The fixed point combinator gives us a very general way of expressing recursive computations. However, we lose many of the desirable properties of our language---for example, we can use the fixed point combinator to encode arbitrary diverging program behavior. In these notes, we consider an alternative approach, based on the recursive structure of data types.
Sections followed by an asterix contain more advanced material. You may find this interesting, but should not expect to see it appear on the exam.
Before we can talk about recursive data types, and recursive computation based on those data types, we need to lay a bit of ground work. We will introduce the idea of type operators, allowing us to abstract over the structure of types. These will play roughly the same role as the "step" functions did in our account of fixed point terms.
Consider several examples of recursive types: the natural numbers Nat, and lists
and trees natural numbers NatList and NatTree. We might express them
using data type declarations as follows.
data Nat = Z | S Nat data NatList = Nil | Cons Nat NatList data NatTree = Leaf | Branch Nat NatTree NatTree
We can distinguish two parts of these recursive definitions. One part is the information
present at each "iteration" of the recursive type. In the case of Nat, that's
simply whether the type represents 0 or the successor of another natural;
for NatList, we capture whether the list is empty, or a cons node, and in the
latter case also capture the value stored in the node. The second part is the recursive
structure.
To separate these two parts, we introduce the idea of type operators. Intuitively speaking, a type operator is a type with a "hole", which we will use to indicate the points of recursion. Formally, we can consider this to be a type with an identified free type variable. For example, the type operators extracted from the recursive definitions about would be:
$$ \begin{align*} N(a) &= 1 + a \\ L_N(a) &= 1 + (N \times a) \\ T_N(a) &= 1 + (N \times a \times a) \end{align*} $$Each of these type operators also gives rise to a term operator, called its map, which modifies the $a$ value while leaving the rest of the structure unchanged. These operators have the following typing rule:
$$ \frac{\Gamma \vdash f : t \to u} {\Gamma \vdash \Map F f : F(t) \to F(u)} $$For example, $\Map N {even}$ would have type $1 + \Nat \to 1 + \Bool$, while $\Map {L_N} {even}$ would have type $1 + (\Nat \times \Nat) \to 1 + (\Nat \times \Bool)$.
Optional exercise. Write out the terms that implement $\Map N - : (t \to u) \to (N(t) \to N(u))$ and $\Map {L_N} - : (t \to u) \to (L_N(t) \to L_N(u)$. Treat $t$ and $u$ as concrete but arbitrary types.
In general, structures that have this kind of transformer are called functors, and the fact that these type operators are functors will be crucial to defining the meaning of computations over the corresponding recursive types. Unfortunately, it turns out that not all type operators describe functors.
Optional exercise. Attempt to write map functions for the following type operators. What goes wrong for $F_{Neg}$?
$$ \begin{align*} F_{Pos}(a) &= \Nat \to a \\ F_{Neg}(a) &= a \to \Nat \\ \end{align*} $$We can formally capture those operators which describe functors by distinguishing between positive and negative occurrences of type variables. We introduce two operators $fv^+(t)$ and $fv^-(t)$ to describe the positive and negative variables in type $t$. They are defined as follows.
$$ \begin{gather*} fv^+ = \Set a \qquad fv^- = \emptyset \qquad fv^\pm(t \to u) = fv^\mp(t) \cup fv^\pm(u) \\ fv^\pm(t \times u) = fv^\pm(t) \cup fv^\pm(u) \qquad fv^\pm(t + u) = fv^\pm(t) \cup fv^\pm(u) \end{gather*} $$The definitions of $fv^\pm(t)$ abbreviate both the positive and negative definitions for type $t$; subsequent references to $\mp$ mean the opposite polarity as $\pm$. For example, the definition for functions abbreviates the following two definitions.
$$ fv^+(t \to u) = fv^-(t) \cup fv^+(u) \qquad fv^-(t \to u) = fv^+(t) \cup fv^-(u) $$Now, we can say that the type operator $F(a) = t$ defines a functor exactly when $a \not\in fv^-(t)$.
Optional exercise. Write down the map function for $F_{NN}(a) = (a \to \Nat) \to \Nat$.
We build a recursive type out of a type operator the same way we built a recursive term out of a
step function: by applying a (least) fixed point operator. The least fixed point operator for
types is conventionally written with the green letter $\mu$. We would recover the
original Nat and NatList types as the least fixed points of the
corresponding type operators:
As for fixed points of step functions, the intuition of these types is that they give the infinite iteration of the type operator. For example, the two fixed points above are intuitively equivalent to:
$$ \begin{align*} \Nat &= 1 + (1 + (1 + (1 + \dots))) \\ \mathtt{NatList} &= 1 + (\Nat \times (1 + (\Nat \times (1 + \dots)))) \end{align*} $$In this intuitive understanding, zero would be represented by $\Inl{()}$, while two would be represented by $\Inr{(\Inr{(\Inl{()})})}$. Unfortunately, this are not quite this simple. In particular, this intuitive view gives us no way to define computation over values of recursive data types.
To get our actual understanding of these types, we need to introduce explicit introduction and elimination forms for least fixed point types. This will get us back in familiar ground, where each type comes with its own introduce and elimination form; it will also allow us to define recursive computations over these types.
The introduction rule for type $\mu F$ is called $\mathsf{in}_F$, while the elimination form is called $\mathtt{fold}_F$. Their typing rules are as follows.
$$ \frac{\Gamma \vdash e : F(\mu F)} {\Gamma \vdash \In F e : \mu F} \qquad \frac{\Gamma \vdash f : F(t) \to t} {\Gamma \vdash \Fold F f : \mu F \to t} $$The $\mathtt{In}_F$ term "wraps" one expansion of a recursive type into an instance of the recursive type. For example, the term $\Inl{()}$ is an instance of type $N(\mu N)$, so the term $\In N {(\Inl{()})}$ is an instance of type $\mu N$. Similarly, two would be represented by the following term.
$$ \In N {(\Inr {(\In N {(\Inr {(\In N {(\Inl{()})})})})})} $$Intuitively, the term $\mathtt{fold}_F\,f$ replaces each instance of the $\mathtt{In}_F$ constructor with an application of the $f$ function. To see this in action, we consider several sample folds, defining simple operations.
We begin with a simple predicate, testing for even numbers. This can be defined as follows.
$$ even = \Fold N {\backslash x : 1 + \Bool \to \CCase x z {\mathtt{True}} b {\mathtt{not}\,b}} $$Consider the action of $even$ on 0 (encoded by $\In N {(\Inl {()})}$). We want to replace the (one) instance of the $\mathtt{In}_F$ constructor with the function in the fold.
$$ \begin{align*} &(\backslash x : 1 + \Bool \to \CCase x z {\mathtt{True}} b {\mathtt{not}\,b}) \, (\Inl{()}) \\ & = \CCase {\Inl{()}} z {\mathtt{True}} b {\mathtt{not}\,b}) \\ & = \mathtt{True} \end{align*} $$Now, consider the action of $even$ in 1 (encoded by $\In N {(\Inr {(\In N {(\Inl {()})})})}$). Again, we want to replace the two instance of the $\mathtt{In}_N$ constructor with the function in the fold; to simplify the example, we abbreviate that function as $e$.
$$ \begin{align*} & e \, (\Inr {(e \, (\Inl {()}))}) \\ & = e \, (\Inr {\mathtt{True}}) \tag{*}\\ & = (\backslash x : 1 + \Bool \to \CCase x z {\mathtt{True}} b {\mathtt{not}\,b}) \, (\Inr {\mathtt{True}}) \\ & = \CCase {\Inr {\mathtt{True}}} z {\mathtt{True}} b {\mathtt{not}\,b} \\ & = \mathtt{not} \, \mathtt{True} \\ & = \mathtt{False} \end{align*} $$In the $*$ labeled step, we rely on the previous example to replace $e\,(\Inl{()})$ with $\mathtt{True}$. The remainder of the evaluation is unsurprising.
Here are several other examples of simple folds: two encodings of addition, and one encoding of tree sum. (We rely on familiar notation for naturals in the tree sum function purely for convenience.
$$ \begin{align*} plus_1 &= \backslash m : \mu N \to \Fold N {\backslash x : 1 + \mu N \to \CCase x z m p {\In N {(\Inr{p})}}} \\ plus_2 &= \mathtt{fold}_N \, (\begin{array}[t]{@{}l} \backslash x : 1 + (\mu N \to \mu N) \to \\ \quad \CCase x z {(\backslash n : \mu N \to n)} f {(\backslash n : \mu N \to \In N {(\Inr {(f\,n)})})}) \end{array} \\ treeSum &= \mathtt{fold}_{T_N} \, (\begin{array}[t]{@{}l} \backslash x : 1 + (\mu N \times \mu N \times \mu N) \to \\ \quad \CCase x z 0 p {\Let{(x,y,z)}{p}{x + y + z}}) \end{array} \end{align*} $$Finally, we can give formal evaluation rules for recursive types.
$$ \frac{e \Eval v} {\In F e \Eval \Inv v} \qquad \frac{f \Eval \lambda x. e_1 \quad e \Eval \Inv w \quad f (\Map F {\Fold F f} \, w) \Eval v} {\Fold F f \, e \Eval v} $$As is hopefully unsurprising at this point, the computational content is all in the elimination form. When eliminating a fold, we begin by evaluating both the fold function and the value. Then, intuitively, we want to replace the outer $\mathsf{in}$ with an application of the fold function. However, before doing so, we need to account for the recursion. We do this by relying on the $\mathsf{map}$ defined for type operator $F$.