language, GHC extensions, GHC As with all known Haskell systems, GHC implements some extensions to the language. They can all be enabled or disabled by commandline flags or language pragmas. By default GHC understands the most recent Haskell version it supports, plus a handful of extensions. Some of the Glasgow extensions serve to give you access to the underlying facilities with which we implement Haskell. Thus, you can get at the Raw Iron, if you are willing to write some non-portable code at a more primitive level. You need not be “stuck” on performance because of the implementation costs of Haskell's “high-level” features—you can always code “under” them. In an extreme case, you can write all your time-critical code in C, and then just glue it together with Haskell! Before you get too carried away working at the lowest level (e.g., sloshing MutableByteArray#s around your program), you may wish to check if there are libraries that provide a “Haskellised veneer” over the features you want. The separate libraries documentation describes all the libraries that come with GHC. languageoption optionslanguage extensionsoptions controlling The language option flags control what variation of the language are permitted. Language options can be controlled in two ways: Every language option can switched on by a command-line flag "-X..." (e.g. -XTemplateHaskell), and switched off by the flag "-XNo..."; (e.g. -XNoTemplateHaskell). Language options recognised by Cabal can also be enabled using the LANGUAGE pragma, thus {-# LANGUAGE TemplateHaskell #-} (see ). The flag -fglasgow-exts -fglasgow-exts is equivalent to enabling the following extensions: &what_glasgow_exts_does; Enabling these options is the only effect of -fglasgow-exts. We are trying to move away from this portmanteau flag, and towards enabling features individually. GHC is built on a raft of primitive data types and operations; "primitive" in the sense that they cannot be defined in Haskell itself. While you really can use this stuff to write fast code, we generally find it a lot less painful, and more satisfying in the long run, to use higher-level language features and libraries. With any luck, the code you write will be optimised to the efficient unboxed version in any case. And if it isn't, we'd like to know about it. All these primitive data types and operations are exported by the library GHC.Prim, for which there is detailed online documentation. (This documentation is generated from the file compiler/prelude/primops.txt.pp.) If you want to mention any of the primitive data types or operations in your program, you must first import GHC.Prim to bring them into scope. Many of them have names ending in "#", and to mention such names you need the -XMagicHash extension (). The primops make extensive use of unboxed types and unboxed tuples, which we briefly summarise here. Unboxed types (Glasgow extension) Most types in GHC are boxed, which means that values of that type are represented by a pointer to a heap object. The representation of a Haskell Int, for example, is a two-word heap object. An unboxed type, however, is represented by the value itself, no pointers or heap allocation are involved. Unboxed types correspond to the “raw machine” types you would use in C: Int# (long int), Double# (double), Addr# (void *), etc. The primitive operations (PrimOps) on these types are what you might expect; e.g., (+#) is addition on Int#s, and is the machine-addition that we all know and love—usually one instruction. Primitive (unboxed) types cannot be defined in Haskell, and are therefore built into the language and compiler. Primitive types are always unlifted; that is, a value of a primitive type cannot be bottom. We use the convention (but it is only a convention) that primitive types, values, and operations have a # suffix (see ). For some primitive types we have special syntax for literals, also described in the same section. Primitive values are often represented by a simple bit-pattern, such as Int#, Float#, Double#. But this is not necessarily the case: a primitive value might be represented by a pointer to a heap-allocated object. Examples include Array#, the type of primitive arrays. A primitive array is heap-allocated because it is too big a value to fit in a register, and would be too expensive to copy around; in a sense, it is accidental that it is represented by a pointer. If a pointer represents a primitive value, then it really does point to that value: no unevaluated thunks, no indirections…nothing can be at the other end of the pointer than the primitive value. A numerically-intensive program using unboxed types can go a lot faster than its “standard” counterpart—we saw a threefold speedup on one example. There are some restrictions on the use of primitive types: The main restriction is that you can't pass a primitive value to a polymorphic function or store one in a polymorphic data type. This rules out things like [Int#] (i.e. lists of primitive integers). The reason for this restriction is that polymorphic arguments and constructor fields are assumed to be pointers: if an unboxed integer is stored in one of these, the garbage collector would attempt to follow it, leading to unpredictable space leaks. Or a seq operation on the polymorphic component may attempt to dereference the pointer, with disastrous results. Even worse, the unboxed value might be larger than a pointer (Double# for instance). You cannot define a newtype whose representation type (the argument type of the data constructor) is an unboxed type. Thus, this is illegal: newtype A = MkA Int# You cannot bind a variable with an unboxed type in a top-level binding. You cannot bind a variable with an unboxed type in a recursive binding. You may bind unboxed variables in a (non-recursive, non-top-level) pattern binding, but you must make any such pattern-match strict. For example, rather than: data Foo = Foo Int Int# f x = let (Foo a b, w) = ..rhs.. in ..body.. you must write: data Foo = Foo Int Int# f x = let !(Foo a b, w) = ..rhs.. in ..body.. since b has type Int#. Unboxed tuples aren't really exported by GHC.Exts; they are a syntactic extension enabled by the language flag -XUnboxedTuples. An unboxed tuple looks like this: (# e_1, ..., e_n #) where e_1..e_n are expressions of any type (primitive or non-primitive). The type of an unboxed tuple looks the same. Unboxed tuples are used for functions that need to return multiple values, but they avoid the heap allocation normally associated with using fully-fledged tuples. When an unboxed tuple is returned, the components are put directly into registers or on the stack; the unboxed tuple itself does not have a composite representation. Many of the primitive operations listed in primops.txt.pp return unboxed tuples. In particular, the IO and ST monads use unboxed tuples to avoid unnecessary allocation during sequences of operations. There are some pretty stringent restrictions on the use of unboxed tuples: Values of unboxed tuple types are subject to the same restrictions as other unboxed types; i.e. they may not be stored in polymorphic data structures or passed to polymorphic functions. No variable can have an unboxed tuple type, nor may a constructor or function argument have an unboxed tuple type. The following are all illegal: data Foo = Foo (# Int, Int #) f :: (# Int, Int #) -> (# Int, Int #) f x = x g :: (# Int, Int #) -> Int g (# a,b #) = a h x = let y = (# x,x #) in ... Unboxed tuples may not be nested. So this is illegal: f :: (# Int, (# Int, Int #), Bool #) The typical use of unboxed tuples is simply to return multiple values, binding those multiple results with a case expression, thus: f x y = (# x+1, y-1 #) g x = case f x x of { (# a, b #) -> a + b } You can have an unboxed tuple in a pattern binding, thus f x = let (# p,q #) = h x in ..body.. If the types of p and q are not unboxed, the resulting binding is lazy like any other Haskell pattern binding. The above example desugars like this: f x = let t = case h x o f{ (# p,q #) -> (p,q) p = fst t q = snd t in ..body.. Indeed, the bindings can even be recursive. The language extension -XUnicodeSyntax-XUnicodeSyntax enables Unicode characters to be used to stand for certain ASCII character sequences. The following alternatives are provided: ASCII Unicode alternative Code point Name :: :: 0x2237 PROPORTION => ⇒ 0x21D2 RIGHTWARDS DOUBLE ARROW forall ∀ 0x2200 FOR ALL -> → 0x2192 RIGHTWARDS ARROW <- ← 0x2190 LEFTWARDS ARROW -< ↢ 0x2919 LEFTWARDS ARROW-TAIL >- ↣ 0x291A RIGHTWARDS ARROW-TAIL -<< 0x291B LEFTWARDS DOUBLE ARROW-TAIL >>- 0x291C RIGHTWARDS DOUBLE ARROW-TAIL * ★ 0x2605 BLACK STAR The language extension -XMagicHash allows "#" as a postfix modifier to identifiers. Thus, "x#" is a valid variable, and "T#" is a valid type constructor or data constructor. The hash sign does not change semantics at all. We tend to use variable names ending in "#" for unboxed values or types (e.g. Int#), but there is no requirement to do so; they are just plain ordinary variables. Nor does the -XMagicHash extension bring anything into scope. For example, to bring Int# into scope you must import GHC.Prim (see ); the -XMagicHash extension then allows you to refer to the Int# that is now in scope. The -XMagicHash also enables some new forms of literals (see ): 'x'# has type Char# "foo"# has type Addr# 3# has type Int#. In general, any Haskell integer lexeme followed by a # is an Int# literal, e.g. -0x3A# as well as 32#. 3## has type Word#. In general, any non-negative Haskell integer lexeme followed by ## is a Word#. 3.2# has type Float#. 3.2## has type Double# GHC supports a small extension to the syntax of module names: a module name is allowed to contain a dot ‘.’. This is also known as the “hierarchical module namespace” extension, because it extends the normally flat Haskell module namespace into a more flexible hierarchy of modules. This extension has very little impact on the language itself; modules names are always fully qualified, so you can just think of the fully qualified module name as the module name. In particular, this means that the full module name must be given after the module keyword at the beginning of the module; for example, the module A.B.C must begin module A.B.C It is a common strategy to use the as keyword to save some typing when using qualified names with hierarchical modules. For example: import qualified Control.Monad.ST.Strict as ST For details on how GHC searches for source and interface files in the presence of hierarchical modules, see . GHC comes with a large collection of libraries arranged hierarchically; see the accompanying library documentation. More libraries to install are available from HackageDB. Pattern guards (Glasgow extension) The discussion that follows is an abbreviated version of Simon Peyton Jones's original proposal. (Note that the proposal was written before pattern guards were implemented, so refers to them as unimplemented.) Suppose we have an abstract data type of finite maps, with a lookup operation: lookup :: FiniteMap -> Int -> Maybe Int The lookup returns Nothing if the supplied key is not in the domain of the mapping, and (Just v) otherwise, where v is the value that the key maps to. Now consider the following definition: clunky env var1 var2 | ok1 && ok2 = val1 + val2 | otherwise = var1 + var2 where m1 = lookup env var1 m2 = lookup env var2 ok1 = maybeToBool m1 ok2 = maybeToBool m2 val1 = expectJust m1 val2 = expectJust m2 The auxiliary functions are maybeToBool :: Maybe a -> Bool maybeToBool (Just x) = True maybeToBool Nothing = False expectJust :: Maybe a -> a expectJust (Just x) = x expectJust Nothing = error "Unexpected Nothing" What is clunky doing? The guard ok1 && ok2 checks that both lookups succeed, using maybeToBool to convert the Maybe types to booleans. The (lazily evaluated) expectJust calls extract the values from the results of the lookups, and binds the returned values to val1 and val2 respectively. If either lookup fails, then clunky takes the otherwise case and returns the sum of its arguments. This is certainly legal Haskell, but it is a tremendously verbose and un-obvious way to achieve the desired effect. Arguably, a more direct way to write clunky would be to use case expressions: clunky env var1 var2 = case lookup env var1 of Nothing -> fail Just val1 -> case lookup env var2 of Nothing -> fail Just val2 -> val1 + val2 where fail = var1 + var2 This is a bit shorter, but hardly better. Of course, we can rewrite any set of pattern-matching, guarded equations as case expressions; that is precisely what the compiler does when compiling equations! The reason that Haskell provides guarded equations is because they allow us to write down the cases we want to consider, one at a time, independently of each other. This structure is hidden in the case version. Two of the right-hand sides are really the same (fail), and the whole expression tends to become more and more indented. Here is how I would write clunky: clunky env var1 var2 | Just val1 <- lookup env var1 , Just val2 <- lookup env var2 = val1 + val2 ...other equations for clunky... The semantics should be clear enough. The qualifiers are matched in order. For a <- qualifier, which I call a pattern guard, the right hand side is evaluated and matched against the pattern on the left. If the match fails then the whole guard fails and the next equation is tried. If it succeeds, then the appropriate binding takes place, and the next qualifier is matched, in the augmented environment. Unlike list comprehensions, however, the type of the expression to the right of the <- is the same as the type of the pattern to its left. The bindings introduced by pattern guards scope over all the remaining guard qualifiers, and over the right hand side of the equation. Just as with list comprehensions, boolean expressions can be freely mixed with among the pattern guards. For example: f x | [y] <- x , y > 3 , Just z <- h y = ... Haskell's current guards therefore emerge as a special case, in which the qualifier list has just one element, a boolean expression. View patterns are enabled by the flag -XViewPatterns. More information and examples of view patterns can be found on the Wiki page. View patterns are somewhat like pattern guards that can be nested inside of other patterns. They are a convenient way of pattern-matching against values of abstract types. For example, in a programming language implementation, we might represent the syntax of the types of the language as follows: type Typ data TypView = Unit | Arrow Typ Typ view :: Type -> TypeView -- additional operations for constructing Typ's ... The representation of Typ is held abstract, permitting implementations to use a fancy representation (e.g., hash-consing to manage sharing). Without view patterns, using this signature a little inconvenient: size :: Typ -> Integer size t = case view t of Unit -> 1 Arrow t1 t2 -> size t1 + size t2 It is necessary to iterate the case, rather than using an equational function definition. And the situation is even worse when the matching against t is buried deep inside another pattern. View patterns permit calling the view function inside the pattern and matching against the result: size (view -> Unit) = 1 size (view -> Arrow t1 t2) = size t1 + size t2 That is, we add a new form of pattern, written expression -> pattern that means "apply the expression to whatever we're trying to match against, and then match the result of that application against the pattern". The expression can be any Haskell expression of function type, and view patterns can be used wherever patterns are used. The semantics of a pattern ( exp -> pat ) are as follows: Scoping: The variables bound by the view pattern are the variables bound by pat. Any variables in exp are bound occurrences, but variables bound "to the left" in a pattern are in scope. This feature permits, for example, one argument to a function to be used in the view of another argument. For example, the function clunky from can be written using view patterns as follows: clunky env (lookup env -> Just val1) (lookup env -> Just val2) = val1 + val2 ...other equations for clunky... More precisely, the scoping rules are: In a single pattern, variables bound by patterns to the left of a view pattern expression are in scope. For example: example :: Maybe ((String -> Integer,Integer), String) -> Bool example Just ((f,_), f -> 4) = True Additionally, in function definitions, variables bound by matching earlier curried arguments may be used in view pattern expressions in later arguments: example :: (String -> Integer) -> String -> Bool example f (f -> 4) = True That is, the scoping is the same as it would be if the curried arguments were collected into a tuple. In mutually recursive bindings, such as let, where, or the top level, view patterns in one declaration may not mention variables bound by other declarations. That is, each declaration must be self-contained. For example, the following program is not allowed: let {(x -> y) = e1 ; (y -> x) = e2 } in x (For some amplification on this design choice see Trac #4061.) Typing: If exp has type T1 -> T2 and pat matches a T2, then the whole view pattern matches a T1. Matching: To the equations in Section 3.17.3 of the Haskell 98 Report, add the following: case v of { (e -> p) -> e1 ; _ -> e2 } = case (e v) of { p -> e1 ; _ -> e2 } That is, to match a variable v against a pattern ( exp -> pat ), evaluate ( exp v ) and match the result against pat. Efficiency: When the same view function is applied in multiple branches of a function definition or a case expression (e.g., in size above), GHC makes an attempt to collect these applications into a single nested case expression, so that the view function is only applied once. Pattern compilation in GHC follows the matrix algorithm described in Chapter 4 of The Implementation of Functional Programming Languages. When the top rows of the first column of a matrix are all view patterns with the "same" expression, these patterns are transformed into a single nested case. This includes, for example, adjacent view patterns that line up in a tuple, as in f ((view -> A, p1), p2) = e1 f ((view -> B, p3), p4) = e2 The current notion of when two view pattern expressions are "the same" is very restricted: it is not even full syntactic equality. However, it does include variables, literals, applications, and tuples; e.g., two instances of view ("hi", "there") will be collected. However, the current implementation does not compare up to alpha-equivalence, so two instances of (x, view x -> y) will not be coalesced. -XNPlusKPatterns n+k pattern support is disabled by default. To enable it, you can use the -XNPlusKPatterns flag. -XNoTraditionalRecordSyntax Traditional record syntax, such as C {f = x}, is enabled by default. To disable it, you can use the -XNoTraditionalRecordSyntax flag. The do-notation of Haskell 98 does not allow recursive bindings, that is, the variables bound in a do-expression are visible only in the textually following code block. Compare this to a let-expression, where bound variables are visible in the entire binding group. It turns out that several applications can benefit from recursive bindings in the do-notation. The -XDoRec flag provides the necessary syntactic support. Here is a simple (albeit contrived) example: {-# LANGUAGE DoRec #-} justOnes = do { rec { xs <- Just (1:xs) } ; return (map negate xs) } As you can guess justOnes will evaluate to Just [-1,-1,-1,.... The background and motivation for recursive do-notation is described in A recursive do for Haskell, by Levent Erkok, John Launchbury, Haskell Workshop 2002, pages: 29-37. Pittsburgh, Pennsylvania. The theory behind monadic value recursion is explained further in Erkok's thesis Value Recursion in Monadic Computations. However, note that GHC uses a different syntax than the one described in these documents. The recursive do-notation is enabled with the flag -XDoRec or, equivalently, the LANGUAGE pragma DoRec. It introduces the single new keyword "rec", which wraps a mutually-recursive group of monadic statements, producing a single statement. Similar to a let statement, the variables bound in the rec are visible throughout the rec group, and below it. For example, compare do { a <- getChar do { a <- getChar ; let { r1 = f a r2 ; rec { r1 <- f a r2 ; r2 = g r1 } ; r2 <- g r1 } ; return (r1 ++ r2) } ; return (r1 ++ r2) } In both cases, r1 and r2 are available both throughout the let or rec block, and in the statements that follow it. The difference is that let is non-monadic, while rec is monadic. (In Haskell let is really letrec, of course.) The static and dynamic semantics of rec can be described as follows: First, similar to let-bindings, the rec is broken into minimal recursive groups, a process known as segmentation. For example: rec { a <- getChar ===> a <- getChar ; b <- f a c rec { b <- f a c ; c <- f b a ; c <- f b a } ; putChar c } putChar c The details of segmentation are described in Section 3.2 of A recursive do for Haskell. Segmentation improves polymorphism, reduces the size of the recursive "knot", and, as the paper describes, also has a semantic effect (unless the monad satisfies the right-shrinking law). Then each resulting rec is desugared, using a call to Control.Monad.Fix.mfix. For example, the rec group in the preceding example is desugared like this: rec { b <- f a c ===> (b,c) <- mfix (\~(b,c) -> do { b <- f a c ; c <- f b a } ; c <- f b a ; return (b,c) }) In general, the statment rec ss is desugared to the statement vs <- mfix (\~vs -> do { ss; return vs }) where vs is a tuple of the variables bound by ss. The original rec typechecks exactly when the above desugared version would do so. For example, this means that the variables vs are all monomorphic in the statements following the rec, because they are bound by a lambda. The mfix function is defined in the MonadFix class, in Control.Monad.Fix, thus: class Monad m => MonadFix m where mfix :: (a -> m a) -> m a Here are some other important points in using the recursive-do notation: It is enabled with the flag -XDoRec. If recursive bindings are required for a monad, then that monad must be declared an instance of the MonadFix class. The following instances of MonadFix are automatically provided: List, Maybe, IO. Furthermore, the Control.Monad.ST and Control.Monad.ST.Lazy modules provide the instances of the MonadFix class for Haskell's internal state monad (strict and lazy, respectively). Like let and where bindings, name shadowing is not allowed within a rec; that is, all the names bound in a single rec must be distinct (Section 3.3 of the paper). It supports rebindable syntax (see ). GHC used to support the flag -XRecursiveDo, which enabled the keyword mdo, precisely as described in A recursive do for Haskell, but this is now deprecated. Instead of mdo { Q; e }, write do { rec Q; e }. Historical note: The old implementation of the mdo-notation (and most of the existing documents) used the name MonadRec for the class and the corresponding library. This name is not supported by GHC. list comprehensionsparallel parallel list comprehensions Parallel list comprehensions are a natural extension to list comprehensions. List comprehensions can be thought of as a nice syntax for writing maps and filters. Parallel comprehensions extend this to include the zipWith family. A parallel list comprehension has multiple independent branches of qualifier lists, each separated by a `|' symbol. For example, the following zips together two lists: [ (x, y) | x <- xs | y <- ys ] The behavior of parallel list comprehensions follows that of zip, in that the resulting list will have the same length as the shortest branch. We can define parallel list comprehensions by translation to regular comprehensions. Here's the basic idea: Given a parallel comprehension of the form: [ e | p1 <- e11, p2 <- e12, ... | q1 <- e21, q2 <- e22, ... ... ] This will be translated to: [ e | ((p1,p2), (q1,q2), ...) <- zipN [(p1,p2) | p1 <- e11, p2 <- e12, ...] [(q1,q2) | q1 <- e21, q2 <- e22, ...] ... ] where `zipN' is the appropriate zip for the given number of branches. list comprehensionsgeneralised extended list comprehensions group sql Generalised list comprehensions are a further enhancement to the list comprehension syntactic sugar to allow operations such as sorting and grouping which are familiar from SQL. They are fully described in the paper Comprehensive comprehensions: comprehensions with "order by" and "group by", except that the syntax we use differs slightly from the paper. The extension is enabled with the flag -XTransformListComp. Here is an example: employees = [ ("Simon", "MS", 80) , ("Erik", "MS", 100) , ("Phil", "Ed", 40) , ("Gordon", "Ed", 45) , ("Paul", "Yale", 60)] output = [ (the dept, sum salary) | (name, dept, salary) <- employees , then group by dept using groupWith , then sortWith by (sum salary) , then take 5 ] In this example, the list output would take on the value: [("Yale", 60), ("Ed", 85), ("MS", 180)] There are three new keywords: group, by, and using. (The functions sortWith and groupWith are not keywords; they are ordinary functions that are exported by GHC.Exts.) There are five new forms of comprehension qualifier, all introduced by the (existing) keyword then: then f This statement requires that f have the type forall a. [a] -> [a]. You can see an example of its use in the motivating example, as this form is used to apply take 5. then f by e This form is similar to the previous one, but allows you to create a function which will be passed as the first argument to f. As a consequence f must have the type forall a. (a -> t) -> [a] -> [a]. As you can see from the type, this function lets f "project out" some information from the elements of the list it is transforming. An example is shown in the opening example, where sortWith is supplied with a function that lets it find out the sum salary for any item in the list comprehension it transforms. then group by e using f This is the most general of the grouping-type statements. In this form, f is required to have type forall a. (a -> t) -> [a] -> [[a]]. As with the then f by e case above, the first argument is a function supplied to f by the compiler which lets it compute e on every element of the list being transformed. However, unlike the non-grouping case, f additionally partitions the list into a number of sublists: this means that at every point after this statement, binders occurring before it in the comprehension refer to lists of possible values, not single values. To help understand this, let's look at an example: -- This works similarly to groupWith in GHC.Exts, but doesn't sort its input first groupRuns :: Eq b => (a -> b) -> [a] -> [[a]] groupRuns f = groupBy (\x y -> f x == f y) output = [ (the x, y) | x <- ([1..3] ++ [1..2]) , y <- [4..6] , then group by x using groupRuns ] This results in the variable output taking on the value below: [(1, [4, 5, 6]), (2, [4, 5, 6]), (3, [4, 5, 6]), (1, [4, 5, 6]), (2, [4, 5, 6])] Note that we have used the the function to change the type of x from a list to its original numeric type. The variable y, in contrast, is left unchanged from the list form introduced by the grouping. then group using f With this form of the group statement, f is required to simply have the type forall a. [a] -> [[a]], which will be used to group up the comprehension so far directly. An example of this form is as follows: output = [ x | y <- [1..5] , x <- "hello" , then group using inits] This will yield a list containing every prefix of the word "hello" written out 5 times: ["","h","he","hel","hell","hello","helloh","hellohe","hellohel","hellohell","hellohello","hellohelloh",...] monad comprehensions Monad comprehensions generalise the list comprehension notation, including parallel comprehensions () and transform comprehensions () to work for any monad. Monad comprehensions support: Bindings: [ x + y | x <- Just 1, y <- Just 2 ] Bindings are translated with the (>>=) and return functions to the usual do-notation: do x <- Just 1 y <- Just 2 return (x+y) Guards: [ x | x <- [1..10], x <= 5 ] Guards are translated with the guard function, which requires a MonadPlus instance: do x <- [1..10] guard (x <= 5) return x Transform statements (as with -XTransformListComp): [ x+y | x <- [1..10], y <- [1..x], then take 2 ] This translates to: do (x,y) <- take 2 (do x <- [1..10] y <- [1..x] return (x,y)) return (x+y) Group statements (as with -XTransformListComp): [ x | x <- [1,1,2,2,3], then group by x using GHC.Exts.groupWith ] [ x | x <- [1,1,2,2,3], then group using myGroup ] Parallel statements (as with -XParallelListComp): [ (x+y) | x <- [1..10] | y <- [11..20] ] Parallel statements are translated using the mzip function, which requires a MonadZip instance defined in Control.Monad.Zip: do (x,y) <- mzip (do x <- [1..10] return x) (do y <- [11..20] return y) return (x+y) All these features are enabled by default if the MonadComprehensions extension is enabled. The types and more detailed examples on how to use comprehensions are explained in the previous chapters and . In general you just have to replace the type [a] with the type Monad m => m a for monad comprehensions. Note: Even though most of these examples are using the list monad, monad comprehensions work for any monad. The base package offers all necessary instances for lists, which make MonadComprehensions backward compatible to built-in, transform and parallel list comprehensions. More formally, the desugaring is as follows. We write D[ e | Q] to mean the desugaring of the monad comprehension [ e | Q]: Expressions: e Declarations: d Lists of qualifiers: Q,R,S -- Basic forms D[ e | ] = return e D[ e | p <- e, Q ] = e >>= \p -> D[ e | Q ] D[ e | e, Q ] = guard e >> \p -> D[ e | Q ] D[ e | let d, Q ] = let d in D[ e | Q ] -- Parallel comprehensions (iterate for multiple parallel branches) D[ e | (Q | R), S ] = mzip D[ Qv | Q ] D[ Rv | R ] >>= \(Qv,Rv) -> D[ e | S ] -- Transform comprehensions D[ e | Q then f, R ] = f D[ Qv | Q ] >>= \Qv -> D[ e | R ] D[ e | Q then f by b, R ] = f (\Qv -> b) D[ Qv | Q ] >>= \Qv -> D[ e | R ] D[ e | Q then group using f, R ] = f D[ Qv | Q ] >>= \ys -> case (fmap selQv1 ys, ..., fmap selQvn ys) of Qv -> D[ e | R ] D[ e | Q then group by b using f, R ] = f (\Qv -> b) D[ Qv | Q ] >>= \ys -> case (fmap selQv1 ys, ..., fmap selQvn ys) of Qv -> D[ e | R ] where Qv is the tuple of variables bound by Q (and used subsequently) selQvi is a selector mapping Qv to the ith component of Qv Operator Standard binding Expected type -------------------------------------------------------------------- return GHC.Base t1 -> m t2 (>>=) GHC.Base m1 t1 -> (t2 -> m2 t3) -> m3 t3 (>>) GHC.Base m1 t1 -> m2 t2 -> m3 t3 guard Control.Monad t1 -> m t2 fmap GHC.Base forall a b. (a->b) -> n a -> n b mzip Control.Monad.Zip forall a b. m a -> m b -> m (a,b) The comprehension should typecheck when its desugaring would typecheck. Monad comprehensions support rebindable syntax (). Without rebindable syntax, the operators from the "standard binding" module are used; with rebindable syntax, the operators are looked up in the current lexical scope. For example, parallel comprehensions will be typechecked and desugared using whatever "mzip" is in scope. The rebindable operators must have the "Expected type" given in the table above. These types are surprisingly general. For example, you can use a bind operator with the type (>>=) :: T x y a -> (a -> T y z b) -> T x z b In the case of transform comprehensions, notice that the groups are parameterised over some arbitrary type n (provided it has an fmap, as well as the comprehension being over an arbitrary monad. -XNoImplicitPrelude option GHC normally imports Prelude.hi files for you. If you'd rather it didn't, then give it a -XNoImplicitPrelude option. The idea is that you can then import a Prelude of your own. (But don't call it Prelude; the Haskell module namespace is flat, and you must not conflict with any Prelude module.) Suppose you are importing a Prelude of your own in order to define your own numeric class hierarchy. It completely defeats that purpose if the literal "1" means "Prelude.fromInteger 1", which is what the Haskell Report specifies. So the -XRebindableSyntax flag causes the following pieces of built-in syntax to refer to whatever is in scope, not the Prelude versions: An integer literal 368 means "fromInteger (368::Integer)", rather than "Prelude.fromInteger (368::Integer)". Fractional literals are handed in just the same way, except that the translation is fromRational (3.68::Rational). The equality test in an overloaded numeric pattern uses whatever (==) is in scope. The subtraction operation, and the greater-than-or-equal test, in n+k patterns use whatever (-) and (>=) are in scope. Negation (e.g. "- (f x)") means "negate (f x)", both in numeric patterns, and expressions. Conditionals (e.g. "if e1 then e2 else e3") means "ifThenElse e1 e2 e3". However case expressions are unaffected. "Do" notation is translated using whatever functions (>>=), (>>), and fail, are in scope (not the Prelude versions). List comprehensions, mdo (), and parallel array comprehensions, are unaffected. Arrow notation (see ) uses whatever arr, (>>>), first, app, (|||) and loop functions are in scope. But unlike the other constructs, the types of these functions must match the Prelude types very closely. Details are in flux; if you want to use this, ask! -XRebindableSyntax implies -XNoImplicitPrelude. In all cases (apart from arrow notation), the static semantics should be that of the desugared form, even if that is a little unexpected. For example, the static semantics of the literal 368 is exactly that of fromInteger (368::Integer); it's fine for fromInteger to have any of the types: fromInteger :: Integer -> Integer fromInteger :: forall a. Foo a => Integer -> a fromInteger :: Num a => a -> Integer fromInteger :: Integer -> Bool -> Bool Be warned: this is an experimental facility, with fewer checks than usual. Use -dcore-lint to typecheck the desugared program. If Core Lint is happy you should be all right. The -XPostfixOperators flag enables a small extension to the syntax of left operator sections, which allows you to define postfix operators. The extension is this: the left section (e !) is equivalent (from the point of view of both type checking and execution) to the expression ((!) e) (for any expression e and operator (!). The strict Haskell 98 interpretation is that the section is equivalent to (\y -> (!) e y) That is, the operator must be a function of two arguments. GHC allows it to take only one argument, and that in turn allows you to write the function postfix. The extension does not extend to the left-hand side of function definitions; you must define such a function in prefix form. The -XTupleSections flag enables Python-style partially applied tuple constructors. For example, the following program (, True) is considered to be an alternative notation for the more unwieldy alternative \x -> (x, True) You can omit any combination of arguments to the tuple, as in the following (, "I", , , "Love", , 1337) which translates to \a b c d -> (a, "I", b, c, "Love", d, 1337) If you have unboxed tuples enabled, tuple sections will also be available for them, like so (# , True #) Because there is no unboxed unit tuple, the following expression (# #) continues to stand for the unboxed singleton tuple data constructor. In record construction and record pattern matching it is entirely unambiguous which field is referred to, even if there are two different data types in scope with a common field name. For example: module M where data S = MkS { x :: Int, y :: Bool } module Foo where import M data T = MkT { x :: Int } ok1 (MkS { x = n }) = n+1 -- Unambiguous ok2 n = MkT { x = n+1 } -- Unambiguous bad1 k = k { x = 3 } -- Ambiguous bad2 k = x k -- Ambiguous Even though there are two x's in scope, it is clear that the x in the pattern in the definition of ok1 can only mean the field x from type S. Similarly for the function ok2. However, in the record update in bad1 and the record selection in bad2 it is not clear which of the two types is intended. Haskell 98 regards all four as ambiguous, but with the -XDisambiguateRecordFields flag, GHC will accept the former two. The rules are precisely the same as those for instance declarations in Haskell 98, where the method names on the left-hand side of the method bindings in an instance declaration refer unambiguously to the method of that class (provided they are in scope at all), even if there are other variables in scope with the same name. This reduces the clutter of qualified names when you import two records from different modules that use the same field name. Some details: Field disambiguation can be combined with punning (see ). For example: module Foo where import M x=True ok3 (MkS { x }) = x+1 -- Uses both disambiguation and punning With -XDisambiguateRecordFields you can use unqualified field names even if the corresponding selector is only in scope qualified For example, assuming the same module M as in our earlier example, this is legal: module Foo where import qualified M -- Note qualified ok4 (M.MkS { x = n }) = n+1 -- Unambiguous Since the constructor MkS is only in scope qualified, you must name it M.MkS, but the field x does not need to be qualified even though M.x is in scope but x is not. (In effect, it is qualified by the constructor.) Record puns are enabled by the flag -XNamedFieldPuns. When using records, it is common to write a pattern that binds a variable with the same name as a record field, such as: data C = C {a :: Int} f (C {a = a}) = a Record punning permits the variable name to be elided, so one can simply write f (C {a}) = a to mean the same pattern as above. That is, in a record pattern, the pattern a expands into the pattern a = a for the same name a. Note that: Record punning can also be used in an expression, writing, for example, let a = 1 in C {a} instead of let a = 1 in C {a = a} The expansion is purely syntactic, so the expanded right-hand side expression refers to the nearest enclosing variable that is spelled the same as the field name. Puns and other patterns can be mixed in the same record: data C = C {a :: Int, b :: Int} f (C {a, b = 4}) = a Puns can be used wherever record patterns occur (e.g. in let bindings or at the top-level). A pun on a qualified field name is expanded by stripping off the module qualifier. For example: f (C {M.a}) = a means f (M.C {M.a = a}) = a (This is useful if the field selector a for constructor M.C is only in scope in qualified form.) Record wildcards are enabled by the flag -XRecordWildCards. This flag implies -XDisambiguateRecordFields. For records with many fields, it can be tiresome to write out each field individually in a record pattern, as in data C = C {a :: Int, b :: Int, c :: Int, d :: Int} f (C {a = 1, b = b, c = c, d = d}) = b + c + d Record wildcard syntax permits a ".." in a record pattern, where each elided field f is replaced by the pattern f = f. For example, the above pattern can be written as f (C {a = 1, ..}) = b + c + d More details: Wildcards can be mixed with other patterns, including puns (); for example, in a pattern C {a = 1, b, ..}). Additionally, record wildcards can be used wherever record patterns occur, including in let bindings and at the top-level. For example, the top-level binding C {a = 1, ..} = e defines b, c, and d. Record wildcards can also be used in expressions, writing, for example, let {a = 1; b = 2; c = 3; d = 4} in C {..} in place of let {a = 1; b = 2; c = 3; d = 4} in C {a=a, b=b, c=c, d=d} The expansion is purely syntactic, so the record wildcard expression refers to the nearest enclosing variables that are spelled the same as the omitted field names. The ".." expands to the missing in-scope record fields. Specifically the expansion of "C {..}" includes f if and only if: f is a record field of constructor C. The record field f is in scope somehow (either qualified or unqualified). In the case of expressions (but not patterns), the variable f is in scope unqualified, apart from the binding of the record selector itself. For example module M where data R = R { a,b,c :: Int } module X where import M( R(a,c) ) f b = R { .. } The R{..} expands to R{M.a=a}, omitting b since the record field is not in scope, and omitting c since the variable c is not in scope (apart from the binding of the record selector c, of course). A careful reading of the Haskell 98 Report reveals that fixity declarations (infix, infixl, and infixr) are permitted to appear inside local bindings such those introduced by let and where. However, the Haskell Report does not specify the semantics of such bindings very precisely. In GHC, a fixity declaration may accompany a local binding: let f = ... infixr 3 `f` in ... and the fixity declaration applies wherever the binding is in scope. For example, in a let, it applies in the right-hand sides of other let-bindings and the body of the letC. Or, in recursive do expressions (), the local fixity declarations of a let statement scope over other statements in the group, just as the bound name does. Moreover, a local fixity declaration *must* accompany a local binding of that name: it is not possible to revise the fixity of name bound elsewhere, as in let infixr 9 $ in ... Because local fixity declarations are technically Haskell 98, no flag is necessary to enable them. With the -XPackageImports flag, GHC allows import declarations to be qualified by the package name that the module is intended to be imported from. For example: import "network" Network.Socket would import the module Network.Socket from the package network (any version). This may be used to disambiguate an import when the same module is available from multiple packages, or is present in both the current package being built and an external package. Note: you probably don't need to use this feature, it was added mainly so that we can build backwards-compatible versions of packages when APIs change. It can lead to fragile dependencies in the common case: modules occasionally move from one package to another, rendering any package-qualified imports broken. With the -XSafe, -XTrustworthy and -XUnsafe language flags, GHC extends the import declaration syntax to take an optional safe keyword after the import keyword. This feature is part of the Safe Haskell GHC extension. For example: import safe qualified Network.Socket as NS would import the module Network.Socket with compilation only succeeding if Network.Socket can be safely imported. For a description of when a import is considered safe see Turning on an option that enables special syntax might cause working Haskell 98 code to fail to compile, perhaps because it uses a variable name which has become a reserved word. This section lists the syntax that is "stolen" by language extensions. We use notation and nonterminal names from the Haskell 98 lexical syntax (see the Haskell 98 Report). We only list syntax changes here that might affect existing working programs (i.e. "stolen" syntax). Many of these extensions will also enable new context-free syntax, but in all cases programs written to use the new syntax would not be compilable without the option enabled. There are two classes of special syntax: New reserved words and symbols: character sequences which are no longer available for use as identifiers in the program. Other special syntax: sequences of characters that have a different meaning when this particular option is turned on. The following syntax is stolen: forall forall Stolen (in types) by: -XExplicitForAll, and hence by -XScopedTypeVariables, -XLiberalTypeSynonyms, -XRank2Types, -XRankNTypes, -XPolymorphicComponents, -XExistentialQuantification mdo mdo Stolen by: -XRecursiveDo, foreign foreign Stolen by: -XForeignFunctionInterface, rec, proc, -<, >-, -<<, >>-, and (|, |) brackets proc Stolen by: -XArrows, ?varid, %varid implicit parameters Stolen by: -XImplicitParams, [|, [e|, [p|, [d|, [t|, $(, $varid Template Haskell Stolen by: -XTemplateHaskell, [:varid| quasi-quotation Stolen by: -XQuasiQuotes, varid{#}, char#, string#, integer#, float#, float##, (#, #), Stolen by: -XMagicHash, With the -XEmptyDataDecls flag (or equivalent LANGUAGE pragma), GHC lets you declare a data type with no constructors. For example: data S -- S :: * data T a -- T :: * -> * Syntactically, the declaration lacks the "= constrs" part. The type can be parameterised over types of any kind, but if the kind is not * then an explicit kind annotation must be used (see ). Such data types have only one value, namely bottom. Nevertheless, they can be useful when defining "phantom types". Haskell allows datatypes to be given contexts, e.g. data Eq a => Set a = NilSet | ConsSet a (Set a) give constructors with types: NilSet :: Set a ConsSet :: Eq a => a -> Set a -> Set a In GHC this feature is an extension called DatatypeContexts, and on by default. GHC allows type constructors, classes, and type variables to be operators, and to be written infix, very much like expressions. More specifically: A type constructor or class can be an operator, beginning with a colon; e.g. :*:. The lexical syntax is the same as that for data constructors. Data type and type-synonym declarations can be written infix, parenthesised if you want further arguments. E.g. data a :*: b = Foo a b type a :+: b = Either a b class a :=: b where ... data (a :**: b) x = Baz a b x type (a :++: b) y = Either (a,b) y Types, and class constraints, can be written infix. For example x :: Int :*: Bool f :: (a :=: b) => a -> b A type variable can be an (unqualified) operator e.g. +. The lexical syntax is the same as that for variable operators, excluding "(.)", "(!)", and "(*)". In a binding position, the operator must be parenthesised. For example: type T (+) = Int + Int f :: T Either f = Left 3 liftA2 :: Arrow (~>) => (a -> b -> c) -> (e ~> a) -> (e ~> b) -> (e ~> c) liftA2 = ... Back-quotes work as for expressions, both for type constructors and type variables; e.g. Int `Either` Bool, or Int `a` Bool. Similarly, parentheses work the same; e.g. (:*:) Int Bool. Fixities may be declared for type constructors, or classes, just as for data constructors. However, one cannot distinguish between the two in a fixity declaration; a fixity declaration sets the fixity for a data constructor and the corresponding type constructor. For example: infixl 7 T, :*: sets the fixity for both type constructor T and data constructor T, and similarly for :*:. Int `a` Bool. Function arrow is infixr with fixity 0. (This might change; I'm not sure what it should be.) Type synonyms are like macros at the type level, but Haskell 98 imposes many rules on individual synonym declarations. With the -XLiberalTypeSynonyms extension, GHC does validity checking on types only after expanding type synonyms. That means that GHC can be very much more liberal about type synonyms than Haskell 98. You can write a forall (including overloading) in a type synonym, thus: type Discard a = forall b. Show b => a -> b -> (a, String) f :: Discard a f x y = (x, show y) g :: Discard Int -> (Int,String) -- A rank-2 type g f = f 3 True If you also use -XUnboxedTuples, you can write an unboxed tuple in a type synonym: type Pr = (# Int, Int #) h :: Int -> Pr h x = (# x, x #) You can apply a type synonym to a forall type: type Foo a = a -> a -> Bool f :: Foo (forall b. b->b) After expanding the synonym, f has the legal (in GHC) type: f :: (forall b. b->b) -> (forall b. b->b) -> Bool You can apply a type synonym to a partially applied type synonym: type Generic i o = forall x. i x -> o x type Id x = x foo :: Generic Id [] After expanding the synonym, foo has the legal (in GHC) type: foo :: forall x. x -> [x] GHC currently does kind checking before expanding synonyms (though even that could be changed.) After expanding type synonyms, GHC does validity checking on types, looking for the following mal-formedness which isn't detected simply by kind checking: Type constructor applied to a type involving for-alls. Unboxed tuple on left of an arrow. Partially-applied type synonym. So, for example, this will be rejected: type Pr = (# Int, Int #) h :: Pr -> Int h x = ... because GHC does not allow unboxed tuples on the left of a function arrow. The idea of using existential quantification in data type declarations was suggested by Perry, and implemented in Hope+ (Nigel Perry, The Implementation of Practical Functional Programming Languages, PhD Thesis, University of London, 1991). It was later formalised by Laufer and Odersky (Polymorphic type inference and abstract data types, TOPLAS, 16(5), pp1411-1430, 1994). It's been in Lennart Augustsson's hbc Haskell compiler for several years, and proved very useful. Here's the idea. Consider the declaration: data Foo = forall a. MkFoo a (a -> Bool) | Nil The data type Foo has two constructors with types: MkFoo :: forall a. a -> (a -> Bool) -> Foo Nil :: Foo Notice that the type variable a in the type of MkFoo does not appear in the data type itself, which is plain Foo. For example, the following expression is fine: [MkFoo 3 even, MkFoo 'c' isUpper] :: [Foo] Here, (MkFoo 3 even) packages an integer with a function even that maps an integer to Bool; and MkFoo 'c' isUpper packages a character with a compatible function. These two things are each of type Foo and can be put in a list. What can we do with a value of type Foo?. In particular, what happens when we pattern-match on MkFoo? f (MkFoo val fn) = ??? Since all we know about val and fn is that they are compatible, the only (useful) thing we can do with them is to apply fn to val to get a boolean. For example: f :: Foo -> Bool f (MkFoo val fn) = fn val What this allows us to do is to package heterogeneous values together with a bunch of functions that manipulate them, and then treat that collection of packages in a uniform manner. You can express quite a bit of object-oriented-like programming this way. What has this to do with existential quantification? Simply that MkFoo has the (nearly) isomorphic type MkFoo :: (exists a . (a, a -> Bool)) -> Foo But Haskell programmers can safely think of the ordinary universally quantified type given above, thereby avoiding adding a new existential quantification construct. An easy extension is to allow arbitrary contexts before the constructor. For example: data Baz = forall a. Eq a => Baz1 a a | forall b. Show b => Baz2 b (b -> b) The two constructors have the types you'd expect: Baz1 :: forall a. Eq a => a -> a -> Baz Baz2 :: forall b. Show b => b -> (b -> b) -> Baz But when pattern matching on Baz1 the matched values can be compared for equality, and when pattern matching on Baz2 the first matched value can be converted to a string (as well as applying the function to it). So this program is legal: f :: Baz -> String f (Baz1 p q) | p == q = "Yes" | otherwise = "No" f (Baz2 v fn) = show (fn v) Operationally, in a dictionary-passing implementation, the constructors Baz1 and Baz2 must store the dictionaries for Eq and Show respectively, and extract it on pattern matching. GHC allows existentials to be used with records syntax as well. For example: data Counter a = forall self. NewCounter { _this :: self , _inc :: self -> self , _display :: self -> IO () , tag :: a } Here tag is a public field, with a well-typed selector function tag :: Counter a -> a. The self type is hidden from the outside; any attempt to apply _this, _inc or _display as functions will raise a compile-time error. In other words, GHC defines a record selector function only for fields whose type does not mention the existentially-quantified variables. (This example used an underscore in the fields for which record selectors will not be defined, but that is only programming style; GHC ignores them.) To make use of these hidden fields, we need to create some helper functions: inc :: Counter a -> Counter a inc (NewCounter x i d t) = NewCounter { _this = i x, _inc = i, _display = d, tag = t } display :: Counter a -> IO () display NewCounter{ _this = x, _display = d } = d x Now we can define counters with different underlying implementations: counterA :: Counter String counterA = NewCounter { _this = 0, _inc = (1+), _display = print, tag = "A" } counterB :: Counter String counterB = NewCounter { _this = "", _inc = ('#':), _display = putStrLn, tag = "B" } main = do display (inc counterA) -- prints "1" display (inc (inc counterB)) -- prints "##" Record update syntax is supported for existentials (and GADTs): setTag :: Counter a -> a -> Counter a setTag obj t = obj{ tag = t } The rule for record update is this: the types of the updated fields may mention only the universally-quantified type variables of the data constructor. For GADTs, the field may mention only types that appear as a simple type-variable argument in the constructor's result type. For example: data T a b where { T1 { f1::a, f2::b, f3::(b,c) } :: T a b } -- c is existential upd1 t x = t { f1=x } -- OK: upd1 :: T a b -> a' -> T a' b upd2 t x = t { f3=x } -- BAD (f3's type mentions c, which is -- existentially quantified) data G a b where { G1 { g1::a, g2::c } :: G a [c] } upd3 g x = g { g1=x } -- OK: upd3 :: G a b -> c -> G c b upd4 g x = g { g2=x } -- BAD (f2's type mentions c, which is not a simple -- type-variable argument in G1's result type) There are several restrictions on the ways in which existentially-quantified constructors can be use. When pattern matching, each pattern match introduces a new, distinct, type for each existential type variable. These types cannot be unified with any other type, nor can they escape from the scope of the pattern match. For example, these fragments are incorrect: f1 (MkFoo a f) = a Here, the type bound by MkFoo "escapes", because a is the result of f1. One way to see why this is wrong is to ask what type f1 has: f1 :: Foo -> a -- Weird! What is this "a" in the result type? Clearly we don't mean this: f1 :: forall a. Foo -> a -- Wrong! The original program is just plain wrong. Here's another sort of error f2 (Baz1 a b) (Baz1 p q) = a==q It's ok to say a==b or p==q, but a==q is wrong because it equates the two distinct types arising from the two Baz1 constructors. You can't pattern-match on an existentially quantified constructor in a let or where group of bindings. So this is illegal: f3 x = a==b where { Baz1 a b = x } Instead, use a case expression: f3 x = case x of Baz1 a b -> a==b In general, you can only pattern-match on an existentially-quantified constructor in a case expression or in the patterns of a function definition. The reason for this restriction is really an implementation one. Type-checking binding groups is already a nightmare without existentials complicating the picture. Also an existential pattern binding at the top level of a module doesn't make sense, because it's not clear how to prevent the existentially-quantified type "escaping". So for now, there's a simple-to-state restriction. We'll see how annoying it is. You can't use existential quantification for newtype declarations. So this is illegal: newtype T = forall a. Ord a => MkT a Reason: a value of type T must be represented as a pair of a dictionary for Ord t and a value of type t. That contradicts the idea that newtype should have no concrete representation. You can get just the same efficiency and effect by using data instead of newtype. If there is no overloading involved, then there is more of a case for allowing an existentially-quantified newtype, because the data version does carry an implementation cost, but single-field existentially quantified constructors aren't much use. So the simple restriction (no existential stuff on newtype) stands, unless there are convincing reasons to change it. You can't use deriving to define instances of a data type with existentially quantified data constructors. Reason: in most cases it would not make sense. For example:; data T = forall a. MkT [a] deriving( Eq ) To derive Eq in the standard way we would need to have equality between the single component of two MkT constructors: instance Eq T where (MkT a) == (MkT b) = ??? But a and b have distinct types, and so can't be compared. It's just about possible to imagine examples in which the derived instance would make sense, but it seems altogether simpler simply to prohibit such declarations. Define your own instances! When the GADTSyntax extension is enabled, GHC allows you to declare an algebraic data type by giving the type signatures of constructors explicitly. For example: data Maybe a where Nothing :: Maybe a Just :: a -> Maybe a The form is called a "GADT-style declaration" because Generalised Algebraic Data Types, described in , can only be declared using this form. Notice that GADT-style syntax generalises existential types (). For example, these two declarations are equivalent: data Foo = forall a. MkFoo a (a -> Bool) data Foo' where { MKFoo :: a -> (a->Bool) -> Foo' } Any data type that can be declared in standard Haskell-98 syntax can also be declared using GADT-style syntax. The choice is largely stylistic, but GADT-style declarations differ in one important respect: they treat class constraints on the data constructors differently. Specifically, if the constructor is given a type-class context, that context is made available by pattern matching. For example: data Set a where MkSet :: Eq a => [a] -> Set a makeSet :: Eq a => [a] -> Set a makeSet xs = MkSet (nub xs) insert :: a -> Set a -> Set a insert a (MkSet as) | a `elem` as = MkSet as | otherwise = MkSet (a:as) A use of MkSet as a constructor (e.g. in the definition of makeSet) gives rise to a (Eq a) constraint, as you would expect. The new feature is that pattern-matching on MkSet (as in the definition of insert) makes available an (Eq a) context. In implementation terms, the MkSet constructor has a hidden field that stores the (Eq a) dictionary that is passed to MkSet; so when pattern-matching that dictionary becomes available for the right-hand side of the match. In the example, the equality dictionary is used to satisfy the equality constraint generated by the call to elem, so that the type of insert itself has no Eq constraint. For example, one possible application is to reify dictionaries: data NumInst a where MkNumInst :: Num a => NumInst a intInst :: NumInst Int intInst = MkNumInst plus :: NumInst a -> a -> a -> a plus MkNumInst p q = p + q Here, a value of type NumInst a is equivalent to an explicit (Num a) dictionary. All this applies to constructors declared using the syntax of . For example, the NumInst data type above could equivalently be declared like this: data NumInst a = Num a => MkNumInst (NumInst a) Notice that, unlike the situation when declaring an existential, there is no forall, because the Num constrains the data type's universally quantified type variable a. A constructor may have both universal and existential type variables: for example, the following two declarations are equivalent: data T1 a = forall b. (Num a, Eq b) => MkT1 a b data T2 a where MkT2 :: (Num a, Eq b) => a -> b -> T2 a All this behaviour contrasts with Haskell 98's peculiar treatment of contexts on a data type declaration (Section 4.2.1 of the Haskell 98 Report). In Haskell 98 the definition data Eq a => Set' a = MkSet' [a] gives MkSet' the same type as MkSet above. But instead of making available an (Eq a) constraint, pattern-matching on MkSet' requires an (Eq a) constraint! GHC faithfully implements this behaviour, odd though it is. But for GADT-style declarations, GHC's behaviour is much more useful, as well as much more intuitive. The rest of this section gives further details about GADT-style data type declarations. The result type of each data constructor must begin with the type constructor being defined. If the result type of all constructors has the form T a1 ... an, where a1 ... an are distinct type variables, then the data type is ordinary; otherwise is a generalised data type (). As with other type signatures, you can give a single signature for several data constructors. In this example we give a single signature for T1 and T2: data T a where T1,T2 :: a -> T a T3 :: T a The type signature of each constructor is independent, and is implicitly universally quantified as usual. In particular, the type variable(s) in the "data T a where" header have no scope, and different constructors may have different universally-quantified type variables: data T a where -- The 'a' has no scope T1,T2 :: b -> T b -- Means forall b. b -> T b T3 :: T a -- Means forall a. T a A constructor signature may mention type class constraints, which can differ for different constructors. For example, this is fine: data T a where T1 :: Eq b => b -> b -> T b T2 :: (Show c, Ix c) => c -> [c] -> T c When pattern matching, these constraints are made available to discharge constraints in the body of the match. For example: f :: T a -> String f (T1 x y) | x==y = "yes" | otherwise = "no" f (T2 a b) = show a Note that f is not overloaded; the Eq constraint arising from the use of == is discharged by the pattern match on T1 and similarly the Show constraint arising from the use of show. Unlike a Haskell-98-style data type declaration, the type variable(s) in the "data Set a where" header have no scope. Indeed, one can write a kind signature instead: data Set :: * -> * where ... or even a mixture of the two: data Bar a :: (* -> *) -> * where ... The type variables (if given) may be explicitly kinded, so we could also write the header for Foo like this: data Bar a (b :: * -> *) where ... You can use strictness annotations, in the obvious places in the constructor type: data Term a where Lit :: !Int -> Term Int If :: Term Bool -> !(Term a) -> !(Term a) -> Term a Pair :: Term a -> Term b -> Term (a,b) You can use a deriving clause on a GADT-style data type declaration. For example, these two declarations are equivalent data Maybe1 a where { Nothing1 :: Maybe1 a ; Just1 :: a -> Maybe1 a } deriving( Eq, Ord ) data Maybe2 a = Nothing2 | Just2 a deriving( Eq, Ord ) The type signature may have quantified type variables that do not appear in the result type: data Foo where MkFoo :: a -> (a->Bool) -> Foo Nil :: Foo Here the type variable a does not appear in the result type of either constructor. Although it is universally quantified in the type of the constructor, such a type variable is often called "existential". Indeed, the above declaration declares precisely the same type as the data Foo in . The type may contain a class context too, of course: data Showable where MkShowable :: Show a => a -> Showable You can use record syntax on a GADT-style data type declaration: data Person where Adult :: { name :: String, children :: [Person] } -> Person Child :: Show a => { name :: !String, funny :: a } -> Person As usual, for every constructor that has a field f, the type of field f must be the same (modulo alpha conversion). The Child constructor above shows that the signature may have a context, existentially-quantified variables, and strictness annotations, just as in the non-record case. (NB: the "type" that follows the double-colon is not really a type, because of the record syntax and strictness annotations. A "type" of this form can appear only in a constructor signature.) Record updates are allowed with GADT-style declarations, only fields that have the following property: the type of the field mentions no existential type variables. As in the case of existentials declared using the Haskell-98-like record syntax (), record-selector functions are generated only for those fields that have well-typed selectors. Here is the example of that section, in GADT-style syntax: data Counter a where NewCounter { _this :: self , _inc :: self -> self , _display :: self -> IO () , tag :: a } :: Counter a As before, only one selector function is generated here, that for tag. Nevertheless, you can still use all the field names in pattern matching and record construction. Generalised Algebraic Data Types generalise ordinary algebraic data types by allowing constructors to have richer return types. Here is an example: data Term a where Lit :: Int -> Term Int Succ :: Term Int -> Term Int IsZero :: Term Int -> Term Bool If :: Term Bool -> Term a -> Term a -> Term a Pair :: Term a -> Term b -> Term (a,b) Notice that the return type of the constructors is not always Term a, as is the case with ordinary data types. This generality allows us to write a well-typed eval function for these Terms: eval :: Term a -> a eval (Lit i) = i eval (Succ t) = 1 + eval t eval (IsZero t) = eval t == 0 eval (If b e1 e2) = if eval b then eval e1 else eval e2 eval (Pair e1 e2) = (eval e1, eval e2) The key point about GADTs is that pattern matching causes type refinement. For example, in the right hand side of the equation eval :: Term a -> a eval (Lit i) = ... the type a is refined to Int. That's the whole point! A precise specification of the type rules is beyond what this user manual aspires to, but the design closely follows that described in the paper Simple unification-based type inference for GADTs, (ICFP 2006). The general principle is this: type refinement is only carried out based on user-supplied type annotations. So if no type signature is supplied for eval, no type refinement happens, and lots of obscure error messages will occur. However, the refinement is quite general. For example, if we had: eval :: Term a -> a -> a eval (Lit i) j = i+j the pattern match causes the type a to be refined to Int (because of the type of the constructor Lit), and that refinement also applies to the type of j, and the result type of the case expression. Hence the addition i+j is legal. These and many other examples are given in papers by Hongwei Xi, and Tim Sheard. There is a longer introduction on the wiki, and Ralf Hinze's Fun with phantom types also has a number of examples. Note that papers may use different notation to that implemented in GHC. The rest of this section outlines the extensions to GHC that support GADTs. The extension is enabled with -XGADTs. The -XGADTs flag also sets -XRelaxedPolyRec. A GADT can only be declared using GADT-style syntax (); the old Haskell-98 syntax for data declarations always declares an ordinary data type. The result type of each constructor must begin with the type constructor being defined, but for a GADT the arguments to the type constructor can be arbitrary monotypes. For example, in the Term data type above, the type of each constructor must end with Term ty, but the ty need not be a type variable (e.g. the Lit constructor). It is permitted to declare an ordinary algebraic data type using GADT-style syntax. What makes a GADT into a GADT is not the syntax, but rather the presence of data constructors whose result type is not just T a b. You cannot use a deriving clause for a GADT; only for an ordinary data type. As mentioned in , record syntax is supported. For example: data Term a where Lit { val :: Int } :: Term Int Succ { num :: Term Int } :: Term Int Pred { num :: Term Int } :: Term Int IsZero { arg :: Term Int } :: Term Bool Pair { arg1 :: Term a , arg2 :: Term b } :: Term (a,b) If { cnd :: Term Bool , tru :: Term a , fls :: Term a } :: Term a However, for GADTs there is the following additional constraint: every constructor that has a field f must have the same result type (modulo alpha conversion) Hence, in the above example, we cannot merge the num and arg fields above into a single name. Although their field types are both Term Int, their selector functions actually have different types: num :: Term Int -> Term Int arg :: Term Bool -> Term Int When pattern-matching against data constructors drawn from a GADT, for example in a case expression, the following rules apply: The type of the scrutinee must be rigid. The type of the entire case expression must be rigid. The type of any free variable mentioned in any of the case alternatives must be rigid. A type is "rigid" if it is completely known to the compiler at its binding site. The easiest way to ensure that a variable a rigid type is to give it a type signature. For more precise details see Simple unification-based type inference for GADTs . The criteria implemented by GHC are given in the Appendix. The Haskell Report is vague about exactly when a deriving clause is legal. For example: data T0 f a = MkT0 a deriving( Eq ) data T1 f a = MkT1 (f a) deriving( Eq ) data T2 f a = MkT2 (f (f a)) deriving( Eq ) The natural generated Eq code would result in these instance declarations: instance Eq a => Eq (T0 f a) where ... instance Eq (f a) => Eq (T1 f a) where ... instance Eq (f (f a)) => Eq (T2 f a) where ... The first of these is obviously fine. The second is still fine, although less obviously. The third is not Haskell 98, and risks losing termination of instances. GHC takes a conservative position: it accepts the first two, but not the third. The rule is this: each constraint in the inferred instance context must consist only of type variables, with no repetitions. This rule is applied regardless of flags. If you want a more exotic context, you can write it yourself, using the standalone deriving mechanism. GHC now allows stand-alone deriving declarations, enabled by -XStandaloneDeriving: data Foo a = Bar a | Baz String deriving instance Eq a => Eq (Foo a) The syntax is identical to that of an ordinary instance declaration apart from (a) the keyword deriving, and (b) the absence of the where part. Note the following points: You must supply an explicit context (in the example the context is (Eq a)), exactly as you would in an ordinary instance declaration. (In contrast, in a deriving clause attached to a data type declaration, the context is inferred.) A deriving instance declaration must obey the same rules concerning form and termination as ordinary instance declarations, controlled by the same flags; see . Unlike a deriving declaration attached to a data declaration, the instance can be more specific than the data type (assuming you also use -XFlexibleInstances, ). Consider for example data Foo a = Bar a | Baz String deriving instance Eq a => Eq (Foo [a]) deriving instance Eq a => Eq (Foo (Maybe a)) This will generate a derived instance for (Foo [a]) and (Foo (Maybe a)), but other types such as (Foo (Int,Bool)) will not be an instance of Eq. Unlike a deriving declaration attached to a data declaration, GHC does not restrict the form of the data type. Instead, GHC simply generates the appropriate boilerplate code for the specified class, and typechecks it. If there is a type error, it is your problem. (GHC will show you the offending code if it has a type error.) The merit of this is that you can derive instances for GADTs and other exotic data types, providing only that the boilerplate code does indeed typecheck. For example: data T a where T1 :: T Int T2 :: T Bool deriving instance Show (T a) In this example, you cannot say ... deriving( Show ) on the data type declaration for T, because T is a GADT, but you can generate the instance declaration using stand-alone deriving. The stand-alone syntax is generalised for newtypes in exactly the same way that ordinary deriving clauses are generalised (). For example: newtype Foo a = MkFoo (State Int a) deriving instance MonadState Int Foo GHC always treats the last parameter of the instance (Foo in this example) as the type whose instance is being derived. Haskell 98 allows the programmer to add "deriving( Eq, Ord )" to a data type declaration, to generate a standard instance declaration for classes specified in the deriving clause. In Haskell 98, the only classes that may appear in the deriving clause are the standard classes Eq, Ord, Enum, Ix, Bounded, Read, and Show. GHC extends this list with several more classes that may be automatically derived: With -XDeriveDataTypeable, you can derive instances of the classes Typeable, and Data, defined in the library modules Data.Typeable and Data.Generics respectively. An instance of Typeable can only be derived if the data type has seven or fewer type parameters, all of kind *. The reason for this is that the Typeable class is derived using the scheme described in Scrap More Boilerplate: Reflection, Zips, and Generalised Casts . (Section 7.4 of the paper describes the multiple Typeable classes that are used, and only Typeable1 up to Typeable7 are provided in the library.) In other cases, there is nothing to stop the programmer writing a TypeableX class, whose kind suits that of the data type constructor, and then writing the data type instance by hand. With -XDeriveGeneric, you can derive instances of the class Generic, defined in GHC.Generics. You can use these to define generic functions, as described in . With -XDeriveFunctor, you can derive instances of the class Functor, defined in GHC.Base. With -XDeriveFoldable, you can derive instances of the class Foldable, defined in Data.Foldable. With -XDeriveTraversable, you can derive instances of the class Traversable, defined in Data.Traversable. In each case the appropriate class must be in scope before it can be mentioned in the deriving clause. When you define an abstract type using newtype, you may want the new type to inherit some instances from its representation. In Haskell 98, you can inherit instances of Eq, Ord, Enum and Bounded by deriving them, but for any other classes you have to write an explicit instance declaration. For example, if you define newtype Dollars = Dollars Int and you want to use arithmetic on Dollars, you have to explicitly define an instance of Num: instance Num Dollars where Dollars a + Dollars b = Dollars (a+b) ... All the instance does is apply and remove the newtype constructor. It is particularly galling that, since the constructor doesn't appear at run-time, this instance declaration defines a dictionary which is wholly equivalent to the Int dictionary, only slower! GHC now permits such instances to be derived instead, using the flag -XGeneralizedNewtypeDeriving, so one can write newtype Dollars = Dollars Int deriving (Eq,Show,Num) and the implementation uses the same Num dictionary for Dollars as for Int. Notionally, the compiler derives an instance declaration of the form instance Num Int => Num Dollars which just adds or removes the newtype constructor according to the type. We can also derive instances of constructor classes in a similar way. For example, suppose we have implemented state and failure monad transformers, such that instance Monad m => Monad (State s m) instance Monad m => Monad (Failure m) In Haskell 98, we can define a parsing monad by type Parser tok m a = State [tok] (Failure m) a which is automatically a monad thanks to the instance declarations above. With the extension, we can make the parser type abstract, without needing to write an instance of class Monad, via newtype Parser tok m a = Parser (State [tok] (Failure m) a) deriving Monad In this case the derived instance declaration is of the form instance Monad (State [tok] (Failure m)) => Monad (Parser tok m) Notice that, since Monad is a constructor class, the instance is a partial application of the new type, not the entire left hand side. We can imagine that the type declaration is "eta-converted" to generate the context of the instance declaration. We can even derive instances of multi-parameter classes, provided the newtype is the last class parameter. In this case, a ``partial application'' of the class appears in the deriving clause. For example, given the class class StateMonad s m | m -> s where ... instance Monad m => StateMonad s (State s m) where ... then we can derive an instance of StateMonad for Parsers by newtype Parser tok m a = Parser (State [tok] (Failure m) a) deriving (Monad, StateMonad [tok]) The derived instance is obtained by completing the application of the class to the new type: instance StateMonad [tok] (State [tok] (Failure m)) => StateMonad [tok] (Parser tok m) As a result of this extension, all derived instances in newtype declarations are treated uniformly (and implemented just by reusing the dictionary for the representation type), except Show and Read, which really behave differently for the newtype and its representation. Derived instance declarations are constructed as follows. Consider the declaration (after expansion of any type synonyms) newtype T v1...vn = T' (t vk+1...vn) deriving (c1...cm) where The ci are partial applications of classes of the form C t1'...tj', where the arity of C is exactly j+1. That is, C lacks exactly one type argument. The k is chosen so that ci (T v1...vk) is well-kinded. The type t is an arbitrary type. The type variables vk+1...vn do not occur in t, nor in the ci, and None of the ci is Read, Show, Typeable, or Data. These classes should not "look through" the type or its constructor. You can still derive these classes for a newtype, but it happens in the usual way, not via this new mechanism. Then, for each ci, the derived instance declaration is: instance ci t => ci (T v1...vk) As an example which does not work, consider newtype NonMonad m s = NonMonad (State s m s) deriving Monad Here we cannot derive the instance instance Monad (State s m) => Monad (NonMonad m) because the type variable s occurs in State s m, and so cannot be "eta-converted" away. It is a good thing that this deriving clause is rejected, because NonMonad m is not, in fact, a monad --- for the same reason. Try defining >>= with the correct type: you won't be able to. Notice also that the order of class parameters becomes important, since we can only derive instances for the last one. If the StateMonad class above were instead defined as class StateMonad m s | m -> s where ... then we would not have been able to derive an instance for the Parser type above. We hypothesise that multi-parameter classes usually have one "main" parameter for which deriving new instances is most interesting. Lastly, all of this applies only for classes other than Read, Show, Typeable, and Data, for which the built-in derivation applies (section 4.3.3. of the Haskell Report). (For the standard classes Eq, Ord, Ix, and Bounded it is immaterial whether the standard method is used or the one described here.) This section, and the next one, documents GHC's type-class extensions. There's lots of background in the paper Type classes: exploring the design space (Simon Peyton Jones, Mark Jones, Erik Meijer). Multi-parameter type classes are permitted, with flag -XMultiParamTypeClasses. For example: class Collection c a where union :: c a -> c a -> c a ...etc. In Haskell 98 the context of a class declaration (which introduces superclasses) must be simple; that is, each predicate must consist of a class applied to type variables. The flag -XFlexibleContexts () lifts this restriction, so that the only restriction on the context in a class declaration is that the class hierarchy must be acyclic. So these class declarations are OK: class Functor (m k) => FiniteMap m k where ... class (Monad m, Monad (t m)) => Transform t m where lift :: m a -> (t m) a As in Haskell 98, The class hierarchy must be acyclic. However, the definition of "acyclic" involves only the superclass relationships. For example, this is OK: class C a where { op :: D b => a -> b -> b } class C a => D a where { ... } Here, C is a superclass of D, but it's OK for a class operation op of C to mention D. (It would not be OK for D to be a superclass of C.) With the extension that adds a kind of constraints, you can write more exotic superclass definitions. The superclass cycle check is even more liberal in these case. For example, this is OK: class A cls c where meth :: cls c => c -> c class A B c => B c where A superclass context for a class C is allowed if, after expanding type synonyms to their right-hand-sides, and uses of classes (other than C) to their superclasses, C does not occur syntactically in the context. Haskell 98 prohibits class method types to mention constraints on the class type variable, thus: class Seq s a where fromList :: [a] -> s a elem :: Eq a => a -> s a -> Bool The type of elem is illegal in Haskell 98, because it contains the constraint Eq a, constrains only the class type variable (in this case a). GHC lifts this restriction (flag -XConstrainedClassMethods). Haskell 98 allows you to define a default implementation when declaring a class: class Enum a where enum :: [a] enum = [] The type of the enum method is [a], and this is also the type of the default method. You can lift this restriction and give another type to the default method using the flag -XDefaultSignatures. For instance, if you have written a generic implementation of enumeration in a class GEnum with method genum in terms of GHC.Generics, you can specify a default method that uses that generic implementation: class Enum a where enum :: [a] default enum :: (Generic a, GEnum (Rep a)) => [a] enum = map to genum We reuse the keyword default to signal that a signature applies to the default method only; when defining instances of the Enum class, the original type [a] of enum still applies. When giving an empty instance, however, the default implementation map to0 genum is filled-in, and type-checked with the type (Generic a, GEnum (Rep a)) => [a]. We use default signatures to simplify generic programming in GHC (). Functional dependencies are implemented as described by Mark Jones in “Type Classes with Functional Dependencies”, Mark P. Jones, In Proceedings of the 9th European Symposium on Programming, ESOP 2000, Berlin, Germany, March 2000, Springer-Verlag LNCS 1782, . Functional dependencies are introduced by a vertical bar in the syntax of a class declaration; e.g. class (Monad m) => MonadState s m | m -> s where ... class Foo a b c | a b -> c where ... There should be more documentation, but there isn't (yet). Yell if you need it. In a class declaration, all of the class type variables must be reachable (in the sense mentioned in ) from the free variables of each method type. For example: class Coll s a where empty :: s insert :: s -> a -> s is not OK, because the type of empty doesn't mention a. Functional dependencies can make the type variable reachable: class Coll s a | s -> a where empty :: s insert :: s -> a -> s Alternatively Coll might be rewritten class Coll s a where empty :: s a insert :: s a -> a -> s a which makes the connection between the type of a collection of a's (namely (s a)) and the element type a. Occasionally this really doesn't work, in which case you can split the class like this: class CollE s where empty :: s class CollE s => Coll s a where insert :: s -> a -> s The following description of the motivation and use of functional dependencies is taken from the Hugs user manual, reproduced here (with minor changes) by kind permission of Mark Jones. Consider the following class, intended as part of a library for collection types: class Collects e ce where empty :: ce insert :: e -> ce -> ce member :: e -> ce -> Bool The type variable e used here represents the element type, while ce is the type of the container itself. Within this framework, we might want to define instances of this class for lists or characteristic functions (both of which can be used to represent collections of any equality type), bit sets (which can be used to represent collections of characters), or hash tables (which can be used to represent any collection whose elements have a hash function). Omitting standard implementation details, this would lead to the following declarations: instance Eq e => Collects e [e] where ... instance Eq e => Collects e (e -> Bool) where ... instance Collects Char BitSet where ... instance (Hashable e, Collects a ce) => Collects e (Array Int ce) where ... All this looks quite promising; we have a class and a range of interesting implementations. Unfortunately, there are some serious problems with the class declaration. First, the empty function has an ambiguous type: empty :: Collects e ce => ce By "ambiguous" we mean that there is a type variable e that appears on the left of the => symbol, but not on the right. The problem with this is that, according to the theoretical foundations of Haskell overloading, we cannot guarantee a well-defined semantics for any term with an ambiguous type. We can sidestep this specific problem by removing the empty member from the class declaration. However, although the remaining members, insert and member, do not have ambiguous types, we still run into problems when we try to use them. For example, consider the following two functions: f x y = insert x . insert y g = f True 'a' for which GHC infers the following types: f :: (Collects a c, Collects b c) => a -> b -> c -> c g :: (Collects Bool c, Collects Char c) => c -> c Notice that the type for f allows the two parameters x and y to be assigned different types, even though it attempts to insert each of the two values, one after the other, into the same collection. If we're trying to model collections that contain only one type of value, then this is clearly an inaccurate type. Worse still, the definition for g is accepted, without causing a type error. As a result, the error in this code will not be flagged at the point where it appears. Instead, it will show up only when we try to use g, which might even be in a different module. Faced with the problems described above, some Haskell programmers might be tempted to use something like the following version of the class declaration: class Collects e c where empty :: c e insert :: e -> c e -> c e member :: e -> c e -> Bool The key difference here is that we abstract over the type constructor c that is used to form the collection type c e, and not over that collection type itself, represented by ce in the original class declaration. This avoids the immediate problems that we mentioned above: empty has type Collects e c => c e, which is not ambiguous. The function f from the previous section has a more accurate type: f :: (Collects e c) => e -> e -> c e -> c e The function g from the previous section is now rejected with a type error as we would hope because the type of f does not allow the two arguments to have different types. This, then, is an example of a multiple parameter class that does actually work quite well in practice, without ambiguity problems. There is, however, a catch. This version of the Collects class is nowhere near as general as the original class seemed to be: only one of the four instances for Collects given above can be used with this version of Collects because only one of them---the instance for lists---has a collection type that can be written in the form c e, for some type constructor c, and element type e. To get a more useful version of the Collects class, Hugs provides a mechanism that allows programmers to specify dependencies between the parameters of a multiple parameter class (For readers with an interest in theoretical foundations and previous work: The use of dependency information can be seen both as a generalization of the proposal for `parametric type classes' that was put forward by Chen, Hudak, and Odersky, or as a special case of Mark Jones's later framework for "improvement" of qualified types. The underlying ideas are also discussed in a more theoretical and abstract setting in a manuscript [implparam], where they are identified as one point in a general design space for systems of implicit parameterization.). To start with an abstract example, consider a declaration such as: class C a b where ... which tells us simply that C can be thought of as a binary relation on types (or type constructors, depending on the kinds of a and b). Extra clauses can be included in the definition of classes to add information about dependencies between parameters, as in the following examples: class D a b | a -> b where ... class E a b | a -> b, b -> a where ... The notation a -> b used here between the | and where symbols --- not to be confused with a function type --- indicates that the a parameter uniquely determines the b parameter, and might be read as "a determines b." Thus D is not just a relation, but actually a (partial) function. Similarly, from the two dependencies that are included in the definition of E, we can see that E represents a (partial) one-one mapping between types. More generally, dependencies take the form x1 ... xn -> y1 ... ym, where x1, ..., xn, and y1, ..., yn are type variables with n>0 and m>=0, meaning that the y parameters are uniquely determined by the x parameters. Spaces can be used as separators if more than one variable appears on any single side of a dependency, as in t -> a b. Note that a class may be annotated with multiple dependencies using commas as separators, as in the definition of E above. Some dependencies that we can write in this notation are redundant, and will be rejected because they don't serve any useful purpose, and may instead indicate an error in the program. Examples of dependencies like this include a -> a , a -> a a , a -> , etc. There can also be some redundancy if multiple dependencies are given, as in a->b, b->c , a->c , and in which some subset implies the remaining dependencies. Examples like this are not treated as errors. Note that dependencies appear only in class declarations, and not in any other part of the language. In particular, the syntax for instance declarations, class constraints, and types is completely unchanged. By including dependencies in a class declaration, we provide a mechanism for the programmer to specify each multiple parameter class more precisely. The compiler, on the other hand, is responsible for ensuring that the set of instances that are in scope at any given point in the program is consistent with any declared dependencies. For example, the following pair of instance declarations cannot appear together in the same scope because they violate the dependency for D, even though either one on its own would be acceptable: instance D Bool Int where ... instance D Bool Char where ... Note also that the following declaration is not allowed, even by itself: instance D [a] b where ... The problem here is that this instance would allow one particular choice of [a] to be associated with more than one choice for b, which contradicts the dependency specified in the definition of D. More generally, this means that, in any instance of the form: instance D t s where ... for some particular types t and s, the only variables that can appear in s are the ones that appear in t, and hence, if the type t is known, then s will be uniquely determined. The benefit of including dependency information is that it allows us to define more general multiple parameter classes, without ambiguity problems, and with the benefit of more accurate types. To illustrate this, we return to the collection class example, and annotate the original definition of Collects with a simple dependency: class Collects e ce | ce -> e where empty :: ce insert :: e -> ce -> ce member :: e -> ce -> Bool The dependency ce -> e here specifies that the type e of elements is uniquely determined by the type of the collection ce. Note that both parameters of Collects are of kind *; there are no constructor classes here. Note too that all of the instances of Collects that we gave earlier can be used together with this new definition. What about the ambiguity problems that we encountered with the original definition? The empty function still has type Collects e ce => ce, but it is no longer necessary to regard that as an ambiguous type: Although the variable e does not appear on the right of the => symbol, the dependency for class Collects tells us that it is uniquely determined by ce, which does appear on the right of the => symbol. Hence the context in which empty is used can still give enough information to determine types for both ce and e, without ambiguity. More generally, we need only regard a type as ambiguous if it contains a variable on the left of the => that is not uniquely determined (either directly or indirectly) by the variables on the right. Dependencies also help to produce more accurate types for user defined functions, and hence to provide earlier detection of errors, and less cluttered types for programmers to work with. Recall the previous definition for a function f: f x y = insert x y = insert x . insert y for which we originally obtained a type: f :: (Collects a c, Collects b c) => a -> b -> c -> c Given the dependency information that we have for Collects, however, we can deduce that a and b must be equal because they both appear as the second parameter in a Collects constraint with the same first parameter c. Hence we can infer a shorter and more accurate type for f: f :: (Collects a c) => a -> a -> c -> c In a similar way, the earlier definition of g will now be flagged as a type error. Although we have given only a few examples here, it should be clear that the addition of dependency information can help to make multiple parameter classes more useful in practice, avoiding ambiguity problems, and allowing more general sets of instance declarations. An instance declaration has the form instance ( assertion1, ..., assertionn) => class type1 ... typem where ... The part before the "=>" is the context, while the part after the "=>" is the head of the instance declaration. In Haskell 98 the head of an instance declaration must be of the form C (T a1 ... an), where C is the class, T is a data type constructor, and the a1 ... an are distinct type variables. GHC relaxes these rules in two ways. The -XFlexibleInstances flag allows the head of the instance declaration to mention arbitrary nested types. For example, this becomes a legal instance declaration instance C (Maybe Int) where ... See also the rules on overlap. With the -XTypeSynonymInstances flag, instance heads may use type synonyms. As always, using a type synonym is just shorthand for writing the RHS of the type synonym definition. For example: type Point = (Int,Int) instance C Point where ... instance C [Point] where ... is legal. However, if you added instance C (Int,Int) where ... as well, then the compiler will complain about the overlapping (actually, identical) instance declarations. As always, type synonyms must be fully applied. You cannot, for example, write: type P a = [[a]] instance Monad P where ... In Haskell 98, the assertions in the context of the instance declaration must be of the form C a where a is a type variable that occurs in the head. The -XFlexibleContexts flag relaxes this rule, as well as the corresponding rule for type signatures (see ). With this flag the context of the instance declaration can each consist of arbitrary (well-kinded) assertions (C t1 ... tn) subject only to the following rules: The Paterson Conditions: for each assertion in the context No type variable has more occurrences in the assertion than in the head The assertion has fewer constructors and variables (taken together and counting repetitions) than the head The Coverage Condition. For each functional dependency, tvsleft -> tvsright, of the class, every type variable in S(tvsright) must appear in S(tvsleft), where S is the substitution mapping each type variable in the class declaration to the corresponding type in the instance declaration. These restrictions ensure that context reduction terminates: each reduction step makes the problem smaller by at least one constructor. Both the Paterson Conditions and the Coverage Condition are lifted if you give the -XUndecidableInstances flag (). You can find lots of background material about the reason for these restrictions in the paper Understanding functional dependencies via Constraint Handling Rules. For example, these are OK: instance C Int [a] -- Multiple parameters instance Eq (S [a]) -- Structured type in head -- Repeated type variable in head instance C4 a a => C4 [a] [a] instance Stateful (ST s) (MutVar s) -- Head can consist of type variables only instance C a instance (Eq a, Show b) => C2 a b -- Non-type variables in context instance Show (s a) => Show (Sized s a) instance C2 Int a => C3 Bool [a] instance C2 Int a => C3 [a] b But these are not: -- Context assertion no smaller than head instance C a => C a where ... -- (C b b) has more occurrences of b than the head instance C b b => Foo [b] where ... The same restrictions apply to instances generated by deriving clauses. Thus the following is accepted: data MinHeap h a = H a (h a) deriving (Show) because the derived instance instance (Show a, Show (h a)) => Show (MinHeap h a) conforms to the above rules. A useful idiom permitted by the above rules is as follows. If one allows overlapping instance declarations then it's quite convenient to have a "default instance" declaration that applies if something more specific does not: instance C a where op = ... -- Default Sometimes even the rules of are too onerous. For example, sometimes you might want to use the following to get the effect of a "class synonym": class (C1 a, C2 a, C3 a) => C a where { } instance (C1 a, C2 a, C3 a) => C a where { } This allows you to write shorter signatures: f :: C a => ... instead of f :: (C1 a, C2 a, C3 a) => ... The restrictions on functional dependencies () are particularly troublesome. It is tempting to introduce type variables in the context that do not appear in the head, something that is excluded by the normal rules. For example: class HasConverter a b | a -> b where convert :: a -> b data Foo a = MkFoo a instance (HasConverter a b,Show b) => Show (Foo a) where show (MkFoo value) = show (convert value) This is dangerous territory, however. Here, for example, is a program that would make the typechecker loop: class D a class F a b | a->b instance F [a] [[a]] instance (D c, F a c) => D [a] -- 'c' is not mentioned in the head Similarly, it can be tempting to lift the coverage condition: class Mul a b c | a b -> c where (.*.) :: a -> b -> c instance Mul Int Int Int where (.*.) = (*) instance Mul Int Float Float where x .*. y = fromIntegral x * y instance Mul a b c => Mul a [b] [c] where x .*. v = map (x.*.) v The third instance declaration does not obey the coverage condition; and indeed the (somewhat strange) definition: f = \ b x y -> if b then x .*. [y] else y makes instance inference go into a loop, because it requires the constraint (Mul a [b] b). Nevertheless, GHC allows you to experiment with more liberal rules. If you use the experimental flag -XUndecidableInstances -XUndecidableInstances, both the Paterson Conditions and the Coverage Condition (described in ) are lifted. Termination is ensured by having a fixed-depth recursion stack. If you exceed the stack depth you get a sort of backtrace, and the opportunity to increase the stack depth with -fcontext-stack=N. In general, GHC requires that that it be unambiguous which instance declaration should be used to resolve a type-class constraint. This behaviour can be modified by two flags: -XOverlappingInstances -XOverlappingInstances and -XIncoherentInstances -XIncoherentInstances , as this section discusses. Both these flags are dynamic flags, and can be set on a per-module basis, using an OPTIONS_GHC pragma if desired (). When GHC tries to resolve, say, the constraint C Int Bool, it tries to match every instance declaration against the constraint, by instantiating the head of the instance declaration. For example, consider these declarations: instance context1 => C Int a where ... -- (A) instance context2 => C a Bool where ... -- (B) instance context3 => C Int [a] where ... -- (C) instance context4 => C Int [Int] where ... -- (D) The instances (A) and (B) match the constraint C Int Bool, but (C) and (D) do not. When matching, GHC takes no account of the context of the instance declaration (context1 etc). GHC's default behaviour is that exactly one instance must match the constraint it is trying to resolve. It is fine for there to be a potential of overlap (by including both declarations (A) and (B), say); an error is only reported if a particular constraint matches more than one. The -XOverlappingInstances flag instructs GHC to allow more than one instance to match, provided there is a most specific one. For example, the constraint C Int [Int] matches instances (A), (C) and (D), but the last is more specific, and hence is chosen. If there is no most-specific match, the program is rejected. However, GHC is conservative about committing to an overlapping instance. For example: f :: [b] -> [b] f x = ... Suppose that from the RHS of f we get the constraint C Int [b]. But GHC does not commit to instance (C), because in a particular call of f, b might be instantiate to Int, in which case instance (D) would be more specific still. So GHC rejects the program. (If you add the flag -XIncoherentInstances, GHC will instead pick (C), without complaining about the problem of subsequent instantiations.) Notice that we gave a type signature to f, so GHC had to check that f has the specified type. Suppose instead we do not give a type signature, asking GHC to infer it instead. In this case, GHC will refrain from simplifying the constraint C Int [b] (for the same reason as before) but, rather than rejecting the program, it will infer the type f :: C Int [b] => [b] -> [b] That postpones the question of which instance to pick to the call site for f by which time more is known about the type b. You can write this type signature yourself if you use the -XFlexibleContexts flag. Exactly the same situation can arise in instance declarations themselves. Suppose we have class Foo a where f :: a -> a instance Foo [b] where f x = ... and, as before, the constraint C Int [b] arises from f's right hand side. GHC will reject the instance, complaining as before that it does not know how to resolve the constraint C Int [b], because it matches more than one instance declaration. The solution is to postpone the choice by adding the constraint to the context of the instance declaration, thus: instance C Int [b] => Foo [b] where f x = ... (You need -XFlexibleInstances to do this.) Warning: overlapping instances must be used with care. They can give rise to incoherence (ie different instance choices are made in different parts of the program) even without -XIncoherentInstances. Consider: {-# LANGUAGE OverlappingInstances #-} module Help where class MyShow a where myshow :: a -> String instance MyShow a => MyShow [a] where myshow xs = concatMap myshow xs showHelp :: MyShow a => [a] -> String showHelp xs = myshow xs {-# LANGUAGE FlexibleInstances, OverlappingInstances #-} module Main where import Help data T = MkT instance MyShow T where myshow x = "Used generic instance" instance MyShow [T] where myshow xs = "Used more specific instance" main = do { print (myshow [MkT]); print (showHelp [MkT]) } In function showHelp GHC sees no overlapping instances, and so uses the MyShow [a] instance without complaint. In the call to myshow in main, GHC resolves the MyShow [T] constraint using the overlapping instance declaration in module Main. As a result, the program prints "Used more specific instance" "Used generic instance" (An alternative possible behaviour, not currently implemented, would be to reject module Help on the grounds that a later instance declaration might overlap the local one.) The willingness to be overlapped or incoherent is a property of the instance declaration itself, controlled by the presence or otherwise of the -XOverlappingInstances and -XIncoherentInstances flags when that module is being defined. Specifically, during the lookup process: If the constraint being looked up matches two instance declarations IA and IB, and IB is a substitution instance of IA (but not vice versa); that is, IB is strictly more specific than IA either IA or IB was compiled with -XOverlappingInstances then the less-specific instance IA is ignored. Suppose an instance declaration does not match the constraint being looked up, but does unify with it, so that it might match when the constraint is further instantiated. Usually GHC will regard this as a reason for not committing to some other constraint. But if the instance declaration was compiled with -XIncoherentInstances, GHC will skip the "does-it-unify?" check for that declaration. These rules make it possible for a library author to design a library that relies on overlapping instances without the library client having to know. The -XIncoherentInstances flag implies the -XOverlappingInstances flag, but not vice versa. GHC supports overloaded string literals. Normally a string literal has type String, but with overloaded string literals enabled (with -XOverloadedStrings) a string literal has type (IsString a) => a. This means that the usual string syntax can be used, e.g., for ByteString, Text, and other variations of string like types. String literals behave very much like integer literals, i.e., they can be used in both expressions and patterns. If used in a pattern the literal with be replaced by an equality test, in the same way as an integer literal is. The class IsString is defined as: class IsString a where fromString :: String -> a The only predefined instance is the obvious one to make strings work as usual: instance IsString [Char] where fromString cs = cs The class IsString is not in scope by default. If you want to mention it explicitly (for example, to give an instance declaration for it), you can import it from module GHC.Exts. Haskell's defaulting mechanism is extended to cover string literals, when -XOverloadedStrings is specified. Specifically: Each type in a default declaration must be an instance of Num or of IsString. The standard defaulting rule (Haskell Report, Section 4.3.4) is extended thus: defaulting applies when all the unresolved constraints involve standard classes or IsString; and at least one is a numeric class or IsString. A small example: module Main where import GHC.Exts( IsString(..) ) newtype MyString = MyString String deriving (Eq, Show) instance IsString MyString where fromString = MyString greet :: MyString -> MyString greet "hello" = "world" greet other = other main = do print $ greet "hello" print $ greet "fool" Note that deriving Eq is necessary for the pattern matching to work since it gets translated into an equality comparison. Indexed type families are a new GHC extension to facilitate type-level programming. Type families are a generalisation of associated data types (“Associated Types with Class”, M. Chakravarty, G. Keller, S. Peyton Jones, and S. Marlow. In Proceedings of “The 32nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL'05)”, pages 1-13, ACM Press, 2005) and associated type synonyms (“Type Associated Type Synonyms”. M. Chakravarty, G. Keller, and S. Peyton Jones. In Proceedings of “The Tenth ACM SIGPLAN International Conference on Functional Programming”, ACM Press, pages 241-253, 2005). Type families themselves are described in the paper “Type Checking with Open Type Functions”, T. Schrijvers, S. Peyton-Jones, M. Chakravarty, and M. Sulzmann, in Proceedings of “ICFP 2008: The 13th ACM SIGPLAN International Conference on Functional Programming”, ACM Press, pages 51-62, 2008. Type families essentially provide type-indexed data types and named functions on types, which are useful for generic programming and highly parameterised library interfaces as well as interfaces with enhanced static information, much like dependent types. They might also be regarded as an alternative to functional dependencies, but provide a more functional style of type-level programming than the relational style of functional dependencies. Indexed type families, or type families for short, are type constructors that represent sets of types. Set members are denoted by supplying the type family constructor with type parameters, which are called type indices. The difference between vanilla parametrised type constructors and family constructors is much like between parametrically polymorphic functions and (ad-hoc polymorphic) methods of type classes. Parametric polymorphic functions behave the same at all type instances, whereas class methods can change their behaviour in dependence on the class type parameters. Similarly, vanilla type constructors imply the same data representation for all type instances, but family constructors can have varying representation types for varying type indices. Indexed type families come in two flavours: data families and type synonym families. They are the indexed family variants of algebraic data types and type synonyms, respectively. The instances of data families can be data types and newtypes. Type families are enabled by the flag -XTypeFamilies. Additional information on the use of type families in GHC is available on the Haskell wiki page on type families. Data families appear in two flavours: (1) they can be defined on the toplevel or (2) they can appear inside type classes (in which case they are known as associated types). The former is the more general variant, as it lacks the requirement for the type-indexes to coincide with the class parameters. However, the latter can lead to more clearly structured code and compiler warnings if some type instances were - possibly accidentally - omitted. In the following, we always discuss the general toplevel form first and then cover the additional constraints placed on associated types. Indexed data families are introduced by a signature, such as data family GMap k :: * -> * The special family distinguishes family from standard data declarations. The result kind annotation is optional and, as usual, defaults to * if omitted. An example is data family Array e Named arguments can also be given explicit kind signatures if needed. Just as with [http://www.haskell.org/ghc/docs/latest/html/users_guide/gadt.html GADT declarations] named arguments are entirely optional, so that we can declare Array alternatively with data family Array :: * -> * Instance declarations of data and newtype families are very similar to standard data and newtype declarations. The only two differences are that the keyword data or newtype is followed by instance and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed in type parameters, and type synonyms are allowed as long as they are fully applied and expand to a type that is itself admissible - exactly as this is required for occurrences of type synonyms in class instance parameters. For example, the Either instance for GMap is data instance GMap (Either a b) v = GMapEither (GMap a v) (GMap b v) In this example, the declaration has only one variant. In general, it can be any number. Data and newtype instance declarations are only permitted when an appropriate family declaration is in scope - just as a class instance declaration requires the class declaration to be visible. Moreover, each instance declaration has to conform to the kind determined by its family declaration. This implies that the number of parameters of an instance declaration matches the arity determined by the kind of the family. A data family instance declaration can use the full expressiveness of ordinary data or newtype declarations: Although, a data family is introduced with the keyword "data", a data family instance can use either data or newtype. For example: data family T a data instance T Int = T1 Int | T2 Bool newtype instance T Char = TC Bool A data instance can use GADT syntax for the data constructors, and indeed can define a GADT. For example: data family G a b data instance G [a] b where G1 :: c -> G [Int] b G2 :: G [a] Bool You can use a deriving clause on a data instance or newtype instance declaration. Even if type families are defined as toplevel declarations, functions that perform different computations for different family instances may still need to be defined as methods of type classes. In particular, the following is not possible: data family T a data instance T Int = A data instance T Char = B foo :: T a -> Int foo A = 1 -- WRONG: These two equations together... foo B = 2 -- ...will produce a type error. Instead, you would have to write foo as a class operation, thus: class Foo a where foo :: T a -> Int instance Foo Int where foo A = 1 instance Foo Char where foo B = 2 (Given the functionality provided by GADTs (Generalised Algebraic Data Types), it might seem as if a definition, such as the above, should be feasible. However, type families are - in contrast to GADTs - are open; i.e., new instances can always be added, possibly in other modules. Supporting pattern matching across different data instances would require a form of extensible case construct.) The instance declarations of a data family used in a single program may not overlap at all, independent of whether they are associated or not. In contrast to type class instances, this is not only a matter of consistency, but one of type safety. Type families appear in two flavours: (1) they can be defined on the toplevel or (2) they can appear inside type classes (in which case they are known as associated type synonyms). The former is the more general variant, as it lacks the requirement for the type-indexes to coincide with the class parameters. However, the latter can lead to more clearly structured code and compiler warnings if some type instances were - possibly accidentally - omitted. In the following, we always discuss the general toplevel form first and then cover the additional constraints placed on associated types. Indexed type families are introduced by a signature, such as type family Elem c :: * The special family distinguishes family from standard type declarations. The result kind annotation is optional and, as usual, defaults to * if omitted. An example is type family Elem c Parameters can also be given explicit kind signatures if needed. We call the number of parameters in a type family declaration, the family's arity, and all applications of a type family must be fully saturated w.r.t. to that arity. This requirement is unlike ordinary type synonyms and it implies that the kind of a type family is not sufficient to determine a family's arity, and hence in general, also insufficient to determine whether a type family application is well formed. As an example, consider the following declaration: type family F a b :: * -> * -- F's arity is 2, -- although its overall kind is * -> * -> * -> * Given this declaration the following are examples of well-formed and malformed types: F Char [Int] -- OK! Kind: * -> * F Char [Int] Bool -- OK! Kind: * F IO Bool -- WRONG: kind mismatch in the first argument F Bool -- WRONG: unsaturated application Instance declarations of type families are very similar to standard type synonym declarations. The only two differences are that the keyword type is followed by instance and that some or all of the type arguments can be non-variable types, but may not contain forall types or type synonym families. However, data families are generally allowed, and type synonyms are allowed as long as they are fully applied and expand to a type that is admissible - these are the exact same requirements as for data instances. For example, the [e] instance for Elem is type instance Elem [e] = e Type family instance declarations are only legitimate when an appropriate family declaration is in scope - just like class instances require the class declaration to be visible. Moreover, each instance declaration has to conform to the kind determined by its family declaration, and the number of type parameters in an instance declaration must match the number of type parameters in the family declaration. Finally, the right-hand side of a type instance must be a monotype (i.e., it may not include foralls) and after the expansion of all saturated vanilla type synonyms, no synonyms, except family synonyms may remain. Here are some examples of admissible and illegal type instances: type family F a :: * type instance F [Int] = Int -- OK! type instance F String = Char -- OK! type instance F (F a) = a -- WRONG: type parameter mentions a type family type instance F (forall a. (a, b)) = b -- WRONG: a forall type appears in a type parameter type instance F Float = forall a.a -- WRONG: right-hand side may not be a forall type type family G a b :: * -> * type instance G Int = (,) -- WRONG: must be two type parameters type instance G Int Char Float = Double -- WRONG: must be two type parameters The instance declarations of a type family used in a single program may only overlap if the right-hand sides of the overlapping instances coincide for the overlapping types. More formally, two instance declarations overlap if there is a substitution that makes the left-hand sides of the instances syntactically the same. Whenever that is the case, the right-hand sides of the instances must also be syntactically equal under the same substitution. This condition is independent of whether the type family is associated or not, and it is not only a matter of consistency, but one of type safety. Here are two example to illustrate the condition under which overlap is permitted. type instance F (a, Int) = [a] type instance F (Int, b) = [b] -- overlap permitted type instance G (a, Int) = [a] type instance G (Char, a) = [a] -- ILLEGAL overlap, as [Char] /= [Int] In order to guarantee that type inference in the presence of type families decidable, we need to place a number of additional restrictions on the formation of type instance declarations (c.f., Definition 5 (Relaxed Conditions) of “Type Checking with Open Type Functions”). Instance declarations have the general form type instance F t1 .. tn = t where we require that for every type family application (G s1 .. sm) in t, s1 .. sm do not contain any type family constructors, the total number of symbols (data type constructors and type variables) in s1 .. sm is strictly smaller than in t1 .. tn, and for every type variable a, a occurs in s1 .. sm at most as often as in t1 .. tn. These restrictions are easily verified and ensure termination of type inference. However, they are not sufficient to guarantee completeness of type inference in the presence of, so called, ''loopy equalities'', such as a ~ [F a], where a recursive occurrence of a type variable is underneath a family application and data constructor application - see the above mentioned paper for details. If the option -XUndecidableInstances is passed to the compiler, the above restrictions are not enforced and it is on the programmer to ensure termination of the normalisation of type families during type inference. A data or type synonym family can be declared as part of a type class, thus: class GMapKey k where data GMap k :: * -> * ... class Collects ce where type Elem ce :: * ... When doing so, we drop the "family" keyword. The type parameters must all be type variables, of course, and some (but not necessarily all) of then can be the class parameters. Each class parameter may only be used at most once per associated type, but some may be omitted and they may be in an order other than in the class head. Hence, the following contrived example is admissible: class C a b c where type T c a x :: * Here c and a are class parameters, but the type is also indexed on a third parameter x. When an associated data or type synonym family instance is declared within a type class instance, we drop the instance keyword in the family instance: instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where data GMap (Either a b) v = GMapEither (GMap a v) (GMap b v) ... instance (Eq (Elem [e])) => Collects ([e]) where type Elem [e] = e ... The most important point about associated family instances is that the type indexes corresponding to class parameters must be identical to the type given in the instance head; here this is the first argument of GMap, namely Either a b, which coincides with the only class parameter. Instances for an associated family can only appear as part of instance declarations of the class in which the family was declared - just as with the equations of the methods of a class. Also in correspondence to how methods are handled, declarations of associated types can be omitted in class instances. If an associated family instance is omitted, the corresponding instance type is not inhabited; i.e., only diverging expressions, such as undefined, can assume the type. Although it is unusual, there can be multiple instances for an associated family in a single instance declaration. For example, this is legitimate: instance GMapKey Flob where data GMap Flob [v] = G1 v data GMap Flob Int = G2 Int ... Here we give two data instance declarations, one in which the last parameter is [v], and one for which it is Int. Since you cannot give any subsequent instances for (GMap Flob ...), this facility is most useful when the free indexed parameter is of a kind with a finite number of alternatives (unlike *). It is possible for the class defining the associated type to specify a default for associated type instances. So for example, this is OK: class IsBoolMap v where type Key v type Key v = Int lookupKey :: Key v -> v -> Maybe Bool instance IsBoolMap [(Int, Bool)] where lookupKey = lookup There can also be multiple defaults for a single type, as long as they do not overlap: class C a where type F a b type F a Int = Bool type F a Bool = Int A default declaration is not permitted for an associated data type. The visibility of class parameters in the right-hand side of associated family instances depends solely on the parameters of the family. As an example, consider the simple class declaration class C a b where data T a Only one of the two class parameters is a parameter to the data family. Hence, the following instance declaration is invalid: instance C [c] d where data T [c] = MkT (c, d) -- WRONG!! 'd' is not in scope Here, the right-hand side of the data instance mentions the type variable d that does not occur in its left-hand side. We cannot admit such data instances as they would compromise type safety. The rules for export lists (Haskell Report Section 5.2) needs adjustment for type families: The form T(..), where T is a data family, names the family T and all the in-scope constructors (whether in scope qualified or unqualified) that are data instances of T. The form T(.., ci, .., fj, ..), where T is a data family, names T and the specified constructors ci and fields fj as usual. The constructors and field names must belong to some data instance of T, but are not required to belong to the same instance. The form C(..), where C is a class, names the class C and all its methods and associated types. The form C(.., mi, .., type Tj, ..), where C is a class, names the class C, and the specified methods mi and associated types Tj. The types need a keyword "type" to distinguish them from data constructors. Recall our running GMapKey class example: class GMapKey k where data GMap k :: * -> * insert :: GMap k v -> k -> v -> GMap k v lookup :: GMap k v -> k -> Maybe v empty :: GMap k v instance (GMapKey a, GMapKey b) => GMapKey (Either a b) where data GMap (Either a b) v = GMapEither (GMap a v) (GMap b v) ...method declartions... Here are some export lists and their meaning: module GMap( GMapKey ): Exports just the class name. module GMap( GMapKey(..) ): Exports the class, the associated type GMap and the member functions empty, lookup, and insert. The data constructors of GMap (in this case GMapEither) are not exported. module GMap( GMapKey( type GMap, empty, lookup, insert ) ): Same as the pevious item. Note the "type" keyword. module GMap( GMapKey(..), GMap(..) ): Same as previous item, but also exports all the data constructors for GMap, namely GMapEither. module GMap ( GMapKey( empty, lookup, insert), GMap(..) ): Same as previous item. module GMap ( GMapKey, empty, lookup, insert, GMap(..) ): Same as previous item. Two things to watch out for: You cannot write GMapKey(type GMap(..)) — i.e., sub-component specifications cannot be nested. To specify GMap's data constructors, you have to list it separately. Consider this example: module X where data family D module Y where import X data instance D Int = D1 | D2 Module Y exports all the entities defined in Y, namely the data constructrs D1 and D2, but not the data family D. That (annoyingly) means that you cannot selectively import Y selectively, thus "import Y( D(D1,D2) )", because Y does not export D. Instead you should list the exports explicitly, thus: module Y( D(..) ) where ... or module Y( module Y, D ) where ... Family instances are implicitly exported, just like class instances. However, this applies only to the heads of instances, not to the data constructors an instance defines. Type families require us to extend the rules for the form of instance heads, which are given in . Specifically: Data type families may appear in an instance head Type synonym families may not appear (at all) in an instance head The reason for the latter restriction is that there is no way to check for instance matching. Consider type family F a type instance F Bool = Int class C a instance C Int instance C (F a) Now a constraint (C (F Bool)) would match both instances. The situation is especially bad because the type instance for F Bool might be in another module, or even in a module that is not yet written. However, type class instances of instances of data families can be defined much like any other data type. For example, we can say data instance T Int = T1 Int | T2 Bool instance Eq (T Int) where (T1 i) == (T1 j) = i==j (T2 i) == (T2 j) = i==j _ == _ = False Note that class instances are always for particular instances of a data family and never for an entire family as a whole. This is for essentially the same reasons that we cannot define a toplevel function that performs pattern matching on the data constructors of different instances of a single type family. It would require a form of extensible case construct. Data instance declarations can also have deriving clauses. For example, we can write data GMap () v = GMapUnit (Maybe v) deriving Show which implicitly defines an instance of the form instance Show v => Show (GMap () v) where ... A type context can include equality constraints of the form t1 ~ t2, which denote that the types t1 and t2 need to be the same. In the presence of type families, whether two types are equal cannot generally be decided locally. Hence, the contexts of function signatures may include equality constraints, as in the following example: sumCollects :: (Collects c1, Collects c2, Elem c1 ~ Elem c2) => c1 -> c2 -> c2 where we require that the element type of c1 and c2 are the same. In general, the types t1 and t2 of an equality constraint may be arbitrary monotypes; i.e., they may not contain any quantifiers, independent of whether higher-rank types are otherwise enabled. Equality constraints can also appear in class and instance contexts. The former enable a simple translation of programs using functional dependencies into programs using family synonyms instead. The general idea is to rewrite a class declaration of the form class C a b | a -> b to class (F a ~ b) => C a b where type F a That is, we represent every functional dependency (FD) a1 .. an -> b by an FD type family F a1 .. an and a superclass context equality F a1 .. an ~ b, essentially giving a name to the functional dependency. In class instances, we define the type instances of FD families in accordance with the class head. Method signatures are not affected by that process. Normally, constraints (which appear in types to the left of the => arrow) have a very restricted syntax. They can only be: Class constraints, e.g. Show a Implicit parameter constraints, e.g. ?x::Int (with the -XImplicitParams flag) Equality constraints, e.g. a ~ Int (with the -XTypeFamilies or -XGADTs flag) With the -XConstraintKinds flag, GHC becomes more liberal in what it accepts as constraints in your program. To be precise, with this flag any type of the new kind Constraint can be used as a constraint. The following things have kind Constraint: Anything which is already valid as a constraint without the flag: saturated applications to type classes, implicit parameter and equality constraints. Tuples, all of whose component types have kind Constraint. So for example the type (Show a, Ord a) is of kind Constraint. Anything whose form is not yet know, but the user has declared to have kind Constraint. So for example type Foo (f :: * -> Constraint) = forall b. f b => b -> b is allowed, as well as examples involving type families: type family Typ a b :: Constraint type instance Typ Int b = Show b type instance Typ Bool b = Num b func :: Typ a b => a -> b -> b func = ... Note that because constraints are just handled as types of a particular kind, this extension allows type constraint synonyms: type Stringy a = (Read a, Show a) foo :: Stringy a => a -> (String, String -> a) foo x = (show x, read) Presently, only standard constraints, tuples and type synonyms for those two sorts of constraint are permitted in instance contexts and superclasses (without extra flags). The reason is that permitting more general constraints can cause type checking to loop, as it would with these two programs: type family Clsish u a type instance Clsish () a = Cls a class Clsish () a => Cls a where class OkCls a where type family OkClsish u a type instance OkClsish () a = OkCls a instance OkClsish () a => OkCls a where You may write programs that use exotic sorts of constraints in instance contexts and superclasses, but to do so you must use -XUndecidableInstances to signal that you don't mind if the type checker fails to terminate. Haskell type signatures are implicitly quantified. When the language option -XExplicitForAll is used, the keyword forall allows us to say exactly what this means. For example: g :: b -> b means this: g :: forall b. (b -> b) The two are treated identically. Of course forall becomes a keyword; you can't use forall as a type variable any more! The -XFlexibleContexts flag lifts the Haskell 98 restriction that the type-class constraints in a type signature must have the form (class type-variable) or (class (type-variable type-variable ...)). With -XFlexibleContexts these type signatures are perfectly OK g :: Eq [a] => ... g :: Ord (T a ()) => ... The flag -XFlexibleContexts also lifts the corresponding restriction on class declarations () and instance declarations (). GHC imposes the following restrictions on the constraints in a type signature. Consider the type: forall tv1..tvn (c1, ...,cn) => type (Here, we write the "foralls" explicitly, although the Haskell source language omits them; in Haskell 98, all the free type variables of an explicit source-language type signature are universally quantified, except for the class type variables in a class declaration. However, in GHC, you can give the foralls if you want. See ). Each universally quantified type variable tvi must be reachable from type. A type variable a is "reachable" if it appears in the same constraint as either a type variable free in type, or another reachable type variable. A value with a type that does not obey this reachability restriction cannot be used without introducing ambiguity; that is why the type is rejected. Here, for example, is an illegal type: forall a. Eq a => Int When a value with this type was used, the constraint Eq tv would be introduced where tv is a fresh type variable, and (in the dictionary-translation implementation) the value would be applied to a dictionary for Eq tv. The difficulty is that we can never know which instance of Eq to use because we never get any more information about tv. Note that the reachability condition is weaker than saying that a is functionally dependent on a type variable free in type (see ). The reason for this is there might be a "hidden" dependency, in a superclass perhaps. So "reachable" is a conservative approximation to "functionally dependent". For example, consider: class C a b | a -> b where ... class C a b => D a b where ... f :: forall a b. D a b => a -> a This is fine, because in fact a does functionally determine b but that is not immediately apparent from f's type. Every constraint ci must mention at least one of the universally quantified type variables tvi. For example, this type is OK because C a b mentions the universally quantified type variable b: forall a. C a b => burble The next type is illegal because the constraint Eq b does not mention a: forall a. Eq b => burble The reason for this restriction is milder than the other one. The excluded types are never useful or necessary (because the offending context doesn't need to be witnessed at this point; it can be floated out). Furthermore, floating them out increases sharing. Lastly, excluding them is a conservative choice; it leaves a patch of territory free in case we need it later. Implicit parameters are implemented as described in "Implicit parameters: dynamic scoping with static types", J Lewis, MB Shields, E Meijer, J Launchbury, 27th ACM Symposium on Principles of Programming Languages (POPL'00), Boston, Jan 2000. (Most of the following, still rather incomplete, documentation is due to Jeff Lewis.) Implicit parameter support is enabled with the option -XImplicitParams. A variable is called dynamically bound when it is bound by the calling context of a function and statically bound when bound by the callee's context. In Haskell, all variables are statically bound. Dynamic binding of variables is a notion that goes back to Lisp, but was later discarded in more modern incarnations, such as Scheme. Dynamic binding can be very confusing in an untyped language, and unfortunately, typed languages, in particular Hindley-Milner typed languages like Haskell, only support static scoping of variables. However, by a simple extension to the type class system of Haskell, we can support dynamic binding. Basically, we express the use of a dynamically bound variable as a constraint on the type. These constraints lead to types of the form (?x::t') => t, which says "this function uses a dynamically-bound variable ?x of type t'". For example, the following expresses the type of a sort function, implicitly parameterized by a comparison function named cmp. sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] The dynamic binding constraints are just a new form of predicate in the type class system. An implicit parameter occurs in an expression using the special form ?x, where x is any valid identifier (e.g. ord ?x is a valid expression). Use of this construct also introduces a new dynamic-binding constraint in the type of the expression. For example, the following definition shows how we can define an implicitly parameterized sort function in terms of an explicitly parameterized sortBy function: sortBy :: (a -> a -> Bool) -> [a] -> [a] sort :: (?cmp :: a -> a -> Bool) => [a] -> [a] sort = sortBy ?cmp Dynamic binding constraints behave just like other type class constraints in that they are automatically propagated. Thus, when a function is used, its implicit parameters are inherited by the function that called it. For example, our sort function might be used to pick out the least value in a list: least :: (?cmp :: a -> a -> Bool) => [a] -> a least xs = head (sort xs) Without lifting a finger, the ?cmp parameter is propagated to become a parameter of least as well. With explicit parameters, the default is that parameters must always be explicit propagated. With implicit parameters, the default is to always propagate them. An implicit-parameter type constraint differs from other type class constraints in the following way: All uses of a particular implicit parameter must have the same type. This means that the type of (?x, ?x) is (?x::a) => (a,a), and not (?x::a, ?x::b) => (a, b), as would be the case for type class constraints. You can't have an implicit parameter in the context of a class or instance declaration. For example, both these declarations are illegal: class (?x::Int) => C a where ... instance (?x::a) => Foo [a] where ... Reason: exactly which implicit parameter you pick up depends on exactly where you invoke a function. But the ``invocation'' of instance declarations is done behind the scenes by the compiler, so it's hard to figure out exactly where it is done. Easiest thing is to outlaw the offending types. Implicit-parameter constraints do not cause ambiguity. For example, consider: f :: (?x :: [a]) => Int -> Int f n = n + length ?x g :: (Read a, Show a) => String -> String g s = show (read s) Here, g has an ambiguous type, and is rejected, but f is fine. The binding for ?x at f's call site is quite unambiguous, and fixes the type a. An implicit parameter is bound using the standard let or where binding forms. For example, we define the min function by binding cmp. min :: [a] -> a min = let ?cmp = (<=) in least A group of implicit-parameter bindings may occur anywhere a normal group of Haskell bindings can occur, except at top level. That is, they can occur in a let (including in a list comprehension, or do-notation, or pattern guards), or a where clause. Note the following points: An implicit-parameter binding group must be a collection of simple bindings to implicit-style variables (no function-style bindings, and no type signatures); these bindings are neither polymorphic or recursive. You may not mix implicit-parameter bindings with ordinary bindings in a single let expression; use two nested lets instead. (In the case of where you are stuck, since you can't nest where clauses.) You may put multiple implicit-parameter bindings in a single binding group; but they are not treated as a mutually recursive group (as ordinary let bindings are). Instead they are treated as a non-recursive group, simultaneously binding all the implicit parameter. The bindings are not nested, and may be re-ordered without changing the meaning of the program. For example, consider: f t = let { ?x = t; ?y = ?x+(1::Int) } in ?x + ?y The use of ?x in the binding for ?y does not "see" the binding for ?x, so the type of f is f :: (?x::Int) => Int -> Int Consider these two definitions: len1 :: [a] -> Int len1 xs = let ?acc = 0 in len_acc1 xs len_acc1 [] = ?acc len_acc1 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc1 xs ------------ len2 :: [a] -> Int len2 xs = let ?acc = 0 in len_acc2 xs len_acc2 :: (?acc :: Int) => [a] -> Int len_acc2 [] = ?acc len_acc2 (x:xs) = let ?acc = ?acc + (1::Int) in len_acc2 xs The only difference between the two groups is that in the second group len_acc is given a type signature. In the former case, len_acc1 is monomorphic in its own right-hand side, so the implicit parameter ?acc is not passed to the recursive call. In the latter case, because len_acc2 has a type signature, the recursive call is made to the polymorphic version, which takes ?acc as an implicit parameter. So we get the following results in GHCi: Prog> len1 "hello" 0 Prog> len2 "hello" 5 Adding a type signature dramatically changes the result! This is a rather counter-intuitive phenomenon, worth watching out for. GHC applies the dreaded Monomorphism Restriction (section 4.5.5 of the Haskell Report) to implicit parameters. For example, consider: f :: Int -> Int f v = let ?x = 0 in let y = ?x + v in let ?x = 5 in y Since the binding for y falls under the Monomorphism Restriction it is not generalised, so the type of y is simply Int, not (?x::Int) => Int. Hence, (f 9) returns result 9. If you add a type signature for y, then y will get type (?x::Int) => Int, so the occurrence of y in the body of the let will see the inner binding of ?x, so (f 9) will return 14. Haskell infers the kind of each type variable. Sometimes it is nice to be able to give the kind explicitly as (machine-checked) documentation, just as it is nice to give a type signature for a function. On some occasions, it is essential to do so. For example, in his paper "Restricted Data Types in Haskell" (Haskell Workshop 1999) John Hughes had to define the data type: data Set cxt a = Set [a] | Unused (cxt a -> ()) The only use for the Unused constructor was to force the correct kind for the type variable cxt. GHC now instead allows you to specify the kind of a type variable directly, wherever a type variable is explicitly bound, with the flag -XKindSignatures. This flag enables kind signatures in the following places: data declarations: data Set (cxt :: * -> *) a = Set [a] type declarations: type T (f :: * -> *) = f Int class declarations: class (Eq a) => C (f :: * -> *) a where ... forall's in type signatures: f :: forall (cxt :: * -> *). Set cxt Int The parentheses are required. Some of the spaces are required too, to separate the lexemes. If you write (f::*->*) you will get a parse error, because "::*->*" is a single lexeme in Haskell. As part of the same extension, you can put kind annotations in types as well. Thus: f :: (Int :: *) -> Int g :: forall a. a -> (a :: *) The syntax is atype ::= '(' ctype '::' kind ') The parentheses are required. GHC's type system supports arbitrary-rank explicit universal quantification in types. For example, all the following types are legal: f1 :: forall a b. a -> b -> a g1 :: forall a b. (Ord a, Eq b) => a -> b -> a f2 :: (forall a. a->a) -> Int -> Int g2 :: (forall a. Eq a => [a] -> a -> Bool) -> Int -> Int f3 :: ((forall a. a->a) -> Int) -> Bool -> Bool f4 :: Int -> (forall a. a -> a) Here, f1 and g1 are rank-1 types, and can be written in standard Haskell (e.g. f1 :: a->b->a). The forall makes explicit the universal quantification that is implicitly added by Haskell. The functions f2 and g2 have rank-2 types; the forall is on the left of a function arrow. As g2 shows, the polymorphic type on the left of the function arrow can be overloaded. The function f3 has a rank-3 type; it has rank-2 types on the left of a function arrow. GHC has three flags to control higher-rank types: -XPolymorphicComponents: data constructors (only) can have polymorphic argument types. -XRank2Types: any function (including data constructors) can have a rank-2 type. -XRankNTypes: any function (including data constructors) can have an arbitrary-rank type. That is, you can nest foralls arbitrarily deep in function arrows. In particular, a forall-type (also called a "type scheme"), including an operational type class context, is legal: On the left or right (see f4, for example) of a function arrow As the argument of a constructor, or type of a field, in a data type declaration. For example, any of the f1,f2,f3,g1,g2 above would be valid field type signatures. As the type of an implicit parameter In a pattern type signature (see ) In a data or newtype declaration one can quantify the types of the constructor arguments. Here are several examples: data T a = T1 (forall b. b -> b -> b) a data MonadT m = MkMonad { return :: forall a. a -> m a, bind :: forall a b. m a -> (a -> m b) -> m b } newtype Swizzle = MkSwizzle (Ord a => [a] -> [a]) The constructors have rank-2 types: T1 :: forall a. (forall b. b -> b -> b) -> a -> T a MkMonad :: forall m. (forall a. a -> m a) -> (forall a b. m a -> (a -> m b) -> m b) -> MonadT m MkSwizzle :: (Ord a => [a] -> [a]) -> Swizzle Notice that you don't need to use a forall if there's an explicit context. For example in the first argument of the constructor MkSwizzle, an implicit "forall a." is prefixed to the argument type. The implicit forall quantifies all type variables that are not already in scope, and are mentioned in the type quantified over. As for type signatures, implicit quantification happens for non-overloaded types too. So if you write this: data T a = MkT (Either a b) (b -> b) it's just as if you had written this: data T a = MkT (forall b. Either a b) (forall b. b -> b) That is, since the type variable b isn't in scope, it's implicitly universally quantified. (Arguably, it would be better to require explicit quantification on constructor arguments where that is what is wanted. Feedback welcomed.) You construct values of types T1, MonadT, Swizzle by applying the constructor to suitable values, just as usual. For example, a1 :: T Int a1 = T1 (\xy->x) 3 a2, a3 :: Swizzle a2 = MkSwizzle sort a3 = MkSwizzle reverse a4 :: MonadT Maybe a4 = let r x = Just x b m k = case m of Just y -> k y Nothing -> Nothing in MkMonad r b mkTs :: (forall b. b -> b -> b) -> a -> [T a] mkTs f x y = [T1 f x, T1 f y] The type of the argument can, as usual, be more general than the type required, as (MkSwizzle reverse) shows. (reverse does not need the Ord constraint.) When you use pattern matching, the bound variables may now have polymorphic types. For example: f :: T a -> a -> (a, Char) f (T1 w k) x = (w k x, w 'c' 'd') g :: (Ord a, Ord b) => Swizzle -> [a] -> (a -> b) -> [b] g (MkSwizzle s) xs f = s (map f (s xs)) h :: MonadT m -> [m a] -> m [a] h m [] = return m [] h m (x:xs) = bind m x $ \y -> bind m (h m xs) $ \ys -> return m (y:ys) In the function h we use the record selectors return and bind to extract the polymorphic bind and return functions from the MonadT data structure, rather than using pattern matching. In general, type inference for arbitrary-rank types is undecidable. GHC uses an algorithm proposed by Odersky and Laufer ("Putting type annotations to work", POPL'96) to get a decidable algorithm by requiring some help from the programmer. We do not yet have a formal specification of "some help" but the rule is this: For a lambda-bound or case-bound variable, x, either the programmer provides an explicit polymorphic type for x, or GHC's type inference will assume that x's type has no foralls in it. What does it mean to "provide" an explicit type for x? You can do that by giving a type signature for x directly, using a pattern type signature (), thus: \ f :: (forall a. a->a) -> (f True, f 'c') Alternatively, you can give a type signature to the enclosing context, which GHC can "push down" to find the type for the variable: (\ f -> (f True, f 'c')) :: (forall a. a->a) -> (Bool,Char) Here the type signature on the expression can be pushed inwards to give a type signature for f. Similarly, and more commonly, one can give a type signature for the function itself: h :: (forall a. a->a) -> (Bool,Char) h f = (f True, f 'c') You don't need to give a type signature if the lambda bound variable is a constructor argument. Here is an example we saw earlier: f :: T a -> a -> (a, Char) f (T1 w k) x = (w k x, w 'c' 'd') Here we do not need to give a type signature to w, because it is an argument of constructor T1 and that tells GHC all it needs to know. GHC performs implicit quantification as follows. At the top level (only) of user-written types, if and only if there is no explicit forall, GHC finds all the type variables mentioned in the type that are not already in scope, and universally quantifies them. For example, the following pairs are equivalent: f :: a -> a f :: forall a. a -> a g (x::a) = let h :: a -> b -> b h x y = y in ... g (x::a) = let h :: forall b. a -> b -> b h x y = y in ... Notice that GHC does not find the innermost possible quantification point. For example: f :: (a -> a) -> Int -- MEANS f :: forall a. (a -> a) -> Int -- NOT f :: (forall a. a -> a) -> Int g :: (Ord a => a -> a) -> Int -- MEANS the illegal type g :: forall a. (Ord a => a -> a) -> Int -- NOT g :: (forall a. Ord a => a -> a) -> Int The latter produces an illegal type, which you might think is silly, but at least the rule is simple. If you want the latter type, you can write your for-alls explicitly. Indeed, doing so is strongly advised for rank-2 types. GHC supports impredicative polymorphism, enabled with -XImpredicativeTypes. This means that you can call a polymorphic function at a polymorphic type, and parameterise data structures over polymorphic types. For example: f :: Maybe (forall a. [a] -> [a]) -> Maybe ([Int], [Char]) f (Just g) = Just (g [3], g "hello") f Nothing = Nothing Notice here that the Maybe type is parameterised by the polymorphic type (forall a. [a] -> [a]). The technical details of this extension are described in the paper Boxy types: type inference for higher-rank types and impredicativity, which appeared at ICFP 2006. GHC supports lexically scoped type variables, without which some type signatures are simply impossible to write. For example: f :: forall a. [a] -> [a] f xs = ys ++ ys where ys :: [a] ys = reverse xs The type signature for f brings the type variable a into scope, because of the explicit forall (). The type variables bound by a forall scope over the entire definition of the accompanying value declaration. In this example, the type variable a scopes over the whole definition of f, including over the type signature for ys. In Haskell 98 it is not possible to declare a type for ys; a major benefit of scoped type variables is that it becomes possible to do so. Lexically-scoped type variables are enabled by -XScopedTypeVariables. This flag implies -XRelaxedPolyRec. Note: GHC 6.6 contains substantial changes to the way that scoped type variables work, compared to earlier releases. Read this section carefully! The design follows the following principles A scoped type variable stands for a type variable, and not for a type. (This is a change from GHC's earlier design.) Furthermore, distinct lexical type variables stand for distinct type variables. This means that every programmer-written type signature (including one that contains free scoped type variables) denotes a rigid type; that is, the type is fully known to the type checker, and no inference is involved. Lexical type variables may be alpha-renamed freely, without changing the program. A lexically scoped type variable can be bound by: A declaration type signature () An expression type signature () A pattern type signature () Class and instance declarations () In Haskell, a programmer-written type signature is implicitly quantified over its free type variables (Section 4.1.2 of the Haskell Report). Lexically scoped type variables affect this implicit quantification rules as follows: any type variable that is in scope is not universally quantified. For example, if type variable a is in scope, then (e :: a -> a) means (e :: a -> a) (e :: b -> b) means (e :: forall b. b->b) (e :: a -> b) means (e :: forall b. a->b) A declaration type signature that has explicit quantification (using forall) brings into scope the explicitly-quantified type variables, in the definition of the named function. For example: f :: forall a. [a] -> [a] f (x:xs) = xs ++ [ x :: a ] The "forall a" brings "a" into scope in the definition of "f". This only happens if: The quantification in f's type signature is explicit. For example: g :: [a] -> [a] g (x:xs) = xs ++ [ x :: a ] This program will be rejected, because "a" does not scope over the definition of "g", so "x::a" means "x::forall a. a" by Haskell's usual implicit quantification rules. The signature gives a type for a function binding or a bare variable binding, not a pattern binding. For example: f1 :: forall a. [a] -> [a] f1 (x:xs) = xs ++ [ x :: a ] -- OK f2 :: forall a. [a] -> [a] f2 = \(x:xs) -> xs ++ [ x :: a ] -- OK f3 :: forall a. [a] -> [a] Just f3 = Just (\(x:xs) -> xs ++ [ x :: a ]) -- Not OK! The binding for f3 is a pattern binding, and so its type signature does not bring a into scope. However f1 is a function binding, and f2 binds a bare variable; in both cases the type signature brings a into scope. An expression type signature that has explicit quantification (using forall) brings into scope the explicitly-quantified type variables, in the annotated expression. For example: f = runST ( (op >>= \(x :: STRef s Int) -> g x) :: forall s. ST s Bool ) Here, the type signature forall s. ST s Bool brings the type variable s into scope, in the annotated expression (op >>= \(x :: STRef s Int) -> g x). A type signature may occur in any pattern; this is a pattern type signature. For example: -- f and g assume that 'a' is already in scope f = \(x::Int, y::a) -> x g (x::a) = x h ((x,y) :: (Int,Bool)) = (y,x) In the case where all the type variables in the pattern type signature are already in scope (i.e. bound by the enclosing context), matters are simple: the signature simply constrains the type of the pattern in the obvious way. Unlike expression and declaration type signatures, pattern type signatures are not implicitly generalised. The pattern in a pattern binding may only mention type variables that are already in scope. For example: f :: forall a. [a] -> (Int, [a]) f xs = (n, zs) where (ys::[a], n) = (reverse xs, length xs) -- OK zs::[a] = xs ++ ys -- OK Just (v::b) = ... -- Not OK; b is not in scope Here, the pattern signatures for ys and zs are fine, but the one for v is not because b is not in scope. However, in all patterns other than pattern bindings, a pattern type signature may mention a type variable that is not in scope; in this case, the signature brings that type variable into scope. This is particularly important for existential data constructors. For example: data T = forall a. MkT [a] k :: T -> T k (MkT [t::a]) = MkT t3 where t3::[a] = [t,t,t] Here, the pattern type signature (t::a) mentions a lexical type variable that is not already in scope. Indeed, it cannot already be in scope, because it is bound by the pattern match. GHC's rule is that in this situation (and only then), a pattern type signature can mention a type variable that is not already in scope; the effect is to bring it into scope, standing for the existentially-bound type variable. When a pattern type signature binds a type variable in this way, GHC insists that the type variable is bound to a rigid, or fully-known, type variable. This means that any user-written type signature always stands for a completely known type. If all this seems a little odd, we think so too. But we must have some way to bring such type variables into scope, else we could not name existentially-bound type variables in subsequent type signatures. This is (now) the only situation in which a pattern type signature is allowed to mention a lexical variable that is not already in scope. For example, both f and g would be illegal if a was not already in scope. The type variables in the head of a class or instance declaration scope over the methods defined in the where part. For example: class C a where op :: [a] -> a op xs = let ys::[a] ys = reverse xs in head ys The Haskell Report specifies that a group of bindings (at top level, or in a let or where) should be sorted into strongly-connected components, and then type-checked in dependency order (Haskell Report, Section 4.5.1). As each group is type-checked, any binders of the group that have an explicit type signature are put in the type environment with the specified polymorphic type, and all others are monomorphic until the group is generalised (Haskell Report, Section 4.5.2). Following a suggestion of Mark Jones, in his paper Typing Haskell in Haskell, GHC implements a more general scheme. If -XRelaxedPolyRec is specified: the dependency analysis ignores references to variables that have an explicit type signature. As a result of this refined dependency analysis, the dependency groups are smaller, and more bindings will typecheck. For example, consider: f :: Eq a => a -> Bool f x = (x == x) || g True || g "Yes" g y = (y <= y) || f True This is rejected by Haskell 98, but under Jones's scheme the definition for g is typechecked first, separately from that for f, because the reference to f in g's right hand side is ignored by the dependency analysis. Then g's type is generalised, to get g :: Ord a => a -> Bool Now, the definition for f is typechecked, with this type for g in the type environment. The same refined dependency analysis also allows the type signatures of mutually-recursive functions to have different contexts, something that is illegal in Haskell 98 (Section 4.5.2, last sentence). With -XRelaxedPolyRec GHC only insists that the type signatures of a refined group have identical type signatures; in practice this means that only variables bound by the same pattern binding must have the same context. For example, this is fine: f :: Eq a => a -> Bool f x = (x == x) || g True g :: Ord a => a -> Bool g y = (y <= y) || f True We are actively thinking of simplifying GHC's type system, by not generalising local bindings. The rationale is described in the paper Let should not be generalised. The experimental new behaviour is enabled by the flag -XMonoLocalBinds. The effect is that local (that is, non-top-level) bindings without a type signature are not generalised at all. You can think of it as an extreme (but much more predictable) version of the Monomorphism Restriction. If you supply a type signature, then the flag has no effect. Template Haskell allows you to do compile-time meta-programming in Haskell. The background to the main technical innovations is discussed in " Template Meta-programming for Haskell" (Proc Haskell Workshop 2002). There is a Wiki page about Template Haskell at http://www.haskell.org/haskellwiki/Template_Haskell, and that is the best place to look for further details. You may also consult the online Haskell library reference material (look for module Language.Haskell.TH). Many changes to the original design are described in Notes on Template Haskell version 2. Not all of these changes are in GHC, however. The first example from that paper is set out below () as a worked example to help get you started. The documentation here describes the realisation of Template Haskell in GHC. It is not detailed enough to understand Template Haskell; see the Wiki page. Template Haskell has the following new syntactic constructions. You need to use the flag -XTemplateHaskell -XTemplateHaskell to switch these syntactic extensions on (-XTemplateHaskell is no longer implied by -fglasgow-exts). A splice is written $x, where x is an identifier, or $(...), where the "..." is an arbitrary expression. There must be no space between the "$" and the identifier or parenthesis. This use of "$" overrides its meaning as an infix operator, just as "M.x" overrides the meaning of "." as an infix operator. If you want the infix operator, put spaces around it. A splice can occur in place of an expression; the spliced expression must have type Q Exp an type; the spliced expression must have type Q Typ a list of top-level declarations; the spliced expression must have type Q [Dec] Note that pattern splices are not supported. Inside a splice you can only call functions defined in imported modules, not functions defined elsewhere in the same module. A expression quotation is written in Oxford brackets, thus: [| ... |], or [e| ... |], where the "..." is an expression; the quotation has type Q Exp. [d| ... |], where the "..." is a list of top-level declarations; the quotation has type Q [Dec]. [t| ... |], where the "..." is a type; the quotation has type Q Type. [p| ... |], where the "..." is a pattern; the quotation has type Q Pat. A quasi-quotation can appear in either a pattern context or an expression context and is also written in Oxford brackets: [varid| ... |], where the "..." is an arbitrary string; a full description of the quasi-quotation facility is given in . A name can be quoted with either one or two prefix single quotes: 'f has type Name, and names the function f. Similarly 'C has type Name and names the data constructor C. In general 'thing interprets thing in an expression context. ''T has type Name, and names the type constructor T. That is, ''thing interprets thing in a type context. These Names can be used to construct Template Haskell expressions, patterns, declarations etc. They may also be given as an argument to the reify function. You may omit the $(...) in a top-level declaration splice. Simply writing an expression (rather than a declaration) implies a splice. For example, you can write module Foo where import Bar f x = x $(deriveStuff 'f) -- Uses the $(...) notation g y = y+1 deriveStuff 'g -- Omits the $(...) h z = z-1 This abbreviation makes top-level declaration slices quieter and less intimidating. (Compared to the original paper, there are many differences of detail. The syntax for a declaration splice uses "$" not "splice". The type of the enclosed expression must be Q [Dec], not [Q Dec]. Pattern splices and quotations are not implemented.) The data types and monadic constructor functions for Template Haskell are in the library Language.Haskell.THSyntax. You can only run a function at compile time if it is imported from another module. That is, you can't define a function in a module, and call it from within a splice in the same module. (It would make sense to do so, but it's hard to implement.) You can only run a function at compile time if it is imported from another module that is not part of a mutually-recursive group of modules that includes the module currently being compiled. Furthermore, all of the modules of the mutually-recursive group must be reachable by non-SOURCE imports from the module where the splice is to be run. For example, when compiling module A, you can only run Template Haskell functions imported from B if B does not import A (directly or indirectly). The reason should be clear: to run B we must compile and run A, but we are currently type-checking A. The flag -ddump-splices shows the expansion of all top-level splices as they happen. If you are building GHC from source, you need at least a stage-2 bootstrap compiler to run Template Haskell. A stage-1 compiler will reject the TH constructs. Reason: TH compiles and runs a program, and then looks at the result. So it's important that the program it compiles produces results whose representations are identical to those of the compiler itself. Template Haskell works in any mode (--make, --interactive, or file-at-a-time). There used to be a restriction to the former two, but that restriction has been lifted. To help you get over the confidence barrier, try out this skeletal worked example. First cut and paste the two modules below into "Main.hs" and "Printf.hs": {- Main.hs -} module Main where -- Import our template "pr" import Printf ( pr ) -- The splice operator $ takes the Haskell source code -- generated at compile time by "pr" and splices it into -- the argument of "putStrLn". main = putStrLn ( $(pr "Hello") ) {- Printf.hs -} module Printf where -- Skeletal printf from the paper. -- It needs to be in a separate module to the one where -- you intend to use it. -- Import some Template Haskell syntax import Language.Haskell.TH -- Describe a format string data Format = D | S | L String -- Parse a format string. This is left largely to you -- as we are here interested in building our first ever -- Template Haskell program and not in building printf. parse :: String -> [Format] parse s = [ L s ] -- Generate Haskell source code from a parsed representation -- of the format string. This code will be spliced into -- the module which calls "pr", at compile time. gen :: [Format] -> Q Exp gen [D] = [| \n -> show n |] gen [S] = [| \s -> s |] gen [L s] = stringE s -- Here we generate the Haskell code for the splice -- from an input format string. pr :: String -> Q Exp pr s = gen (parse s) Now run the compiler (here we are a Cygwin prompt on Windows): $ ghc --make -XTemplateHaskell main.hs -o main.exe Run "main.exe" and here is your output: $ ./main Hello profilingwith Template Haskell Template Haskell relies on GHC's built-in bytecode compiler and interpreter to run the splice expressions. The bytecode interpreter runs the compiled expression on top of the same runtime on which GHC itself is running; this means that the compiled code referred to by the interpreted expression must be compatible with this runtime, and in particular this means that object code that is compiled for profiling cannot be loaded and used by a splice expression, because profiled object code is only compatible with the profiling version of the runtime. This causes difficulties if you have a multi-module program containing Template Haskell code and you need to compile it for profiling, because GHC cannot load the profiled object code and use it when executing the splices. Fortunately GHC provides a workaround. The basic idea is to compile the program twice: Compile the program or library first the normal way, without -prof-prof. Then compile it again with -prof, and additionally use -osuf p_o-osuf to name the object files differently (you can choose any suffix that isn't the normal object suffix here). GHC will automatically load the object files built in the first step when executing splice expressions. If you omit the -osuf flag when building with -prof and Template Haskell is used, GHC will emit an error message. Quasi-quotation allows patterns and expressions to be written using programmer-defined concrete syntax; the motivation behind the extension and several examples are documented in "Why It's Nice to be Quoted: Quasiquoting for Haskell" (Proc Haskell Workshop 2007). The example below shows how to write a quasiquoter for a simple expression language. Here are the salient features A quasi-quote has the form [quoter| string |]. The quoter must be the (unqualified) name of an imported quoter; it cannot be an arbitrary expression. The quoter cannot be "e", "t", "d", or "p", since those overlap with Template Haskell quotations. There must be no spaces in the token [quoter|. The quoted string can be arbitrary, and may contain newlines. The quoted string finishes at the first occurrence of the two-character sequence "|]". Absolutely no escaping is performed. If you want to embed that character sequence in the string, you must invent your own escape convention (such as, say, using the string "|~]" instead), and make your quoter function interpret "|~]" as "|]". One way to implement this is to compose your quoter with a pre-processing pass to perform your escape conversion. See the discussion in Trac for details. A quasiquote may appear in place of An expression A pattern A type A top-level declaration (Only the first two are described in the paper.) A quoter is a value of type Language.Haskell.TH.Quote.QuasiQuoter, which is defined thus: data QuasiQuoter = QuasiQuoter { quoteExp :: String -> Q Exp, quotePat :: String -> Q Pat, quoteType :: String -> Q Type, quoteDec :: String -> Q [Dec] } That is, a quoter is a tuple of four parsers, one for each of the contexts in which a quasi-quote can occur. A quasi-quote is expanded by applying the appropriate parser to the string enclosed by the Oxford brackets. The context of the quasi-quote (expression, pattern, type, declaration) determines which of the parsers is called. The example below shows quasi-quotation in action. The quoter expr is bound to a value of type QuasiQuoter defined in module Expr. The example makes use of an antiquoted variable n, indicated by the syntax 'int:n (this syntax for anti-quotation was defined by the parser's author, not by GHC). This binds n to the integer value argument of the constructor IntExpr when pattern matching. Please see the referenced paper for further details regarding anti-quotation as well as the description of a technique that uses SYB to leverage a single parser of type String -> a to generate both an expression parser that returns a value of type Q Exp and a pattern parser that returns a value of type Q Pat. Quasiquoters must obey the same stage restrictions as Template Haskell, e.g., in the example, expr cannot be defined in Main.hs where it is used, but must be imported. {- ------------- file Main.hs --------------- -} module Main where import Expr main :: IO () main = do { print $ eval [expr|1 + 2|] ; case IntExpr 1 of { [expr|'int:n|] -> print n ; _ -> return () } } {- ------------- file Expr.hs --------------- -} module Expr where import qualified Language.Haskell.TH as TH import Language.Haskell.TH.Quote data Expr = IntExpr Integer | AntiIntExpr String | BinopExpr BinOp Expr Expr | AntiExpr String deriving(Show, Typeable, Data) data BinOp = AddOp | SubOp | MulOp | DivOp deriving(Show, Typeable, Data) eval :: Expr -> Integer eval (IntExpr n) = n eval (BinopExpr op x y) = (opToFun op) (eval x) (eval y) where opToFun AddOp = (+) opToFun SubOp = (-) opToFun MulOp = (*) opToFun DivOp = div expr = QuasiQuoter { quoteExp = parseExprExp, quotePat = parseExprPat } -- Parse an Expr, returning its representation as -- either a Q Exp or a Q Pat. See the referenced paper -- for how to use SYB to do this by writing a single -- parser of type String -> Expr instead of two -- separate parsers. parseExprExp :: String -> Q Exp parseExprExp ... parseExprPat :: String -> Q Pat parseExprPat ... Now run the compiler: $ ghc --make -XQuasiQuotes Main.hs -o main Run "main" and here is your output: $ ./main 3 1 Arrows are a generalization of monads introduced by John Hughes. For more details, see “Generalising Monads to Arrows”, John Hughes, in Science of Computer Programming 37, pp67–111, May 2000. The paper that introduced arrows: a friendly introduction, motivated with programming examples. “A New Notation for Arrows”, Ross Paterson, in ICFP, Sep 2001. Introduced the notation described here. “Arrows and Computation”, Ross Paterson, in The Fun of Programming, Palgrave, 2003. “Programming with Arrows”, John Hughes, in 5th International Summer School on Advanced Functional Programming, Lecture Notes in Computer Science vol. 3622, Springer, 2004. This paper includes another introduction to the notation, with practical examples. “Type and Translation Rules for Arrow Notation in GHC”, Ross Paterson and Simon Peyton Jones, September 16, 2004. A terse enumeration of the formal rules used (extracted from comments in the source code). The arrows web page at http://www.haskell.org/arrows/. With the -XArrows flag, GHC supports the arrow notation described in the second of these papers, translating it using combinators from the Control.Arrow module. What follows is a brief introduction to the notation; it won't make much sense unless you've read Hughes's paper. The extension adds a new kind of expression for defining arrows: exp10 ::= ... | proc apat -> cmd where proc is a new keyword. The variables of the pattern are bound in the body of the proc-expression, which is a new sort of thing called a command. The syntax of commands is as follows: cmd ::= exp10 -< exp | exp10 -<< exp | cmd0 with cmd0 up to cmd9 defined using infix operators as for expressions, and cmd10 ::= \ apat ... apat -> cmd | let decls in cmd | if exp then cmd else cmd | case exp of { calts } | do { cstmt ; ... cstmt ; cmd } | fcmd fcmd ::= fcmd aexp | ( cmd ) | (| aexp cmd ... cmd |) cstmt ::= let decls | pat <- cmd | rec { cstmt ; ... cstmt [;] } | cmd where calts are like alts except that the bodies are commands instead of expressions. Commands produce values, but (like monadic computations) may yield more than one value, or none, and may do other things as well. For the most part, familiarity with monadic notation is a good guide to using commands. However the values of expressions, even monadic ones, are determined by the values of the variables they contain; this is not necessarily the case for commands. A simple example of the new notation is the expression proc x -> f -< x+1 We call this a procedure or arrow abstraction. As with a lambda expression, the variable x is a new variable bound within the proc-expression. It refers to the input to the arrow. In the above example, -< is not an identifier but an new reserved symbol used for building commands from an expression of arrow type and an expression to be fed as input to that arrow. (The weird look will make more sense later.) It may be read as analogue of application for arrows. The above example is equivalent to the Haskell expression arr (\ x -> x+1) >>> f That would make no sense if the expression to the left of -< involves the bound variable x. More generally, the expression to the left of -< may not involve any local variable, i.e. a variable bound in the current arrow abstraction. For such a situation there is a variant -<<, as in proc x -> f x -<< x+1 which is equivalent to arr (\ x -> (f x, x+1)) >>> app so in this case the arrow must belong to the ArrowApply class. Such an arrow is equivalent to a monad, so if you're using this form you may find a monadic formulation more convenient. Another form of command is a form of do-notation. For example, you can write proc x -> do y <- f -< x+1 g -< 2*y let z = x+y t <- h -< x*z returnA -< t+z You can read this much like ordinary do-notation, but with commands in place of monadic expressions. The first line sends the value of x+1 as an input to the arrow f, and matches its output against y. In the next line, the output is discarded. The arrow returnA is defined in the Control.Arrow module as arr id. The above example is treated as an abbreviation for arr (\ x -> (x, x)) >>> first (arr (\ x -> x+1) >>> f) >>> arr (\ (y, x) -> (y, (x, y))) >>> first (arr (\ y -> 2*y) >>> g) >>> arr snd >>> arr (\ (x, y) -> let z = x+y in ((x, z), z)) >>> first (arr (\ (x, z) -> x*z) >>> h) >>> arr (\ (t, z) -> t+z) >>> returnA Note that variables not used later in the composition are projected out. After simplification using rewrite rules (see ) defined in the Control.Arrow module, this reduces to arr (\ x -> (x+1, x)) >>> first f >>> arr (\ (y, x) -> (2*y, (x, y))) >>> first g >>> arr (\ (_, (x, y)) -> let z = x+y in (x*z, z)) >>> first h >>> arr (\ (t, z) -> t+z) which is what you might have written by hand. With arrow notation, GHC keeps track of all those tuples of variables for you. Note that although the above translation suggests that let-bound variables like z must be monomorphic, the actual translation produces Core, so polymorphic variables are allowed. It's also possible to have mutually recursive bindings, using the new rec keyword, as in the following example: counter :: ArrowCircuit a => a Bool Int counter = proc reset -> do rec output <- returnA -< if reset then 0 else next next <- delay 0 -< output+1 returnA -< output The translation of such forms uses the loop combinator, so the arrow concerned must belong to the ArrowLoop class. In the previous example, we used a conditional expression to construct the input for an arrow. Sometimes we want to conditionally execute different commands, as in proc (x,y) -> if f x y then g -< x+1 else h -< y+2 which is translated to arr (\ (x,y) -> if f x y then Left x else Right y) >>> (arr (\x -> x+1) >>> f) ||| (arr (\y -> y+2) >>> g) Since the translation uses |||, the arrow concerned must belong to the ArrowChoice class. There are also case commands, like case input of [] -> f -< () [x] -> g -< x+1 x1:x2:xs -> do y <- h -< (x1, x2) ys <- k -< xs returnA -< y:ys The syntax is the same as for case expressions, except that the bodies of the alternatives are commands rather than expressions. The translation is similar to that of if commands. As we're seen, arrow notation provides constructs, modelled on those for expressions, for sequencing, value recursion and conditionals. But suitable combinators, which you can define in ordinary Haskell, may also be used to build new commands out of existing ones. The basic idea is that a command defines an arrow from environments to values. These environments assign values to the free local variables of the command. Thus combinators that produce arrows from arrows may also be used to build commands from commands. For example, the ArrowPlus class includes a combinator ArrowPlus a => (<+>) :: a e c -> a e c -> a e c so we can use it to build commands: expr' = proc x -> do returnA -< x <+> do symbol Plus -< () y <- term -< () expr' -< x + y <+> do symbol Minus -< () y <- term -< () expr' -< x - y (The do on the first line is needed to prevent the first <+> ... from being interpreted as part of the expression on the previous line.) This is equivalent to expr' = (proc x -> returnA -< x) <+> (proc x -> do symbol Plus -< () y <- term -< () expr' -< x + y) <+> (proc x -> do symbol Minus -< () y <- term -< () expr' -< x - y) It is essential that this operator be polymorphic in e (representing the environment input to the command and thence to its subcommands) and satisfy the corresponding naturality property arr k >>> (f <+> g) = (arr k >>> f) <+> (arr k >>> g) at least for strict k. (This should be automatic if you're not using seq.) This ensures that environments seen by the subcommands are environments of the whole command, and also allows the translation to safely trim these environments. The operator must also not use any variable defined within the current arrow abstraction. We could define our own operator untilA :: ArrowChoice a => a e () -> a e Bool -> a e () untilA body cond = proc x -> b <- cond -< x if b then returnA -< () else do body -< x untilA body cond -< x and use it in the same way. Of course this infix syntax only makes sense for binary operators; there is also a more general syntax involving special brackets: proc x -> do y <- f -< x+1 (|untilA (increment -< x+y) (within 0.5 -< x)|) Some operators will need to pass additional inputs to their subcommands. For example, in an arrow type supporting exceptions, the operator that attaches an exception handler will wish to pass the exception that occurred to the handler. Such an operator might have a type handleA :: ... => a e c -> a (e,Ex) c -> a e c where Ex is the type of exceptions handled. You could then use this with arrow notation by writing a command body `handleA` \ ex -> handler so that if an exception is raised in the command body, the variable ex is bound to the value of the exception and the command handler, which typically refers to ex, is entered. Though the syntax here looks like a functional lambda, we are talking about commands, and something different is going on. The input to the arrow represented by a command consists of values for the free local variables in the command, plus a stack of anonymous values. In all the prior examples, this stack was empty. In the second argument to handleA, this stack consists of one value, the value of the exception. The command form of lambda merely gives this value a name. More concretely, the values on the stack are paired to the right of the environment. So operators like handleA that pass extra inputs to their subcommands can be designed for use with the notation by pairing the values with the environment in this way. More precisely, the type of each argument of the operator (and its result) should have the form a (...(e,t1), ... tn) t where e is a polymorphic variable (representing the environment) and ti are the types of the values on the stack, with t1 being the top. The polymorphic variable e must not occur in a, ti or t. However the arrows involved need not be the same. Here are some more examples of suitable operators: bracketA :: ... => a e b -> a (e,b) c -> a (e,c) d -> a e d runReader :: ... => a e c -> a' (e,State) c runState :: ... => a e c -> a' (e,State) (c,State) We can supply the extra input required by commands built with the last two by applying them to ordinary expressions, as in proc x -> do s <- ... (|runReader (do { ... })|) s which adds s to the stack of inputs to the command built using runReader. The command versions of lambda abstraction and application are analogous to the expression versions. In particular, the beta and eta rules describe equivalences of commands. These three features (operators, lambda abstraction and application) are the core of the notation; everything else can be built using them, though the results would be somewhat clumsy. For example, we could simulate do-notation by defining bind :: Arrow a => a e b -> a (e,b) c -> a e c u `bind` f = returnA &&& u >>> f bind_ :: Arrow a => a e b -> a e c -> a e c u `bind_` f = u `bind` (arr fst >>> f) We could simulate if by defining cond :: ArrowChoice a => a e b -> a e b -> a (e,Bool) b cond f g = arr (\ (e,b) -> if b then Left e else Right e) >>> f ||| g Instead of a single form of arrow application (arrow tail) with two translations, the implementation provides two forms -< (first-order) and -<< (higher-order). User-defined operators are flagged with banana brackets instead of a new form keyword. Although only GHC implements arrow notation directly, there is also a preprocessor (available from the arrows web page) that translates arrow notation into Haskell 98 for use with other Haskell systems. You would still want to check arrow programs with GHC; tracing type errors in the preprocessor output is not easy. Modules intended for both GHC and the preprocessor must observe some additional restrictions: The module must import Control.Arrow. The preprocessor cannot cope with other Haskell extensions. These would have to go in separate modules. Because the preprocessor targets Haskell (rather than Core), let-bound variables are monomorphic. GHC supports an extension of pattern matching called bang patterns, written !pat. Bang patterns are under consideration for Haskell Prime. The Haskell prime feature description contains more discussion and examples than the material below. The key change is the addition of a new rule to the semantics of pattern matching in the Haskell 98 report. Add new bullet 10, saying: Matching the pattern !pat against a value v behaves as follows: if v is bottom, the match diverges otherwise, pat is matched against v Bang patterns are enabled by the flag -XBangPatterns. The main idea is to add a single new production to the syntax of patterns: pat ::= !pat Matching an expression e against a pattern !p is done by first evaluating e (to WHNF) and then matching the result against p. Example: f1 !x = True This definition makes f1 is strict in x, whereas without the bang it would be lazy. Bang patterns can be nested of course: f2 (!x, y) = [x,y] Here, f2 is strict in x but not in y. A bang only really has an effect if it precedes a variable or wild-card pattern: f3 !(x,y) = [x,y] f4 (x,y) = [x,y] Here, f3 and f4 are identical; putting a bang before a pattern that forces evaluation anyway does nothing. There is one (apparent) exception to this general rule that a bang only makes a difference when it precedes a variable or wild-card: a bang at the top level of a let or where binding makes the binding strict, regardless of the pattern. (We say "apparent" exception because the Right Way to think of it is that the bang at the top of a binding is not part of the pattern; rather it is part of the syntax of the binding, creating a "bang-pattern binding".) For example: let ![x,y] = e in b is a bang-pattern binding. Operationally, it behaves just like a case expression: case e of [x,y] -> b Like a case expression, a bang-pattern binding must be non-recursive, and is monomorphic. However, nested bangs in a pattern binding behave uniformly with all other forms of pattern matching. For example let (!x,[y]) = e in b is equivalent to this: let { t = case e of (x,[y]) -> x `seq` (x,y) x = fst t y = snd t } in b The binding is lazy, but when either x or y is evaluated by b the entire pattern is matched, including forcing the evaluation of x. Bang patterns work in case expressions too, of course: g5 x = let y = f x in body g6 x = case f x of { y -> body } g7 x = case f x of { !y -> body } The functions g5 and g6 mean exactly the same thing. But g7 evaluates (f x), binds y to the result, and then evaluates body. We add a single new production to the syntax of patterns: pat ::= !pat There is one problem with syntactic ambiguity. Consider: f !x = 3 Is this a definition of the infix function "(!)", or of the "f" with a bang pattern? GHC resolves this ambiguity in favour of the latter. If you want to define (!) with bang-patterns enabled, you have to do so using prefix notation: (!) f x = 3 The semantics of Haskell pattern matching is described in Section 3.17.2 of the Haskell Report. To this description add one extra item 10, saying: Matching the pattern !pat against a value v behaves as follows: if v is bottom, the match diverges otherwise, pat is matched against v Similarly, in Figure 4 of Section 3.17.3, add a new case (t): case v of { !pat -> e; _ -> e' } = v `seq` case v of { pat -> e; _ -> e' } That leaves let expressions, whose translation is given in Section 3.12 of the Haskell Report. In the translation box, first apply the following transformation: for each pattern pi that is of form !qi = ei, transform it to (xi,!qi) = ((),ei), and replace e0 by (xi `seq` e0). Then, when none of the left-hand-side patterns have a bang at the top, apply the rules in the existing box. The effect of the let rule is to force complete matching of the pattern qi before evaluation of the body is begun. The bang is retained in the translated form in case qi is a variable, thus: let !y = f x in b The let-binding can be recursive. However, it is much more common for the let-binding to be non-recursive, in which case the following law holds: (let !p = rhs in body) is equivalent to (case rhs of !p -> body) A pattern with a bang at the outermost level is not allowed at the top level of a module. If you want to make use of assertions in your standard Haskell code, you could define a function like the following: assert :: Bool -> a -> a assert False x = error "assertion failed!" assert _ x = x which works, but gives you back a less than useful error message -- an assertion failed, but which and where? One way out is to define an extended assert function which also takes a descriptive string to include in the error message and perhaps combine this with the use of a pre-processor which inserts the source location where assert was used. Ghc offers a helping hand here, doing all of this for you. For every use of assert in the user's source: kelvinToC :: Double -> Double kelvinToC k = assert (k >= 0.0) (k+273.15) Ghc will rewrite this to also include the source location where the assertion was made, assert pred val ==> assertError "Main.hs|15" pred val The rewrite is only performed by the compiler when it spots applications of Control.Exception.assert, so you can still define and use your own versions of assert, should you so wish. If not, import Control.Exception to make use assert in your code. GHC ignores assertions when optimisation is turned on with the -O-O flag. That is, expressions of the form assert pred e will be rewritten to e. You can also disable assertions using the -fignore-asserts option-fignore-asserts . Assertion failures can be caught, see the documentation for the Control.Exception library for the details. pragma GHC supports several pragmas, or instructions to the compiler placed in the source code. Pragmas don't normally affect the meaning of the program, but they might affect the efficiency of the generated code. Pragmas all take the form {-# word ... #-} where word indicates the type of pragma, and is followed optionally by information specific to that type of pragma. Case is ignored in word. The various values for word that GHC understands are described in the following sections; any pragma encountered with an unrecognised word is ignored. The layout rule applies in pragmas, so the closing #-} should start in a column to the right of the opening {-#. Certain pragmas are file-header pragmas: A file-header pragma must precede the module keyword in the file. There can be as many file-header pragmas as you please, and they can be preceded or followed by comments. File-header pragmas are read once only, before pre-processing the file (e.g. with cpp). The file-header pragmas are: {-# LANGUAGE #-}, {-# OPTIONS_GHC #-}, and {-# INCLUDE #-}. LANGUAGEpragma pragmaLANGUAGE The LANGUAGE pragma allows language extensions to be enabled in a portable way. It is the intention that all Haskell compilers support the LANGUAGE pragma with the same syntax, although not all extensions are supported by all compilers, of course. The LANGUAGE pragma should be used instead of OPTIONS_GHC, if possible. For example, to enable the FFI and preprocessing with CPP: {-# LANGUAGE ForeignFunctionInterface, CPP #-} LANGUAGE is a file-header pragma (see ). Every language extension can also be turned into a command-line flag by prefixing it with "-X"; for example -XForeignFunctionInterface. (Similarly, all "-X" flags can be written as LANGUAGE pragmas. A list of all supported language extensions can be obtained by invoking ghc --supported-extensions (see ). Any extension from the Extension type defined in Language.Haskell.Extension may be used. GHC will report an error if any of the requested extensions are not supported. OPTIONS_GHC pragmaOPTIONS_GHC The OPTIONS_GHC pragma is used to specify additional options that are given to the compiler when compiling this source file. See for details. Previous versions of GHC accepted OPTIONS rather than OPTIONS_GHC, but that is now deprecated. OPTIONS_GHC is a file-header pragma (see ). The INCLUDE used to be necessary for specifying header files to be included when using the FFI and compiling via C. It is no longer required for GHC, but is accepted (and ignored) for compatibility with other compilers. WARNING DEPRECATED The WARNING pragma allows you to attach an arbitrary warning to a particular function, class, or type. A DEPRECATED pragma lets you specify that a particular function, class, or type is deprecated. There are two ways of using these pragmas. You can work on an entire module thus: module Wibble {-# DEPRECATED "Use Wobble instead" #-} where ... Or: module Wibble {-# WARNING "This is an unstable interface." #-} where ... When you compile any module that import Wibble, GHC will print the specified message. You can attach a warning to a function, class, type, or data constructor, with the following top-level declarations: {-# DEPRECATED f, C, T "Don't use these" #-} {-# WARNING unsafePerformIO "This is unsafe; I hope you know what you're doing" #-} When you compile any module that imports and uses any of the specified entities, GHC will print the specified message. You can only attach to entities declared at top level in the module being compiled, and you can only use unqualified names in the list of entities. A capitalised name, such as T refers to either the type constructor T or the data constructor T, or both if both are in scope. If both are in scope, there is currently no way to specify one without the other (c.f. fixities ). Warnings and deprecations are not reported for (a) uses within the defining module, and (b) uses in an export list. The latter reduces spurious complaints within a library in which one module gathers together and re-exports the exports of several others. You can suppress the warnings with the flag -fno-warn-warnings-deprecations. These pragmas control the inlining of function definitions. INLINE GHC (with -O, as always) tries to inline (or “unfold”) functions/values that are “small enough,” thus avoiding the call overhead and possibly exposing other more-wonderful optimisations. Normally, if GHC decides a function is “too expensive” to inline, it will not do so, nor will it export that unfolding for other modules to use. The sledgehammer you can bring to bear is the INLINEINLINE pragma pragma, used thusly: key_function :: Int -> String -> (Bool, Double) {-# INLINE key_function #-} The major effect of an INLINE pragma is to declare a function's “cost” to be very low. The normal unfolding machinery will then be very keen to inline it. However, an INLINE pragma for a function "f" has a number of other effects: While GHC is keen to inline the function, it does not do so blindly. For example, if you write map key_function xs there really isn't any point in inlining key_function to get map (\x -> body) xs In general, GHC only inlines the function if there is some reason (no matter how slight) to suppose that it is useful to do so. Moreover, GHC will only inline the function if it is fully applied, where "fully applied" means applied to as many arguments as appear (syntactically) on the LHS of the function definition. For example: comp1 :: (b -> c) -> (a -> b) -> a -> c {-# INLINE comp1 #-} comp1 f g = \x -> f (g x) comp2 :: (b -> c) -> (a -> b) -> a -> c {-# INLINE comp2 #-} comp2 f g x = f (g x) The two functions comp1 and comp2 have the same semantics, but comp1 will be inlined when applied to two arguments, while comp2 requires three. This might make a big difference if you say map (not `comp1` not) xs which will optimise better than the corresponding use of `comp2`. It is useful for GHC to optimise the definition of an INLINE function f just like any other non-INLINE function, in case the non-inlined version of f is ultimately called. But we don't want to inline the optimised version of f; a major reason for INLINE pragmas is to expose functions in f's RHS that have rewrite rules, and it's no good if those functions have been optimised away. So GHC guarantees to inline precisely the code that you wrote, no more and no less. It does this by capturing a copy of the definition of the function to use for inlining (we call this the "inline-RHS"), which it leaves untouched, while optimising the ordinarily RHS as usual. For externally-visible functions the inline-RHS (not the optimised RHS) is recorded in the interface file. An INLINE function is not worker/wrappered by strictness analysis. It's going to be inlined wholesale instead. GHC ensures that inlining cannot go on forever: every mutually-recursive group is cut by one or more loop breakers that is never inlined (see Secrets of the GHC inliner, JFP 12(4) July 2002). GHC tries not to select a function with an INLINE pragma as a loop breaker, but when there is no choice even an INLINE function can be selected, in which case the INLINE pragma is ignored. For example, for a self-recursive function, the loop breaker can only be the function itself, so an INLINE pragma is always ignored. Syntactically, an INLINE pragma for a function can be put anywhere its type signature could be put. INLINE pragmas are a particularly good idea for the then/return (or bind/unit) functions in a monad. For example, in GHC's own UniqueSupply monad code, we have: {-# INLINE thenUs #-} {-# INLINE returnUs #-} See also the NOINLINE () and INLINABLE () pragmas. Note: the HBC compiler doesn't like INLINE pragmas, so if you want your code to be HBC-compatible you'll have to surround the pragma with C pre-processor directives #ifdef __GLASGOW_HASKELL__...#endif. An {-# INLINABLE f #-} pragma on a function f has the following behaviour: While INLINE says "please inline me", the INLINABLE says "feel free to inline me; use your discretion". In other words the choice is left to GHC, which uses the same rules as for pragma-free functions. Unlike INLINE, that decision is made at the call site, and will therefore be affected by the inlining threshold, optimisation level etc. Like INLINE, the INLINABLE pragma retains a copy of the original RHS for inlining purposes, and persists it in the interface file, regardless of the size of the RHS. One way to use INLINABLE is in conjunction with the special function inline (). The call inline f tries very hard to inline f. To make sure that f can be inlined, it is a good idea to mark the definition of f as INLINABLE, so that GHC guarantees to expose an unfolding regardless of how big it is. Moreover, by annotating f as INLINABLE, you ensure that f's original RHS is inlined, rather than whatever random optimised version of f GHC's optimiser has produced. The INLINABLE pragma also works with SPECIALISE: if you mark function f as INLINABLE, then you can subsequently SPECIALISE in another module (see ). Unlike INLINE, it is OK to use an INLINABLE pragma on a recursive function. The principal reason do to so to allow later use of SPECIALISE NOINLINE NOTINLINE The NOINLINE pragma does exactly what you'd expect: it stops the named function from being inlined by the compiler. You shouldn't ever need to do this, unless you're very cautious about code size. NOTINLINE is a synonym for NOINLINE (NOINLINE is specified by Haskell 98 as the standard way to disable inlining, so it should be used if you want your code to be portable). CONLIKE An INLINE or NOINLINE pragma may have a CONLIKE modifier, which affects matching in RULEs (only). See . Sometimes you want to control exactly when in GHC's pipeline the INLINE pragma is switched on. Inlining happens only during runs of the simplifier. Each run of the simplifier has a different phase number; the phase number decreases towards zero. If you use -dverbose-core2core you'll see the sequence of phase numbers for successive runs of the simplifier. In an INLINE pragma you can optionally specify a phase number, thus: "INLINE[k] f" means: do not inline f until phase k, but from phase k onwards be very keen to inline it. "INLINE[~k] f" means: be very keen to inline f until phase k, but from phase k onwards do not inline it. "NOINLINE[k] f" means: do not inline f until phase k, but from phase k onwards be willing to inline it (as if there was no pragma). "NOINLINE[~k] f" means: be willing to inline f until phase k, but from phase k onwards do not inline it. The same information is summarised here: -- Before phase 2 Phase 2 and later {-# INLINE [2] f #-} -- No Yes {-# INLINE [~2] f #-} -- Yes No {-# NOINLINE [2] f #-} -- No Maybe {-# NOINLINE [~2] f #-} -- Maybe No {-# INLINE f #-} -- Yes Yes {-# NOINLINE f #-} -- No No By "Maybe" we mean that the usual heuristic inlining rules apply (if the function body is small, or it is applied to interesting-looking arguments etc). Another way to understand the semantics is this: For both INLINE and NOINLINE, the phase number says when inlining is allowed at all. The INLINE pragma has the additional effect of making the function body look small, so that when inlining is allowed it is very likely to happen. The same phase-numbering control is available for RULES (). LINEpragma pragmaLINE This pragma is similar to C's #line pragma, and is mainly for use in automatically generated Haskell code. It lets you specify the line number and filename of the original code; for example {-# LINE 42 "Foo.vhs" #-} if you'd generated the current file from something called Foo.vhs and this line corresponds to line 42 in the original. GHC will adjust its error messages to refer to the line/file named in the LINE pragma. The RULES pragma lets you specify rewrite rules. It is described in . SPECIALIZE pragma pragma, SPECIALIZE overloading, death to (UK spelling also accepted.) For key overloaded functions, you can create extra versions (NB: more code space) specialised to particular types. Thus, if you have an overloaded function: hammeredLookup :: Ord key => [(key, value)] -> key -> value If it is heavily used on lists with Widget keys, you could specialise it as follows: {-# SPECIALIZE hammeredLookup :: [(Widget, value)] -> Widget -> value #-} A SPECIALIZE pragma for a function can be put anywhere its type signature could be put. Moreover, you can also SPECIALIZE an imported function provided it was given an INLINABLE pragma at its definition site (). A SPECIALIZE has the effect of generating (a) a specialised version of the function and (b) a rewrite rule (see ) that rewrites a call to the un-specialised function into a call to the specialised one. Moreover, given a SPECIALIZE pragma for a function f, GHC will automatically create specialisations for any type-class-overloaded functions called by f, if they are in the same module as the SPECIALIZE pragma, or if they are INLINABLE; and so on, transitively. The type in a SPECIALIZE pragma can be any type that is less polymorphic than the type of the original function. In concrete terms, if the original function is f then the pragma {-# SPECIALIZE f :: <type> #-} is valid if and only if the definition f_spec :: <type> f_spec = f is valid. Here are some examples (where we only give the type signature for the original function, not its code): f :: Eq a => a -> b -> b {-# SPECIALISE f :: Int -> b -> b #-} g :: (Eq a, Ix b) => a -> b -> b {-# SPECIALISE g :: (Eq a) => a -> Int -> Int #-} h :: Eq a => a -> a -> a {-# SPECIALISE h :: (Eq a) => [a] -> [a] -> [a] #-} The last of these examples will generate a RULE with a somewhat-complex left-hand side (try it yourself), so it might not fire very well. If you use this kind of specialisation, let us know how well it works. A SPECIALIZE pragma can optionally be followed with a INLINE or NOINLINE pragma, optionally followed by a phase, as described in . The INLINE pragma affects the specialised version of the function (only), and applies even if the function is recursive. The motivating example is this: -- A GADT for arrays with type-indexed representation data Arr e where ArrInt :: !Int -> ByteArray# -> Arr Int ArrPair :: !Int -> Arr e1 -> Arr e2 -> Arr (e1, e2) (!:) :: Arr e -> Int -> e {-# SPECIALISE INLINE (!:) :: Arr Int -> Int -> Int #-} {-# SPECIALISE INLINE (!:) :: Arr (a, b) -> Int -> (a, b) #-} (ArrInt _ ba) !: (I# i) = I# (indexIntArray# ba i) (ArrPair _ a1 a2) !: i = (a1 !: i, a2 !: i) Here, (!:) is a recursive function that indexes arrays of type Arr e. Consider a call to (!:) at type (Int,Int). The second specialisation will fire, and the specialised function will be inlined. It has two calls to (!:), both at type Int. Both these calls fire the first specialisation, whose body is also inlined. The result is a type-based unrolling of the indexing function. Warning: you can make GHC diverge by using SPECIALISE INLINE on an ordinarily-recursive function. Generally, you can only give a SPECIALIZE pragma for a function defined in the same module. However if a function f is given an INLINABLE pragma at its definition site, then it can subsequently be specialised by importing modules (see ). For example module Map( lookup, blah blah ) where lookup :: Ord key => [(key,a)] -> key -> Maybe a lookup = ... {-# INLINABLE lookup #-} module Client where import Map( lookup ) data T = T1 | T2 deriving( Eq, Ord ) {-# SPECIALISE lookup :: [(T,a)] -> T -> Maybe a Here, lookup is declared INLINABLE, but it cannot be specialised for type T at its definition site, because that type does not exist yet. Instead a client module can define T and then specialise lookup at that type. Moreover, every module that imports Client (or imports a module that imports Client, transitively) will "see", and make use of, the specialised version of lookup. You don't need to put a SPECIALIZE pragma in every module. Moreover you often don't even need the SPECIALIZE pragma in the first place. When compiling a module M, GHC's optimiser (with -O) automatically considers each top-level overloaded function declared in M, and specialises it for the different types at which it is called in M. The optimiser also considers each imported INLINABLE overloaded function, and specialises it for the different types at which it is called in M. So in our example, it would be enough for lookup to be called at type T: module Client where import Map( lookup ) data T = T1 | T2 deriving( Eq, Ord ) findT1 :: [(T,a)] -> Maybe a findT1 m = lookup m T1 -- A call of lookup at type T However, sometimes there are no such calls, in which case the pragma can be useful. Note: In earlier versions of GHC, it was possible to provide your own specialised function for a given type: {-# SPECIALIZE hammeredLookup :: [(Int, value)] -> Int -> value = intLookup #-} This feature has been removed, as it is now subsumed by the RULES pragma (see ). SPECIALIZE pragma overloading, death to Same idea, except for instance declarations. For example: instance (Eq a) => Eq (Foo a) where { {-# SPECIALIZE instance Eq (Foo [(Int, Bar)]) #-} ... usual stuff ... } The pragma must occur inside the where part of the instance declaration. Compatible with HBC, by the way, except perhaps in the placement of the pragma. UNPACK The UNPACK indicates to the compiler that it should unpack the contents of a constructor field into the constructor itself, removing a level of indirection. For example: data T = T {-# UNPACK #-} !Float {-# UNPACK #-} !Float will create a constructor T containing two unboxed floats. This may not always be an optimisation: if the T constructor is scrutinised and the floats passed to a non-strict function for example, they will have to be reboxed (this is done automatically by the compiler). Unpacking constructor fields should only be used in conjunction with -Oin fact, UNPACK has no effect without -O, for technical reasons (see tick 5252), in order to expose unfoldings to the compiler so the reboxing can be removed as often as possible. For example: f :: T -> Float f (T f1 f2) = f1 + f2 The compiler will avoid reboxing f1 and f2 by inlining + on floats, but only when -O is on. Any single-constructor data is eligible for unpacking; for example data T = T {-# UNPACK #-} !(Int,Int) will store the two Ints directly in the T constructor, by flattening the pair. Multi-level unpacking is also supported: data T = T {-# UNPACK #-} !S data S = S {-# UNPACK #-} !Int {-# UNPACK #-} !Int will store two unboxed Int#s directly in the T constructor. The unpacker can see through newtypes, too. See also the -funbox-strict-fields flag, which essentially has the effect of adding {-# UNPACK #-} to every strict constructor field. NOUNPACK The NOUNPACK pragma indicates to the compiler that it should not unpack the contents of a constructor field. Example: data T = T {-# NOUNPACK #-} !(Int,Int) Even with the flags -funbox-strict-fields and -O, the field of the constructor T is not unpacked. SOURCE The {-# SOURCE #-} pragma is used only in import declarations, to break a module loop. It is described in detail in . The programmer can specify rewrite rules as part of the source program (in a pragma). Here is an example: {-# RULES "map/map" forall f g xs. map f (map g xs) = map (f.g) xs #-} Use the debug flag -ddump-simpl-stats to see what rules fired. If you need more information, then -ddump-rule-firings shows you each individual rule firing and -ddump-rule-rewrites also shows what the code looks like before and after the rewrite. From a syntactic point of view: There may be zero or more rules in a RULES pragma, separated by semicolons (which may be generated by the layout rule). The layout rule applies in a pragma. Currently no new indentation level is set, so if you put several rules in single RULES pragma and wish to use layout to separate them, you must lay out the starting in the same column as the enclosing definitions. {-# RULES "map/map" forall f g xs. map f (map g xs) = map (f.g) xs "map/append" forall f xs ys. map f (xs ++ ys) = map f xs ++ map f ys #-} Furthermore, the closing #-} should start in a column to the right of the opening {-#. Each rule has a name, enclosed in double quotes. The name itself has no significance at all. It is only used when reporting how many times the rule fired. A rule may optionally have a phase-control number (see ), immediately after the name of the rule. Thus: {-# RULES "map/map" [2] forall f g xs. map f (map g xs) = map (f.g) xs #-} The "[2]" means that the rule is active in Phase 2 and subsequent phases. The inverse notation "[~2]" is also accepted, meaning that the rule is active up to, but not including, Phase 2. Each variable mentioned in a rule must either be in scope (e.g. map), or bound by the forall (e.g. f, g, xs). The variables bound by the forall are called the pattern variables. They are separated by spaces, just like in a type forall. A pattern variable may optionally have a type signature. If the type of the pattern variable is polymorphic, it must have a type signature. For example, here is the foldr/build rule: "fold/build" forall k z (g::forall b. (a->b->b) -> b -> b) . foldr k z (build g) = g k z Since g has a polymorphic type, it must have a type signature. The left hand side of a rule must consist of a top-level variable applied to arbitrary expressions. For example, this is not OK: "wrong1" forall e1 e2. case True of { True -> e1; False -> e2 } = e1 "wrong2" forall f. f True = True In "wrong1", the LHS is not an application; in "wrong2", the LHS has a pattern variable in the head. A rule does not need to be in the same module as (any of) the variables it mentions, though of course they need to be in scope. All rules are implicitly exported from the module, and are therefore in force in any module that imports the module that defined the rule, directly or indirectly. (That is, if A imports B, which imports C, then C's rules are in force when compiling A.) The situation is very similar to that for instance declarations. Inside a RULE "forall" is treated as a keyword, regardless of any other flag settings. Furthermore, inside a RULE, the language extension -XScopedTypeVariables is automatically enabled; see . Like other pragmas, RULE pragmas are always checked for scope errors, and are typechecked. Typechecking means that the LHS and RHS of a rule are typechecked, and must have the same type. However, rules are only enabled if the -fenable-rewrite-rules flag is on (see ). From a semantic point of view: Rules are enabled (that is, used during optimisation) by the -fenable-rewrite-rules flag. This flag is implied by -O, and may be switched off (as usual) by -fno-enable-rewrite-rules. (NB: enabling -fenable-rewrite-rules without -O may not do what you expect, though, because without -O GHC ignores all optimisation information in interface files; see -fignore-interface-pragmas, .) Note that -fenable-rewrite-rules is an optimisation flag, and has no effect on parsing or typechecking. Rules are regarded as left-to-right rewrite rules. When GHC finds an expression that is a substitution instance of the LHS of a rule, it replaces the expression by the (appropriately-substituted) RHS. By "a substitution instance" we mean that the LHS can be made equal to the expression by substituting for the pattern variables. GHC makes absolutely no attempt to verify that the LHS and RHS of a rule have the same meaning. That is undecidable in general, and infeasible in most interesting cases. The responsibility is entirely the programmer's! GHC makes no attempt to make sure that the rules are confluent or terminating. For example: "loop" forall x y. f x y = f y x This rule will cause the compiler to go into an infinite loop. If more than one rule matches a call, GHC will choose one arbitrarily to apply. GHC currently uses a very simple, syntactic, matching algorithm for matching a rule LHS with an expression. It seeks a substitution which makes the LHS and expression syntactically equal modulo alpha conversion. The pattern (rule), but not the expression, is eta-expanded if necessary. (Eta-expanding the expression can lead to laziness bugs.) But not beta conversion (that's called higher-order matching). Matching is carried out on GHC's intermediate language, which includes type abstractions and applications. So a rule only matches if the types match too. See below. GHC keeps trying to apply the rules as it optimises the program. For example, consider: let s = map f t = map g in s (t xs) The expression s (t xs) does not match the rule "map/map", but GHC will substitute for s and t, giving an expression which does match. If s or t was (a) used more than once, and (b) large or a redex, then it would not be substituted, and the rule would not fire. Ordinary inlining happens at the same time as rule rewriting, which may lead to unexpected results. Consider this (artificial) example f x = x g y = f y h z = g True {-# RULES "f" f True = False #-} Since f's right-hand side is small, it is inlined into g, to give g y = y Now g is inlined into h, but f's RULE has no chance to fire. If instead GHC had first inlined g into h then there would have been a better chance that f's RULE might fire. The way to get predictable behaviour is to use a NOINLINE pragma, or an INLINE[phase] pragma, on f, to ensure that it is not inlined until its RULEs have had a chance to fire. GHC is very cautious about duplicating work. For example, consider f k z xs = let xs = build g in ...(foldr k z xs)...sum xs... {-# RULES "foldr/build" forall k z g. foldr k z (build g) = g k z #-} Since xs is used twice, GHC does not fire the foldr/build rule. Rightly so, because it might take a lot of work to compute xs, which would be duplicated if the rule fired. Sometimes, however, this approach is over-cautious, and we do want the rule to fire, even though doing so would duplicate redex. There is no way that GHC can work out when this is a good idea, so we provide the CONLIKE pragma to declare it, thus: {-# INLINE[1] CONLIKE f #-} f x = blah CONLIKE is a modifier to an INLINE or NOINLINE pragma. It specifies that an application of f to one argument (in general, the number of arguments to the left of the '=' sign) should be considered cheap enough to duplicate, if such a duplication would make rule fire. (The name "CONLIKE" is short for "constructor-like", because constructors certainly have such a property.) The CONLIKE pragma is a modifier to INLINE/NOINLINE because it really only makes sense to match f on the LHS of a rule if you are sure that f is not going to be inlined before the rule has a chance to fire. The RULES mechanism is used to implement fusion (deforestation) of common list functions. If a "good consumer" consumes an intermediate list constructed by a "good producer", the intermediate list should be eliminated entirely. The following are good producers: List comprehensions Enumerations of Int and Char (e.g. ['a'..'z']). Explicit lists (e.g. [True, False]) The cons constructor (e.g 3:4:[]) ++ map take, filter iterate, repeat zip, zipWith The following are good consumers: List comprehensions array (on its second argument) ++ (on its first argument) foldr map take, filter concat unzip, unzip2, unzip3, unzip4 zip, zipWith (but on one argument only; if both are good producers, zip will fuse with one but not the other) partition head and, or, any, all sequence_ msum sortBy So, for example, the following should generate no intermediate lists: array (1,10) [(i,i*i) | i <- map (+ 1) [0..9]] This list could readily be extended; if there are Prelude functions that you use a lot which are not included, please tell us. If you want to write your own good consumers or producers, look at the Prelude definitions of the above functions to see how to do so. Rewrite rules can be used to get the same effect as a feature present in earlier versions of GHC. For example, suppose that: genericLookup :: Ord a => Table a b -> a -> b intLookup :: Table Int b -> Int -> b where intLookup is an implementation of genericLookup that works very fast for keys of type Int. You might wish to tell GHC to use intLookup instead of genericLookup whenever the latter was called with type Table Int b -> Int -> b. It used to be possible to write {-# SPECIALIZE genericLookup :: Table Int b -> Int -> b = intLookup #-} This feature is no longer in GHC, but rewrite rules let you do the same thing: {-# RULES "genericLookup/Int" genericLookup = intLookup #-} This slightly odd-looking rule instructs GHC to replace genericLookup by intLookup whenever the types match. What is more, this rule does not need to be in the same file as genericLookup, unlike the SPECIALIZE pragmas which currently do (so that they have an original definition available to specialise). It is Your Responsibility to make sure that intLookup really behaves as a specialised version of genericLookup!!! An example in which using RULES for specialisation will Win Big: toDouble :: Real a => a -> Double toDouble = fromRational . toRational {-# RULES "toDouble/Int" toDouble = i2d #-} i2d (I# i) = D# (int2Double# i) -- uses Glasgow prim-op directly The i2d function is virtually one machine instruction; the default conversion—via an intermediate Rational—is obscenely expensive by comparison. Use -ddump-rules to see the rules that are defined in this module. This includes rules generated by the specialisation pass, but excludes rules imported from other modules. Use -ddump-simpl-stats to see what rules are being fired. If you add -dppr-debug you get a more detailed listing. Use -ddump-rule-firings or -ddump-rule-rewrites to see in great detail what rules are being fired. If you add -dppr-debug you get a still more detailed listing. The definition of (say) build in GHC/Base.lhs looks like this: build :: forall a. (forall b. (a -> b -> b) -> b -> b) -> [a] {-# INLINE build #-} build g = g (:) [] Notice the INLINE! That prevents (:) from being inlined when compiling PrelBase, so that an importing module will “see” the (:), and can match it on the LHS of a rule. INLINE prevents any inlining happening in the RHS of the INLINE thing. I regret the delicacy of this. In libraries/base/GHC/Base.lhs look at the rules for map to see how to write rules that will do fusion and yet give an efficient program even if fusion doesn't happen. More rules in GHC/List.lhs. CORE pragma pragma, CORE core, annotation The external core format supports Note annotations; the CORE pragma gives a way to specify what these should be in your Haskell source code. Syntactically, core annotations are attached to expressions and take a Haskell string literal as an argument. The following function definition shows an example: f x = ({-# CORE "foo" #-} show) ({-# CORE "bar" #-} x) Semantically, this is equivalent to: g x = show x However, when external core is generated (via -fext-core), there will be Notes attached to the expressions show and x. The core function declaration for f is: f :: %forall a . GHCziShow.ZCTShow a -> a -> GHCziBase.ZMZN GHCziBase.Char = \ @ a (zddShow::GHCziShow.ZCTShow a) (eta::a) -> (%note "foo" %case zddShow %of (tpl::GHCziShow.ZCTShow a) {GHCziShow.ZCDShow (tpl1::GHCziBase.Int -> a -> GHCziBase.ZMZN GHCziBase.Char -> GHCziBase.ZMZN GHCziBase.Cha r) (tpl2::a -> GHCziBase.ZMZN GHCziBase.Char) (tpl3::GHCziBase.ZMZN a -> GHCziBase.ZMZN GHCziBase.Char -> GHCziBase.ZMZN GHCziBase.Cha r) -> tpl2}) (%note "bar" eta); Here, we can see that the function show (which has been expanded out to a case expression over the Show dictionary) has a %note attached to it, as does the expression eta (which used to be called x). GHC has a few built-in functions with special behaviour. These are now described in the module GHC.Prim in the library documentation. In particular: inline allows control over inlining on a per-call-site basis. lazy restrains the strictness analyser. unsafeCoerce# allows you to fool the type checker. GHC used to have an implementation of generic classes as defined in the paper "Derivable type classes", Ralf Hinze and Simon Peyton Jones, Haskell Workshop, Montreal Sept 2000, pp94-105. These have been removed and replaced by the more general support for generic programming. Using a combination of -XDeriveGeneric () and -XDefaultSignatures (), you can easily do datatype-generic programming using the GHC.Generics framework. This section gives a very brief overview of how to do it. Generic programming support in GHC allows defining classes with methods that do not need a user specification when instantiating: the method body is automatically derived by GHC. This is similar to what happens for standard classes such as Read and Show, for instance, but now for user-defined classes. The first thing we need is generic representations. The GHC.Generics module defines a couple of primitive types that are used to represent Haskell datatypes: -- | Unit: used for constructors without arguments data U1 p = U1 -- | Constants, additional parameters and recursion of kind * newtype K1 i c p = K1 { unK1 :: c } -- | Meta-information (constructor names, etc.) newtype M1 i c f p = M1 { unM1 :: f p } -- | Sums: encode choice between constructors infixr 5 :+: data (:+:) f g p = L1 (f p) | R1 (g p) -- | Products: encode multiple arguments to constructors infixr 6 :*: data (:*:) f g p = f p :*: g p The Generic class mediates between user-defined datatypes and their internal representation as a sum-of-products: class Generic a where -- Encode the representation of a user datatype type Rep a :: * -> * -- Convert from the datatype to its representation from :: a -> (Rep a) x -- Convert from the representation to the datatype to :: (Rep a) x -> a Instances of this class can be derived by GHC with the -XDeriveGeneric (), and are necessary to be able to define generic instances automatically. For example, a user-defined datatype of trees data UserTree a = Node a (UserTree a) (UserTree a) | Leaf gets the following representation: instance Generic (UserTree a) where -- Representation type type Rep (UserTree a) = M1 D D1UserTree ( M1 C C1_0UserTree ( M1 S NoSelector (K1 P a) :*: M1 S NoSelector (K1 R (UserTree a)) :*: M1 S NoSelector (K1 R (UserTree a))) :+: M1 C C1_1UserTree U1) -- Conversion functions from (Node x l r) = M1 (L1 (M1 (M1 (K1 x) :*: M1 (K1 l) :*: M1 (K1 r)))) from Leaf = M1 (R1 (M1 U1)) to (M1 (L1 (M1 (M1 (K1 x) :*: M1 (K1 l) :*: M1 (K1 r))))) = Node x l r to (M1 (R1 (M1 U1))) = Leaf -- Meta-information data D1UserTree data C1_0UserTree data C1_1UserTree instance Datatype D1UserTree where datatypeName _ = "UserTree" moduleName _ = "Main" instance Constructor C1_0UserTree where conName _ = "Node" instance Constructor C1_1UserTree where conName _ = "Leaf" This representation is generated automatically if a deriving Generic clause is attached to the datatype. Standalone deriving can also be used. A generic function is defined by creating a class and giving instances for each of the representation types of GHC.Generics. As an example we show generic serialization: data Bin = O | I class GSerialize f where gput :: f a -> [Bin] instance GSerialize U1 where gput U1 = [] instance (GSerialize a, GSerialize b) => GSerialize (a :*: b) where gput (x :*: y) = gput x ++ gput y instance (GSerialize a, GSerialize b) => GSerialize (a :+: b) where gput (L1 x) = O : gput x gput (R1 x) = I : gput x instance (GSerialize a) => GSerialize (M1 i c a) where gput (M1 x) = gput x instance (Serialize a) => GSerialize (K1 i a) where gput (K1 x) = put x Typically this class will not be exported, as it only makes sense to have instances for the representation types. The only thing left to do now is to define a "front-end" class, which is exposed to the user: class Serialize a where put :: a -> [Bin] default put :: (Generic a, GSerialize (Rep a)) => a -> [Bit] put = gput . from Here we use a default signature to specify that the user does not have to provide an implementation for put, as long as there is a Generic instance for the type to instantiate. For the UserTree type, for instance, the user can just write: instance (Serialize a) => Serialize (UserTree a) The default method for put is then used, corresponding to the generic implementation of serialization. For more detail please refer to the HaskellWiki page or the original paper: Jose Pedro Magalhaes, Atze Dijkstra, Johan Jeuring, and Andres Loeh. A generic deriving mechanism for Haskell. Proceedings of the third ACM Haskell symposium on Haskell (Haskell'2010), pp. 37-48, ACM, 2010. Note: the current support for generic programming in GHC is preliminary. In particular, we only allow deriving instances for the Generic class. Support for deriving Generic1 (and thus enabling generic functions of kind * -> * such as fmap) will come at a later stage. GHC supports two flags that control the way in which generalisation is carried out at let and where bindings. -XNoMonomorphismRestriction Haskell's monomorphism restriction (see Section 4.5.5 of the Haskell Report) can be completely switched off by -XNoMonomorphismRestriction. -XNoMonoPatBinds -XMonoPatBinds As an experimental change, we are exploring the possibility of making pattern bindings monomorphic; that is, not generalised at all. A pattern binding is a binding whose LHS has no function arguments, and is not a simple variable. For example: f x = x -- Not a pattern binding f = \x -> x -- Not a pattern binding f :: Int -> Int = \x -> x -- Not a pattern binding (g,h) = e -- A pattern binding (f) = e -- A pattern binding [x] = e -- A pattern binding Experimentally, GHC now makes pattern bindings monomorphic by default. Use -XNoMonoPatBinds to recover the standard behaviour.