In this chapter, we describe the syntax and informal semantics of Haskell declarations.
| module | -> | module modid [exports] where body | |
| | | body | ||
| body | -> | { impdecls ; topdecls } | |
| | | { impdecls } | ||
| | | { topdecls } | ||
| topdecls | -> | topdecl1 ; ... ; topdecln | (n>=1) |
| topdecl | -> | type simpletype = type | |
| | | data [context =>] simpletype = constrs [deriving] | ||
| | | newtype [context =>] simpletype = newconstr [deriving] | ||
| | | class [scontext =>] tycls tyvar [where cdecls] | ||
| | | instance [scontext =>] qtycls inst [where idecls] | ||
| | | default (type1 , ... , typen) | (n>=0) | |
| | | decl | ||
| decls | -> | { decl1 ; ... ; decln } | (n>=0) |
| decl | -> | gendecl | |
| | | (funlhs | pat0) rhs | ||
| cdecls | -> | { cdecl1 ; ... ; cdecln } | (n>=0) |
| cdecl | -> | gendecl | |
| | | (funlhs | var) rhs | ||
| idecls | -> | { idecl1 ; ... ; idecln } | (n>=0) |
| idecl | -> | (funlhs | var) rhs | |
| | | (empty) | ||
| gendecl | -> | vars :: [context =>] type | (type signature) |
| | | fixity [integer] ops | (fixity declaration) | |
| | | (empty declaration) | ||
| ops | -> | op1 , ... , opn | (n>=1) |
| vars | -> | var1 , ... , varn | (n>=1) |
| fixity | -> | infixl | infixr | infix |
The declarations in the syntactic category topdecls are only allowed at the top level of a Haskell module (see Chapter 5), whereas decls may be used either at the top level or in nested scopes (i.e. those within a let or where construct).
For exposition, we divide the declarations into three groups: user-defined datatypes, consisting of type, newtype, and data declarations (Section 4.2); type classes and overloading, consisting of class, instance, and default declarations (Section 4.3); and nested declarations, consisting of value bindings, type signatures, and fixity declarations (Section 4.4).
Haskell has several primitive datatypes that are "hard-wired" (such as integers and floating-point numbers), but most "built-in" datatypes are defined with normal Haskell code, using normal type and data declarations. These "built-in" datatypes are described in detail in Section 6.1.
Haskell uses a traditional Hindley-Milner polymorphic type system to provide a static type semantics [3, 5], but the type system has been extended with type classes (or just classes) that provide a structured way to introduce overloaded functions.
A class declaration (Section 4.3.1) introduces a new type class and the overloaded operations that must be supported by any type that is an instance of that class. An instance declaration (Section 4.3.2) declares that a type is an instance of a class and includes the definitions of the overloaded operations---called class methods---instantiated on the named type.
For example, suppose we wish to overload the operations (+) and
negate on types Int and Float. We introduce a new
type class called Num:
class Num a where -- simplified class declaration for Num
(+) :: a -> a -> a -- (Num is defined in the Prelude)
negate :: a -> a
This declaration may be read "a type a is an instance of the class
Num if there are class methods (+) and negate, of the
given types, defined on it."
We may then declare Int and Float to be instances of this class:
instance Num Int where -- simplified instance of Num Int
x + y = addInt x y
negate x = negateInt x
instance Num Float where -- simplified instance of Num Float
x + y = addFloat x y
negate x = negateFloat x
where addInt, negateInt, addFloat, and negateFloat are assumed
in this case to be primitive functions, but in general could be any
user-defined function. The first declaration above may be read
"Int is an instance of the class Num as witnessed by these
definitions (i.e. class methods) for (+) and negate."
More examples of type classes can be found in the papers by Jones [7] or Wadler and Blott [12]. The term `type class' was used to describe the original Haskell 1.0 type system; `constructor class' was used to describe an extension to the original type classes. There is no longer any reason to use two different terms: in this report, `type class' includes both the original Haskell type classes and the constructor classes introduced by Jones.
To ensure that they are valid, type expressions are classified into different kinds, which take one of two possible forms:
| type | -> | btype [-> type] | (function type) |
| btype | -> | [btype] atype | (type application) |
| atype | -> | gtycon | |
| | | tyvar | ||
| | | ( type1 , ... , typek ) | (tuple type, k>=2) | |
| | | [ type ] | (list type) | |
| | | ( type ) | (parenthesised constructor) | |
| gtycon | -> | qtycon | |
| | | () | (unit type) | |
| | | [] | (list constructor) | |
| | | (->) | (function constructor) | |
| | | (,{,}) | (tupling constructors) |
The syntax for Haskell type expressions is given above. Just as data values are built using data constructors, type values are built from type constructors. As with data constructors, the names of type constructors start with uppercase letters. Unlike data constructors, infix type constructors are not allowed (other than (->)).
The main forms of type expression are as follows:
Special syntax is provided to allow certain type expressions to be written in a more traditional style:
Although the list and tuple types have special syntax, their semantics is the same as the equivalent user-defined algebraic data types.
Notice that expressions and types have a consistent syntax. If ti is the type of expression or pattern ei, then the expressions (\ e1 -> e2), [e1], and (e1,e2) have the types (t1 -> t2), [t1], and (t1,t2), respectively.
With one exception (that of the distinguished type variable in a class declaration (Section 4.3.1)), the type variables in a Haskell type expression are all assumed to be universally quantified; there is no explicit syntax for universal quantification [3]. For example, the type expression a -> a denotes the type forall a. a ->a. For clarity, however, we often write quantification explicitly when discussing the types of Haskell programs. When we write an explicitly quantified type, the scope of the forall extends as far to the right as possible; for example, forall a. a ->a means forall a. (a ->a).
| context | -> | class | |
| | | ( class1 , ... , classn ) | (n>=0) | |
| class | -> | qtycls tyvar | |
| | | qtycls ( tyvar atype1 ... atypen ) | (n>=1) | |
| qtycls | -> | [ modid . ] tycls | |
| tycls | -> | conid | |
| tyvar | -> | varid |
A class assertion has form qtycls tyvar, and indicates the membership of the type tyvar in the class qtycls. A class identifier begins with an uppercase letter. A context consists of zero or more class assertions, and has the general form
( C1 u1, ..., Cn un )
where C1, ..., Cn are class identifiers, and each of the u1, ..., un is either a type variable, or the application of type variable to one or more types. The outer parentheses may be omitted when n=1. In general, we use cx to denote a context and we write cx => t to indicate the type t restricted by the context cx. The context cx must only contain type variables referenced in t. For convenience, we write cx => t even if the context cx is empty, although in this case the concrete syntax contains no =>.
In this section, we provide informal details of the type system. (Wadler and Blott [12] and Jones [7] discuss type and constructor classes, respectively, in more detail.)
The Haskell type system attributes a type to each
expression in the program. In general, a type is of the form
forall u. cx =>t,
where u is a set of type variables u1, ..., un.
In any such type, any of the universally-quantified type variables ui
that are free in cx must also be free in t.
Furthermore, the context cx must be of the form given above in
Section 4.1.3. For example, here are some
valid types:
Eq a => a -> a
(Eq a, Show a, Eq b) => [a] -> [b] -> String
(Eq (f a), Functor f) => (a -> b) -> f a -> f b -> Bool
In the third type, the constraint Eq (f a) cannot be made
simpler because f is universally quantified.
The type of an expression e depends on a type environment that gives types for the free variables in e, and a class environment that declares which types are instances of which classes (a type becomes an instance of a class only via the presence of an instance declaration or a deriving clause).
Types are related by a generalization preorder (specified below); the most general type, up to the equivalence induced by the generalization preorder, that can be assigned to a particular expression (in a given environment) is called its principal type. Haskell 's extended Hindley-Milner type system can infer the principal type of all expressions, including the proper use of overloaded class methods (although certain ambiguous overloadings could arise, as described in Section 4.3.4). Therefore, explicit typings (called type signatures) are usually optional (see Sections 3.16 and 4.4.1).
The type forall u. cx1 =>t1 is more general than the type forall w. cx2 =>t2 if and only if there is a substitution S whose domain is u such that:
A value of type
forall u. cx =>t,
may be instantiated at types s if and only if
the context cx[s/u] holds.
For example, consider the function double:
double x = x + x
The most general type of double is
forall a. Num a =>a ->a.
double may be applied to values of type Int (instantiating a to
Int), since Num Int holds, because Int is an instance of the class Num.
However, double may not normally be applied to values
of type Char, because Char is not normally an instance of class Num. The
user may choose to declare such an instance, in which case double may
indeed be applied to a Char.
In this section, we describe algebraic datatypes (data declarations), renamed datatypes (newtype declarations), and type synonyms (type declarations). These declarations may only appear at the top level of a module.
| topdecl | -> | data [context =>] simpletype = constrs [deriving] | |
| simpletype | -> | tycon tyvar1 ... tyvark | (k>=0) |
| constrs | -> | constr1 | ... | constrn | (n>=1) |
| constr | -> | con [!] atype1 ... [!] atypek | (arity con = k, k>=0) |
| | | (btype | ! atype) conop (btype | ! atype) | (infix conop) | |
| | | con { fielddecl1 , ... , fielddecln } | (n>=0) | |
| fielddecl | -> | vars :: (type | ! atype) | |
| deriving | -> | deriving (dclass | (dclass1, ... , dclassn)) | (n>=0) |
| dclass | -> | qtycls |
An algebraic datatype declaration has the form:
data cx => T u1 ... uk = K1 t11 ... t1k1 | ...| Kn tn1 ... tnkn
where cx is a context. This declaration introduces a new type constructor T with one or more constituent data constructors K1, ..., Kn. In this Report, the unqualified term "constructor" always means "data constructor".
The types of the data constructors are given by:
Ki :: forall u1 ... uk. cxi =>ti1 ->...->tiki ->(T u1 ... uk)
where cxi is the largest subset of cx that constrains only those type variables free in the types ti1, ..., tiki. The type variables u1 through uk must be distinct and may appear in cx and the tij; it is a static error for any other type variable to appear in cx or on the right-hand-side. The new type constant T has a kind of the form k1->...->kk->* where the kinds ki of the argument variables ui are determined by kind inference as described in Section 4.6. This means that T may be used in type expressions with anywhere between 0 and k arguments.
For example, the declaration
data Eq a => Set a = NilSet | ConsSet a (Set a)
introduces a type constructor Set of kind *->*, and constructors NilSet and
ConsSet with types
| NilSet | :: forall a. Set a |
| ConsSet | :: forall a. Eq a =>a ->Set a ->Set a |
In the example given, the overloaded
type for ConsSet ensures that ConsSet can only be applied to values whose
type is an instance of the class Eq.
Pattern matching against ConsSet also gives rise to an Eq a constraint.
For example:
f (ConsSet a s) = a
the function f has inferred type Eq a => Set a -> a.
The context in the data declaration has no other effect whatsoever.
The visibility of a datatype's constructors (i.e. the "abstractness" of the datatype) outside of the module in which the datatype is defined is controlled by the form of the datatype's name in the export list as described in Section 5.8.
The optional deriving part of a data declaration has to do with derived instances, and is described in Section 4.3.3.
A constructor definition in a data declaration may assign labels to the
fields of the constructor, using the record syntax (C { ... }).
Constructors using field labels may be freely mixed with constructors
without them.
A constructor with associated field labels may still be used as an
ordinary constructor; features using labels are
simply a shorthand for operations using an underlying positional
constructor. The arguments to the positional constructor occur in the
same order as the labeled fields. For example, the declaration
data C = F { f1,f2 :: Int, f3 :: Bool }
defines a type and constructor identical to the one produced by
data C = F Int Int Bool
Operations using field labels are described in Section 3.15.
A data declaration may use the same field label in multiple
constructors as long as the typing of the field is the same in all
cases after type synonym expansion. A label cannot be shared by
more than one type in scope. Field names share the top level namespace
with ordinary variables and class methods and must not conflict with
other top level names in scope.
The pattern F {} matches any value built with constructor F, whether or not F was declared with record syntax.
Translation:A declaration of the formdata cx => T u1 ... uk = ... | K s1 ... sn | ... where each si is either of the form ! ti or ti, replaces every occurrence of K in an expression by (\ x1 ... xn -> ( ((K op1 x1) op2 x2) ... ) opn xn) where opi is the non-strict apply function $ if si is of the form ti, and opi is the strict apply function $! (see Section 6.2) if si is of the form ! ti. Pattern matching on K is not affected by strictness flags. |
| topdecl | -> | type simpletype = type | |
| simpletype | -> | tycon tyvar1 ... tyvark | (k>=0) |
type T u1 ... uk = t
which introduces a new type constructor, T. The type (T t1 ...
tk) is equivalent to the type t[t1/u1, ..., tk/uk]. The type
variables u1 through uk must be distinct and are scoped only
over t; it is a static error for any other type variable to appear
in t. The kind of the new type constructor T is of the form
k1->...->kk->k where
the kinds ki of the arguments ui and k of the right hand
side t are determined by kind inference as described in
Section 4.6.
For example, the following definition can be used to provide an alternative
way of writing the list type constructor:
type List = []
Type constructor symbols T introduced by type synonym declarations cannot
be partially applied; it is a static error to use T without the full number
of arguments.
Although recursive and mutually recursive datatypes are allowed,
this is not so for type synonyms, unless an algebraic datatype
intervenes. For example,
type Rec a = [Circ a]
data Circ a = Tag [Rec a]
is allowed, whereas
type Rec a = [Circ a] -- invalid
type Circ a = [Rec a] -- invalid
is not. Similarly, type Rec a = [Rec a] is not allowed.
Type synonyms are a convenient, but strictly syntactic, mechanism to make type signatures more readable. A synonym and its definition are completely interchangeable, except in the instance type of an instance declaration (Section 4.3.2).
| topdecl | -> | newtype [context =>] simpletype = newconstr [deriving] | |
| newconstr | -> | con atype | |
| | | con { var :: type } | ||
| simpletype | -> | tycon tyvar1 ... tyvark | (k>=0) |
A declaration of the form
newtype cx => T u1 ... uk = N t
introduces a new type whose representation is the same as an existing type. The type (T u1 ... uk) renames the datatype t. It differs from a type synonym in that it creates a distinct type that must be explicitly coerced to or from the original type. Also, unlike type synonyms, newtype may be used to define recursive types. The constructor N in an expression coerces a value from type t to type (T u1 ... uk). Using N in a pattern coerces a value from type (T u1 ... uk) to type t. These coercions may be implemented without execution time overhead; newtype does not change the underlying representation of an object.
New instances (see Section 4.3.2) can be defined for a type defined by newtype but may not be defined for a type synonym. A type created by newtype differs from an algebraic datatype in that the representation of an algebraic datatype has an extra level of indirection. This difference may make access to the representation less efficient. The difference is reflected in different rules for pattern matching (see Section 3.17). Unlike algebraic datatypes, the newtype constructor N is unlifted, so that N _|_ is the same as _|_.
The following examples clarify the differences between data (algebraic
datatypes), type (type synonyms), and newtype (renaming types.)
Given the declarations
data D1 = D1 Int
data D2 = D2 !Int
type S = Int
newtype N = N Int
d1 (D1 i) = 42
d2 (D2 i) = 42
s i = 42
n (N i) = 42
the expressions ( d1 _|_), ( d2 _|_) and
(d2 (D2 _|_) ) are all
equivalent to _|_, whereas ( n _|_), ( n ( N
_|_) ), ( d1 ( D1 _|_) ) and ( s _|_)
are all equivalent to 42. In particular, ( N _|_) is equivalent to
_|_ while ( D1 _|_) is not equivalent to _|_.
The optional deriving part of a newtype declaration is treated in the same way as the deriving component of a data declaration; see Section 4.3.3.
A newtype declaration may use field-naming syntax, though of course
there may only be one field. Thus:
newtype Age = Age { unAge :: Int }
brings into scope both a constructor and a de-constructor:
Age :: Int -> Age
unAge :: Age -> Int
| topdecl | -> | class [scontext =>] tycls tyvar [where cdecls] | |
| scontext | -> | simpleclass | |
| | | ( simpleclass1 , ... , simpleclassn ) | (n>=0) | |
| simpleclass | -> | qtycls tyvar | |
| cdecls | -> | { cdecl1 ; ... ; cdecln } | (n>=0) |
| cdecl | -> | gendecl | |
| | | (funlhs | var) rhs |
A class declaration introduces a new class and the operations (class methods) on it. A class declaration has the general form:
| class cx => C u where cdecls |
This introduces a new class name C; the type variable u is scoped only over the class method signatures in the class body. The context cx specifies the superclasses of C, as described below; the only type variable that may be referred to in cx is u.
The superclass relation must not be cyclic; i.e. it must form a directed acyclic graph.
The cdecls part of a class declaration contains three kinds of declarations:
vi :: cxi => ti
in cdecls. Class methods share the top level namespace with variable bindings and field names; they must not conflict with other top level bindings in scope. That is, a class method can not have the same name as a top level definition, a field name, or another class method.
The type of the top-level class method vi is:
vi :: forall u,w. (C u, cxi) =>ti
The ti must mention u; it may mention type variables
w other than u, in which case the type of vi is
polymorphic in both u and w.
The cxi may constrain only w; in particular,
the cxi may not constrain u.
For example:
class Foo a where
op :: Num b => a -> b -> a
Here the type of op is
forall a, b. (Foo a, Num b) =>a ->b ->a.
A class
declaration with no where part
may be useful for combining a
collection of classes into a larger one that inherits all of the
class methods in the original ones. For example:
class (Read a, Show a) => Textual a
In such a case, if a type is an instance of all
superclasses, it is
not automatically an instance of the subclass, even though the
subclass has no immediate class methods. The instance declaration must be
given explicitly with no where part.
| topdecl | -> | instance [scontext =>] qtycls inst [where idecls] | |
| inst | -> | gtycon | |
| | | ( gtycon tyvar1 ... tyvark ) | (k>=0, tyvars distinct) | |
| | | ( tyvar1 , ... , tyvark ) | (k>=2, tyvars distinct) | |
| | | [ tyvar ] | ||
| | | ( tyvar1 -> tyvar2 ) | (tyvar1 and tyvar2 distinct) | |
| idecls | -> | { idecl1 ; ... ; idecln } | (n>=0) |
| idecl | -> | (funlhs | var) rhs | |
| | | (empty) |
class cx => C u where { cbody }
be a class declaration. The general form of the corresponding instance declaration is:
instance cx' => C (T u1 ... uk) where { d }
where k>=0. The type (T u1 ... uk) must take the form of a type constructor T applied to simple type variables u1, ... uk; furthermore, T must not be a type synonym, and the ui must all be distinct.
This prohibits instance declarations
such as:
instance C (a,a) where ...
instance C (Int,a) where ...
instance C [[a]] where ...
The declarations d may contain bindings only for the class
methods of C. It is illegal to give a
binding for a class method that is not in scope, but the name under
which it is in scope is immaterial; in particular, it may be a qualified
name. (This rule is identical to that used for subordinate names in
export lists --- Section 5.2.)
For example, this is legal, even though range is in scope only
with the qualified name Ix.range.
module A where
import qualified Ix
instance Ix.Ix T where
range = ...
The declarations may not contain any type
signatures or fixity declarations,
since these have already been given in the class
declaration. As in the case of default class methods
(Section 4.3.1), the method declarations must take the form of
a variable or function definition.
If no binding is given for some class method then the corresponding default class method in the class declaration is used (if present); if such a default does not exist then the class method of this instance is bound to undefined and no compile-time error results.
An instance declaration that makes the type T to be an instance of class C is called a C-T instance declaration and is subject to these static restrictions:
In fact, except in pathological cases it is possible to infer from the instance declaration the most general instance context cx' satisfying the above two constraints, but it is nevertheless mandatory to write an explicit instance context.
If the two instance declarations instead read like this:
instance Num a => Foo [a] where ...
instance (Eq a, Show a) => Bar [a] where ...
then the program would be invalid. The second instance declaration is
valid only if [a] is an instance of Foo under the assumptions
(Eq a, Show a). But this does not hold, since [a] is only an
instance of Foo under the stronger assumption Num a.
Further examples of instance declarations may be found in Chapter 8.
As mentioned in Section 4.2.1, data and newtype declarations contain an optional deriving form. If the form is included, then derived instance declarations are automatically generated for the datatype in each of the named classes. These instances are subject to the same restrictions as user-defined instances. When deriving a class C for a type T, instances for all superclasses of C must exist for T, either via an explicit instance declaration or by including the superclass in the deriving clause.
Derived instances provide convenient commonly-used operations for user-defined datatypes. For example, derived instances for datatypes in the class Eq define the operations == and /=, freeing the programmer from the need to define them.
The only classes in the Prelude for which derived instances are allowed are Eq, Ord, Enum, Bounded, Show, and Read, all mentioned in Figure 6.1, page . The precise details of how the derived instances are generated for each of these classes are provided in Chapter 10, including a specification of when such derived instances are possible. Classes defined by the standard libraries may also be derivable.
A static error results if it is not possible to derive an instance declaration over a class named in a deriving form. For example, not all datatypes can properly support class methods in Enum. It is also a static error to give an explicit instance declaration for a class that is also derived.
If the deriving form is omitted from a data or newtype declaration, then no instance declarations are derived for that datatype; that is, omitting a deriving form is equivalent to including an empty deriving form: deriving ().
| topdecl | -> | default (type1 , ... , typen) | (n>=0) |
A problem inherent with Haskell -style overloading is the
possibility of an ambiguous type.
For example, using the
read and show functions defined in Chapter 10,
and supposing that just Int and Bool are members of Read and
Show, then the expression
let x = read "..." in show x -- invalid
is ambiguous, because the types for show and read,
| show | :: forall a. Show a =>a ->String |
| read | :: forall a. Read a =>String ->a |
could be satisfied by instantiating a as either Int in both cases, or Bool. Such expressions are considered ill-typed, a static error.
We say that an expression e has an ambiguous type if, in its type forall u. cx =>t, there is a type variable u in u that occurs in cx but not in t. Such types are invalid.
For example, the earlier expression involving show and read has an ambiguous type since its type is forall a. Show a, Read a =>String.
Ambiguous types can only be circumvented by
input from the user. One way is through the use of expression
type-signatures
as described in Section 3.16.
For example, for the ambiguous expression given earlier, one could
write:
let x = read "..." in show (x::Bool)
which disambiguates the type.
Occasionally, an otherwise ambiguous expression needs to be made
the same type as some variable, rather than being given a fixed
type with an expression type-signature. This is the purpose
of the function asTypeOf (Chapter 8):
x `asTypeOf` y has the value of x, but x and y are
forced to have the same type. For example,
approxSqrt x = encodeFloat 1 (exponent x `div` 2) `asTypeOf` x
(See Section 6.4.6 for a description of encodeFloat and exponent.)
Ambiguities in the class Num are most common, so Haskell provides another way to resolve them---with a default declaration:
default (t1 , ... , tn)
where n>=0, and each ti must be a type for which Num ti holds. In situations where an ambiguous type is discovered, an ambiguous type variable, v, is defaultable if:
Only one default declaration is permitted per module, and its effect
is limited to that module. If no default declaration is given in a
module then it assumed to be:
default (Integer, Double)
The empty default declaration, default (), turns off all defaults in a module.
The following declarations may be used in any declaration list, including the top level of a module.
| gendecl | -> | vars :: [context =>] type | |
| vars | -> | var1 , ..., varn | (n>=1) |
v1, ..., vn :: cx => t
which is equivalent to asserting vi :: cx => t for each i from 1 to n. Each vi must have a value binding in the same declaration list that contains the type signature; i.e. it is invalid to give a type signature for a variable bound in an outer scope. Moreover, it is invalid to give more than one type signature for one variable, even if the signatures are identical.
As mentioned in Section 4.1.2,
every type variable appearing in a signature
is universally quantified over that signature, and hence
the scope of a type variable is limited to the type
signature that contains it. For example, in the following
declarations
f :: a -> a
f x = x :: a -- invalid
the a's in the two type signatures are quite distinct. Indeed,
these declarations contain a static error, since x does not have
type forall a. a. (The type of x is dependent on the type of
f; there is currently no way in Haskell to specify a signature
for a variable with a dependent type; this is explained in Section
4.5.4.)
If a given program includes a signature for a variable f, then each use of f is treated as having the declared type. It is a static error if the same type cannot also be inferred for the defining occurrence of f.
If a variable f is defined without providing a corresponding type signature declaration, then each use of f outside its own declaration group (see Section 4.5) is treated as having the corresponding inferred, or principal type . However, to ensure that type inference is still possible, the defining occurrence, and all uses of f within its declaration group must have the same monomorphic type (from which the principal type is obtained by generalization, as described in Section 4.5.2).
For example, if we define
sqr x = x*x
then the principal type is
sqr :: forall a. Num a =>a ->a,
which allows
applications such as sqr 5 or sqr 0.1. It is also valid to declare
a more specific type, such as
sqr :: Int -> Int
but now applications such as sqr 0.1 are invalid. Type signatures such as
sqr :: (Num a, Num b) => a -> b -- invalid
sqr :: a -> a -- invalid
are invalid, as they are more general than the principal type of sqr.
Type signatures can also be used to support
polymorphic recursion.
The following definition is pathological, but illustrates how a type
signature can be used to specify a type more general than the one that
would be inferred:
data T a = K (T Int) (T a)
f :: T a -> a
f (K x y) = if f x == 1 then f y else undefined
If we remove the signature declaration, the type of f will be
inferred as T Int -> Int due to the first recursive call for which
the argument to f is T Int. Polymorphic recursion allows the user
to supply the more general type signature, T a -> a.
| gendecl | -> | fixity [integer] ops | |
| fixity | -> | infixl | infixr | infix | |
| ops | -> | op1 , ... , opn | (n>=1) |
| op | -> | varop | conop |
There are three kinds of fixity, non-, left- and right-associativity (infix, infixl, and infixr, respectively), and ten precedence levels, 0 to 9 inclusive (level 0 binds least tightly, and level 9 binds most tightly). If the digit is omitted, level 9 is assumed. Any operator lacking a fixity declaration is assumed to be infixl 9 (See Section 3 for more on the use of fixities). Table 4.1 lists the fixities and precedences of the operators defined in the Prelude.
| Prec- | Left associative | Non-associative | Right associative |
| edence | operators | operators | operators |
| 9 | !! | . | |
| 8 | ^, ^^, ** | ||
| 7 | *, /, `div`, | ||
| `mod`, `rem`, `quot` | |||
| 6 | +, - | ||
| 5 | :, ++ | ||
| 4 | ==, /=, <, <=, >, >=, | ||
| `elem`, `notElem` | |||
| 3 | && | ||
| 2 | || | ||
| 1 | >>, >>= | ||
| 0 | $, $!, `seq` | ||
Fixity is a property of a particular entity (constructor or variable), just like
its type; fixity is not a property of that entity's name.
For example:
module Bar( op ) where
infixr 7 `op`
op = ...
module Foo where
import qualified Bar
infix 3 `op`
a `op` b = (a `Bar.op` b) + 1
f x = let
p `op` q = (p `Foo.op` q) * 2
in ...
Here, `Bar.op` is infixr 7, `Foo.op` is infix 3, and
the nested definition of op in f's right-hand side has the
default fixity of infixl 9. (It would also be possible
to give a fixity to the nested definition of `op` with a nested
fixity declaration.)
| decl | -> | (funlhs | pat0) rhs |
| funlhs | -> | var apat {apat } |
| | | pati+1 varop(a,i) pati+1 | |
| | | lpati varop(l,i) pati+1 | |
| | | pati+1 varop(r,i) rpati | |
| | | ( funlhs ) apat {apat } | |
| rhs | -> | = exp [where decls] |
| | | gdrhs [where decls] | |
| gdrhs | -> | gd = exp [gdrhs] |
| gd | -> | | exp0 |
| x | p11 ... p1k | match1 |
| ... | ||
| x | pn1 ... pnk | matchn |
where each pij is a pattern, and where each matchi is of the general form:
| = ei where { declsi } |
or
| | gi1 | = ei1 |
| ... | |
| | gimi | = eimi |
| where { declsi } |
and where n>=1, 1<=i<=n, mi>=1. The former is treated as shorthand for a particular case of the latter, namely:
| | True = ei where { declsi } |
Note that all clauses defining a function must be contiguous, and the number of patterns in each clause must be the same. The set of patterns corresponding to each match must be linear---no variable is allowed to appear more than once in the entire set.
Alternative syntax is provided for binding functional values to infix
operators. For example, these three function
definitions are all equivalent:
plus x y z = x+y+z
x `plus` y = \ z -> x+y+z
(x `plus` y) z = x+y+z
Translation:The general binding form for functions is semantically equivalent to the equation (i.e. simple pattern binding):x = \ x1 ... xk -> case (x1, ..., xk) of
where the xi are new identifiers. |
A pattern binding binds variables to values. A simple pattern binding has form p = e. The pattern p is matched "lazily" as an irrefutable pattern, as if there were an implicit ~ in front of it. See the translation in Section 3.12.
The general form of a pattern binding is p match, where a match is the same structure as for function bindings above; in other words, a pattern binding is:
| p | | g1 | = e1 |
| | g2 | = e2 | |
| ... | ||
| | gm | = em | |
| where { decls } |
Translation:The pattern binding above is semantically equivalent to this simple pattern binding:
|
The static semantics of the function and pattern bindings of a let expression or where clause are discussed in this section.
In general the static semantics are given by the normal Hindley-Milner inference rules. A dependency analysis transformation is first performed to increase polymorphism. Two variables bound by value declarations are in the same declaration group if either
The Hindley-Milner type system assigns types to a let-expression in two stages. First, the right-hand side of the declaration is typed, giving a type with no universal quantification. Second, all type variables that occur in this type are universally quantified unless they are associated with bound variables in the type environment; this is called generalization. Finally, the body of the let-expression is typed.
For example, consider the declaration
f x = let g y = (y,y)
in ...
The type of g's definition is
a ->(a,a). The generalization step
attributes to g the polymorphic type
forall a. a ->(a,a),
after which the typing of the "..." part can proceed.
When typing overloaded definitions, all the overloading
constraints from a single declaration group are collected together,
to form the context for the type of each variable declared in the group.
For example, in the definition:
f x = let g1 x y = if x>y then show x else g2 y x
g2 p q = g1 q p
in ...
The types of the definitions of g1 and g2 are both
a ->a ->String, and the accumulated constraints are
Ord a (arising from the use of >), and Show a (arising from the
use of show).
The type variables appearing in this collection of constraints are
called the constrained type variables.
The generalization step attributes to both g1 and g2 the type
forall a. (Ord a, Show a) =>a ->a ->String
Notice that g2 is overloaded in the same way as g1 even though the occurrences of > and show are in the definition of g1.
If the programmer supplies explicit type signatures for more than one variable in a declaration group, the contexts of these signatures must be identical up to renaming of the type variables.
Here is an example that shows the need for a
constraint of the form C (m t) where m is one of the type
variables being generalized; that is, where the class C applies to a type
expression that is not a type variable or a type constructor.
Consider:
f :: (Monad m, Eq (m a)) => a -> m a -> Bool
f x y = return x == y
The type of return is Monad m => a -> m a; the type of (==) is
Eq a => a -> a -> Bool. The type of f should be
therefore (Monad m, Eq (m a)) => a -> m a -> Bool, and the context
cannot be simplified further.
The instance declaration derived from a data type deriving clause
(see Section 4.3.3)
must, like any instance declaration, have a simple context; that is,
all the constraints must be of the form C a, where a is a type variable.
For example, in the type
data Apply a b = App (a b) deriving Show
the derived Show instance will produce a context Show (a b), which
cannot be reduced and is not simple; thus a static error results.
The effect of such monomorphism is that the first argument of all
applications of g must be of a single type.
For example, it would be valid for
the "..." to be
(g True, g False)
(which would, incidentally, force x to have type Bool) but invalid
for it to be
(g True, g 'c')
In general, a type forall u. cx =>t
is said to be monomorphic
in the type variable a if a is free in
forall u. cx =>t.
It is worth noting that the explicit type signatures provided by Haskell
are not powerful enough to express types that include monomorphic type
variables. For example, we cannot write
f x = let
g :: a -> b -> ([a],b)
g y z = ([x,y], z)
in ...
because that would claim that g was polymorphic in both a and b
(Section 4.4.1). In this program, g can only be given
a type signature if its first argument is restricted to a type not involving
type variables; for example
g :: Int -> b -> ([Int],b)
This signature would also cause x to have type Int.
Haskell places certain extra restrictions on the generalization step, beyond the standard Hindley-Milner restriction described above, which further reduces polymorphism in particular cases.
The monomorphism restriction depends on the binding syntax of a variable. Recall that a variable is bound by either a function binding or a pattern binding, and that a simple pattern binding is a pattern binding in which the pattern consists of only a single variable (Section 4.4.3).
The following two rules define the monomorphism restriction:
The monomorphism restriction
|
Rule 1 is required for two reasons, both of which are fairly subtle.
The same constraint applies to pattern-bound functions. For example, in
(f,g) = ((+),(-))
both f and g are monomorphic regardless of any type
signatures supplied for f or g.
Rule 2 is required because there is no way to enforce monomorphic use
of an exported binding, except by performing type inference on modules
outside the current module. Rule 2 states that the exact types of all
the variables bound in a module must be determined by that module alone, and not
by any modules that import it.
module M1(len1) where
default( Int, Double )
len1 = genericLength "Hello"
module M2 where
import M1(len1)
len2 = (2*len1) :: Rational
When type inference on module M1 is complete, len1 has the
monomorphic type Num a => a (by Rule 1). Rule 2 now states that
the monomorphic type variable a is ambiguous, and must be resolved using
the defaulting rules of Section 4.3.4.
Hence, len1 gets type Int, and its use in len2 is type-incorrect.
(If the above code is actually what is wanted, a type signature on
len1 would solve the problem.)
This issue does not arise for nested bindings, because their entire scope is visible to the compiler.
The monomorphism rule has a number of consequences for the programmer.
Anything defined with function syntax usually
generalizes as a function is expected to. Thus in
f x y = x+y
the function f may be used at any overloading in class Num.
There is no danger of recomputation here. However, the same function
defined with pattern syntax:
f = \x -> \y -> x+y
requires a type signature if f is to be fully overloaded.
Many functions are most naturally defined using simple pattern
bindings; the user must be careful to affix these with type signatures
to retain full overloading. The standard prelude contains many
examples of this:
sum :: (Num a) => [a] -> a
sum = foldl (+) 0
Rule 1 applies to both top-level and nested definitions. Consider
module M where
len1 = genericLength "Hello"
len2 = (2*len1) :: Rational
Here, type inference finds that len1 has the monomorphic type (Num a => a);
and the type variable a is resolved to Rational when performing type
inference on len2.
This section describes the rules that are used to perform kind inference, i.e. to calculate a suitable kind for each type constructor and class appearing in a given program.
The first step in the kind inference process is to arrange the set of
datatype, synonym, and class definitions into dependency groups. This can
be achieved in much the same way as the dependency analysis for value
declarations that was described in Section 4.5.
For example, the following program fragment includes the definition
of a datatype constructor D, a synonym S and a class C, all of
which would be included in the same dependency group:
data C a => D a = Foo (S a)
type S a = [D a]
class C a where
bar :: a -> D a -> Bool
The kinds of variables, constructors, and classes within each group
are determined using standard techniques of type inference and
kind-preserving unification [7]. For example, in the
definitions above, the parameter a appears as an argument of the
function constructor (->) in the type of bar and hence must
have kind *. It follows that both D and S must have
kind *->* and that every instance of class C must
have kind *.
It is possible that some parts of an inferred kind may not be fully
determined by the corresponding definitions; in such cases, a default
of * is assumed. For example, we could assume an arbitrary kind
k for the a parameter in each of the following examples:
data App f a = A (f a)
data Tree a = Leaf | Fork (Tree a) (Tree a)
This would give kinds
(k->*)->k->* and
k->* for App and Tree, respectively, for any
kind k, and would require an extension to allow polymorphic
kinds. Instead, using the default binding k=*, the
actual kinds for these two constructors are
(*->*)->*->* and
*->*, respectively.
Defaults are applied to each dependency group without consideration of
the ways in which particular type constructor constants or classes are
used in later dependency groups or elsewhere in the program. For example,
adding the following definition to those above does not influence the
kind inferred for Tree (by changing it to
(*->*)->*, for instance), and instead
generates a static error because the kind of [], *->*,
does not match the kind * that is expected for an argument of Tree:
type FunnyTree = Tree [] -- invalid
This is important because it ensures that each constructor and class are
used consistently with the same kind whenever they are in scope.