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. (We would, however, 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 doesn't 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 the specification in this paper could easily be recast in a more 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 complex, but modestly sized Haskell programs.
Another goal for this paper was 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 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; by comparison, the core parts of the Hugs typechecker takes some 90+ pages of C code.
Regrettably, length restrictions have prevented us from including many examples in this paper to illustrate the definitions at each stage. For the same reason, definitions of a few 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 not included in the typeset version of this 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.
module TypingHaskellInHaskell where import List(nub, (\\), intersect, union, partition)
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 |
predicate | p, q, ... | Pred |
- `deferred' | d, ... | |
- `retained' | r, ... | |
qualified type | qt, ... | QualType |
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 |
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 new identifiers dynamically using the enumId function:
type Id = String enumId :: Int -> Id enumId n = "v" ++ show n
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 [3,7].
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
The following examples show how standard primitive datatypes are represented as type constants:
tChar = TCon (Tycon "Char" Star) tArrow = 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.
Types of the form TGen n represent `generic', or quantified type variables; their role is described in Section 8.
We do not provide a representation for type synonyms, assuming instead that they have been fully expanded before typechecking. It is always possible for an implementation to do this 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 two helper functions. The first provides a way to construct function types:
infixr 4 `fn` fn :: Type -> Type -> Type a `fn` b = TAp (TAp tArrow a) b
The second introduces 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.
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 can be 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)^{2} 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-or to anything containing 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, specified by apply (s1 @@ s2) = apply s1 . apply s2, can be defined more concretely 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 a result of type Maybe Subst:
merge :: Subst -> Subst -> Maybe Subst merge s1 s2 = if agree then Just s else Nothing where dom s = map fst s s = s1 ++ s2 agree = all (\v -> apply s1 (TVar v) == apply s2 (TVar v)) (dom s1 `intersect` dom s2)
It is easy to check that both (@@) and merge produce kind-preserving results from kind-preserving arguments.
The syntax of Haskell types has been carefully chosen to ensure that, if two types have any unifying substitutions, then they also have a most general unifier, which can be calculated by a simple variant of Robinson's algorithm [11]. One of the main 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 any other mechanism that allows reductions or rewriting in the type language, will often make unification undecidable, non-unitary (meaning that there may not be most general unifiers), or both. This, for example, is why it is not possible to 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 :: Type -> Type -> Maybe Subst varBind :: Tyvar -> Type -> Maybe Subst
Both of these return results using Maybe because unification is a partial function. However, because Maybe is a monad, the programming task can be simplified by using Haskell's monadic do-notation. The main unification algorithm is described by mgu, which uses 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') Just (s2 @@ s1) mgu (TVar u) t = varBind u t mgu t (TVar u) = varBind u t mgu (TCon tc1) (TCon tc2) | tc1==tc2 = Just nullSubst mgu t1 t2 = Nothing
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 = Just nullSubst | u `elem` tv t = Nothing | kind u == kind t = Just (u +-> t) | otherwise = Nothing
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 :: Type -> Type -> Maybe Subst
Matching follows the same pattern as unification, except that it uses merge rather than @@ in the case for type applications, 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 = Just (u +-> t) match (TCon tc1) (TCon tc2) | tc1==tc2 = Just nullSubst match t1 t2 = Nothing
data Qual t = [Pred] :=> t deriving Eq
Predicates themselves consist of a class name, and a type:
data Pred = IsIn Class Type deriving Eq
Haskell's classes represent sets of types. For example, a predicate IsIn c t asserts that t is a member of the class c. It would be easy to extend the Pred datatype to allow other forms of predicate, as is done with Trex records in Hugs [7]. 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 c t) = IsIn c (apply s t) tv (IsIn c t) = tv t
data Class = Class { name :: Id, super :: [Class], insts :: [Inst] } type Inst = Qual Pred
Values of type Class and Inst correspond to source level class and instance declarations, respectively. 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 member functions for the class, or their implementations in a particular instance.
A derived equality on Class is not useful because the data structures may be cyclic and so a test for structural equality might not terminate when applied to equal arguments. Instead, we use the name field to define an equality:
instance Eq Class where c == c' = name c == name c'
Apart from using a different keyword, Haskell class and instance declarations begin in the same way, with a clause of the form preds => pred for some (possibly empty) `context' preds, and a `head' predicate pred. In a class declaration, the context is used to specify the immediate superclasses, which we represent more directly by the list of classes in the field super: If a type is an instance of a class c, then it must also be an instance of any superclasses of c. 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 :: Pred -> [Pred] bySuper p@(IsIn c t) = p : concat (map bySuper supers) where supers = [ IsIn c' t | c' <- super c ]
The list bySuper p may contain duplicates, but it will always be finite because restrictions in Haskell ensure that the superclass hierarchy is acyclic.
The final field in each Class structure, insts, is the list of instance declarations for that particular class. Each such instance declaration is represented by a clause ps :=> h. Here, h is a predicate that describes the form of instances that the declaration can produce, while the context ps lists any constraints that it requires. We can use the following function to see if a particular predicate p can be deduced using a given instance. The result is either Just ps, where ps is a list of subgoals that must be established to complete the proof, or Nothing if the instance does not apply:
byInst :: Pred -> Inst -> Maybe [Pred] byInst p (ps :=> h) = do u <- matchPred h p Just (map (apply u) ps)
To see if an instance applies, we use one-way matching on predicates, which is implemented as follows:
matchPred :: Pred -> Pred -> Maybe Subst matchPred (IsIn c t) (IsIn c' t') | c == c' = match t t' | otherwise = Nothing
We can find all the relevant instances for a given predicate p = IsIn c t in insts c. So, if there are any ways to apply an instance to p, then we can find one using:
reducePred :: Pred -> Maybe [Pred] reducePred p@(IsIn c t) = foldr (|||) Nothing poss where poss = map (byInst p) (insts c) Nothing ||| y = y Just x ||| y = Just x
In fact, because Haskell prevents the definition of overlapping instances, we can be sure that, if reducePreds succeeds, then we have actually found the only applicable instance.
(||-) :: [Pred] -> Pred -> Bool ps ||- p = any (p `elem`) (map bySuper ps) || case reducePred p of Nothing -> False Just qs -> all (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-are enough to guarantee that tests for entailment will terminate. Completeness of the algorithm is also important: will 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 such that ps ||- q. 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 [10,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 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.
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 enforced statically by a careful choice of the representation for types and type schemes. However, after considering several other alternatives, we eventually settled for the representation shown here because it allows for simple implementations of equality and substitution. For example, because the implementation of substitution on Type ignores TGen values, 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 very 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.
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)
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 :: Id -> [Assump] -> Scheme find i as = head [ sc | (i':>:sc) <- as, i==i' ]
We do not make any allowance here for the possibility that the variable i might not appear in as, and assume instead that a previous compiler pass will have detected any occurrences of unbound variables.
newtype TI a = TI (Subst -> Int -> (Subst, Int, a)) instance Monad TI where return x = TI (\s n -> (s,n,x)) TI c >>= f = TI (\s n -> let (s',m,x) = c s n TI fx = f x in fx s' m) runTI :: TI a -> a runTI (TI c) = result where (s,n,result) = c nullSubst 0
We provide two operations that deal with the current substitution: getSubst 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 case mgu (apply s t1) (apply s t2) of Just u -> extSubst u Nothing -> error "unification"
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 or fresh 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. Although we use it at only this one place, it is still 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)
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 our typing rules are expressed in terms of the following type synonym:
type Infer e t = [Assump] -> e -> TI ([Pred], t)
In more theoretical treatments, it would not be surprising to see the rules expressed in terms of judgments P | A\vdash e:t, where P is a set of predicates, A is a set of assumptions, e is an expression, and t is a corresponding type [6]. Judgments like this can be thought of as 4-tuples, and the typing rules themselves just correspond to a 4-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.
data Literal = LitInt Integer | LitChar Char
Type inference for literals is straightforward. For characters, we just return typeChar. 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.
tiLit :: Literal -> TI ([Pred],Type) tiLit (LitChar _) = return ([], tChar) tiLit (LitInt _) = do v <- newTVar Star return ([IsIn cNum v], v)
For this last case, we assume the existence of a constant cNum :: Class to represent the Haskell class Num, but, for reasons of space, we do not present the definition here. It is straightforward to add additional cases for Haskell's floating point and String literals.
data Pat = PVar Id | PLit Literal | PCon Assump [Pat]
A PVar i pattern matches any value, and binds the result to the variable i. A PLit l pattern matches only the particular value denoted by the literal l. 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 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 of Haskell 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 it would need to be extended to account for patterns using labeled fields, and for (n+k) patterns. Neither of these cause any substantial problems, but they do add a little 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 occurence 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.
For literal patterns, we use tiLit from the previous section:
tiPat (PLit l) = do (ps, t) <- tiLit l return (ps, [], 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 = [ p | (ps,_,_) <- psasts, p<-ps ] as = [ a | (_,as,_) <- psasts, a<-as ] 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.
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.
Haskell has a much richer syntax of expressions, but they can all be translated into Expr values. For example, a lambda expression like \x->e can be rewritten using a local definition as let f x = e in f, where f is a new variable. Similar translations are used for case expressions.
Type inference for expressions is quite straightforward:
tiExpr :: Infer Expr Type tiExpr as (Var i) = do let sc = find i as (ps :=> t) <- freshInst sc return (ps, t) tiExpr as (Const (i:>:sc)) = do (ps :=> t) <- freshInst sc return (ps, t) tiExpr as (Lit l) = do (ps,t) <- tiLit l return (ps, t) tiExpr as (Ap e f) = do (ps,te) <- tiExpr as e (qs,tf) <- tiExpr as f t <- newTVar Star unify (fn tf t) te return (ps++qs, t) tiExpr as (Let bg e) = do (ps, as') <- tiBindGroup as bg (qs, t) <- tiExpr (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.
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 building a new list as' of assumptions for any bound variables, and use it to infer types for each of the patterns, 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 as (pats, e) = do (ps, as', ts) <- tiPats pats (qs,t) <- tiExpr (as'++as) e return (ps++qs, foldr fn t ts)
In practice, we will often need to 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 :: [Assump] -> [Alt] -> Type -> TI [Pred] tiAlts as alts t = do psts <- mapM (tiAlt 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 prominent role. Even in an unextended Haskell implementation, this could still help to improve the quality of type error messages.
To understand the basic problem, suppose that we have run tiExpr over the body of a function f to obtain a set of predicates ps and a type t. At this point we could infer a type for f 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:
reduce :: [Tyvar] -> [Tyvar] -> [Pred] -> ([Pred], [Pred])
reduce fs gs ps = (ds,rs') where (ds, rs) = split fs ps rs' = useDefaults (fs++gs) rs
The first stage of this algorithm, which we call context splitting, is implemented by split and is described in Section 11.5.1. Its purpose is to separate the deferred predicates from the retained predicates, using reducePred as necessary. The second stage implemented by useDefaults, is described in Section 11.5.2. Its purpose is to eliminate ambiguities in the retained predicates, whenever possible. The fs and gs parameters specify appropriate sets of `fixed' and `generic' type variables, respectively. The former is just the set of variables appearing free in the assumptions, while the latter is the set of variables over which we would like to quantify. Any variable in ps that is not in either fs or gs may cause ambiguity, as we describe in Section 11.5.2.
split :: [Tyvar] -> [Pred] -> ([Pred], [Pred]) split fs = partition (all (`elem` fs) . tv) . simplify [] . toHnfs
The first stage of this pipeline, implemented by toHnfs, uses reducePred to break down any inferred predicates into the form that Haskell requires:
toHnfs :: [Pred] -> [Pred] toHnfs = concat . map toHnf
toHnf :: Pred -> [Pred] toHnf p = if inHnf p then [p] else case reducePred p of Nothing -> error "context reduction" Just ps -> toHnfs ps
The name toHnfs is motivated by similarities with the concept of head-normal forms in l-calculus. The test to determine whether a given predicate is in the appropriate form is implemented by the following function:
inHnf :: Pred -> Bool inHnf (IsIn c t) = hnf t where hnf (TVar v) = True hnf (TCon tc) = False hnf (TAp t _) = hnf t
The second stage of the pipeline uses information about superclasses to eliminate redundant predicates. More precisely, if the list produced by toHnfs contains some predicate p, then we can eliminate any occurrence of a predicate from bySuper p in the rest of the list. As a special case, this also means that we can eliminate any repeated occurrences of p, which always appears as the first element in bySuper p. This process is implemented by the simplify function, using an accumulating parameter to bulid up the final result:
simplify :: [Pred] -> [Pred] -> [Pred] simplify rs [] = rs simplify rs (p:ps) = simplify (p:(rs\\qs)) (ps\\qs) where qs = bySuper p rs \\ qs = [ r | r<-rs, r `notElem` qs ]
Note that we have used a modified version of the (\\) operator; with the standard Haskell semantics for (\\), we could not guarantee that all duplicate entries would be removed.
The third stage of context reduction uses partition to separate deferred predicates-i.e., those containing only fixed variables-from retained predicates. The set of fixed variables is passed in as the fs argument to split.
Suppose 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. To determine whether defaults can be applied, we compute a triple (v,qs,ts) for each ambiguous variable v. In each case, qs is the list of predicates in ps that involve v, and ts is the list of types that could be used as a default value for v:
ambig :: [Tyvar] -> [Pred] -> [(Tyvar,[Pred],[Type])]
ambig vs ps = [ (v, qs, defs v qs) | v <- tv ps \\ vs, let qs = [ p | p<-ps, v `elem` tv p ] ]
If the ts component of any one of these triples turns out to be empty, then defaulting cannot be applied to the corresponding variable, and the ambiguity cannot be avoided. On the other hand, if ts is non-empty, then we will be able to substitute head ts for v and remove the predicates in qs from ps.
Given one of these triples (v,qs,ts), and as specified by the Haskell report [10,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 third component of each triple:
defs :: Tyvar -> [Pred] -> [Type] defs v qs = [ t | all ((TVar v)==) ts, all (`elem` stdClasses) cs, any (`elem` numClasses) cs, t <- defaults, and [ [] ||- IsIn c t | c <- cs ]] where cs = [ c | (IsIn c t) <- qs ] ts = [ t | (IsIn c t) <- qs ]
The defaulting process can now be described by the following function, which generates an error if there are any ambiguous type variables that cannot be defaulted:
useDefaults :: [Tyvar] -> [Pred] -> [Pred] useDefaults vs ps | any null tss = error "ambiguity" | otherwise = ps \\ ps' where ams = ambig vs ps tss = [ ts | (v,qs,ts) <- ams ] ps' = [ p | (v,qs,ts) <- ams, p<-qs ]
A modified version of this process is required at the top-level, when type inference for an entire module is complete [10,Section 4.5.5, Rule 2]. In this case, any remaining type variables are considered ambiguous, and we need to arrange for defaulting to return a substitution mapping any such variables to their defaulted types:
topDefaults :: [Pred] -> Maybe Subst topDefaults ps | any null tss = Nothing | otherwise = Just (zip vs (map head tss)) where ams = ambig [] ps tss = [ ts | (v,qs,ts) <- ams ] vs = [ v | (v,qs,ts) <- ams ]
type Expl = (Id, Scheme, [Alt])
Haskell requires that each Alt in the definition of any given value has the same number of arguments in each left-hand side, but we do not need to enforce that restriction 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 [4,8], we will assume that the declared typing for i is already included in the assumptions when we call the following function:
tiExpl :: [Assump] -> Expl -> TI [Pred] tiExpl as (i, sc, alts) = do (qs :=> t) <- freshInst sc ps <- tiAlts as alts t s <- getSubst let qs' = apply s qs t' = apply s t ps' = [ p | p <- apply s ps, not (qs' ||- p) ] fs = tv (apply s as) gs = tv t' \\ fs (ds,rs) = reduce fs gs ps' sc' = quantify gs (qs':=>t') if sc /= sc' then error "signature too general" else if not (null rs) then error "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 reduce 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.
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 simple way to test for this condition:
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 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 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) = reduce 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 [4,8].) Next we use the process of context reduction 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 reduce, 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.
foldr f a (x:xs) = f x (foldr f a xs) foldr f a [] = a and xs = foldr (&&) True xsIf 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] -> BoolTo 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 TrueAlthough 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 -> Booland 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, while the second component breaks down any remaining bindings into a sequence of smaller, implicitly typed binding groups, arranged in dependency order. In choosing our representation for the abstract syntax, 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 in an appropriate manner. For a correct implementation of the semantics specified in the Haskell report, we must place all of the implicitly typed bindings in a single group, even if a more refined decomposition would be possible. In addition, if that group is restricted, then we must also check that none of the explicitly typed bindings in the same BindGroup have any predicates in their type. With hindsight, these seem like strange restrictions that we might prefer to avoid in any further 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 [10,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) = exprcan be rewritten as:
nv = expr x = fst nv y = snd nvwhere 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, at first glance, 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 as (es,iss) = do let as' = [ v:>:sc | (v,sc,alts) <- es ] (ps, as'') <- tiSeq tiImpls (as'++as) iss qs <- mapM (tiExpl (as''++as'++as)) es return (ps++concat qs, 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 as [] = return ([],[]) tiSeq ti as (bs:bss) = do (ps,as') <- ti as bs (qs,as'') <- tiSeq ti (as'++as) bss return (ps++qs, as''++as')
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 :: [Assump] -> Program -> [Assump] tiProgram as bgs = runTI $ do (ps, as') <- tiSeq tiBindGroup as bgs s <- getSubst let ([], rs) = split [] (apply s ps) case topDefaults rs of Just s' -> return (apply (s'@@s) as') Nothing -> error "top-level ambiguity"
This completes our presentation of the Haskell type system.
The type checker has been developed, type-checked, and tested using the ``Haskell 98 mode'' of Hugs 98 [7]. 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.
^{1}Throughout, we use `Haskell' as an abbreviation for `Haskell 98'.
^{2}The typeset version of the symbol +-> is written +-> in the concrete syntax of Haskell.
^{3}The typeset version of the symbol :=> is written :=> in the concrete syntax of Haskell, and corresponds directly to the => symbol that is used in instance declarations and in types.
^{4}The typeset version of the symbol ||-
is written ||- in the concrete syntax of Haskell.