Typing Haskell in Haskell1

MARK P. JONES
Pacific Software Research Center
Department of Computer Science and Engineering
Oregon Graduate Institute of Science and Technology
20000 NW Walker Road, Beaverton, OR 97006, USA
mpj@cse.ogi.edu

Version of November 23, 2000.

Abstract

Haskell benefits from a sophisticated type system, but implementors, programmers, and researchers suffer because it has no formal description. To remedy this shortcoming, we present a Haskell program that implements a Haskell typechecker, thus providing a mathematically rigorous specification in a notation that is familiar to Haskell users. We expect this program to fill a serious gap in current descriptions of Haskell, both as a starting point for discussions about existing features of the type system, and as a platform from which to explore new proposals.

1  Introduction

Haskell2 benefits from one of the most sophisticated type systems of any widely used programming language. Unfortunately, it also suffers because there is no formal specification of what the type system should be. As a result: This paper presents a formal description of the Haskell type system using the notation of Haskell itself as a specification language. Indeed, the source code for this paper is itself an executable Haskell program that is passed through a custom preprocessor and then through LATEX to obtain the typeset version. The type checker is available in source form on the Internet at http://www.cse.ogi.edu/~mpj/thih/. We hope that this will serve as a resource for the Haskell community, and that it will be a significant step in addressing the problems described previously.

One audience whose needs may not be particularly well met by this paper are researchers in programming language type systems who do not have experience of Haskell. (Of course, we encourage anyone in that position to learn more about Haskell!) Indeed, we do not follow the traditional route in such settings where the type system might first be presented in its purest form, and then related to a more concrete type inference algorithm by soundness and completeness theorems. Here, we deal only with type inference. It does not even make sense to ask if our algorithm computes `principal' types: such a question requires a comparison between two different presentations of a type system, and we only have one. Nevertheless, we believe that our specification could be recast in a standard, type-theoretic manner and used to develop a presentation of Haskell typing in a more traditional style.

The code presented here can be executed with any Haskell system, but our primary goals have been clarity and simplicity, and the resulting code is not intended to be an efficient implementation of type inference. Indeed, in some places, our choice of representation may lead to significant overheads and duplicated computation. It would be interesting to try to derive a more efficient, but provably correct implementation from the specification given here. We have not attempted to do this because we expect that it would obscure the key ideas that we want to emphasize. It therefore remains as a topic for future work, and as a test to assess the applicability of program transformation and synthesis to modestly sized Haskell programs.

Another goal of this paper is to give as complete a description of the Haskell type system as possible, while also aiming for conciseness. For this to be possible, we have assumed that certain transformations and checks will have been made prior to typechecking, and hence that we can work with a much simpler abstract syntax than the full source-level syntax of Haskell would suggest. As we argue informally at various points in the paper, we do not believe that there would be any significant difficulty in extending our system to deal with the missing constructs. All of the fundamental components, including the thorniest aspects of Haskell typing, are addressed in the framework that we present here. Our specification does not attempt to deal with all of the issues that would occur in the implementation of a type checker in a full Haskell implementation. We do not tackle the problems of interfacing a typechecker with compiler front ends (to track source code locations in error diagnostics, for example) or back ends (to describe the implementation of overloading, for example), nor do we attempt to formalize any of the extensions that are implemented in current Haskell systems. This is one of things that makes our specification relatively concise (429 lines of Haskell code). By comparison, the core parts of the Hugs typechecker take some 90+ pages of C code.

Some examples are included in the paper to illustrate the datatypes and representations that are used. However, for reasons of space, the definitions of some constants that represent entities in the standard prelude, as well as the machinery that we use in testing to display the results of type inference, are included only in the electronic distribution, and not in the typeset version of the paper. Apart from those details, this paper gives the full source code.

We expect the program described here to evolve in at least three different ways.

It will come as no surprise to learn that some knowledge of Haskell will be required to read this paper. That said, we have tried to keep the definitions and code as clear and simple as possible, and although we have made some use of Haskell overloading and do-notation, we have generally avoided using the more esoteric features of Haskell. In addition, some experience with the basics of Hindley-Milner style type inference [ Hindley, 1969, Milner, 1978, Damas & Milner, 1982]will be needed to understand the algorithms presented here. Although we have aimed to keep our presentation as simple as possible, some aspects of the problems that we are trying to address have inherent complexity or technical depth that cannot be side-stepped. In short, this paper will probably not be useful as a tutorial introduction to Hindley-Milner style type inference!

2  Preliminaries

For simplicity, we present the code for our typechecker as a single Haskell module. The program uses only a handful of standard prelude functions, like map, concat, all, any, mapM, etc., and a few operations from the List and Monad libraries:

  module TypingHaskellInHaskell where
  import List(nub, (\\), intersect, union, partition)
  import Monad(msum)

For the most part, our choice of variable names follows the notational conventions set out in Figure 1.

Description Symbol Type
kind k, ... Kind
type constructor tc, ... Tycon
type variable v, ... Tyvar
- `fixed' f, ...
- `generic' g, ...
type t, ... Type
class c, ... Class
instance it, ... Inst
predicate p, q, ... Pred
- `deferred' d, ...
- `retained' r, ...
qualified type qt, ... QualType
class environmentce, ... ClassEnv
scheme sc, ... Scheme
substitution s, ... Subst
unifier u, ... Subst
assumption a, ... Assump
identifier i, ... Id
literal l, ... Literal
pattern pat, ... Pat
expression e, f, ... Expr
alternative alt, ... Alt
binding group bg, ... BindGroup

Figure 1: Notational Conventions

A trailing s on a variable name usually indicates a list. Numeric suffices or primes are used as further decoration where necessary. For example, we use k or k' for a kind, and ks or ks' for a list of kinds. The types and terms appearing in the table are described more fully in later sections. To distinguish the code for the typechecker from program fragments that are used to discuss its behavior, we typeset the former in an italic font, and the latter in a typewriter font.

Throughout this paper, we implement identifiers as strings, and assume that there is a simple way to generate identifiers from integers using the enumId function:

  type Id  = String
 
  enumId  :: Int -> Id
  enumId n = "v" ++ show n

The enumId function will be used in the definition of the newTVar operator in Section 10 to describe the allocation of fresh type variables during type inference. With the simple implementation shown here, we assume that variable names beginning with ``v'' do not appear in input programs.

3  Kinds

To ensure that they are valid, Haskell type constructors are classified into different kinds: the kind * (pronounced `star') represents the set of all simple (i.e., nullary) type expressions, like Int and Char -> Bool; kinds of the form k1 -> k2 represent type constructors that take an argument type of kind k1 to a result type of kind k2. For example, the standard list, Maybe and IO constructors all have kind * -> *. Here, we will represent kinds as values of the following datatype:

  data Kind  = Star | Kfun Kind Kind
               deriving Eq

Kinds play essentially the same role for type constructors as types do for values, but the kind system is clearly very primitive. There are a number of extensions that would make interesting topics for future research, including polymorphic kinds, subkinding, and record/product kinds. A simple extension of the kind system-adding a new row kind-has already proved to be useful for the Trex implementation of extensible records in Hugs [ Gaster & Jones, 1996, Jones & Peterson, 1999].

4  Types

The next step is to define a representation for types. Stripping away syntactic sugar, Haskell type expressions are either type variables or constants (each of which has an associated kind), or applications of one type to another: applying a type of kind k1 -> k2 to a type of kind k1 produces a type of kind k2:

  data Type  = TVar Tyvar | TCon Tycon | TAp  Type Type | TGen Int
               deriving Eq
 
  data Tyvar = Tyvar Id Kind
               deriving Eq

  data Tycon = Tycon Id Kind
               deriving Eq

This definition also includes types of the form TGen n, which represent `generic' or quantified type variables. The only place where TGen values are used is in the representation of type schemes, which will be described in Section 8.

The following examples show how standard primitive datatypes are represented as type constants:

  tUnit    = TCon (Tycon "()" Star)
  tChar    = TCon (Tycon "Char" Star)
  tInt     = TCon (Tycon "Int" Star)
  tInteger = TCon (Tycon "Integer" Star)
  tFloat   = TCon (Tycon "Float" Star)
  tDouble  = TCon (Tycon "Double" Star)
 
  tList    = TCon (Tycon "[]" (Kfun Star Star))
  tArrow   = TCon (Tycon "(->)" (Kfun Star (Kfun Star Star)))
  tTuple2  = TCon (Tycon "(,)" (Kfun Star (Kfun Star Star)))

A full Haskell compiler or interpreter might store additional information with each type constant-such as the the list of constructor functions for an algebraic datatype-but such details are not needed during typechecking.

More complex types are built up from constants and variables using the TAp constructor. For example, the representation for the type Int -> [a] is as follows:

  TAp (TAp tArrow tInt) (TAp tList (TVar (Tyvar "a" Star)))

We do not provide a representation for type synonyms, assuming instead that they have been fully expanded before typechecking. For example, the String type-a synonym for [Char]-is represented as:

  tString    :: Type
  tString     = list tChar

It is always possible for an implementation to expand synonyms in this way because Haskell prevents the use of a synonym without its full complement of arguments. Moreover, the process is guaranteed to terminate because recursive synonym definitions are prohibited. In practice, however, implementations are likely to expand synonyms more lazily: in some cases, type error diagnostics may be easier to understand if they display synonyms rather than expansions.

We end this section with the definition of a few helper functions. The first three provide simple ways to construct function, list, and pair types, respectively:

  infixr      4 `fn`
  fn         :: Type -> Type -> Type
  a `fn` b    = TAp (TAp tArrow a) b
 
  list       :: Type -> Type
  list t      = TAp tList t

  pair       :: Type -> Type -> Type
  pair a b    = TAp (TAp tTuple2 a) b

We also define an overloaded function, kind, that can be used to determine the kind of a type variable, type constant, or type expression:

  class HasKind t where
    kind :: t -> Kind
  instance HasKind Tyvar where
    kind (Tyvar v k) = k
  instance HasKind Tycon where
    kind (Tycon v k) = k
  instance HasKind Type where
    kind (TCon tc) = kind tc
    kind (TVar u)  = kind u
    kind (TAp t _) = case (kind t) of
                       (Kfun _ k) -> k

Most of the cases here are straightforward. Notice, however, that we can calculate the kind of an application (TAp t t') using only the kind of its first argument t: Assuming that the type is well-formed, t must have a kind k'->k, where k' is the kind of t' and k is the kind of the whole application. This shows that we need only traverse the leftmost spine of a type expression to calculate its kind.

5  Substitutions

Substitutions-finite functions, mapping type variables to types-play a major role in type inference. In this paper, we represent substitutions using association lists:

  type Subst  = [(Tyvar, Type)]

To ensure that we work only with well-formed type expressions, we will be careful to construct only kind-preserving substitutions in which variables are mapped only to types of the same kind. The simplest substitution is the null substitution, represented by the empty list, which is obviously kind-preserving:

  nullSubst  :: Subst
  nullSubst   = []

Almost as simple are the substitutions (u +-> t)3 that map a single variable u to a type t of the same kind:

  (+->)      :: Tyvar -> Type -> Subst
  u +-> t     = [(u, t)]

This is kind-preserving if, and only if, kind u = kind t.

Substitutions can be applied to types-and, in fact, to any other value with type components-in a natural way. This suggests that we overload the operation to apply a substitution so that it can work on different types of object:

  class Types t where
    apply :: Subst -> t -> t
    tv    :: t -> [Tyvar]

In each case, the purpose of applying a substitution is the same: To replace every occurrence of a type variable in the domain of the substitution with the corresponding type. We also include a function tv that returns the set of type variables (i.e., Tyvars) appearing in its argument, listed in order of first occurrence (from left to right), with no duplicates. The definitions of these operations for Type are as follows:

  instance Types Type where
    apply s (TVar u)  = case lookup u s of
                         Just t  -> t
                         Nothing -> TVar u
    apply s (TAp l r) = TAp (apply s l) (apply s r)
    apply s t         = t
 
    tv (TVar u)  = [u]
    tv (TAp l r) = tv l `union` tv r
    tv t         = []

It is straightforward (and useful!) to extend these operations to work on lists:

  instance Types a => Types [a] where
    apply s = map (apply s)
    tv      = nub . concat . map tv

The apply function can be used to build more complex substitutions. For example, composition of substitutions, satisfying apply (s1 @@ s2) = apply s1 . apply s2, can be defined using:

  infixr 4 @@
  (@@)       :: Subst -> Subst -> Subst
  s1 @@ s2    = [ (u, apply s1 t) | (u,t) <- s2 ] ++ s1

We can also form a `parallel' composition s1++s2 of two substitutions s1 and s2, but the result is left-biased because bindings in s1 take precedence over any bindings for the same variables in s2. For a more symmetric version of this operation, we use a merge function, which checks that the two substitutions agree at every variable in the domain of both and hence guarantees that apply (s1++s2) = apply (s2++s1). Clearly, this is a partial function, which we reflect by arranging for merge to return its result in a monad, using the standard fail function to provide a string diagnostic in cases where the function is undefined.

  merge      :: Monad m => Subst -> Subst -> m Subst
  merge s1 s2 = if agree then return (s1++s2) else fail "merge fails"
   where agree = all (\v -> apply s1 (TVar v) == apply s2 (TVar v))
                     (map fst s1 `intersect` map fst s2)

It is easy to check that both (@@) and merge produce kind-preserving results from kind-preserving arguments. In the next section, we will see how the first of these composition operators is used to describe unification, while the second is used in the formulation of a matching operation.

6  Unification and Matching

The goal of unification is to find a substitution that makes two types equal-for example, to ensure that the domain type of a function matches up with the type of an argument value. However, it is also important for unification to find as `small' a substitution as possible because that will lead to most general types. More formally, a substitution s is a unifier of two types t1 and t2 if apply s t1 = apply s t2. A most general unifier, or mgu, of two such types is a unifier u with the property that any other unifier s can be written as s'@@u, for some substitution s'.

The syntax of Haskell types has been chosen to ensure that, if two types have any unifying substitutions, then they have a most general unifier, which can be calculated by a simple variant of Robinson's algorithm [ Robinson, 1965]. One of the reasons for this is that there are no non-trivial equalities on types. Extending the type system with higher-order features (such as lambda expressions on types), or with other mechanisms that allow reductions or rewriting in the type language, could make unification undecidable, non-unitary (meaning that there may not be most general unifiers), or both. This, for example, is why Haskell does not allow type synonyms to be partially applied (and interpreted as some restricted kind of lambda expression).

The calculation of most general unifiers is implemented by a pair of functions:

  mgu     :: Monad m => Type -> Type -> m Subst
  varBind :: Monad m => Tyvar -> Type -> m Subst

These functions return results in a monad, capturing the fact that unification is a partial function. The main algorithm is described by mgu, using the structure of its arguments to guide the calculation:

  mgu (TAp l r) (TAp l' r') = do s1 <- mgu l l'
                                 s2 <- mgu (apply s1 r) (apply s1 r')
                                 return (s2 @@ s1)
  mgu (TVar u) t        = varBind u t
  mgu t (TVar u)        = varBind u t
  mgu (TCon tc1) (TCon tc2)
             | tc1==tc2 = return nullSubst
  mgu t1 t2             = fail "types do not unify"

The varBind function is used for the special case of unifying a variable u with a type t. At first glance, one might think that we could just use the substitution (u+->t) for this. In practice, however, tests are required to ensure that this is valid, including an `occurs check' (u `elem` tv t) and a test to ensure that the substitution is kind-preserving:

  varBind u t | t == TVar u      = return nullSubst
              | u `elem` tv t    = fail "occurs check fails"
              | kind u /= kind t = fail "kinds do not match"
              | otherwise        = return (u +-> t)

In the following sections, we will also make use of an operation called matching that is closely related to unification. Given two types t1 and t2, the goal of matching is to find a substitution s such that apply s t1 = t2. Because the substitution is applied only to one type, this operation is often described as one-way matching. The calculation of matching substitutions is implemented by a function:

  match :: Monad m => Type -> Type -> m Subst

Matching follows the same pattern as unification, except that it uses merge rather than @@ to combine substitutions, and it does not allow binding of variables in t2:

  match (TAp l r) (TAp l' r') = do sl <- match l l'
                                   sr <- match r r'
                                   merge sl sr
  match (TVar u)   t | kind u == kind t = return (u +-> t)
  match (TCon tc1) (TCon tc2)
           | tc1==tc2         = return nullSubst
  match t1 t2                 = fail "types do not match"

7  Type Classes, Predicates and Qualified Types

One of the most unusual features of the Haskell type system, at least in comparison to those of other polymorphically typed languages like ML, is the support that it provides for type classes. Described by Wadler and Blott [ Wadler & Blott, 1989] as a general mechanism that subsumes several ad-hoc forms of overloading, type classes have found many uses (and, sometimes, abuses!) in the ten years since they were introduced. A significant portion of the code presented in this paper, particularly in this section, is needed to describe the handling of type classes in Haskell. (Of course, type classes are not the only source of complexity. The treatment of mixed implicit and explicit typing, mutually recursive bindings, and pattern matching-which are often elided in more theoretical presentations-are also significant contributors, as is the extra level of detail and precision that is needed in executable code.)

7.1  Basic Definitions

A Haskell type class can be thought of as a set of types (of some particular kind), each of which supports a certain collection of member functions that are specified as part of the class declaration. The types in each class (known as instances) are specified by a collection of instance declarations. Haskell types can be qualified by adding a (possibly empty) list of predicates, or class constraints, to restrict the ways in which type variables are instantiated4:

  data Qual t = [Pred] :=> t
                deriving Eq

In a value of the form ps :=> t, we refer to ps as the context and to t as the head. Predicates themselves consist of a class identifier and a type; a predicate of the form IsIn i t asserts that t is a member of the class named i:

  data Pred   = IsIn Id Type
                deriving Eq

For example, using the Qual and Pred datatypes, the type (Num a) => a -> Int can be represented by:

  [IsIn "Num" (TVar (Tyvar "a" Star))] :=> (TVar (Tyvar "a" Star) `fn` tInt)

It would be easy to extend the Pred datatype to allow other forms of predicate, as is done with Trex records in Hugs [ Jones & Peterson, 1999]. Another frequently requested extension is to allow classes to accept multiple parameters, which would require a list of Types rather than the single Type in the definition above.

The extension of Types to the Qual and Pred datatypes is straightforward:

  instance Types t => Types (Qual t) where
    apply s (ps :=> t) = apply s ps :=> apply s t
    tv (ps :=> t)      = tv ps `union` tv t
 
  instance Types Pred where
    apply s (IsIn i t) = IsIn i (apply s t)
    tv (IsIn i t)      = tv t

The tasks of calculating most general unifiers and matching substitutions on types also extend naturally to predicates:

  mguPred, matchPred :: Pred -> Pred -> Maybe Subst
  mguPred             = lift mgu
  matchPred           = lift match
 
  lift m (IsIn i t) (IsIn i' t')
           | i == i'   = m t t'
           | otherwise = fail "classes differ"

We will represent each class by a pair of lists, one containing the name of each superclass, and another containing an entry for each instance declaration:

  type Class    = ([Id], [Inst])
  type Inst     = Qual Pred

For example, a simplified version of the standard Haskell class Ord might be described by the following value of type Class:

  (["Eq"], [[] :=> IsIn "Ord" tUnit,
            [] :=> IsIn "Ord" tChar,
            [] :=> IsIn "Ord" tInt,
            [IsIn "Ord" (TVar (Tyvar "a" Star)),
             IsIn "Ord" (TVar (Tyvar "b" Star))]
               :=> IsIn "Ord" (pair (TVar (Tyvar "a" Star))
                                    (TVar (Tyvar "b" Star)))])

This structure captures the fact that Eq is a superclass of Ord (the only one in fact), and lists four instance declarations for the unit, character, integer, and pair types (if a and b are in Ord, then (a,b) is also in Ord). Of course, this is only a fraction of the list of Ord instances that are defined in the full Haskell prelude. Only the details that are needed for type inference are included in these representations. A full Haskell implementation would need to store additional information for each declaration, such as the list of member functions for each class and details of their implementations in each particular instance.

7.2  Class Environments

The information provided by the class and instance declarations in a given program can be captured by a class environment of type:

  data ClassEnv = ClassEnv { classes  :: Id -> Maybe Class,
                             defaults :: [Type] }

The classes component in a ClassEnv value is a partial function that maps identifiers to Class values (or to Nothing if there is no class corresponding to the specified identifier). We define helper functions super and insts to extract the list of superclass identifiers, and the list of instances, respectively, for a class name i in a class environment ce:

  super     :: ClassEnv -> Id -> [Id]
  super ce i = case classes ce i of Just (is, its) -> is
 
  insts     :: ClassEnv -> Id -> [Inst]
  insts ce i = case classes ce i of Just (is, its) -> its

These functions are intended to be used only in cases where it is known that the class i is defined in the environment ce. In some cases, this condition might be guaranteed by static analysis prior to type checking. Alternatively, we can resort to a dynamic check by testing defined (classes ce i) before applying either function. The function defined used here is defined as follows5:

  defined :: Maybe a -> Bool
  defined (Just x) = True
  defined Nothing  = False

We will also define a helper function, modify, to describe how a class environment can be updated to reflect a new binding of a Class value to a given identifier:

  modify       :: ClassEnv -> Id -> Class -> ClassEnv
  modify ce i c = ce{classes = \j -> if i==j then Just c
                                             else classes ce j}

The defaults component of a ClassEnv value is used to provide a list of types for defaulting, as described in Section 11.5.1. Haskell allows programmers to specify a value for this list using a default declaration; if no explicit declaration is given, then a default (Integer,Double) declaration is assumed. It is easy to describe this using the ClassEnv type. For example, cedefaults=[tInt] is the result of modifying a class environment ce to reflect the presence of a default (Int) declaration. Further discussion of defaulting is deferred to Section 11.5.1.

In the remainder of this section, we will show how to build an appropriate class environment for a given program, starting from an (almost) empty class environment, and extending it as necessary to reflect the effect of each class or instance declaration in the program. The initial class environment is defined as follows:

  initialEnv :: ClassEnv
  initialEnv  = ClassEnv { classes  = \i -> fail "class not defined",
                           defaults = [tInteger, tDouble] }

As we process each class or instance declaration in a program, we transform the initial class environment to add entries, either for a new class, or for a new instance, respectively. In either case, there is a possibility that the new declaration might be incompatible with the previous declarations, attempting, for example, to redefine an existing class or instance. For this reason, we will describe transformations of a class environment as functions of the EnvTransformer type, using a Maybe type to allow for the possibility of errors:

  type EnvTransformer = ClassEnv -> Maybe ClassEnv

The sequencing of multiple transformers can be described by a (forward) composition operator (<:>):

  infixr 5 <:>
  (<:>)       :: EnvTransformer -> EnvTransformer -> EnvTransformer
  (f <:> g) ce = do ce' <- f ce
                    g ce'

Some readers will recognize this as a special case of the more general Kleisli composition operator; without the type declaration, the definition given here would work for any monad and for any element types, not just for Maybe and ClassEnv.

To add a new class to an environment, we must check that there is not already a class with the same name, and that all of the named superclasses are already defined. This is a simple way of enforcing Haskell's restriction that the superclass hierarchy be acyclic. Of course, in practice, it will be necessary to topologically sort the set of class declarations in a program to determine a suitable ordering; any cycles in the hierarchy will typically be detected at this stage.

  addClass                              :: Id -> [Id] -> EnvTransformer
  addClass i is ce
   | defined (classes ce i)              = fail "class already defined"
   | any (not . defined . classes ce) is = fail "superclass not defined"
   | otherwise                           = return (modify ce i (is, []))

For example, we can describe the effect of the class declarations in the Haskell prelude using the following transformer:

  addPreludeClasses :: EnvTransformer
  addPreludeClasses  = addCoreClasses <:> addNumClasses

This definition breaks down the set of standard Haskell classes into two separate pieces. The core classes are described as follows:

  addCoreClasses ::   EnvTransformer
  addCoreClasses  =   addClass "Eq" []
                  <:> addClass "Ord" ["Eq"]
                  <:> addClass "Show" []
                  <:> addClass "Read" []
                  <:> addClass "Bounded" []
                  <:> addClass "Enum" []
                  <:> addClass "Functor" []
                  <:> addClass "Monad" []

The hierarchy of numeric classes is captured separately in the following definition:

  addNumClasses  ::   EnvTransformer
  addNumClasses   =   addClass "Num" ["Eq", "Show"]
                  <:> addClass "Real" ["Num", "Ord"]
                  <:> addClass "Fractional" ["Num"]
                  <:> addClass "Integral" ["Real", "Enum"]
                  <:> addClass "RealFrac" ["Real", "Fractional"]
                  <:> addClass "Floating" ["Fractional"]
                  <:> addClass "RealFloat" ["RealFrac", "Floating"]

To add a new instance to a class, we must check that the class to which the instance applies is defined, and that the new instance does not overlap with any previously declared instance:

  addInst                        :: [Pred] -> Pred -> EnvTransformer
  addInst ps p@(IsIn i _) ce
   | not (defined (classes ce i)) = fail "no class for instance"
   | any (overlap p) qs           = fail "overlapping instance"
   | otherwise                    = return (modify ce i c)
     where its = insts ce i
           qs  = [ q | (_ :=> q) <- its ]
           c   = (super ce i, (ps:=>p) : its)

Two instances for a class are said to overlap if there is some predicate that is a substitution instance of the heads of both instance declarations. It is easy to test for overlapping predicates using the functions that we have defined previously:

  overlap       :: Pred -> Pred -> Bool
  overlap p q    = defined (mguPred p q)

This test covers simple cases where a program provides two instance declarations for the same type (for example, two declarations for Eq Int), but it also covers cases where more interesting overlaps occur (for example, between the predicates Eq [Int] and Eq [a], or between predicates Eq (a,Bool) and Eq (Int,b)). In each case, the existence of an overlap indicates the possibility of a semantic ambiguity, with two applicable instance declarations, and no clear reason to prefer one over the other. This is why Haskell treats such overlaps as an error. Extensions to Haskell to support overlapping instances in certain special cases have been considered elsewhere; they appear to have interesting applications, but also have some potentially troublesome impact on program semantics [ Peyton Jones et al. , 1997]. We will not consider such issues further in this paper.

To illustrate how the addInst function might be used, the following definition shows how the standard prelude class environment can be extended to include the four instances for Ord from the example in Section 7.1:

  exampleInsts ::  EnvTransformer
  exampleInsts =   addPreludeClasses
               <:> addInst [] (IsIn "Ord" tUnit)
               <:> addInst [] (IsIn "Ord" tChar)
               <:> addInst [] (IsIn "Ord" tInt)
               <:> addInst [IsIn "Ord" (TVar (Tyvar "a" Star)),
                            IsIn "Ord" (TVar (Tyvar "b" Star))]
                           (IsIn "Ord" (pair (TVar (Tyvar "a" Star))
                                             (TVar (Tyvar "b" Star))))

The Haskell report imposes some further restrictions on class and instance declarations that are not enforced by the definitions of addClass and addInst. For example, the superclasses of a class should have the same kind as the class itself; the parameters of any predicates in an instance context should be type variables, each of which should appear in the head of the instance; and the type appearing in the head of an instance should consist of a type constructor applied to a sequence of distinct type variable arguments. Because these conditions have no direct impact on type checking, and because they are straightforward but tedious to verify, we have chosen not to include tests for them here, and instead assume that they have been checked during static analysis prior to type checking.

7.3  Entailment

In this section, we describe how class environments can be used to answer questions about which types are instances of particular classes. More generally, we consider the treatment of entailment: given a predicate p and a list of predicates ps, our goal is to determine whether p will hold whenever all of the predicates in ps are satisfied. In the special case where p = IsIn i t and ps = [], this amounts to determining whether t is an instance of the class i. In the theory of qualified types [ Jones, 1992], assertions like this are captured using judgements of the form ps ||- p; we use a different notation here-the entail function that is defined at the end of this section-to make the dependence on a class environment explicit.

As a first step, we can ask how information about superclasses and instances can be used independently to help reason about entailments. For example, if a type is an instance of a class i, then it must also be an instance of any superclasses of i. Hence, using only superclass information, we can be sure that, if a given predicate p holds, then so too must all of the predicates in the list bySuper p:

  bySuper :: ClassEnv -> Pred -> [Pred]
  bySuper ce p@(IsIn i t)
   = p : concat [ bySuper ce (IsIn i' t) | i' <- super ce i ]

The list bySuper ce p may contain duplicates, but it will always be finite because of the restriction that the superclass hierarchy is acyclic.

Next we consider how information about instances can be used. Of course, for a given predicate p = IsIn i t, we can find all the directly relevant instances in a class environment ce by looking in insts ce i. As we have seen, individual instance declarations are mapped into clauses of the form ps :=> h. The head predicate h describes the general form of instances that can be constructed from this declaration, and we can use matchPred to determine whether this instance is applicable to the given predicate p. If it is applicable, then matching will return a substitution u, and the remaining subgoals are the elements of map (apply u) ps. The following function uses these ideas to determine the list of subgoals for a given predicate:

  byInst                   :: ClassEnv -> Pred -> Maybe [Pred]
  byInst ce p@(IsIn i t)    = msum [ tryInst it | it <- insts ce i ]
   where tryInst (ps :=> h) = do u <- matchPred h p
                                 Just (map (apply u) ps)

The msum function used here comes from the standard Monad library, and returns the first defined element in a list of Maybe values; if there are no defined elements in the list, then it returns Nothing. Because Haskell prevents overlapping instances, there is at most one applicable instance for any given p, and we can be sure that the first defined element will actually be the only defined element in this list.

The bySuper and byInst functions can be used in combination to define a general entailment operator, entail. Given a particular class environment ce, the intention here is that entail ce ps p will be True if, and only if, the predicate p will hold whenever all of the predicates in ps are satisfied:

  entail        :: ClassEnv -> [Pred] -> Pred -> Bool
  entail ce ps p = any (p `elem`) (map (bySuper ce) ps) ||
                   case byInst ce p of
                     Nothing -> False
                     Just qs -> all (entail ce ps) qs

The first step here is to determine whether p can be deduced from ps using only superclasses. If that fails, we look for a matching instance and generate a list of predicates qs as a new goal, each of which must, in turn, follow from ps.

Conditions specified in the Haskell report-namely that the class hierarchy is acyclic and that the types in any instance declaration are strictly smaller than those in the head-translate into conditions on the values for the ClassEnv that can be passed in as ce, and these are enough to guarantee that tests for entailment will terminate. Completeness of the algorithm is also important: will entail ce ps p always return True whenever there is a way to prove p from ps? In fact our algorithm does not cover all possible cases: it does not test to see if p is a superclass of some other predicate q for which entail ce ps q is True. Extending the algorithm to test for this would be very difficult because there is no obvious way to choose a particular q, and, in general, there will be infinitely many potential candidates to consider. Fortunately, a technical condition in the Haskell report [ Peyton Jones & Hughes, 1999,Condition 1 on Page 47] reassures us that this is not necessary: if p can be obtained as an immediate superclass of some predicate q that was built using an instance declaration in an entailment entail ce ps q, then ps must already be strong enough to deduce p. Thus, although we have not formally proved these properties, we believe that our algorithm is sound, complete, and guaranteed to terminate.

7.4  Context Reduction

Class environments also play an important role in an aspect of the Haskell type system that is known as context reduction. The basic goal of context reduction is to reduce a list of predicates to an equivalent but, in some sense, simpler list. The Haskell report [ Peyton Jones & Hughes, 1999]provides only informal hints about this aspect of the Haskell typing, where both pragmatics and theory have important parts to play. We believe therefore that this is one of the areas where a more formal specification will be particularly valuable.

One way to simplify a list of predicates is to simplify the type components of individual predicates in the list. For example, given the instance declarations in the Haskell standard prelude, we could replace any occurrences of predicates like Eq [a], Eq (a,a), or Eq ([a],Int) with Eq a. This is valid because, for any choice of a, each one of these predicates holds if, and only if, Eq a holds. Notice that, in some cases, an attempt to simplify type components-for example, by replacing Eq (a, b) with (Eq a, Eq b)-may increase the number of predicates in the list. The extent to which simplifications like this are used in a system of qualified types has an impact on the implementation and performance of overloading in practical systems [ Jones, 1992,Chapter 7]. In Haskell, however, the decisions are made for us by a syntactic restriction that forces us to simplify predicates until we obtain types in a kind of `head-normal form'. This terminology is motivated by similarities with the concept of head-normal forms in l-calculus. More precisely, the syntax of Haskell requires class arguments to be of the form v t1 ... tn, where v is a type variable, and t1,...,tn are types (and n 0). The following function allows us to determine whether a given predicate meets these restrictions:

  inHnf       :: Pred -> Bool
  inHnf (IsIn c t) = hnf t
   where hnf (TVar v)  = True
         hnf (TCon tc) = False
         hnf (TAp t _) = hnf t

Predicates that do not fit this pattern must be broken down using byInst. In some cases, this will result in predicates being eliminated altogether. In others, where byInst fails, it will indicate that a predicate is unsatisfiable, and will trigger an error diagnostic. This process is captured in the following definition:

  toHnfs      :: Monad m => ClassEnv -> [Pred] -> m [Pred]
  toHnfs ce ps = do pss <- mapM (toHnf ce) ps
                    return (concat pss)
 
  toHnf                 :: Monad m => ClassEnv -> Pred -> m [Pred]
  toHnf ce p | inHnf p   = return [p]
             | otherwise = case byInst ce p of
                             Nothing -> fail "context reduction"
                             Just ps -> toHnfs ce ps

Another way to simplify a list of predicates is to reduce the number of elements that it contains. There are several ways that this might be achieved: by removing duplicates (e.g., reducing (Eq a, Eq a) to Eq a); by eliminating predicates that are already known to hold (e.g., removing any occurrences of Num Int); or by using superclass information (e.g., reducing (Eq a, Ord a) to Ord a). In each case, the reduced list of predicates, is equivalent to the initial list, meaning that all the predicates in the first will be satisfied if, and only if, all of the predicates in the second are satisfied. The simplification algorithm that we will use here is based on the observation that a predicate p in a list of predicates (p:ps) can be eliminated if p is entailed by ps. As a special case, this will eliminate duplicated predicates: if p is repeated in ps, then it will also be entailed by ps. These ideas are used in the following definition of the simplify function, which loops through each predicate in the list and uses an accumulating parameter to build up the final result. Each time we encounter a predicate that is entailed by the others, we remove it from the list.

  simplify   :: ClassEnv -> [Pred] -> [Pred]
  simplify ce = loop []
   where loop rs []                            = rs
         loop rs (p:ps) | entail ce (rs++ps) p = loop rs ps
                        | otherwise            = loop (p:rs) ps

Now we can describe the particular form of context reduction used in Haskell as a combination of toHnfs and simplify. Specifically, we use toHnfs to reduce the list of predicates to head-normal form, and then simplify the result:

  reduce      :: Monad m => ClassEnv -> [Pred] -> m [Pred]
  reduce ce ps = do qs <- toHnfs ce ps
                    return (simplify ce qs)

As a technical aside, we note that there is some redundancy in the definition of reduce. The simplify function is defined in terms of entail, which makes use of the information provided by both superclass and instance declarations. The predicates in qs, however, are guaranteed to be in head-normal form, and hence will not match instance declarations that satisfy the syntactic restrictions of Haskell. It follows that we could make do with a version of simplify that used only the following function in determining (superclass) entailments:

  scEntail        :: ClassEnv -> [Pred] -> Pred -> Bool
  scEntail ce ps p = any (p `elem`) (map (bySuper ce) ps)

8  Type Schemes

Type schemes are used to describe polymorphic types, and are represented using a list of kinds and a qualified type:

  data Scheme = Forall [Kind] (Qual Type)
                deriving Eq

There is no direct equivalent of Forall in the syntax of Haskell. Instead, implicit quantifiers are inserted as necessary to bind free type variables.

In a type scheme Forall ks qt, each type of the form TGen n that appears in the qualified type qt represents a generic, or universally quantified type variable whose kind is given by ks!!n. This is the only place where we will allow TGen values to appear in a type. We had originally hoped that this restriction could be captured statically by a careful choice of the representation for types and type schemes. Unfortunately, we have not yet found a satisfactory way to enforce this, and, after considering several alternatives, we have settled for the representation shown here because it allows for simple implementations of equality and substitution. For example, the implementation of apply on Type values ignores TGen values, so we can be sure that there will be no variable capture problems in the following definition:

  instance Types Scheme where
    apply s (Forall ks qt) = Forall ks (apply s qt)
    tv (Forall ks qt)      = tv qt

Type schemes are constructed by quantifying a qualified type qt with respect to a list of type variables vs:

  quantify      :: [Tyvar] -> Qual Type -> Scheme
  quantify vs qt = Forall ks (apply s qt)
   where vs' = [ v | v <- tv qt, v `elem` vs ]
         ks  = map kind vs'
         s   = zip vs' (map TGen [0..])

Note that the order of the kinds in ks is determined by the order in which the variables v appear in tv qt, and not by the order in which they appear in vs. So, for example, the leftmost quantified variable in a type scheme will always be represented by TGen 0. By insisting that type schemes are constructed in this way, we obtain a unique canonical form for Scheme values. This is important because it means that we can test whether two type schemes are the same-for example, to determine whether an inferred type agrees with a declared type-using Haskell's derived equality, and without having to implement more complex tests for a-equivalence.

In practice, we sometimes need to convert a Type into a Scheme without adding any qualifying predicates or quantified variables. For this special case, we can use the following function instead of quantify:

  toScheme      :: Type -> Scheme
  toScheme t     = Forall [] ([] :=> t)

To complete our description of type schemes, we need to be able to instantiate the quantified variables in Scheme values. In fact, for the purposes of type inference, we only need the special case that instantiates a type scheme with fresh type variables. We therefore defer further description of instantiation to Section 10 where the mechanisms for generating fresh type variables are introduced.

9  Assumptions

Assumptions about the type of a variable are represented by values of the Assump datatype, each of which pairs a variable name with a type scheme:

  data Assump = Id :>: Scheme

Once again, we can extend the Types class to allow the application of a substitution to an assumption:

  instance Types Assump where
    apply s (i :>: sc) = i :>: (apply s sc)
    tv (i :>: sc)      = tv sc

Thanks to the instance definition for Types on lists (Section 5), we can also use the apply and tv operators on the lists of assumptions that are used to record the type of each program variable during type inference. We will also use the following function to find the type of a particular variable in a given set of assumptions:

  find                 :: Monad m => Id -> [Assump] -> m Scheme
  find i []             = fail ("unbound identifier: " ++ i)
  find i ((i':>:sc):as) = if i==i' then return sc else find i as

This definition allows for the possibility that the variable i might not appear in as. In practice, occurrences of unbound variables will probably have been detected in earlier compiler passes.

10  A Type Inference Monad

It is now quite standard to use monads as a way to hide certain aspects of `plumbing' and to draw attention instead to more important aspects of a program's design [ Wadler, 1992]. The purpose of this section is to define the monad that will be used in the description of the main type inference algorithm in Section 11. Our choice of monad is motivated by the needs of maintaining a `current substitution' and of generating fresh type variables during typechecking. In a more realistic implementation, we might also want to add error reporting facilities, but in this paper the crude but simple fail function from the Haskell prelude is all that we require. It follows that we need a simple state monad with only a substitution and an integer (from which we can generate new type variables) as its state:

  newtype TI a = TI (Subst -> Int -> (Subst, Int, a))
 
  instance Monad TI where
    return x   = TI (\s n -> (s,n,x))
    TI f >>= g = TI (\s n -> case f s n of
                              (s',m,x) -> let TI gx = g x
                                          in  gx s' m)
 
  runTI       :: TI a -> a
  runTI (TI f) = x where (s,n,x) = f nullSubst 0

The getSubst operation returns the current substitution, while unify extends it with a most general unifier of its arguments:

  getSubst   :: TI Subst
  getSubst    = TI (\s n -> (s,n,s))
 
  unify      :: Type -> Type -> TI ()
  unify t1 t2 = do s <- getSubst
                   u <- mgu (apply s t1) (apply s t2)
                   extSubst u

For clarity, we define the operation that extends the substitution as a separate function, even though it is used only here in the definition of unify:

  extSubst   :: Subst -> TI ()
  extSubst s' = TI (\s n -> (s'@@s, n, ()))

Overall, the decision to hide the current substitution in the TI monad makes the presentation of type inference much clearer. In particular, it avoids heavy use of apply every time an extension is (or might have been) computed.

There is only one primitive that deals with the integer portion of the state, using it in combination with enumId to generate a new type variable of a specified kind:

  newTVar    :: Kind -> TI Type
  newTVar k   = TI (\s n -> let v = Tyvar (enumId n) k
                            in  (s, n+1, TVar v))

One place where newTVar is useful is in instantiating a type scheme with new type variables of appropriate kinds:

  freshInst               :: Scheme -> TI (Qual Type)
  freshInst (Forall ks qt) = do ts <- mapM newTVar ks
                                return (inst ts qt)

The structure of this definition guarantees that ts has exactly the right number of type variables, and each with the right kind, to match ks. Hence, if the type scheme is well-formed, then the qualified type returned by freshInst will not contain any unbound generics of the form TGen n. The definition relies on an auxiliary function inst, which is a variation of apply that works on generic variables. In other words, inst ts t replaces each occurrence of a generic variable TGen n in t with ts!!n. It is convenient to build up the definition of inst using overloading:

  class Instantiate t where
    inst  :: [Type] -> t -> t
  instance Instantiate Type where
    inst ts (TAp l r) = TAp (inst ts l) (inst ts r)
    inst ts (TGen n)  = ts !! n
    inst ts t         = t
  instance Instantiate a => Instantiate [a] where
    inst ts = map (inst ts)
  instance Instantiate t => Instantiate (Qual t) where
    inst ts (ps :=> t) = inst ts ps :=> inst ts t
  instance Instantiate Pred where
    inst ts (IsIn c t) = IsIn c (inst ts t)

11  Type Inference

With this section we have reached the heart of the paper, detailing our algorithm for type inference. It is here that we finally see how the machinery that has been built up in earlier sections is actually put to use. We develop the complete algorithm in stages, working through the abstract syntax of the input language from the simplest part (literals) to the most complex (binding groups). Most of the typing rules are expressed by functions whose types are simple variants of the following synonym:

  type Infer e t = ClassEnv -> [Assump] -> e -> TI ([Pred], t)

In more theoretical treatments, it would not be surprising to see the rules expressed in terms of judgments G;P | A\vdash e:t, where G is a class environment, P is a set of predicates, A is a set of assumptions, e is an expression, and t is a corresponding type [ Jones, 1992]. Judgments like this can be thought of as 5-tuples, and the typing rules themselves just correspond to a 5-place relation. Exactly the same structure shows up in types of the form Infer e t, except that, by using functions, we distinguish very clearly between input and output parameters.

11.1  Literals

Like other languages, Haskell provides special syntax for constant values of certain primitive datatypes, including numerics, characters, and strings. We will represent these literal expressions as values of the Literal datatype:

  data Literal = LitInt  Integer
               | LitChar Char
               | LitRat  Rational
               | LitStr  String

Type inference for literals is straightforward. For characters, we just return tChar. For integers, we return a new type variable v together with a predicate to indicate that v must be an instance of the Num class. The cases for String and floating point literals follow similar patterns:

  tiLit            :: Literal -> TI ([Pred],Type)
  tiLit (LitChar _) = return ([], tChar)
  tiLit (LitInt _)  = do v <- newTVar Star
                         return ([IsIn "Num" v], v)
  tiLit (LitStr _)  = return ([], tString)
  tiLit (LitRat _)  = do v <- newTVar Star
                         return ([IsIn "Fractional" v], v)

11.2  Patterns

Patterns are used to inspect and deconstruct data values in lambda abstractions, function and pattern bindings, list comprehensions, do notation, and case expressions. We will represent patterns using values of the Pat datatype:

  data Pat        = PVar Id
                  | PWildcard
                  | PAs  Id Pat
                  | PLit Literal
                  | PNpk Id Integer
                  | PCon Assump [Pat]

A PVar i pattern matches any value and binds the result to the variable i. A PWildcard pattern, corresponding to an underscore _ in Haskell syntax, matches any value, but does not bind any variables. A pattern of the form (PAs i pat), known as an ``as-pattern'' and written using the syntax i@pat in Haskell, binds the variable i to any value that matches the pattern pat, while also binding any variables that appear in pat. A PLit l pattern matches only the particular value denoted by the literal l. A pattern (PNpk i k) is an (n+k) pattern, which matches any positive integral value m that is greater than or equal to k, and binds the variable i to the difference (m-k). Finally, a pattern of the form PCon a pats matches only values that were built using the constructor function a with a sequence of arguments that matches pats. We use values a of type Assump to represent constructor functions; all that we really need for typechecking is the type, although the name is useful for debugging. A full implementation would store additional details, such as arity, and use this to check that constructor functions in patterns are always fully applied.

Most Haskell patterns have a direct representation in Pat, but extensions would be needed to account for patterns using labeled fields. This is not difficult, but adds some complexity, which we prefer to avoid in this presentation.

Type inference for patterns has two goals: To calculate a type for each bound variable, and to determine what type of values the whole pattern might match. This leads us to look for a function:

  tiPat :: Pat -> TI ([Pred], [Assump], Type)

Note that we do not need to pass in a list of assumptions here; by definition, any occurrence of a variable in a pattern would hide rather than refer to a variable of the same name in an enclosing scope.

For a variable pattern, PVar i, we just return a new assumption, binding i to a fresh type variable.

  tiPat (PVar i) = do v <- newTVar Star
                      return ([], [i :>: toScheme v], v)

Haskell does not allow multiple use of any variable in a pattern, so we can be sure that this is the first and only occurrence of i that we will encounter in the pattern. Wildcards are typed in the same way except that we do not need to create a new assumption:

  tiPat PWildcard   = do v <- newTVar Star
                         return ([], [], v)

To type an as-pattern PAs i pat, we calculate a set of assumptions and a type for the pat pattern, and then add an extra assumption to bind i:

  tiPat (PAs i pat) = do (ps, as, t) <- tiPat pat
                         return (ps, (i:>:toScheme t):as, t)

For literal patterns, we use tiLit from the previous section:

  tiPat (PLit l) = do (ps, t) <- tiLit l
                      return (ps, [], t)

The rule for (n+k) patterns does not fix a type for the bound variable, but adds a predicate to constrain the choice to instances of the Integral class:

  tiPat (PNpk i k)  = do t <- newTVar Star
                         return ([IsIn "Integral" t], [i:>:toScheme t], t)

The case for constructed patterns is slightly more complex:

  tiPat (PCon (i:>:sc) pats) = do (ps,as,ts) <- tiPats pats
                                  t'         <- newTVar Star
                                  (qs :=> t) <- freshInst sc
                                  unify t (foldr fn t' ts)
                                  return (ps++qs, as, t')

First we use the tiPats function, defined below, to calculate types ts for each subpattern in pats together with corresponding lists of assumptions in as and predicates in ps. Next, we generate a new type variable t' that will be used to capture the (as yet unknown) type of the whole pattern. From this information, we would expect the constructor function at the head of the pattern to have type foldr fn t' ts. We can check that this is possible by instantiating the known type sc of the constructor and unifying.

The tiPats function is a variation of tiPat that takes a list of patterns as input, and returns a list of types (together with a list of predicates and a list of assumptions) as its result.

  tiPats     :: [Pat] -> TI ([Pred], [Assump], [Type])
  tiPats pats = do psasts <- mapM tiPat pats
                   let ps = concat [ ps' | (ps',_,_) <- psasts ]
                       as = concat [ as' | (_,as',_) <- psasts ]
                       ts = [ t | (_,_,t) <- psasts ]
                   return (ps, as, ts)

We have already seen how tiPats was used in the treatment of PCon patterns above. It is also useful in other situations where lists of patterns are used, such as on the left hand side of an equation in a function definition.

11.3  Expressions

Next we describe type inference for expressions, represented by the Expr datatype:

  data Expr = Var   Id
            | Lit   Literal
            | Const Assump
            | Ap    Expr Expr
            | Let   BindGroup Expr

The Var and Lit constructors are used to represent variables and literals, respectively. The Const constructor is used to deal with named constants, such as the constructor or selector functions associated with a particular datatype or the member functions that are associated with a particular class. We use values of type Assump to supply a name and type scheme, which is all the information that we need for the purposes of type inference. Function application is represented using the Ap constructor, while Let is used to represent let expressions. (Note that the definition of the BindGroup type, used here to represent binding groups, will be delayed to Section 11.6.3.) Of course, Haskell has a much richer syntax of expressions-which includes l-abstractions, case expressions, conditionals, list comprehensions, and do-notation-but they all have simple translations into Expr values. For example, a l-expression like \x->e can be rewritten using a local definition as let f x = e in f, where f is a new variable.

Type inference for expressions is quite straightforward:

  tiExpr                       :: Infer Expr Type
  tiExpr ce as (Var i)          = do sc         <- find i as
                                     (ps :=> t) <- freshInst sc
                                     return (ps, t)
  tiExpr ce as (Const (i:>:sc)) = do (ps :=> t) <- freshInst sc
                                     return (ps, t)
  tiExpr ce as (Lit l)          = do (ps,t) <- tiLit l
                                     return (ps, t)
  tiExpr ce as (Ap e f)         = do (ps,te) <- tiExpr ce as e
                                     (qs,tf) <- tiExpr ce as f
                                     t       <- newTVar Star
                                     unify (tf `fn` t) te
                                     return (ps++qs, t)
  tiExpr ce as (Let bg e)       = do (ps, as') <- tiBindGroup ce as bg
                                     (qs, t)   <- tiExpr ce (as' ++ as) e
                                     return (ps ++ qs, t)

The final case here, for Let expressions, uses the function tiBindGroup presented in Section 11.6.3, to generate a list of assumptions as' for the variables defined in bg. All of these variables are in scope when we calculate a type t for the body e, which also serves as the type of the whole expression.

11.4  Alternatives

The representation of function bindings in following sections uses alternatives, represented by values of type Alt:

  type Alt = ([Pat], Expr)

An Alt specifies the left and right hand sides of a function definition. With a more complete syntax for Expr, values of type Alt might also be used in the representation of lambda and case expressions.

For type inference, we begin by using tiPats to infer a type for each of the patterns, and to build a new list as' of assumptions for any bound variables, as described in Section 11.2. Next, we calculate the type of the body in the scope of the bound variables, and combine this with the types of each pattern to obtain a single (function) type for the whole Alt:

  tiAlt                :: Infer Alt Type
  tiAlt ce as (pats, e) = do (ps, as', ts) <- tiPats pats
                             (qs,t)  <- tiExpr ce (as'++as) e
                             return (ps++qs, foldr fn t ts)

In practice, we will often run the typechecker over a list of alternatives, alts, and check that the returned type in each case agrees with some known type t. This process can be packaged up in the following definition:

  tiAlts             :: ClassEnv -> [Assump] -> [Alt] -> Type -> TI [Pred]
  tiAlts ce as alts t = do psts <- mapM (tiAlt ce as) alts
                           mapM (unify t) (map snd psts)
                           return (concat (map fst psts))

Although we do not need it here, the signature for tiAlts would allow an implementation to push the type argument inside the checking of each Alt, interleaving unification with type inference instead of leaving it to the end. This is essential in extensions like the support for rank-2 polymorphism in Hugs where explicit type information plays a key role. Even in an unextended Haskell implementation, this could still help to improve the quality of type error messages. Of course, we can still use tiAlts to infer a type from scratch. All this requires is that we generate and pass in a fresh type variable v in the parameter t to tiAlts, and then inspect the value of v under the current substitution when it returns.

11.5  From Types to Type Schemes

We have seen how lists of predicates are accumulated during type inference; now we will describe how those predicates are used to construct inferred types. This process is sometimes referred to as generalization because the goal is always to calculate the most general types that are possible. In a standard Hindley-Milner system, we can usually calculate most general types by quantifying over all relevant type variables that do not appear in the assumptions. In this section, we will describe how this process is modified to deal with the predicates in Haskell types.

To understand the basic problem, suppose that we have run the type checker over the body of a function h to obtain a list of predicates ps and a type t. At this point, to obtain the most general result, we could infer a type for h by forming the qualified type qt = (ps :=> t) and then quantifying over any variables in qt that do not appear in the assumptions. While this is permitted by the theory of qualified types, it is often not the best thing to do in practice. For example:

In this paper we use a function called split to address these issues. For the situation described previously where we have inferred a type t and a list of predicates ps for a function h, we can use split to rewrite and break ps into a pair (ds, rs) of deferred predicates ds and `retained' predicates rs. The predicates in rs will be used to form an inferred type (rs :=> t) for h, while the predicates in ds will be passed out as constraints to the enclosing scope. We use the following definition for split:

  split :: Monad m => ClassEnv -> [Tyvar] -> [Tyvar] -> [Pred]
                        -> m ([Pred], [Pred])
  split ce fs gs ps = do ps' <- reduce ce ps
                         let (ds, rs) = partition (all (`elem` fs) . tv) ps'
                         rs' <- defaultedPreds ce (fs++gs) rs
                         return (ds, rs \\ rs')

In addition to a list of predicates ps, the split function is parameterized by two lists of type variables. The first, fs, specifies the set of `fixed' variables, which are just the variables appearing free in the assumptions. The second, gs, specifies the set of variables over which we would like to quantify; for the example above, it would just be the variables in (tv t \\ fs). It is possible for ps to contain variables that are not in either fs or gs (and hence not in the parameter (fs++gs) that is passed to defaultedPreds). In Section 11.5.1 we will see that this is an indication of ambiguity.

There are three stages in the split function, corresponding directly to the three points listed previously. The first stage uses reduce to perform context reduction. The second stage uses the standard prelude function partition to identify the deferred predicates, ds; these are just the predicates in ps' that contain only fixed type variables. The third stage determines whether any of the predicates in rs should be eliminated using Haskell's defaulting mechanism, and produces a list of all such predicates in rs'. Hence the final set of retained predicates is produced by the expression rs \\ rs' in the last line of the definition.

11.5.1  Ambiguity and Defaults

In the terminology of Haskell [ Peyton Jones & Hughes, 1999,Section 4.3.4], a type scheme ps => t is ambiguous if ps contains generic variables that do not also appear in t. This condition is important because theoretical studies [ Blott, 1991, Jones, 1992]have shown that, in the general case, we can only guarantee a well-defined semantics for a term if its most general type is not ambiguous. As a result, expressions with ambiguous types are considered ill-typed in Haskell and will result in a static error. The following definition shows a fairly typical example illustrating how ambiguity problems can occur:
   stringInc x = show (read x + 1)
The intention here is that a string representation of a number will be parsed (using the prelude function read), incremented, and converted back to a string (using the prelude function show). But there is a genuine ambiguity because there is nothing to specify which type of number is intended, and because different choices can lead to different semantics. For example, stringInc "1.5" might produce a result of "2.5" if floating point numbers are used, or a parse error (or perhaps a result of "2") if integers are used. This semantic ambiguity is reflected by a syntactic ambiguity in the inferred type of stringInc:
   stringInc :: (Read a, Num a) => String -> String
(There is no Show a constraint here because Show is a superclass of Num.) A programmer can fix this particular problem quite easily by picking a particular type for a, and by adding an appropriate type annotation:
   stringInc x = show (read x + 1 :: Int)
Practical experience suggests that ambiguities like this tend to occur quite infrequently in real Haskell code. Moreover, when ambiguities are detected, the error diagnostics that are generated can often be useful in guiding programmers to genuine problems in their code. However, the designers of Haskell felt that, in some situations involving numeric types-and particularly involving overloaded numeric literals-the potential for ambiguity was significant enough to become quite a burden on programmers. Haskell's default mechanism was therefore introduced as a pragmatic compromise that is convenient-because it automates the task of picking types for otherwise ambiguous variables-but also dangerous-because it involves making choices about the semantics of a program in ways that are not always directly visible to programmers. For this latter reason, the use of defaulting is restricted so that it will only apply under certain, fairly restrictive circumstances.

The remainder of this section explains in more detail how ambiguities in Haskell programs can be detected and, when appropriate, eliminated by a suitable choice of defaults. The first step is to identify any sources of ambiguity. Suppose, for example, that we are about to qualify a type with a list of predicates ps and that vs lists all known variables, both fixed and generic. An ambiguity occurs precisely if there is a type variable that appears in ps but not in vs (i.e., in tv ps \\ vs). The goal of defaulting is to bind each ambiguous type variable v to a monotype t. The type t must be chosen so that all of the predicates in ps that involve v will be satisfied once t has been substituted for v. The following function calculates the list of ambiguous variables and pairs each one with the list of predicates that must be satisfied by any choice of a default:

  type Ambiguity       = (Tyvar, [Pred])
 
  ambiguities         :: ClassEnv -> [Tyvar] -> [Pred] -> [Ambiguity]
  ambiguities ce vs ps = [ (v, filter (elem v . tv) ps) | v <- tv ps \\ vs ]

Given one of these pairs (v,qs), and as specified by the Haskell report [ Peyton Jones & Hughes, 1999,Section 4.3.4], defaulting is permitted if, and only if, all of the following conditions are satisfied:

These conditions are captured rather more succinctly in the following definition, which we use to calculate the candidates for resolving a particular ambiguity:

  candidates           :: ClassEnv -> Ambiguity -> [Type]
  candidates ce (v, qs) = [ t' | let is = [ i | IsIn i t <- qs ]
                                     ts = [ t | IsIn i t <- qs ],
                                 all ((TVar v)==) ts,
                                 any (`elem` numClasses) is,
                                 all (`elem` stdClasses) is,
                                 t' <- defaults ce,
                                 all (entail ce []) [ IsIn i t' | i <- is ] ]

If candidates returns an empty list for a given ambiguity, then defaulting cannot be applied to the corresponding variable, and the ambiguity cannot be avoided. On the other hand, if the result is a non-empty list ts, then we will be able to substitute head ts for v and remove the predicates in qs from ps. The calculations for the defaulting substitution, and for the list of predicates that it eliminates follow very similar patterns, which we capture by defining them in terms of a single, higher-order function:

  withDefaults :: Monad m => ([Ambiguity] -> [Type] -> a)
                    -> ClassEnv -> [Tyvar] -> [Pred] -> m a
  withDefaults f ce vs ps
      | any null tss  = fail "cannot resolve ambiguity"
      | otherwise     = return (f vps (map head tss))
        where vps = ambiguities ce vs ps
              tss = map (candidates ce) vps

The withDefaults function takes care of picking suitable defaults, and of checking whether there are any ambiguities that cannot be eliminated. If defaulting succeeds, then the list of predicates that can be eliminated is obtained by concatenating the predicates in each Ambiguity pair:

  defaultedPreds :: Monad m => ClassEnv -> [Tyvar] -> [Pred] -> m [Pred]
  defaultedPreds  = withDefaults (\vps ts -> concat (map snd vps))

In a similar way, the defaulting substitution can be obtained by zipping the list of variables together with the list of defaults:

  defaultSubst   :: Monad m => ClassEnv -> [Tyvar] -> [Pred] -> m Subst
  defaultSubst    = withDefaults (\vps ts -> zip (map fst vps) ts)

One might wonder why the defaulting substitution is useful to us here; if the ambiguous variables don't appear anywhere in the assumptions or in the inferred types, then applying this substitution to those components would have no effect. In fact, we will only need defaultSubst at the top-level, when type inference for an entire module is complete [ Peyton Jones & Hughes, 1999,Section 4.5.5, Rule 2]. In this case, it is possible that Haskell's infamous `monomorphism restriction' (see Section 11.6.2) may prevent generalization over some type variables. But Haskell does not allow the types of top-level functions to contain unbound type variables. Instead, any remaining variables are considered ambiguous, even if they appear in inferred types; the substitution is needed to ensure that they are bound correctly.

11.6  Binding Groups

Our last remaining technical challenge is to describe typechecking for binding groups. This area is neglected in most theoretical treatments of type inference, often being regarded as a simple exercise in extending basic ideas. In Haskell, at least, nothing could be further from the truth! With interactions between overloading, polymorphic recursion, and the mixing of both explicitly and implicitly typed bindings, this is the most complex, and most subtle component of type inference. We will start by describing the treatment of explicitly typed bindings and implicitly typed bindings as separate cases, and then show how these can be combined.

11.6.1  Explicitly Typed Bindings

The simplest case is for explicitly typed bindings, each of which is described by the name of the function that is being defined, the declared type scheme, and the list of alternatives in its definition:

  type Expl = (Id, Scheme, [Alt])

Haskell requires that each Alt in the definition of a given identifier has the same number of left-hand side arguments, but we do not need to enforce that here.

Type inference for an explicitly typed binding is fairly easy; we need only check that the declared type is valid, and do not need to infer a type from first principles. To support the use of polymorphic recursion [ Henglein, 1993, Kfoury et al. , 1993], we will assume that the declared typing for i is already included in the assumptions when we call the following function:

  tiExpl :: ClassEnv -> [Assump] -> Expl -> TI [Pred]
  tiExpl ce as (i, sc, alts)
          = do (qs :=> t) <- freshInst sc
               ps         <- tiAlts ce as alts t
               s          <- getSubst
               let qs'     = apply s qs
                   t'      = apply s t
                   fs      = tv (apply s as)
                   gs      = tv t' \\ fs
                   sc'     = quantify gs (qs':=>t')
                   ps'     = filter (not . entail ce qs') (apply s ps)
               (ds,rs)    <- split ce fs gs ps'
               if sc /= sc' then
                   fail "signature too general"
                 else if not (null rs) then
                   fail "context too weak"
                 else
                   return ds

This code begins by instantiating the declared type scheme sc and checking each alternative against the resulting type t. When all of the alternatives have been processed, the inferred type for i is qs' :=> t'. If the type declaration is accurate, then this should be the same, up to renaming of generic variables, as the original type qs:=>t. If the type signature is too general, then the calculation of sc' will result in a type scheme that is more specific than sc and an error will be reported.

In the meantime, we must discharge any predicates that were generated while checking the list of alternatives. Predicates that are entailed by the context qs' can be eliminated without further ado. Any remaining predicates are collected in ps' and passed as arguments to split along with the appropriate sets of fixed and generic variables. If there are any retained predicates after context reduction, then an error is reported, indicating that the declared context is too weak.

11.6.2  Implicitly Typed Bindings

Two complications occur when we deal with implicitly typed bindings. The first is that we must deal with groups of mutually recursive bindings as a single unit rather than inferring types for each binding one at a time. The second is Haskell's monomorphism restriction, which restricts the use of overloading in certain cases.

A single implicitly typed binding is described by a pair containing the name of the variable and a list of alternatives:

  type Impl   = (Id, [Alt])

The monomorphism restriction is invoked when one or more of the entries in a list of implicitly typed bindings is simple, meaning that it has an alternative with no left-hand side patterns. The following function provides a way to test for this:

  restricted   :: [Impl] -> Bool
  restricted bs = any simple bs
   where simple (i,alts) = any (null . fst) alts

Type inference for groups of mutually recursive, implicitly typed bindings is described by the following function:

  tiImpls         :: Infer [Impl] [Assump]
  tiImpls ce as bs = do ts <- mapM (\_ -> newTVar Star) bs
                        let is    = map fst bs
                            scs   = map toScheme ts
                            as'   = zipWith (:>:) is scs ++ as
                            altss = map snd bs
                        pss <- sequence (zipWith (tiAlts ce as') altss ts)
                        s   <- getSubst
                        let ps'     = apply s (concat pss)
                            ts'     = apply s ts
                            fs      = tv (apply s as)
                            vss     = map tv ts'
                            gs      = foldr1 union vss \\ fs
                        (ds,rs) <- split ce fs (foldr1 intersect vss) ps'
                        if restricted bs then
                            let gs'  = gs \\ tv rs
                                scs' = map (quantify gs' . ([]:=>)) ts'
                            in return (ds++rs, zipWith (:>:) is scs')
                          else
                            let scs' = map (quantify gs . (rs:=>)) ts'
                            in return (ds, zipWith (:>:) is scs')

In the first part of this process, we extend as with assumptions binding each identifier defined in bs to a new type variable, and use these to type check each alternative in each binding. This is necessary to ensure that each variable is used with the same type at every occurrence within the defining list of bindings. (Lifting this restriction makes type inference undecidable [ Henglein, 1993, Kfoury et al. , 1993].) Next we use split to break the inferred predicates in ps' into a list of deferred predicates ds and retained predicates rs. The list gs collects all the generic variables that appear in one or more of the inferred types ts', but not in the list fs of fixed variables. Note that a different list is passed to split, including only variables that appear in all of the inferred types. This is important because all of those types will eventually be qualified by the same set of predicates, and we do not want any of the resulting type schemes to be ambiguous. The final step begins with a test to see if the monomorphism restriction should be applied, and then continues to calculate an assumption containing the principal types for each of the defined values. For an unrestricted binding, this is simply a matter of qualifying over the retained predicates in rs and quantifying over the generic variables in gs. If the binding group is restricted, then we must defer the predicates in rs as well as those in ds, and hence we can only quantify over variables in gs that do not appear in rs.

11.6.3  Combined Binding Groups

Haskell requires a process of dependency analysis to break down complete sets of bindings-either at the top-level of a program, or within a local definition-into the smallest possible groups of mutually recursive definitions, and ordered so that no group depends on the values defined in later groups. This is necessary to obtain the most general types possible. For example, consider the following fragment from a standard prelude for Haskell:
   foldr f a (x:xs) = f x (foldr f a xs)
   foldr f a []     = a
   and xs           = foldr (&&) True xs
If these definitions were placed in the same binding group, then we would not obtain the most general possible type for foldr; all occurrences of a variable are required to have the same type at each point within the defining binding group, which would lead to the following type for foldr:
   (Bool -> Bool -> Bool) -> Bool -> [Bool] -> Bool
To avoid this problem, we need only notice that the definition of foldr does not depend in any way on &&, and hence we can place the two functions in separate binding groups, inferring first the most general type for foldr, and then the correct type for and.

In the presence of explicitly typed bindings, we can refine the dependency analysis process a little further. For example, consider the following pair of bindings:

   f   :: Eq a => a -> Bool
   f x  = (x==x) || g True
   g y  = (y<=y) || f True
Although these bindings are mutually recursive, we do not need to infer types for f and g at the same time. Instead, we can use the declared type of f to infer a type:
   g   :: Ord a => a -> Bool
and then use this to check the body of f, ensuring that its declared type is correct.

Motivated by these observations, we will represent Haskell binding groups using the following datatype:

  type BindGroup  = ([Expl], [[Impl]])

The first component in each such pair lists any explicitly typed bindings in the group. The second component provides an opportunity to break down the list of any implicitly typed bindings into several smaller lists, arranged in dependency order. In other words, if a binding group is represented by a pair (es,[is_1,...,is_n]), then the implicitly typed bindings in each is_i should depend only on the bindings in es, is_1, ..., is_i, and not on any bindings in is_j when j>i. (Bindings in es could depend on any of the bindings in the group, but will presumably depend on at least those in is_n, or else the group would not be minimal. Note also that if es is empty, then n must be 1.) In choosing this representation, we have assumed that dependency analysis has been carried out prior to type checking, and that the bindings in each group have been organized into values of type BindGroup as appropriate. In particular, by separating out implicitly typed bindings as much as possible, we can potentially increase the degree of polymorphism in inferred types. For a correct implementation of the semantics specified in the Haskell report, a simpler but less flexible approach is required: all implicitly typed bindings must be placed in a single list, even if a more refined decomposition would be possible. In addition, if the group is restricted, then we must also ensure that none of the explicitly typed bindings in the same BindGroup have any predicates in their type, even though this is not strictly necessary. With hindsight, these are restrictions that we might prefer to avoid in any future revision of Haskell.

A more serious concern is that the Haskell report does not indicate clearly whether the previous example defining f and g should be valid. At the time of writing, some implementations accept it, while others do not. This is exactly the kind of problem that can occur when there is no precise, formal specification! Curiously, however, the report does indicate that a modification of the example to include an explicit type for g would be illegal. This is a consequence of a throw-away comment specifying that all explicit type signatures in a binding group must have the same context up to renaming of variables [ Peyton Jones & Hughes, 1999,Section 4.5.2]. This is a syntactic restriction that can easily be checked prior to type checking. Our comments here, however, suggest that it is unnecessarily restrictive.

In addition to the function bindings that we have seen already, Haskell allows variables to be defined using pattern bindings of the form pat = expr. We do not need to deal directly with such bindings because they are easily translated into the simpler framework used in this paper. For example, a binding:

   (x,y) = expr
can be rewritten as:
   nv = expr
   x  = fst nv
   y  = snd nv
where nv is a new variable. The precise definition of the monomorphism restriction in Haskell makes specific reference to pattern bindings, treating any binding group that includes one as restricted. So it may seem that the definition of restricted binding groups in this paper is not quite accurate. However, if we use translations as suggested here, then it turns out to be equivalent: even if the programmer supplies explicit type signatures for x and y in the original program, the translation will still contain an implicitly typed binding for the new variable nv.

Now, at last, we are ready to present the algorithm for type inference of a complete binding group, as implemented by the following function:

  tiBindGroup :: Infer BindGroup [Assump]
  tiBindGroup ce as (es,iss) =
    do let as' = [ v:>:sc | (v,sc,alts) <- es ]
       (ps, as'') <- tiSeq tiImpls ce (as'++as) iss
       qss        <- mapM (tiExpl ce (as''++as'++as)) es
       return (ps++concat qss, as''++as')

The structure of this definition is quite straightforward. First we form a list of assumptions as' for each of the explicitly typed bindings in the group. Next, we use this to check each group of implicitly typed bindings, extending the assumption set further at each stage. Finally, we return to the explicitly typed bindings to verify that each of the declared types is acceptable. In dealing with the list of implicitly typed binding groups, we use the following utility function, which typechecks a list of binding groups and accumulates assumptions as it runs through the list:

  tiSeq                  :: Infer bg [Assump] -> Infer [bg] [Assump]
  tiSeq ti ce as []       = return ([],[])
  tiSeq ti ce as (bs:bss) = do (ps,as')  <- ti ce as bs
                               (qs,as'') <- tiSeq ti ce (as'++as) bss
                               return (ps++qs, as''++as')

11.6.4  Top-level Binding Groups

At the top-level, a Haskell program can be thought of as a list of binding groups:

  type Program = [BindGroup]

Even the definitions of member functions in class and instance declarations can be included in this representation; they can be translated into top-level, explicitly typed bindings. The type inference process for a program takes a list of assumptions giving the types of any primitives, and returns a set of assumptions for any variables.

  tiProgram :: ClassEnv -> [Assump] -> Program -> [Assump]
  tiProgram ce as bgs = runTI $
                        do (ps, as') <- tiSeq tiBindGroup ce as bgs
                           s         <- getSubst
                           rs        <- reduce ce (apply s ps)
                           s'        <- defaultSubst ce [] rs
                           return (apply (s'@@s) as')

This completes our presentation of the Haskell type system.

12  Conclusions

We have presented a complete Haskell program that implements a type checker for the Haskell language. In the process, we have clarified certain aspects of the current design, as well as identifying some ambiguities in the existing, informal specification.

The type checker has been developed, type-checked, and tested using the ``Haskell 98 mode'' of Hugs 98 [ Jones & Peterson, 1999]. The full program includes many additional functions, not shown in this paper, to ease the task of testing, debugging, and displaying results. We have also translated several large Haskell programs-including the Standard Prelude, the Maybe and List libraries, and the source code for the type checker itself-into the representations described in Section 11, and successfully passed these through the type checker. As a result of these and other experiments we have good evidence that the type checker is working as intended, and in accordance with the expectations of Haskell programmers.

We believe that this typechecker can play a useful role, both as a formal specification for the Haskell type system, and as a testbed for experimenting with future extensions.

Acknowledgments

This paper has benefited from feedback from Johan Nordlander, Tom Pledger, Lennart Augustsson, Stephen Eldridge, Tim Sheard, Andy Gordon, and from an anonymous referee. The research reported in this paper was supported, in part, by the USAF Air Materiel Command, contract # F19628-96-C-0161, and by the National Science Foundation grants CCR-9703218 and CCR-9974980.

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Footnotes:

1An earlier version of this paper was presented at the Haskell workshop in Paris, France, on October 1, 2000. Both papers describe the same type system, but some significant parts of this version have been rewritten, restructured, or expanded to clarify the presentation.

2Throughout, we use `Haskell' as an abbreviation for `Haskell 98'.

3The `maps to' symbol +-> is written as +-> in the concrete syntax of Haskell.

4The symbol :=> (pronounced `then') is written as :=> in the concrete syntax of Haskell, and corresponds directly to the => symbol that is used in instance declarations and in types.

5The same function appears in the standard Maybe library, but with a less intuitive name: isJust.


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