# Interpretations of Fold

or
Fold as a Recursion Scheme

2018-07-16

What is fold? There are several ways to think about it, and several interpretations to explore.

## First Interpretation

One interpretation of fold is that it is a way to apply an operation “between” each two consequtive elements in a list. Fold-right applies them in a right-associative manner:

```List.fold_right (^) ["a"; "b"; "c"] "z"
⇒ ("a" ^ ("b" ^ ("c" ^ "z")))
⇒ "abcz"
```

Fold-left applies them in a left-associative manner:

```List.fold_left (^) "z" ["a"; "b"; "c"]
⇒ ((("z" ^ "a") ^ "b") ^ "c")
⇒ "zabc"
```

The `List` module tries to be helpful and orders the parameters to `fold_right` and `fold_left` to hint where the initial parameter goes, `abcz` vs. `zabc`. Still, it is hard to remember the parameters of fold.

OCaml standard library has `StdLabels` module with labeled versions of functions, which helps:

```open StdLabels

List.fold_right ~init:"z" ~f:(^) ["a"; "b"; "c"]
⇒ ("a" ^ ("b" ^ ("c" ^ "z")))
⇒ "abcz"
```

Jane Street Base uses the same labeled parameters.

Still, it is hard to remember the parameters of fold. Does `~f` take the accumulator first, or a list item?

## Second Interpretation

Another interpretation of fold is that it is a recursion scheme with each parameter being a callback for each variant constructor.

Let me explain. First, let’s revisit the list type:

```type 'a list =
| []
| (::) of 'a * 'a list
```

Nil constructor (`[]`) takes no parameters. Cons constructor `(::)` takes two parameters: an element of the list and the tail.

Now consider `fold_right` signature:

```val fold_right : f:('a -> 'b -> 'b) -> 'a list -> init:'b -> 'b
```

The `init` parameter is a value (takes no parameters). While `f` takes two parameters: an element of the list and the result of applying the fold to the tail of the list.

The correspondence between the two parameters of fold and the two list constructors is clear when you look at the implementation:

```let rec fold_right ~init ~f = function
| [] -> init
```

The fold recurses on the tail parameter similar to how the list type definition is recursive in the tail parameter.

Let’s rename the parameters to make the relation between parameters and constructors even more clear:

```let rec fold ~nil ~cons = function
| [] -> nil
```

We use `nil` value for the nil constructor, `cons` callback for the cons constructor. Fold is called recursively on the parameter that is recursive in the type definition.

Now, it is easy to remember parameters of fold:

The fold definition is dual to the type.

### Then how do you explain fold-left?

While fold-right is the natural fold for lists, dual to the type, fold-left is an unnatural fold for the list type. It is a fold that pretends that the list type is defined as left-heavy tree, as opposed to a standard right-heavy tree. It pretends that the cons constructor takes `'a list * 'a` instead of `'a * 'a list`. Sometimes it is useful to pretend, as we’ll see soon.

## Example I: Binary Tree

The power and generality of fold is very well known for lists. Now you can get the same power for any variant data type.

Let’s define a binary tree, a natural fold for it, and then implement a few functions using the fold.

First, the type:

```type 'a t =
| Leaf of 'a
| Node of 'a t * 'a t
```

Now, the fold:

```let rec fold ~leaf ~node = function
| Leaf a -> leaf a
| Node (left, right) ->
node (fold ~leaf ~node left) (fold ~leaf ~node right)
```

The fold recurses on both branches, similar to how the type is defined as recursive on both node parameters.

### The Recipe

Here’s how to define a natural fold for your variant type.

Fold takes one “callback” parameter for each variant. Labelled parameters work best for this.

Pattern-match on your variant type. For each constructor pattern, call the corresponding callback with the payload of each variant constructor, if any. One exception to that are the recursive values. For those, recursively call fold with the same callback parameters.

Let’s define functional constructors. They will get handy later.

```let leaf a = Leaf a
let node left right = Node (left, right)
```

Now, let’s get down to business.

The size of a tree is the sum of the sizes of each node, whereas a leaf node counts as one (regardless of the value):

```let size = fold ~leaf:(fun _ -> 1) ~node:(+)
```

The height of a singe leaf is one (regardless of the value), while the height of a node is one more than the height of its heighest branch:

```let height =
fold ~leaf:(fun _ -> 1) ~node:(fun a b -> 1 + max a b)
```

Let’s define `to_string` function which serializes a tree as an s-expression. As a parameter it takes a function that knows how to serialize a leaf value. We pass this parameter with label punning:

```let to_string leaf = fold ~leaf ~node:(sprintf "(%s %s)")
```

Let’s define and observe the relation between `map`, `bind`, and `iter`. The map uses the `leaf` constructor which we defined previously. Both the map and the bind pass the `node` constructor with label punning.

```let map f = fold ~leaf:(f >> leaf) ~node
let bind f = fold ~leaf:f ~node
let iter f = fold ~leaf:f ~node:(fun _ _ -> ())
```

Finally, `for_all` and `exists`:

```let for_all predicate = fold ~leaf:predicate ~node:(&&)
let exists predicate = fold ~leaf:predicate ~node:(||)
```

Not all of these functions are efficient. For example, in a strict language like OCaml, by passing `(&&)` and `(||)` to `for_all` and `exists` (as opposed to inlining them) we lost the short-circuiting power. However, the natural fold allowed us to quickly build a lot of functionality, and it reveals interesting relations between the functions that we implement.

### It’s useful to pretend

Let’s implement `fold_right` for our binary tree by pretending that it is a right-heavy linked list:

```let rec fold_right ~f ~init = function
| Leaf a -> f a init
| Node (left, right) ->
fold_right ~f ~init:(fold_right ~f ~init right) left
```

This allows us to express some list-centric functions more naturally:

```let to_list = fold_right ~f:List.cons ~init:[]

let map_to_list f = fold_right ~f:(f >> List.cons) ~init:[]
```

## Example II: Compiler Pass

Let’s re-implement the dead code elimination pass from the previous post, while using fold as our recursion scheme, instead of map.

First we implement fold by following the recipe. Then the pass itself:

```module Dead_code_elimination = struct
open Syntax

let pass =
fold ~unit ~boolean ~number ~id
~divide ~sequence ~let_ ~if_:(function
| {conditional=Boolean true; consequence; _} ->
consequence
| {conditional=Boolean false; alternative; _} ->
alternative
| other -> If other)
end
```

All the punned labled parameters look kinda funny. We could have made them optional with default implementations, but then the fold type would be more monomorphic, being able to only return the same type it folds on.

For comparison, here is original code:

```module Dead_code_elimination = struct
open Syntax

let rec pass = function
| If {conditional=Boolean true; consequence; _} ->
pass consequence
| If {conditional=Boolean false; alternative; _} ->
pass alternative
| other -> map pass other
end
```

Notice how in case of fold we did not need to recursively apply the pass, as the fold does it for us. On one hand, this saves us from mistakes: we can’t accidentally forget to recur in one of the branches. On the other hand, we loose some control: ability to get access to a value both before and after the transformation, and ability not to recur.

## Summary

Fold is a powerful recursion scheme and an elegant tool that allows to quickly implememnt a rich programming interface for a variant type.

Both natural folds and unnatural folds are useful. Similar to how lists implement natural `fold_right` and unnatural `fold_left`, for my variant types I would often implement a natural fold called `fold` with labelled parameters, and one or more unnatural folds, like `fold_right` which pretends to fold a list. Often folds are useful for non-variant types too, like the `Array.fold_right` that pretends that it folds a list.

## Code

Full code, including the ommitted pieces for the compiler pass is available in a GitHub gist.