LambdaCalculus chapter: Church numerals

frap_book.tex

@@ -2022,14 +2022,14 @@ In fact, flow-insensitive and flow-sensitive interval analysis with this widenin
 
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-\chapter{Lambda-Calculus and Simple Type Safety}
+\chapter{Lambda Calculus and Simple Type Safety}
 
 We'll now take a break from the imperative language we've been studying for the last two chapters, instead looking at a classic sort of small language that distills the essence of \emph{functional} programming\index{functional programming}.
 That's the language paradigm that we've been using throughout this book, as we coded executable versions of algorithms.
 Its distinctive characteristics are first, a computation style based on simplifying terms instead of running step-by-step instructions that modify state; and second, use of functions as first-class values.
 Functional programming went mainstream in the early 21st century, influencing widely adopted languages from JavaScript\index{JavaScript}, where first-class functions are routinely used as callbacks in asynchronous event processing; to Scala\index{Scala}, a hybrid language that melds functional-programming ideas with object-oriented programming for the Java platform; to Haskell\index{Haskell}, a purely functional language that has become popular with programming hobbyists and is seeing increasing adoption in industry.
 
-The heart of functional programming persists even in \emph{$\lambda$-calculus}\index{$\lambda$-calculus} (or lambda-calculus\index{lambda-calculus}), the simplest version of which contains just three syntactic forms, but which provides probably the simplest of the widely known Turing-complete languages that is (nearly!) pleasant to program in directly.
+The heart of functional programming persists even in \emph{$\lambda$-calculus}\index{$\lambda$-calculus} (or lambda calculus\index{lambda calculus}), the simplest version of which contains just three syntactic forms, but which provides probably the simplest of the widely known Turing-complete languages that is (nearly!) pleasant to program in directly.
 
 
 \section{Untyped Lambda Calculus}
@@ -2097,6 +2097,80 @@ Note that we only ever need to evaluate \emph{closed} terms\index{closed terms},
 It may be surprising that these two rules are enough to define the full semantics of a Turing-complete language!
 
 
+\section{A Quick Case Study in Program Verification: Church Numerals}
+
+\newcommand{\church}[1]{\underline{#1}}
+
+Since $\lambda$-calculus is Turing-complete, it must be able to represent numbers and all the usual arithmetic operations.
+The classic representation is \emph{Church numerals}\index{Church numerals}, where every natural number $n$ is represented as a particular $\lambda$-term $\church{n}$ that, when passed a function $f$ as input, returns $f^n$, the $n$-way self-composition of $f$.
+In some sense, repeating a process is the fundamental use of a natural number, and it turns out that we can recover all of the usual operations atop this primitive.
+
+\newcommand{\lc}[1]{\mathsf{#1}}
+
+Two $\lambda$-calculus functions are sufficient to build up all the naturals as Church numerals.
+\begin{eqnarray*}
+  \lc{zero} &=& \lambda f. \; \lambda x. \; x \\
+  \lc{plus1} &=& \lambda n. \; \lambda f. \; \lambda x. \; f \; (n \; f \; x)
+\end{eqnarray*}
+Our representation of 0 returns an identity function, no matter which $f$ it is passed.
+Our successor operation takes in a number $n$ and returns a new one that first runs $n$ and then applies $f$ one extra time.
+Now we have $\church{0} = \lc{zero}$, $\church{1} = \lc{plus1} \; \lc{zero}$, $\church{2} = \lc{plus1} \; (\lc{plus1} \; \lc{zero})$, and so on.
+
+\newcommand{\prechurch}[1]{\left \lfloor #1 \right \rfloor}
+
+These Church numerals are not values yet.
+Let us formalize which values they evaluate to and tweak the encoding to use the values instead.
+We write $\prechurch{n}$ for the body of a $\lambda$-abstraction that we are building to represent $n$, where variables $f$ and $x$ are in scope.
+\begin{eqnarray*}
+  \prechurch{0} &=& x \\
+  \prechurch{n+1} &=& f \; ((\lambda f. \; \lambda x. \; \prechurch{n}) \; f \; x)
+\end{eqnarray*}
+The $n+1$ case may seem wastefully large, but, in fact, this is the precise form of the values produced by evaluating repeated applications of $\lc{plus1}$ to $\lc{zero}$, as the reader can verify using the big-step semantics.
+We define $\church{n} = \lambda f. \; \lambda x. \; \prechurch{n}$, giving a canonical encoding for each number.
+
+Now we notate correctness of an encoding $e$ for number $n$ by $e \sim n$, defining it as $\bigstep{e}{\church{n}}$, meaning that $e$ evaluates to the Church encoding of $n$.
+Two first easy results show that our primitive constructors are correct.
+
+\begin{theorem}
+  $\lc{zero} \sim 0$.
+\end{theorem}
+
+\begin{theorem}
+  If $e_n \sim n$, then $\lc{plus1} \; e_n \sim n+1$.
+\end{theorem}
+
+Things get more interesting as we start to code up the arithmetic operations.
+\begin{eqnarray*}
+  \lc{add} &=& \lambda n. \; \lambda m. \; n \; \lc{plus1} \; m
+\end{eqnarray*}
+
+That is, addition of $n$ to $m$ is calculated by applying $n$ $\lc{plus1}$ operations to $m$.
+
+\begin{theorem}\label{church_add}
+  If $e_n \sim n$ and $e_m \sim m$, then $\lc{add} \; e_n \; e_m \sim n + m$.
+\end{theorem}
+\begin{proof}
+  After a few steps applying the big-step rules directly, we finish by induction on $n$.
+  A silly-seeming but necessary lemma proves that $\subst{\prechurch{n}}{m}{e} = \prechurch{n}$, since $\prechurch{n}$ does not contain free occurrences of $m$.
+\end{proof}
+
+Multiplication proceeds in much the same way.
+\begin{eqnarray*}
+  \lc{mult} &=& \lambda n. \; \lambda m. \; n \; (\lc{add} \; m) \; \lc{zero}
+\end{eqnarray*}
+
+\begin{theorem}
+  If $e_n \sim n$ and $e_m \sim m$, then $\lc{mult} \; e_n \; e_m \sim n \times m$.
+\end{theorem}
+\begin{proof}
+  After a few steps applying the big-step rules directly, we finish by induction on $n$, within which we appeal to Theorem \ref{church_add}.
+\end{proof}
+
+An enjoyable (though not entirely trivial) exercise for the reader is to generalize the methods of Church encoding to encoding of other inductive datatypes, including the syntax of $\lambda$-calculus itself.
+A hallmark of a Turing-complete language is that it can host an interpreter for itself, and $\lambda$-calculus is no exception!
+
+
+
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 \appendix