This post explains how the visitor pattern is essentially the same thing as Church encoding (or Böhm-Berarducci encoding). This post also explains how you can usefully employ the visitor pattern / Church encoding / Böhm-Berarducci encoding to expand your programming toolbox.

#### Background

Church encoding is named after Alonzo Church, who discovered that you could model any type of data structure in the untyped lambda calculus using only functions. The context for this was that he was trying to show that lambda calculus could be treated as a universal computational engine, even though the only features it supported were functions.

Note:Later on, Corrado Böhm and Alessandro Berarducci devised the equivalent solution in a typed lambda calculus (specifically, System F):… so I’ll use “Church encoding” when talking about this trick in the context of an untyped language and use “Böhm-Berarducci” encoding when talking about the same trick in the context of a typed language. If we’re not talking about any specific language then I’ll use “Church encoding”.

In particular, you can model the following types of data structures using language support for functions and nothing else:

records / structs (known as “product types” if you want to get fancy)

The “product” of two types

`A`

and`B`

is a type that stores both an`A`

**and**a`B`

(e.g. a record with two fields, where the first field has type`A`

and the second has type`B`

)enums / tagged unions (known as “sum types”)

The “sum” of two types

`A`

and`B`

is a type that stores either an`A`

**or**a`B`

. (e.g. a tagged union where the first tag stores a value of type`A`

and the second tag stores a value of type`B`

)recursive data structures

… and if you can precisely model product types, sum types, and recursion, then you can essentially model any data structure. I’m oversimplifying things, but that’s close enough to true for our purposes.

#### Relevance

The reason we care about Church-encoding is because not all programming languages natively support sum types or recursion (although most programming languages support product types in the form of records / structs).

However, most programming languages *do* support functions, so if we have functions then we can use them as a “backdoor” to introduce support for sum types or recursion into our language. This is the essence of the visitor pattern: using functions to Church-encode sum types or recursion into a language that does not natively support sum types or recursion.

To illustrate this, suppose that we begin from the following Haskell code:

```
data Shape
= Circle{ x :: Double, y :: Double, r :: Double }
| Rectangle{ x :: Double, y :: Double, w :: Double, h :: Double }
exampleCircle :: Shape
= Circle 2.0 1.4 4.5
exampleCircle
exampleRectangle :: Shape
= Rectangle 1.3 3.1 10.3 7.7
exampleRectangle
area :: Shape -> Double
= case shape of
area shape Circle x y r -> pi * r ^ 2
Rectangle x y w h -> w * h
main :: IO ()
= do
main print (area exampleCircle)
print (area exampleRectangle)
```

… but then we hypothetically disable Haskell’s support for algebraic data types. How would we amend our example to still work in such a restricted subset of the language?

We’d use Böhm-Berarducci encoding (the typed version of Church-encoding), and the solution would look like this:

```
{-# LANGUAGE RankNTypes #-}
-- | This plays the same role as the old `Shape` type
type Shape = forall shape
. (Double -> Double -> Double -> shape)
-> (Double -> Double -> Double -> Double -> shape)
-> shape
-- | This plays the same role as the old `Circle` constructor
_Circle :: Double -> Double -> Double -> Shape
= \_Circle _Rectangle -> _Circle x y r
_Circle x y r
-- | This plays the same role as the old `Rectangle` constructor
_Rectangle :: Double -> Double -> Double -> Double -> Shape
= \_Circle _Rectangle -> _Rectangle x y w h
_Rectangle x y w h
exampleCircle :: Shape
= _Circle 2.0 1.4 4.5
exampleCircle
exampleRectangle :: Shape
= _Rectangle 1.3 3.1 10.3 7.7
exampleRectangle
area :: Shape -> Double
= shape
area shape -> pi * r ^ 2)
(\x y r -> w * h)
(\x y w h
main :: IO ()
= do
main print (area exampleCircle)
print (area exampleRectangle)
```

The key is the new representation of the `Shape`

type, which is the type of a higher-order function. In fact, if we squint we might recognize that the `Shape`

type synonym:

```
type Shape = forall shape
. (Double -> Double -> Double -> shape)
-> (Double -> Double -> Double -> Double -> shape)
-> shape
```

… looks an awful lot like a GADT-style definition for the `Shape`

type:

```
{-# LANGUAGE GADTs #-}
data Shape where
Circle :: Double -> Double -> Double -> Shape
Rectangle :: Double -> Double -> Double -> Double -> Shape
```

This is not a coincidence! Essentially, Böhm-Berarducci encoding models a type as a function that expects each “constructor” as a function argument that has the same type as that constructor. I put “constructor” in quotes since we never actually use a real constructor. Those function arguments are place-holders that will remain abstract until we attempt to “pattern match” on a value of type `Shape`

.

In the `area`

function we “pattern match” on `Shape`

by supplying handlers instead of constructors. To make this explicit, let’s use equational reasoning to see what happens when we evaluate `area exampleCircle`

:

```
area exampleCircle
-- Substitute the `area` function with its definition
= exampleCircle
-> pi * r ^ 2)
(\x y r -> w * h)
(\x y w h
-- Substitute `exampleCircle` with its definition
= _Circle 2.0 1.4 4.5
-> pi * r ^ 2)
(\x y r -> w * h)
(\x y w h
-- Substitute the `_Circle` function with its definition
= (\_Circle _Rectangle -> _Circle 2.0 1.4 4.5)
-> pi * r ^ 2)
(\x y r -> w * h)
(\x y w h
-- Evaluate the outer-most anonymous function
= (\x y r -> pi * r ^ 2) 2.0 1.4 4.5
-- Evaluate the anonymous function
= pi * 4.5 ^ 2
```

In other words, Church encoding / Böhm-Berarducci encoding both work by maintaining a fiction that eventually somebody will provide us the “real” constructors right up until we actually need them. Then when we “pattern match” on the value we pull a last-minute bait-and-switch and use each “handler” of the pattern match where the constructor would normally go and everything works out so that we don’t need the constructor after all. Church-encoding is sort of like the functional programming equivalent of “fake it until you make it”.

The same trick works for recursive data structures as well. For example, the way that we Böhm-Berarducci-encode this Haskell data structure:

```
data Tree = Node Int Tree Tree | Leaf
exampleTree :: Tree
= Node 1 (Node 2 Leaf Leaf) (Node 3 Leaf Leaf)
exampleTree
preorder :: Tree -> [Int]
= case tree of
preorder tree Node value left right -> value : preorder left ++ preorder right
Leaf -> []
main :: IO ()
= print (preorder exampleTree) main
```

… is like this:

```
{-# LANGUAGE RankNTypes #-}
type Tree = forall tree
. (Int -> tree -> tree -> tree) -- Node :: Int -> Tree -> Tree -> Tree
-> tree -- Leaf :: Tree
-> tree
_Node :: Int -> Tree -> Tree -> Tree
=
_Node value left right -> _Node value (left _Node _Leaf) (right _Node _Leaf)
\_Node _Leaf
_Leaf :: Tree
= \_Node _Leaf -> _Leaf
_Leaf
exampleTree :: Tree
= _Node 1 (_Node 2 _Leaf _Leaf) (_Node 3 _Leaf _Leaf)
exampleTree
preorder :: Tree -> [Int]
= tree
preorder tree -> value : left ++ right)
(\value left right
[]
main :: IO ()
= print (preorder exampleTree) main
```

This time the translation is not quite as mechanical as before, due to the introduction of recursion. In particular, two differences stand out.

First, the way we encode the `_Node`

constructor is not as straightforward as we thought:

```
_Node :: Int -> Tree -> Tree -> Tree
=
_Node value left right -> _Node value (left _Node _Leaf) (right _Node _Leaf) \_Node _Leaf
```

This is because we need to thread through the `_Node`

/ `_Leaf`

function arguments through to the node’s children.

Second, the way we consume the `Tree`

is also different. Compare the original code:

```
preorder :: Tree -> [Int]
= case tree of
preorder tree Node value left right -> value : preorder left ++ preorder right
Leaf -> []
```

… to the Böhm-Berarducci-encoded version:

```
preorder :: Tree -> [Int]
= tree
preorder tree -> value : left ++ right)
(\value left right []
```

The latter version doesn’t require the `preorder`

function to recursively call itself. The `preorder`

function is performing a task that is morally recursive but the `preorder`

function is, strictly speaking, not recursive at all.

In fact, if we look at the Böhm-Berarducci-encoded solution closely we see that we never use recursion anywhere within the code! There are no recursive datatypes and there are also no recursive functions, yet somehow we still managed to encode a recursive data type and recursive functions on that type. This is what I mean when I say that Church encoding / Böhm-Berarducci encoding let you encode recursion in a language that does not natively support recursion. Our code would work just fine in a recursion-free subset of Haskell!

For example, Dhall is a real example of a language that does not natively support recursion and Dhall uses this same trick to model recursive data types and recursive functions:

That post goes into more detail about the algorithm for Böhm-Berarducci-encoding Haskell types, so you might find that post useful if the above examples were not sufficiently intuitive or clear.

#### Visitor pattern

The visitor pattern is a special case of Church encoding / Böhm Berarducci encoding. I’m not going to provide a standalone explanation of the visitor pattern since the linked Wikipedia page already does that. This section will focus on explaining the correspondence between Church encoding / Böhm-Berarducci encoding and the visitor pattern.

The exact correspondence goes like this. Given:

a Church-encoded / Böhm-Berarducci-encoded type

`T`

e.g.

`Shape`

in the first example… with constructors

`C₀`

,`C₁`

,`C₂`

, …e.g.

`Circle`

,`Rectangle`

… and values of type

`T`

named`v₀`

,`v₁`

,`v₂`

, …e.g.

`exampleCircle`

,`exampleRectangle`

… then the correspondence (using terminology from the Wikipedia article) is:

The “element” class corresponds to the type

`T`

e.g.

`Shape`

A “concrete element” (i.e. an object of the “element” class) corresponds to a constructor for the type

`T`

e.g.

`Circle`

,`Rectangle`

The

`accept`

method of the element selects which handler from the visitor to use, in the same way that our Church-encoded constructors would select one handler (named after the matching constructor) out of all the handler functions supplied to them.`_Circle :: Double -> Double -> Double -> Shape = \_Circle _Rectangle -> _Circle x y r _Circle x y r _Rectangle :: Double -> Double -> Double -> Double -> Shape = \_Circle _Rectangle -> _Rectangle x y w h _Rectangle x y w h`

A “visitor” class corresponds to the type of a function that pattern matches on a value of type

`T`

Specifically, a “visitor” class is equivalent to the following Haskell type:

`T -> IO ()`

This is more restrictive than Böhm-Berarducci encoding, which permits pattern matches that return any type of value, like our

`area`

function, which returns a`Double`

. In other words, Böhm-Berarducci encoding is not limited to just performing side effects when “visiting” constructors.(Edit: Travis Brown notes that the visitor pattern is not restricted to performing side effects. This might be an idiosyncracy of how Wikipedia presents the design pattern)

A “concrete visitor” (i.e. an object of the “visitor” class) corresponds to a function that “pattern matches” on a value of type

`T`

e.g.

`area`

… where each overloaded

`visit`

method of the visitor corresponds to a branch of our Church-encoded “pattern match”:`area :: Shape -> Double = shape area shape -> pi * r ^ 2) (\x y r -> w * h) (\x y w h`

The “client” corresponds to a value of type

`T`

e.g.

`exampleCircle`

,`exampleRectangle`

:`exampleCircle :: Shape = _Circle 2.0 1.4 4.5 exampleCircle exampleRectangle :: Shape = _Rectangle 1.3 3.1 10.3 7.7 exampleRectangle`

The Wikipedia explanation of the visitor pattern adds the wrinkle that the client can represent more than one such value. In my opinion, what the visitor pattern should say is that the client can be a recursive value which may have self-similar children (like our example

`Tree`

). This small change would improve the correspondence between the visitor pattern and Church-encoding.

#### Limitations of Böhm-Berarducci encoding

Church encoding works in any untyped language, but Böhm-Berarducci encoding does not work in all typed languages!

Specifically, Böhm-Berarducci only works in general for languages that support polymorphic types (a.k.a. generic programming). This is because the type of a Böhm-Berarducci-encoded value is a polymorphic type:

```
type Shape = forall shape
. (Double -> Double -> Double -> shape)
-> (Double -> Double -> Double -> Double -> shape)
-> shape
```

… but such a type cannot be represented in a language that lacks polymorphism. So what the visitor pattern commonly does to work around this limitation is to pick a specific `result`

type, and since there isn’t a one-size-fits-all type, they’ll usually make the result a side effect, as if we had specialized the universally quantified type to `IO ()`

:

```
type Shape =
. (Double -> Double -> Double -> IO ())
-> (Double -> Double -> Double -> Double -> IO ())
-> IO ()
```

This is why Go has great difficulty modeling sum types accurately, because Go does not support polymorphism (“generics”) and therefore Böhm-Berarducci encoding does not work in general for introducing sum types in Go. This is also why people with programming language theory backgrounds make a bigger deal out of Go’s lack of generics than Go’s lack of sum types, because if Go had generics then people could work around the lack of sum types using a Böhm-Berarducci encoding.

#### Conclusions

Hopefully this gives you a better idea of what Church encoding and Böhm-Berarducci encoding are and how they relate to the visitor pattern.

In my opinion, Böhm-Berarducci encoding is a bigger deal in statically-typed languages because it provides a way to introduce sum types and recursion into a language in a type-safe way that makes invalid states unrepresentable. Conversely, Church encoding is not as big of a deal in dynamically-typed languages because a Church-encoded type is still vulnerable to runtime exceptions.

Thanks for this insightful write up. I'm sensing a strong relation of the Böhm-Berarducci encoding of Trees to F-Algebras and catamorphisms:

ReplyDeletepreorder :: Tree -> [Int]

pretty much looks like the F-Algebra of the Tree functor to me. I can't pinpoint this more precisely though for now.

I had exactly the same feeling, but I might be biased because I just spent a week studying recursion-schemes and I know see them everywhere.

DeleteTo add to this: Gabriel noted that the preorder function never recursively calls itself. If I understand correctly, that's because the construction of trees actually bakes in that its constituent parts are provided with the same 'handlers' for each constructor, much like an F-algebra would do for the Base functor?

I'm wondering what implications this would have when we want to write minmax evaluation over these same trees, so without introducing a tag/constructor in the datatype signaling whether a node is a max or a min node.

With a traditional (tagged initial?) datatype encoding, we could write 'minmax' by pattern matching on up to two levels of nesting at once, and then recursively call 'minmax' again.

I'm not so sure how this would translate for the Böhm-Berarducci approach, whether it is possible and how.. I'm might need to do some experimenting this evening!

We can write the pattern functor for Tree:

Deletetype TreeF t = forall r. (Int -> t -> t -> r) -> r -> r

Then we can write the Church-style Fixpoint of functor:

type Fix f = forall t. (f t -> t) -> t

And (Fix TreeF) should be isomorphic to Tree, but I am unable to prove it.

If I recall correctly, if you want to pattern match on more than one level the Church-encoding (or Böhm-Berarducci) is not enough. You need course-of-value recusion aka histomorphism, see the paper "Course-of-Value Induction in Cedille".

DeleteActually...in the "Course-of-Value Induction in Cedille" paper they show that you can pattern match on more than one level even with the Böhm-Berarducci encoding by a technique called "tupling".

DeleteGreat post!

ReplyDeleteMinor typo:

-- Evaluate the anonymous function

= pi * 4.6 ^ 2

-= pi * 4.6 ^ 2

As well as the Church/Böhm-Berarducci encoding, there's a Parigot encoding. For the former, binary trees are expressed by

ReplyDeletetype Tree = forall t . (Int -> t -> t -> t) -> t -> t

so the type is not recursive, but the constructors have to be recursive; this corresponds to what GOF call "internal visitors", where it is the visitor that controls the order of traversal. Whereas for the latter the type is

type Tree = forall t . (Int -> Tree -> Tree -> t) -> t -> t

which is recursive, but now the constructors need not be; this corresponds to GOF's "external visitors", where it is the client that controls the order of traversal.

For more detail, see this paper from OOPSLA 2008, mostly by my former student Bruno Oliveira:

http://www.cs.ox.ac.uk/publications/publication1398-abstract.html

I think:

ReplyDeletetype Tree = forall t . (Int -> Tree -> Tree -> t) -> t -> t

Is actually the Scott encoding (the case match). The Parigot encoding would be:

type Tree = forall t . (Int -> Tree -> Tree -> t -> t -> t) -> t -> t

So that you have access to both the sub-trees and the results of the recursive calls of the sub-trees.

Another interesting encoding is the Mendler-encoding:

type Tree = forall t . (forall r. (r -> t) -> Int -> r -> r -> t) -> t -> t

Here you get to decide if you want to perform the recursive call or not. This is not very interesting for a lazy language, but for a strict language it will matter.

You can also modify the Mendler encoding to allow for unrestricted recursion:

type Tree = forall t . ((Tree -> t) -> Int -> Tree -> Tree -> t) -> t -> t

Basically the Scott encoding with access to the recursive call.

Would be helpful to have an example in some more mainstream language for those of us not versed in Haskell.

ReplyDeleteMy intended audience is just Haskell programmers , but all my posts have a Creative Commons so people can freely translate them to use examples in other programming languages.

DeleteActually, you don't need parametric polymorphism to encode a type-safe algebraic type. ADTs are existential types, and all you need are existential types. You have chosen to encode existentials using universals (presumably because universals are far more fundamental in ML-like languages), but you could have just as well used a Haskell type class or a ML module, which are existential (and then you'd get the so-called tagless final style encoding).

ReplyDeleteIn Go, however, existentials are fundamental and very much "in your face". They are called interfaces. Here is an encoding of an ADT in Go, using only interfaces and no generics: https://play.golang.org/p/IbwK3pKfpCS.

Just like traditional ADTs, this is closed, type-safe, and the pattern matching is exhaustive.

Now that we get generics in Go, we will be able to encode more than ADTs, we will be able to encode a lot of what GADTs can encode.