In this section, we describe the syntax and informal semantics of Haskell expressions, including their translations into the Haskell kernel, where appropriate. Except in the case of let expressions, these translations preserve both the static and dynamic semantics. Some of the names and symbols used in the syntax are not reserved. These are indicated by the `special' productions in the lexical syntax. Examples include ! (used only in data declarations) and as (used in import declarations).
Free variables and constructors used in these translations refer to entities defined by the Prelude. To avoid clutter, we use True instead of Prelude.True or map instead of Prelude.map. (Prelude.True is a qualified name as described in Section 5.1.2.)
In the syntax that follows, there are some families of nonterminals indexed by precedence levels (written as a superscript). Similarly, the nonterminals op, varop, and conop may have a double index: a letter l, r, or n for left-, right- or non-associativity and a precedence level. A precedence-level variable i ranges from 0 to 9; an associativity variable a varies over {l, r, n}. Thus, for example
aexp -> ( expi+1 qop(a,i) )actually stands for 30 productions, with 10 substitutions for i and 3 for a.
exp -> exp0 :: [context =>] type (expression type signature)
| exp0
expi -> expi+1 [qop( n,i) expi+1]
| lexpi
| rexpi
lexpi -> (lexpi | expi+1) qop( l,i) expi+1
lexp6 -> - exp7
rexpi -> expi+1 qop( r,i) (rexpi | expi+1)
exp10 -> \ apat1 ... apatn -> exp (lambda abstraction, n>=1)
| let decllist in exp (let expression)
| if exp then exp else exp (conditional)
| case exp of { alts [;] } (case expression)
| do { stmts [;] } (do expression)
| fexp
fexp -> [fexp] aexp (function application)
aexp -> qvar (variable)
| gcon (general constructor)
| literal
| ( exp ) (parenthesized expression)
| ( exp1 , ... , expk ) (tuple, k>=2)
| [ exp1 , ... , expk ] (list, k>=1)
| [ exp1 [, exp2] .. [exp3] ] (arithmetic sequence)
| [ exp | qual1 , ... , qualn ] (list comprehension, n>=1)
| ( expi+1 qop(a,i) ) (left section)
| ( qop(a,i) expi+1 ) (right section)
| qcon { fbind1 , ... , fbindn } (labeled construction, n>=0)
| aexp{qcon} { fbind1 , ... , fbindn } (labeled update, n >= 1)
| Item | Associativity |
| simple terms, parenthesized terms | -- |
| irrefutable patterns (~) | -- |
| as-patterns ( @) | right |
| function application | left |
| do, if, let, lambda(\), case (leftwards) | right |
| case (rightwards) | right |
| infix operators, prec. 9 | as defined |
| ... | ... |
| infix operators, prec. 0 | as defined |
| function types (->) | right |
| contexts (=>) | -- |
| type constraints (::) | -- |
| do, if, let, lambda(\) (rightwards) | right |
| sequences (..) | -- |
| generators (<-) | -- |
| grouping (,) | n-ary |
| guards (|) | -- |
| case alternatives (->) | -- |
| definitions (=) | -- |
| separation (;) | n-ary |
As an aid to understanding this grammar, Table 1 shows the relative precedence of expressions, patterns and definitions, plus an extended associativity. - indicates that the item is non-associative.
The grammar is ambiguous regarding the extent of lambda abstractions, let expressions, and conditionals. The ambiguity is resolved by the metarule that each of these constructs extends as far to the right as possible. As a consequence, each of these constructs has two precedences, one to its left, which is the precedence used in the grammar; and one to its right which is obtained via the metarule. See the sample parses below.
Expressions involving infix operators are disambiguated by the operator's fixity (see Section 5.6). Consecutive unparenthesised operators with the same precedence must both be either left or right associative to avoid a syntax error. Given an unparenthesised expression "x qop(a,i) y qop(b,j) z", parentheses must be added around either "x qop(a,i) y" or "y qop(b,j) z" when i=j unless a=b= l or a=b= r.
Negation is the only prefix operator in Haskell; it has the same precedence as the infix - operator defined in the Prelude (see Figure 2).
The separation of function arrows from case alternatives solves the ambiguity which otherwise arises when an unparenthesised function type is used in an expression, such as the guard in a case expression.
Sample parses are shown below.
| This | Parses as |
| f x + g y | (f x) + (g y) |
| - f x + y | (- (f x)) + y |
| let { ... } in x + y | let { ... } in (x + y) |
| z + let { ... } in x + y | z + (let { ... } in (x + y)) |
| f x y :: Int | (f x y) :: Int |
| \ x -> a+b :: Int | \ x -> ((a+b) :: Int) |
For the sake of clarity, the rest of this section shows the syntax of expressions without their precedences.
3.1 Errors
Errors during expression evaluation, denoted by bottom,
are indistinguishable from non-termination. Since Haskell is a
lazy language, all Haskell types include bottom. That is, a value
of any type may be bound to a computation that, when demanded, results
in an error. When evaluated, errors cause immediate program
termination and cannot be caught by the user. The Prelude provides
two functions to directly
cause such errors:
error :: String -> a undefined :: aA call to error terminates execution of the program and returns an appropriate error indication to the operating system. It should also display the string in some system-dependent manner. When undefined is used, the error message is created by the compiler.
Translations of Haskell expressions use error and undefined to explicitly indicate where execution time errors may occur. The actual program behavior when an error occurs is up to the implementation. The messages passed to the error function in these translations are only suggestions; implementations may choose to display more or less information when an error occurs.
Alphanumeric operators are formed by enclosing an identifier between
grave accents (backquotes). Any variable or constructor may be used as
an operator in this way. If fun is an identifier (either variable
or constructor), then an expression of the form fun x y is
equivalent to x `fun` y. If no fixity
declaration is given for `fun` then it defaults
to highest precedence and left associativity
(see Section 5.6).
Similarly, any symbolic operator may be used as a (curried) variable
or constructor by enclosing it in parentheses. If op is an infix
operator, then an expression or pattern of the form x op y is
equivalent to (op) x y.
Qualified names may only be used to reference an imported variable or
constructor (see Section 5.1.2)
but not in the definition of a new variable or constructor. Thus
Special syntax is used to name some constructors for some of the
built-in types, as found
in the production for gcon and literal. These are described
in Section 6.1.
An integer literal represents the
application of the function fromInteger to the
appropriate value of type
Integer. Similarly, a floating point literal stands for an application of
fromRational to a value of type Rational (that is,
Ratio Integer).
3.2 Variables, Constructors, and Operators
aexp -> qvar (variable)
| gcon (general constructor)
| literal
gcon -> ()
| []
| (,{,})
| qcon
qvar -> qvarid | ( qvarsym ) (qualified variable)
qcon -> qconid | ( qconsym ) (qualified constructor)
let F.x = 1 in F.x -- invalid
incorrectly uses a qualifier in the definition of x, regardless of
the module containing this definition. Qualification does not affect
the nature of an operator: F.+ is an infix operator just as + is.
|
Translation: The integer literal i is equivalent to fromInteger i, where fromInteger is a method in class Num (see Section 6.3.1). The floating point literal f is equivalent to fromRational (n Ratio.% d), where fromRational is a method in class Fractional and Ratio.% constructs a rational from two integers, as defined in the Ratio library. The integers n and d are chosen so that n/d = f. |
3.3 Curried Applications and Lambda Abstractions
fexp -> [fexp] aexp (function application) exp -> \ apat1 ... apatn -> expFunction application is written e1 e2. Application associates to the left, so the parentheses may be omitted in (f x) y. Because e1 could be a data constructor, partial applications of data constructors are allowed.
Lambda abstractions are written \ p1 ... pn -> e, where the pi are patterns. An expression such as \x:xs->x is syntactically incorrect, and must be rewritten as \(x:xs)->x.
The set of patterns must be linear---no variable may appear more than once in the set.
|
Translation: The lambda abstraction \ p1 ... pn -> e is equivalent to \ x_1 ... x_n -> case (x_1, ..., x_n) of (p_1, ..., p_n) -> e where the xi are new identifiers. Given this translation combined with the semantics of case expressions and pattern matching described in Section 3.17.3, if the pattern fails to match, then the result is bottom. |
exp -> exp1 qop exp2
| - exp (prefix negation)
The form e1 qop e2 is the infix application of binary
operator qop to expressions e1 and e2.
The special form -e denotes prefix negation, the only prefix operator in Haskell, and is syntax for negate (e). The binary - operator does not necessarily refer to the definition of - in the Prelude; it may be rebound by the module system. However, unary - will always refer to the negate function defined in the Prelude. There is no link between the local meaning of the - operator and unary negation.
Prefix negation has the same precedence as the infix operator - defined in the Prelude (see Figure 2). Because e1-e2 parses as an infix application of the binary operator -, one must write e1(-e2) for the alternative parsing. Similarly, (-) is syntax for (\ x y -> x-y), as with any infix operator, and does not denote (\ x -> -x)---one must use negate for that.
|
Translation: e1 op e2 is equivalent to (op) e1 e2. -e is equivalent to negate (e). |
aexp -> ( exp qop )
| ( qop exp )
Sections are written as ( op e ) or ( e op ), where
op is a binary operator and e is an expression. Sections are a
convenient syntax for partial application of binary operators.
The normal rules of syntactic precedence apply to sections; for example, (*a+b) is syntactically invalid, but (+a*b) and (*(a+b)) are valid. Syntactic associativity, however, is not taken into account in sections; thus, (a+b+) must be written ((a+b)+).
Because - is treated specially in the grammar, (- exp) is not a section, but an application of prefix negation, as described in the preceding section. However, there is a subtract function defined in the Prelude such that (subtract exp) is equivalent to the disallowed section. The expression (+ (- exp)) can serve the same purpose.
|
Translation: For binary operator op and expression e, if x is a variable that does not occur free in e, the section (op e) is equivalent to \ x -> x op e, and the section (e op) is equivalent to (op) e. |
exp -> if exp1 then exp2 else exp3A conditional expression has the form if e1 then e2 else e3 and returns the value of e2 if the value of e1 is True, e3 if e1 is False, and bottom otherwise.
|
Translation: if e1 then e2 else e3 is equivalent to: case e_1 of { True -> e_2 ; False -> e_3 } where True and False are the two nullary constructors from the type Bool, as defined in the Prelude. |
aexp -> [ exp1 , ... , expk ] (k>=1)Lists are written [e1, ..., ek], where k>=1; the empty list is written []. Standard operations on lists are given in the Prelude (see Appendix A, notably Section A.1).
|
Translation: [e1, ..., ek] is equivalent to e_1 : (e_2 : ( ... (e_k : []))) where : and [] are constructors for lists, as defined in the Prelude (see Section 6.1.3). The types of e1 through ek must all be the same (call it t), and the type of the overall expression is [t] (see Section 4.1.1). |
aexp -> ( exp1 , ... , expk ) (k>=2)Tuples are written (e1, ..., ek), and may be of arbitrary length k>=2. Standard operations on tuples are given in the Prelude (see Appendix A).
|
Translation: (e1, ..., ek) for k>=2 is an instance of a k-tuple as defined in the Prelude, and requires no translation. If t1 through tk are the types of e1 through ek, respectively, then the type of the resulting tuple is (t1, ..., tk) (see Section 4.1.1). |
3.9 Unit Expressions and Parenthesized Expressions
aexp -> ()
| ( exp )
The form (e) is simply a parenthesized expression, and is
equivalent to e. The unit expression () has type
() (see
Section 4.1.1); it is the only member of that type apart
from bottom (it can
be thought of as the "nullary tuple")---see Section 6.1.5.
|
Translation: (e) is equivalent to e. |
aexp -> [ exp1 [, exp2] .. [exp3] ]The form [e1, e2 .. e3] denotes an arithmetic sequence from e1 in increments of e2-e1 of values not greater than e3 (if the increment is nonnegative) or not less than e3 (if the increment is negative). Thus, the resulting list is empty if the increment is nonnegative and e3 is less than e1 or if the increment is negative and e3 is greater than e1. If the increment is zero, an infinite list of e1s results if e3 is not less than e1. If e3 is omitted, the result is an infinite list, unless the element type is finite, in which case the implied limit is the greatest value of the type if the increment is nonnegative, or the least value, otherwise.
The forms [e1.. e3] and [e1..] are similar to those above, but with an implied increment of one.
Arithmetic sequences may be defined over any type in class Enum, including Char, Int, and Integer (see Figure 5 and Section 4.3.3). For example, ['a'..'z'] denotes the list of lowercase letters in alphabetical order.
|
Translation: Arithmetic sequences satisfy these identities:
|
aexp -> [ exp | qual1 , ... , qualn ] (list comprehension, n>=1)
qual -> pat <- exp
| let decllist
| exp
A list comprehension has the form [ e | q1, ..., qn ], n>=1, where the qi qualifiers are either
Such a list comprehension returns the list of elements produced by evaluating e in the successive environments created by the nested, depth-first evaluation of the generators in the qualifier list. Binding of variables occurs according to the normal pattern matching rules (see Section 3.17), and if a match fails then that element of the list is simply skipped over. Thus:
[ x | xs <- [ [(1,2),(3,4)], [(5,4),(3,2)] ],
(3,x) <- xs ]
yields the list [4,2]. If a qualifier is a guard, it must evaluate
to True for the previous pattern match to succeed.
As usual, bindings in list comprehensions can shadow those in outer
scopes; for example:
| [ x | x <- x, x <- x ] | = | [ z | y <- x, z <- y] |
|
Translation: List comprehensions satisfy these identities, which may be used as a translation into the kernel:
|
exp -> let decllist in expLet expressions have the general form let { d1 ; ... ; dn } in e, and introduce a nested, lexically-scoped, mutually-recursive list of declarations (let is often called letrec in other languages). The scope of the declarations is the expression e and the right hand side of the declarations. Declarations are described in Section 4. Pattern bindings are matched lazily; an implicit ~ makes these patterns irrefutable. For example,
let (x,y) = undefined in edoes not cause an execution-time error until x or y are evaluated.
|
Translation: The dynamic semantics of the expression let { d1 ; ... ; dn } in e0 are captured by this translation: After removing all type signatures, each declaration di is translated into an equation of the form pi = ei, where pi and ei are patterns and expressions respectively, using the translation in Section 4.4.2. Once done, these identities hold, which may be used as a translation into the kernel:
|
exp -> case exp of { alts [;] }
alts -> alt1 ; ... ; altn (n>=1)
alt -> pat -> exp [where decllist]
| pat gdpat [where decllist]
gdpat -> gd -> exp [ gdpat ]
gd -> | exp0
A case expression has the general form
case e of { p_1 match_1 ; ... ; p_n match_n }
where each matchi is of the general form
| | gi1 | -> ei1 | |
| ... | ||
| | gimi | -> eimi | |
| where decllisti |
Each alternative pi matchi consists of a pattern pi and its matches, matchi, which consists of pairs of guards gij and bodies eij (expressions), as well as optional bindings (decllisti) that scope over all of the guards and expressions of the alternative. An alternative of the form
pat -> exp where decllist
is treated as shorthand for:
| pat | True | -> expr | |
| where decllist |
A case expression must have at least one alternative and each alternative must have at least one body. Each body must have the same type, and the type of the whole expression is that type.
A case expression is evaluated by pattern matching the expression e against the individual alternatives. The matches are tried sequentially, from top to bottom. The first successful match causes evaluation of the corresponding alternative body, in the environment of the case expression extended by the bindings created during the matching of that alternative and by the decllisti associated with that alternative. If no match succeeds, the result is bottom. Pattern matching is described in Section 3.17, with the formal semantics of case expressions in Section 3.17.3.
exp -> do { stmts [;]} (do expression)
stmts -> exp [; stmts]
| pat <- exp ; stmts
| let decllist ; stmts
A do expression provides a more readable syntax for monadic
programming.
|
Translation: Do expressions satisfy these identities, which may be used as a translation into the kernel:
|
A failure-free pattern is one that can only be refuted by bottom. Failure-free patterns are defined as follows:
This translation requires a monad in class MonadZero if any pattern bound by <- is not failure-free. Otherwise, only class methods from Monad are generated. Type errors resulting from patterns which are not failure-free can be corrected by using ~ to force the pattern to be failure-free.
As indicated by the translation of do, variables bound by let have fully polymorphic types while those defined by <- are lambda bound and are thus monomorphic.
3.15 Datatypes with Field Labels
A datatype declaration may optionally include field labels for some or
all of the components of the type (see Section 4.2.1).
Readers unfamiliar with datatype declarations in Haskell may wish
to read Section 4.2.1 first.
These field labels can be used to
construct, select from, and update fields in a manner
that is independent of the overall structure of the datatype.
Different datatypes cannot share common field labels in the same scope.
A field label can be used at most once in a constructor.
Within a datatype, however, a field name can be used in more
than one constructor provided the field has the same typing in all
constructors.
3.15.1 Field Selection
aexp -> qvar
Field names are used as selector functions. When used as a variable,
a field name serves as a function which extracts the field from an
object. Selectors are top level bindings and so they
may be shadowed by local variables but cannot conflict with
other top level bindings of the same name. This shadowing only
affects selector functions; in other record constructs, field labels
cannot be confused with ordinary variables.
|
Translation: A field label f introduces a selector function defined as:
|
aexp -> qcon { fbind1 , ... , fbindn } (labeled construction, n>=0)
fbind -> var | var = exp
A constructor with labeled fields may be used to construct a value
in which the components are specified by name rather than by position.
Unlike the braces used in declaration lists, these are not subject to
layout; the { and } characters must be explicit. (This is also
true of field updates and field patterns.)
Construction using field names is subject to the following constraints:
|
Translation: In the binding f = v, the field f labels v. Any binding f that omits the = v is expanded to f = f.
The auxiliary function pickCi bs d is defined as follows: If the ith component of a constructor C has the field name f, and if f=v appears in the binding list bs, then pickCi bs d is v. Otherwise, pickCi bs d is the default value d. |
aexp -> aexp<qcon> { fbind1 , ... , fbindn } (labeled update, n>=1)
Values belonging to a datatype with field names may be
non-destructively updated. This creates a new value in which the
specified field values replace those in the existing value.
Updates are restricted in the following ways:
|
Translation: Using the prior definition of pick,
|
data T = C1 {f1,f2 :: Int}
| C2 {f1 :: Int,
f3,f4 :: Char}
| Expression | Translation |
| C1 {f1 = 3} | C1' 3 undefined |
| C2 {f1 = 1, f4 = 'A', f3 = 'B'} | C2' 1 'B' 'A' |
| x {f1 = 1} | case x of C1' _ f2 -> C1' 1 f2 |
| C2' _ f3 f4 -> C2' 1 f3 f4 | |
The field f1 is common to both constructors in T. The constructors C1' and C2' are `hidden constructors', see the translation in Section 4.2.1. A compile-time error will result if no single constructor defines the set of field names used in an update, such as x {f2 = 1, f3 = 'x'}.
3.16 Expression Type-Signatures
exp -> exp :: [context =>] typeExpression type-signatures have the form e :: t, where e is an expression and t is a type (Section 4.1.1); they are used to type an expression explicitly and may be used to resolve ambiguous typings due to overloading (see Section 4.3.4). The value of the expression is just that of exp. As with normal type signatures (see Section 4.4.1), the declared type may be more specific than the principal type derivable from exp, but it is an error to give a type that is more general than, or not comparable to, the principal type.
Patterns appear in lambda abstractions, function definitions, pattern bindings, list comprehensions, do expressions, and case expressions However, the first five of these ultimately translate into case expressions, so defining the semantics of pattern matching for case expressions is sufficient.
Patterns have this syntax:
pat -> var + integer (successor pattern)
pat | pat0
pati -> pati+1 [qconop( n,i) pati+1]
| lpati
| rpati
lpati -> (lpati | pati+1) qconop( l,i) pati+1
lpat6 -> - (integer | float) (negative literal)
rpati -> pati+1 qconop( r,i) (rpati | pati+1)
pat10 -> apat
| gcon apat1 ... apatk (arity gcon = k, k>=1)
apat -> var [ @ apat] (as pattern)
| gcon (arity gcon = 0)
| qcon { fpat1 , ... , fpatk } (labeled pattern, k>=0)
| literal
| _ (wildcard)
| ( pat ) (parenthesized pattern)
| ( pat1 , ... , patk ) (tuple pattern, k>=2)
| [ pat1 , ... , patk ] (list pattern, k>=1)
| ~ apat (irrefutable pattern)
fpat -> var = pat
| var
The arity of a constructor must match the number of
sub-patterns associated with it; one cannot match against a
partially-applied constructor.
All patterns must be linear ---no variable may appear more than once.
Patterns of the form var @pat are called as-patterns, and allow one to use var as a name for the value being matched by pat. For example,
case e of { xs@(x:rest) -> if x==0 then rest else xs }
is equivalent to:
let { xs = e } in
case xs of { (x:rest) -> if x == 0 then rest else xs }
Patterns of the form _ are wildcards and are useful when some part of a pattern is not referenced on the right-hand-side. It is as if an identifier not used elsewhere were put in its place. For example,
case e of { [x,_,_] -> if x==0 then True else False }
is equivalent to:
case e of { [x,y,z] -> if x==0 then True else False }
In the pattern matching rules given below we distinguish two kinds of patterns: an irrefutable pattern is: a variable, a wildcard, N apat where N is a constructor defined by newtype and apat is irrefutable (see Section 4.2.3), var @apat where apat is irrefutable, or of the form ~apat (whether or not apat is irrefutable). All other patterns are refutable.
Patterns are matched against values. Attempting to match a pattern
can have one of three results: it may fail; it may
succeed, returning a binding for each variable in the pattern; or it
may diverge (i.e. return bottom). Pattern matching proceeds
from left to right, and outside to inside, according to these rules:
Matching any value against the wildcard pattern _ always succeeds
and no binding is done.
Operationally, this means that no matching is done on an
irrefutable pattern until one of the variables in the pattern is used.
At that point the entire pattern is matched against the value, and if
the match fails or diverges, so does the overall computation.
Aside from the obvious static type constraints (for
example, it is a static error to match a character against a
boolean), these static class constraints hold: an integer
literal pattern
can only be matched against a value in the class
Num and a floating literal pattern
can only be matched against a value
in the class Fractional.
and a n+k pattern
can only be matched
against a value in the class Integral.
Many people feel that n+k patterns should not be used. These
patterns may be removed or changed in future versions of Haskell.
Compilers should support a flag which disables the use of these
patterns.
Here are some examples:
3.17.2 Informal Semantics of Pattern Matching
Additional examples illustrating some of the subtleties of pattern matching
may be found in Section 4.2.3.
(\ ~(x,y) -> 0) bottom => 0
(\ (x,y) -> 0) bottom => bottom
(\ ~[x] -> 0) [] => 0
(\ ~[x] -> x) [] => bottom
(\ ~[x,~(a,b)] -> x) [(0,1),bottom] => (0,1)
(\ ~[x, (a,b)] -> x) [(0,1),bottom] => bottom
(\ (x:xs) -> x:x:xs) bottom => bottom
(\ ~(x:xs) -> x:x:xs) bottom => bottom:bottom:bottom
Top level patterns in case expressions and the set of top level patterns in function or pattern bindings may have zero or more associated guards. A guard is a boolean expression that is evaluated only after all of the arguments have been successfully matched, and it must be true for the overall pattern match to succeed. The environment of the guard is the same as the right-hand-side of the case-expression alternative, function definition, or pattern binding to which it is attached.
The guard semantics have an obvious influence on the strictness characteristics of a function or case expression. In particular, an otherwise irrefutable pattern may be evaluated because of a guard. For example, in
f ~(x,y,z) [a] | a==y = 1both a and y will be evaluated by a standard definition of ==.
|
3.17.3 Formal Semantics of Pattern Matching
The semantics of all pattern matching constructs other than case expressions are defined by giving identities that relate those constructs to case expressions. The semantics of case expressions themselves are in turn given as a series of identities, in Figures 3--4. Any implementation should behave so that these identities hold; it is not expected that it will use them directly, since that would generate rather inefficient code.
|
In Figures 3--4: e, e' and ei are expressions; g and gi are boolean-valued expressions; p and pi are patterns; v, x, and xi are variables; K and K' are algebraic datatype (data) constructors (including tuple constructors); N is a newtype constructor;
and k is a character, string, or numeric literal.
Rule (b) matches a general source-language case expression, regardless of whether it actually includes guards---if no guards are written, then True is substituted for the guards gi,j in the matchi forms. Subsequent identities manipulate the resulting case expression into simpler and simpler forms.
Rule (h) in Figure 4 involves the overloaded operator ==; it is this rule that defines the meaning of pattern matching against overloaded constants.
These identities all preserve the static semantics. Rules (d), (e), and (j) use a lambda rather than a let; this indicates that variables bound by case are monomorphically typed (Section 4.1.3).