add one. \\
$H~\langle 0, 1 \rangle$ should reduce to $\langle 1, 2\rangle$ \\
$H~\langle 1, 2 \rangle$ should reduce to $\langle 2, 3\rangle$ \\
$H~\langle 10, 4 \rangle$ should reduce to $\langle 4, 5\rangle$ \\
Given an input pair, it should shift its right argument left, then add one. \par
$H~\langle 0, 1 \rangle$ should reduce to $\langle 1, 2\rangle$ \par
$H~\langle 1, 2 \rangle$ should reduce to $\langle 2, 3\rangle$ \par
$H~\langle 10, 4 \rangle$ should reduce to $\langle 4, 5\rangle$
\begin{solution}
$H = \lm p . \Bigl\langle~(p~F)~,~S(p~F)~\Bigr\rangle$
@ -177,8 +177,8 @@ $H~\langle 10, 4 \rangle$ should reduce to $\langle 4, 5\rangle$ \\
\vfill
\problem{}
Design a function $D$ that un-does $S$. That means \\
$D(1) = 0$, $D(2) = 1$, etc. $D(0)$ should be zero. \\
Design a function $D$ that un-does $S$. That means \par
$D(1) = 0$, $D(2) = 1$, etc. $D(0)$ should be zero. \par
\hint{$H$ will help you make an elegant solution.}
\begin{solution}
@ -186,8 +186,8 @@ $D(1) = 0$, $D(2) = 1$, etc. $D(0)$ should be zero. \\
\end{solution}
\begin{solution}
Here's a different solution. \\
Can you figure out how it works? \\
Here's a different solution. \par
Can you figure out how it works?
\vspace{1ex}