hypertypes: Types parameterised by hypertypes

Hypertypes enable constructing rich recursive types from individual components, and processing them generically with type classes.

They are a solution to the Expression Problem, as described by Phil Wadler (1998):

The goal is to define a data type by cases, where one can add new cases to the data type and new functions over the data type, without recompiling existing code, and while retaining static type safety.

Data types a la carte (DTALC, Swierstra, 2008) offers a solution for the expression problem which is only applicable for simple recursive expressions, without support for mutually recursive types. In practice, programming language ASTs do tend to be mutually recursive. multirec (Rodriguez et al, 2009) uses GADTs to encode mutually recursive types but in comparison to DTALC it lacks in the ability to construct the types from re-usable components.

Hypertypes allow constructing expressions from re-usable terms like DTALC, which can be rich mutually recursive types like in multirec.

The name "Hypertypes" is inspired by Hyperfunctions (S. Krstic et al, FICS 2001), which are a similar construct at the value level.

Introduction to the "field constructor" pattern

To motivate hypertypes and explain the expressive power they allow, this section will discuss the benefits and limitations of other forms of constructing data types, ranging from unparameterized types to more powerful approaches:

• Type - unparameterized types
• Type -> Type - simple parameterized types
• (Type -> Type) -> Type - also known as HKD (Higher-Kinded Data)
• (Index -> Type) -> Index -> Type - multirec's approach

Type: Simple type, simple functionality

Suppose we have the following type in an application:

data Person = Person
    { height :: Double
    , weight :: Double
    }

Let's imagine that we want to let a user fill in a Person via a form, where during the process the record may have missing fields.

We may want a way to represent a state with missing fields, but this type doesn't allow for it.

We can either create an additional type for that, or augment Person to provide more functionality. Augmenting Person is preferred because it will result in less boiler-plate and less types to maintain as we make changes to it.

Type -> Type: Adding a type parameter

A possible solution is to parameterize Person on the field type:

data Person a = Person
    { height :: a
    , weight :: a
    }

This would solve our problem.

We can parameterize with Double for the normal structure, and with Maybe Double for the variant with missing fields.

This approach reaches its limits when the fields have multiple different types, as in:

data Person = Person
    { height :: Double
    , weight :: Double
    , name :: Text
    }

We would now need an additional parameter to parameterize how to store the fields of type Text! Is there a way to use a single type parameter for both types of fields? Yes, there is:

(Type -> Type) -> Type: Higher-Kinded Data

The "Higher-Kinded Data" pattern represents Person like so:

data Person f = Person
    { height :: f Double
    , weight :: f Double
    , name :: f Text
    }

For the plain case we would use Person Identity.

Identity from Data.Functor.Identity is defined as so:

data Identity a = Identity a

And for the variant with missing fields we would use Person Maybe.

The benefit of this parameterization over the previous one is that Person's kind doesn't need to change when adding more field types, so such changes don't propagate all over the code base.

Note that various helper classes such as Rank2.Functor and Rank2.Traversable (from the rank2classes package) allow us to conveniently convert between Person Identity and Person Maybe.

HKD for nested structures

Let's employ the same transformation we did for Person to a more complicated data structure:

data Expr
    = Const Int
    | Add Expr Expr
    | Mul Expr Expr

The HKD form of Expr would be:

data Expr f
    = Const (f Int)
    | Add (f (Expr f)) (f (Expr f))
    | Mul (f (Expr f)) (f (Expr f))

This does allow representing nested structures with missing elements. But classes like Rank2.Functor no longer work for it. To understand why let's look at Rank2.Functor's definition

class Functor f where
    (<$>) :: (forall a. p a -> q a) -> f p -> f q

The rank-2 function argument expects the field type a to stay the same when it changes p to q, however in the above formulation of Expr the field type Expr p change to Expr q when changing the type parameter.

Type -> Type: The DTALC and recursion-schemes approach

Another formulation of Expr is the same as the Type -> Type approach discussed above:

data Expr a
    = Const Int
    | Add a a
    | Mul a a

Notes:

• The recursion-schemes package can generate this type for us from the plain definition of Expr using TemplateHaskell
• DTALC also allows us to construct this type by combining standalone Const, Add, and Mul types with the :+: operator (i.e Const Int :+: Add :+: Mul)

This approach does have the single node type limitation, so we gave up on parameterizing over the Int in Const. This is a big limitation, but as we'll see, we do get several advantages in return.

First, we can represent plain expressions as Fix Expr, using:

newtype Fix f = Fix (f (Fix f))

We can then use useful combinators from recursion-schemes for folding and processing of Exprs.

unification-fd is a good example of the power of this approach. It implements generic unification for ASTs, where it uses the parameterization to represent sub-expressions via unification variables.

In constrast to the HKD approach, we can also use rich fix-points which store several different fix-points within, like Diff:

data Diff f
    = Same (f (Fix f))
    | SameTopLevel (f (Diff f))
    | Different (f (Fix f)) (f (Fix f))

(Note how Diff parameterizes f by both Fix and Diff)

The main drawback of this approach is that in practice ASTs tend to be mutually recursive datatypes. For example:

data Expr
    = Var Text
    | App Expr Expr
    | Lam Text Typ Expr
data Typ
    = IntT
    | FuncT Typ Typ

This type is an example for an AST which DTALC and recursion-schemes cannot represent.

Can the "field constructor" pattern be used to represent such ASTs? Yes:

(Index -> Type) -> Index -> Type: The multirec approach

multirec's way to define the above AST:

data Expr :: Index
data Typ :: Index

data AST :: (Index -> Type) -> Index -> Type where
    Var :: Text -> AST r Expr
    App :: r Expr -> r Expr -> AST r Expr
    Lam :: Text -> r Typ -> r Expr -> AST r Expr
    IntT :: AST r Typ
    FuncT :: r Typ -> r Typ -> AST r Typ

(this is a slight variant of multirec's actual presentation, where for improved legibility Index is used rather than Type)

multirec offers various utilities to process such data types. It offers HFunctor, a variant of Functor for these structures, and various recursive combinators.

But multirec has several limitations:

• Using a single GADT for the data type limits composition and modularity.
• Invocations of HFunctor for a Typ node need to support transforming all indices of AST, including Expr, even though Typ doesn't have Expr child nodes.

hypertypes's approach

The hypertypes representation of the above AST example:

data Expr h
    = EVar Text
    | EApp (h :# Expr) (h :# Expr)
    | ELam Text (h :# Typ) (h :# Expr)
data Typ h
    = TInt
    | TFunc (h :# Typ) (h :# Typ)

Sub-expressions are nested using the :# type operator. On the left side of :# is Expr's type parameter h which is the "nest type", and on the right side Expr and Typ are the nested nodes.

:# is defined as:

-- A type parameterized by a hypertype
type HyperType = AHyperType -> Type

-- A kind for hypertypes
newtype AHyperType = AHyperType { getHyperType :: HyperType }

-- GetHyperType is getHyperType lifted to the type level
type family GetHyperType h where
    GetHyperType ('AHyperType t) = t

type p :# q = (GetHyperType p) ('AHyperType q)
-- AHyperType is DataKinds syntax for using AHyperType in types

The hypertypes library provides:

• Variants of standard classes like Functor for hypertypes with derivations. (Unlike in multirec's HFunctor, only the actual child node types of each node need to be handled)
• Combinators for recursive processing and transformation of nested structures
• Implementations of common AST terms
• A unification implementation for mutually recursive types inspired by unification-fd
• A generic and fast implementation of Hindley-Milner type inference ("Efficient generalization with levels" as described in How OCaml type checker works, Kiselyov, 2013)

Constructing types from individual components

Note that another way to formulate the above expression would be using pre-existing parts, such as:

data RExpr h
    = RVar (Var Text RExpr h)
    | RApp (App RExpr h)
    | RLam (TypedLam Text Typ RExpr h)
    deriving (Generic, Generic1, HNodes, HFunctor, HFoldable, HTraversable, ZipMatch)

This form supports using DeriveAnyClass to derive instances for various HyperType classes such as HFunctor based on Generic1. Note that due to a technical limitation of Generic1 the form of Expr from before, which directly nests values, doesn't have a Generic1 instance (so the instances for Expr are derived using TemplateHaskell instead).

Examples

How do we represent an expression of the example language declared above?

Let's start with the verbose way:

verboseExpr :: Pure # Expr
verboseExpr =
    Pure (ELam "x" (Pure TInt) (Pure (EVar "x")))

Explanations for the above:

• Pure # Expr is a type synonym for Pure ('AHyperType Expr)
• Pure is the simplest "pass-through" nest type
• The above is quite verbose with a lot of instances of Pure and many parentheses
• Writing an expression of the above RExpr would be even more verbose due to additional Var and TypedLam data constructors!

To write it more consicely, the HasHPlain class, along with a TemplateHaskell generator for it, exists:

> let e = hPlain # verboseExpr
-- Note: This (#) comes from Control.Lens

> e
ELamP "x" TIntP (EVarP "x")

> :t e
e :: HPlain Expr

It's now easier to see that e represents λ(x:Int). x

HPlain is a data family of "plain versions" of expressions, generated via TemplateHaskell. Note that it flattens embedded constructors for maximal convinience, so that the plain version of RExpr is as convinient to use as that of Expr!

This is somewhat similar to how recursion-schemes can derive a parameterized version of an AST, but is the other way around: the parameterized type is the source and the plain one is generated.

So now, let's define some example expressions concisely:

exprA, exprB :: HPlain Expr

exprA = ELamP "x" IntTP (EVarP "x")

exprB = ELamP "x" (TFuncP TIntP TIntP) (EVarP "x")

What can we do with these expressions? Let's compute a diff:

> let d = diffP exprA exprB

> d
CommonBodyP
(ELam "x"
    (DifferentP TIntP (TFuncP TIntP TIntP))
    (CommonSubTreeP (EVarP "x"))
)

> :t d
d :: DiffP # Expr
-- (An Expr with the DiffP nest type)

Let's see the type of diffP:

> :t diffP
diffP ::
    ( RTraversable h
    , Recursively ZipMatch h
    , Recursively HasHPlain h
    ) =>
    HPlain h -> HPlain h -> DiffP # h

diffP can compute the diff for any AST that is recursively traversable, can be matched, and has a plain representation.

Now, let's format this diff better:

> let formatDiff _ x y = "- " <> show x <> "\n+ " <> show y <> "\n"

> putStrLn (foldDiffsP formatDiff d)
- TIntP
+ TFuncP TIntP TIntP

> :t foldDiffsP
foldDiffsP ::
    ( Monoid r
    , Recursively HFoldable h
    , Recursively HasHPlain h
    ) =>
    (forall n. HasHPlain n => HRecWitness h n -> HPlain n -> HPlain n -> r) ->
    DiffP # h ->
    r

Why is the ignored argument of formatDiff there? It is the HRecWitness h n from the type of foldDiffsP above. It is a witness that "proves" that the folded node n is a recursive node of h, essentially restricting the forall n. to ns that are recursive nodes of h.

Witness parameters

First, I want to give thanks and credit: We learned of this elegant solution from multirec!

What are witness parameters?

Let's look at how HFunctor is defined:

class HNodes h => HFunctor h where
    -- | 'HFunctor' variant of 'fmap'
    hmap ::
        (forall n. HWitness h n -> p # n -> q # n) ->
        h # p ->
        h # q

HFunctor can change an h's nest-type from p to q.

HWitness is a data family which is a member of HNodes.

For example, let's see the definition of Expr's HWitness:

data instance HWitness Expr n where
    W_Expr_Expr :: HWitness Expr Expr
    W_Expr_Typ :: HWitness Expr Typ

Note that this GADT is automatically generated via TemplateHaskell.

What does the witness give us? It restricts forall n. to the nodes of h. When mapping over an Expr we can:

• Ignore the witness and use a mapping from a p of any n to a q of it
• Pattern match on the witness to handle Expr's specific node types
• Use the #> operator to convert the witness to a class constraint on n.

Understanding HyperTypes

• We want structures to be parameterized by nest-types
• Nest-types are parameterized by the structures, too
• Therefore, structures and their nest-types need to be parameterized by each other
• This results in infinite types, as the structure is parameterized by something which may be parameterized by the structure itself.

multirec ties this knot by using indices to represent types. hypertypes does this by using DataKinds and the AHyperType newtype which is used for both structures and their nest-types. An implication of the two being the same is that the same classes and combinators are re-used for both.

What Haskell is this

hypertypes is implemented with GHC and heavily relies on quite a few language extensions:

• ConstraintKinds and TypeFamilies are needed for the HNodesConstraint type family that lifts a constraint to apply over a value's nodes. Type families are also used to encode term's results in type inference.
• DataKinds allows parameterizing types over AHyperTypes
• DefaultSignatures are used for default methods that return Dicts to avoid undecidable super-classes
• DeriveGeneric, DerivingVia, GeneralizedNewtypeDeriving, StandaloneDeriving and TemplateHaskell are used to derive type-class instances
• EmptyCase is needed for instances of leaf nodes
• FlexibleContexts, FlexibleInstances and UndecidableInstances are required to specify many constraints
• GADTs and RankNTypes enable functions like hmap which get foralled functions with witness parameters
• MultiParamTypeClasses is needed for the Unify and Infer type classes
• ScopedTypeVariables and TypeApplications assist writing short code that type checks

Many harmless syntactic extensions are also used:

• DerivingStrategies, LambdaCase, TupleSections, TypeOperators

How does hypertypes compare/relate to

Note that comparisons to multirec, HKD, recursion-schemes, rank2classes, and unification-fd were discussed above.

In addition:

hyperfunctions

S. Krstic et al [KLP2001] have described the a type which they call a "Hyperfunction". Here is it's definition from the hyperfunctions package:

newtype Hyper a b = Hyper { invoke :: Hyper b a -> b }

AHyperTypes are isomorphic to Hyper Type Type (assuming a PolyKinds variant of Hyper), so they can be seen as type-level "hyperfunctions".

For more info on hyperfunctions and their use cases in the value level see [LKS2013]

References

• [KLP2001] S. Krstic, J. Launchbury, and D. Pavlovic. Hyperfunctions. In Proceeding of Fixed Points in Computer Science, FICS 2001
• [LKS2013] J. Launchbury, S. Krstic, T. E. Sauerwein. Coroutining Folds with Hyperfunctions. In In Proceedings Festschrift for Dave Schmidt, EPTCS 2013

Data Types a la Carte

In addition to the external fix-points described above, Data Types a la Carte (DTALC) also describes how to define ASTs structurally.

I.e, rather than having

data Expr a
    = Val Int
    | Add a a -- "a" stands for a sub-expression (recursion-schemes style)

We can have

newtype Val a = Val Int

data Add a = Add a a

-- Expr is a structural sum of Val and Add
type Expr = Val :+: Add

This enables re-usability of the AST elements Val and Add in various ASTs, where the functionality is shared via type classes. Code using these type classes can work generically for different ASTs.

Like DTALC, hypertypes has:

• Instances type for combinators such as :+: and :*:, so that these can be used to build ASTs
• Implementations of common AST terms in the Hyper.Type.AST module hierarchy (App, Lam, Let, Var, TypeSig and others)
• Classes like HFunctor, HTraversable, Unify, Infer with instances for the provided AST terms

As an example of a reusable term let's look at the definition of App:

-- | A term for function applications.
data App expr h = App
    { _appFunc :: h :# expr
    , _appArg :: h :# expr
    }

Unlike a DTALC-based apply, which would be parameterized by a single type parameter (a :: Type), App is parameterized on two type parameters, (expr :: HyperType) and (h :: AHyperType). expr represents the node type of App expr's child nodes and h is the tree's fix-point. This enables using App in mutually recursive ASTs where it may be parameterized by several different exprs.

Unlike DTALC, in hypertypes one typically needs to explicitly declare the datatypes for their expression types so that they can be used as App's expr type parameter. Similarly, multirec's DTALC variant also requires explicitly declaring type indices.

While it is possible to declare ASTs as newtypes wrapping :+:s of existing terms and deriving all the instances via GeneralizedNewtypeDeriving, our usage and examples declare types in the straight forward way, with named data constructors, as we think that this results in more readable and performant code.

bound

bound is a library for expressing ASTs with type-safe De-Bruijn indices rather than parameter names, via an AST type constructor that is indexed on the variables in scope.

An intereseting aspect of bound's ASTs is that recursively they are made of an infinite amount of types.

When implementing hypertypes we had the explicit goal of making sure that such ASTs are expressible with it, and for this reason the Hyper.Type.AST.NamelessScope module in the tests implementing it is provided, and the test suite includes a language implementation based on it (LangA in the tests).

lens

hypertypes strives to be maximally compatible with lens, and offers Traversals and Setters wherever possible. But unfortunately the RankNTypes nature of many combinators in hypertypes makes them not composable with optics. For the special simpler cases when all child nodes have the same types the htraverse1 traversal and hmapped1 setter are available.