i, de
ne some
terms, give some examples, and then point to the Report for details. We suggest, however, that the
reader completely ignore the details until the Gentle Introduction has been completely read. On the
1
we encourage a thorough reading once this tutorial is completed. This will not only give the reader a feel for what real Haskell code looks like.2 2 VALUES. Haskell's Standard Prelude (in Appendix A of the Report and the standard libraries (found in the Library Report [5]) contain lots of useful examples of Haskell code. but will also familiarize her with Haskell's standard set of prede. TYPES. AND OTHER GOODIES other hand.
ned functions and types. as with this paragraph. this may be a diÆcult adjustment. [We have also taken the course of not laying out a plethora of lexical syntax rules at the outset. or even ML. for those familiar with Java. C. the Haskell web site. For those whose only experience is with relatively \untypeful" languages such as Perl. has a wealth of information about the Haskell language and its implementations. Finally. Tcl.org. we introduce them incrementally as our examples demand.1" refer to sections in the Report).. and the newcomer is best o becoming well aware of the full power and complexity of Haskell's type system from the outset. This is in stark contrast to the organization of the Report. although the Report remains the authoritative source for details (references such as \x2. Modula. the adjustment should be easier but still not insigni. or Scheme. and enclose them in brackets.] Haskell is a typeful programming language:1 types are pervasive. Rather.
cant. all computations are done via the evaluation of expressions (syntactic terms) to yield values (abstract entities that we regard as answers). since Haskell's type system is dierent and somewhat richer than most. \typeful programming" is part of the Haskell programming experience. (Intuitively. 2 Values. and cannot be avoided.2. Just as expressions denote values. Examples of type expressions include the atomic types Integer (in. as well as structured values such as the list [1. In any case. the character 'a'. we can think of types as sets of values. Types.3] and the pair ('b'. and the function \x -> x+1.) Examples of expressions include atomic values such as the integer 5.4). and Other Goodies Because Haskell is a purely functional language. Every value has an associated type. type expressions are syntactic terms that denote type values (or just types).
Char (characters).nite-precision integers).Integer) (character. All Haskell values are \. integer pairs). Integer->Integer (functions mapping Integer to Integer). as well as the structured types [Integer] (homogeneous lists of integers) and (Char. de
the function inc can be de.ned by a series of equations. For example.
ned by the single equation: inc n = n+1 An equation is an example of a declaration.4. Another kind of declaration is a type signature declaration (x4. with which we can declare an explicit typing for inc: inc :: Integer -> Integer We will have much more to say about function de.1).
nitions in Section 3. For pedagogical purposes." to another expression or value e2 . or \reduces. when we wish to indicate that an expression e1 evaluates. note that: ) e2 inc (inc 3) ) 5 Haskell's static type system de. we will write: e1 For example.
the type system . Not all errors are caught by the type system. that is.nes the formal relationship between types and values (x4. that the programmer has not mismatched types in some way. so the expression 'a'+'b' is ill-typed. For example. Still.1.3). we cannot generally add together two characters. The main advantage of statically typed languages is wellknown: All type errors are detected at compile-time. The static type system ensures that Haskell programs are type safe. an expression such as 1/0 is typable but its evaluation will result in an error at execution time.
Haskell's type system is powerful enough to allow us to avoid writing any type signatures at all. no run-time type tags or tests are required). Nevertheless. In fact.2 we say that the type system infers the correct types for us. judicious placement of type signatures such as that we gave for inc is a good idea. [The reader will note that we have capitalized identi. since type signatures are a very eective form of documentation and help bring programming errors to light. aids the user in reasoning about programs.nds many program errors at compile time. and also permits a compiler to generate more eÆcient code (for example. The type system also ensures that user-supplied type signatures are correct.
ers that denote speci.
c types. but not identi. such as Integer and Char.
and fOO are all distinct identi.ers that denote values. This is not just a convention: it is enforced by Haskell's lexical syntax. too: foo. fOo. In fact. such as inc. the case of the other characters matters.
ers.] 2.1 Polymorphic Types Haskell also incorporates polymorphic types|types that are universally quanti.
(8a)[a] is the family of types consisting of. For example. . Polymorphic type expressions essentially describe families of types. the type of lists of a. for every type a. Lists of 2 With a few exceptions to be described later.ed in some way over all types.
.g.'c']). AND OTHER GOODIES integers (e. however. even lists of lists of integers. are all members of this family. lists of characters (['a'.'b'] is not a valid example. since there is no single type that contains both 2 and 'b'. etc.3]).) [Identi. (Note. that [2.2. [1.'b'.4 2 VALUES. TYPES.
and are uncapitalized to distinguish them from speci.ers such as a above are called type variables.
c types such as Int. since Haskell has only universally quanti. Furthermore.
ed types. there is no need to explicitly write out the symbol for universal quanti.
all type variables are implicitly universally quanti. and thus we simply write [a] in the example above. In other words.cation.
2.] Lists are a commonly used data structure in functional languages. and are a good vehicle for explaining the principles of polymorphism. The list [1.3] in Haskell is actually shorthand for the list 1:(2:(3:[])).ed. where [] is the empty list and : is the in.
x operator that adds its .
As an example of a user-de.3 Since : is right associative.rst argument to the front of its second argument (a list). we can also write this list as 1:2:3:[].
consider the problem of counting the number of elements in a list: length length [] length (x:xs) :: [a] -> Integer = 0 = 1 + length xs This de.ned function that operates on lists.
We can read the equations as saying: \The length of the empty list is 0.nition is almost self-explanatory. and the length of a list whose .
rst element is x and remainder is xs is 1 plus the length of xs. In a function application these patterns are matched against actual parameters in a fairly intuitive way ([] only matches the empty list. binding x to the . and should be read that way. this example highlights an important aspect of Haskell that is yet to be explained: pattern matching.) Although intuitive." (Note the naming convention used here. The left-hand sides of the equations contain patterns such as [] and x:xs. and x:xs will successfully match any list with at least one element. xs is the plural of x.
and if all equations fail. the right-hand side is evaluated and returned as the result of the application. De.rst element and xs to the rest of the list). the next equation is tried. If it fails. an error results. If the match succeeds.
[3]] ) ) ) 3 3 3 Here are two other useful polymorphic functions on lists that will be used later. [Char]. Function head returns the .ning functions by pattern matching is quite common in Haskell. we will return to this issue in Section 4. or [[Integer]]. The length function is also an example of a polymorphic function. length [1.'b'.[2].3] length ['a'. It can be applied to a list containing elements of any type.'c'] length [[1]. and the user should become familiar with the various kinds of patterns that are allowed. for example [Integer].2.
function tail returns all but the .rst element of a list.
3 : and [] are like Lisp's cons and nil. respectively.rst. .
5 2.2 User-De.
these functions are not de.ned Types head head (x:xs) :: [a] -> a = x tail tail (x:xs) :: [a] -> [a] = xs Unlike length.
we . With polymorphic types.ned for all possible values of their argument. A runtime error occurs when these functions are applied to an empty list.
nd that some types are in a sense strictly more general than others in the sense that the set of values they de.
With regard to this generalization ordering. For example. every well-typed expression is guaranteed to have a unique principal type (explained below). the principal type can be inferred automatically (x4.1. the latter type can be derived from the former by a suitable substitution for a. and second. Haskell's type system possesses two important properties: First. the reader will . In comparison to a monomorphically typed language such as C. the type [a] is more general than [Char].3). In other words.ne is larger.
a->a. An expression's or function's principal type is the least general type that. For example. the principal type of head is [a]->a. \contains all instances of the expression". but too general. intuitively.nd that polymorphism improves expressiveness. and type inference lessens the burden of types on the programmer. whereas something like [Integer]->Integer is too speci. or even a are correct types. [b]->a.
4 and several other (mostly functional) languages. 2.c.2 User-De. The existence of unique principal types is the hallmark feature of the Hindley-Milner type system. which forms the basis of the type systems of Haskell. Miranda. ML.
ned Types We can de.
2. An important prede.ne our own types in Haskell using a data declaration. which we introduce via a series of examples (x4.1).
ned type in Haskell is that of truth values: data Bool = False | True The type being de.
for short).ned here is Bool. Similarly. we might wish to de. Type Bool is an example of a (nullary) type constructor. and True and False are (also nullary) data constructors (or just constructors. and it has exactly two values: True and False.
since they consist of a .ne a color type: data Color = Red | Green | Blue | Indigo | Violet Both Bool and Color are examples of enumerated types.
. a type like Point is often called a tuple type.nite number of nullary data constructors. Here is an example of a type with just one data constructor: data Point a = Pt a a Because of the single constructor. since it is essentially 4 \Miranda" is a trademark of Research Software. Ltd.
multi-constructor types. More importantly. are called (disjoint) union or sum types. it de. such as Bool and Color.5 In contrast. AND OTHER GOODIES just a cartesian product (in this case binary) of other types. TYPES. however.6 2 VALUES. Point is an example of a polymorphic type: for any type t.
and thus the following typings are valid: Pt 2. and applying a type constructor to yield a type.0 3. whereas the latter happens at compile-time and is part of the type system's process of ensuring type safety. since from the type t it constructs a new type Point t. (In the same sense. The Haskell syntax allows [] t to be written as [t]. Similarly.nes the type of cartesian points that use t as the coordinate type. an expression such as Pt 'a' 1 is ill-typed because 'a' and 1 are of dierent types. It is important to distinguish between applying a data constructor to yield a value.  .) Note that the type of the binary data constructor Pt is a -> a -> Point a. the former happens at run-time and is how we compute things in Haskell. using the list example given earlier. The Point type can now be seen clearly as a unary type constructor. [] is also a type constructor. -> is a type constructor: given two types t and u.0 Pt 'a' 'b' Pt True False :: Point Float :: Point Char :: Point Bool On the other hand. t->u is the type of functions mapping elements of type t to elements of type u. Given any type t we can \apply" [] to yield a new type [t].
1 Recursive Types Types can also be recursive.2.] 2.rst. it serves to make the link between a type and its data constructor more obvious. as in the type of binary trees: data Tree a = Leaf a | Branch (Tree a) (Tree a) Here we have de.
whereas Branch and Leaf are data constructors. When reading data declarations such as this. Aside from establishing a connection between these constructors.ned a polymorphic binary tree type whose elements are either leaf nodes containing a value of type a. remember again that Tree is a type constructor. the above declaration is essentially de. or internal nodes (\branches") containing (recursively) two sub-trees.
.ning the following types for Branch and Leaf: Branch Leaf 5 :: Tree a -> Tree a -> Tree a :: a -> Tree a Tuples are somewhat like records in other languages.
7 2.3 Type Synonyms With this example we have .
maps trees of a into lists of a. for any type a. That is. A suitable de.rst. in this case we see that the type should be Tree a -> [a]. fringe is a polymorphic function that.
nition follows: fringe :: Tree a -> [a] fringe (Leaf x) = [x] fringe (Branch left right) = fringe left ++ fringe right Here ++ is the in.
x operator that concatenates two lists (its full de.
nition will be given in Section 9.1). the fringe function is de. As with the length example given earlier.
ned using pattern matching. except that here we see patterns involving user-de.
[Note that the formal parameters are easily identi.ned constructors: Leaf and Branch.
3 Type Synonyms For convenience.ed as the ones beginning with lower-case letters.] 2. Haskell provides a way to de.
Type synonyms are created using a type declaration (x4.Address) String None | Addr String Type synonyms do not de. Here are several examples: type type type data String Person Name Address = = = = [Char] (Name.e. i.ne type synonyms. names for commonly used types.2.2).
the type Person -> Name is precisely equivalent to (String. but this is not the only purpose of type synonyms: they can also improve readability of programs by being more mnemonic. 2.ne new types.Address) -> String. and characters. We can even give new names to polymorphic types: type AssocList a b = [(a. tuples. integers.b)] This is the type of \association lists" which associate values of type a with those of type b. We have also shown how new user-de. For example. The new names are often shorter than the types they are synonymous with. the above examples highlight this. indeed.4 Built-in Types Are Not Special Earlier we introduced several \built-in" types such as lists. but simply give new names for existing types.
ned types can be de.
are the built-in types in any way more special than the user-de.ned. Aside from special syntax.
but has no semantic consequences. We can emphasize this point by considering what the type declarations would look like for these built-in types if in fact we were allowed to use the special syntax in de. The special syntax is for convenience and for consistency with historical convention.ned ones? The answer is no.
For example. the Char type might be written as: .ning them.
...8 2 VALUES... AND OTHER GOODIES data Char = 'a' | 'b' | 'c' | .. to .This is not valid -. | 'A' | 'B' | 'C' | . . | '1' | '2' | '3' | .. TYPES..Haskell code! These constructor names are not syntactically valid. -.
x them we would have to write something like: data Char = Ca | Cb | Cc | .. they are quite unconventional for representing characters. Thinking of Char in this way makes it clear that we can pattern-match against characters in function de... ... | C1 | C2 | C3 | . Even though these constructors are more concise. writing \pseudo-Haskell" code in this way helps us to see through the special syntax. In any case.. We see now that Char is just an enumerated type consisting of a large number of nullary constructors. | CA | CB | CC | ...
] Similarly.nitions.2). we could de.and all subsequent characters to the end of the line are ignored. just as we would expect to be able to do so for any of a type's constructors. [This example also demonstrates the use of comments in Haskell. Haskell also permits nested comments which have the form {-: : :-} and can appear anywhere (x2. the characters -.
ne Int (.
. where -65532 and 65532..xed precision integers) and Integer by: data Int = -65532 | ... are the maximum and minimum . | 65532 -....more pseudo-code data Integer = .. | -1 | 0 | 1 | . say. -2 | -1 | 0 | 1 | 2 .
but it's still .xed precision integers for a given implementation. Int is a much larger enumeration than Char.
nite! In contrast. the pseudo-code for Integer is intended to convey an in.
nite enumeration. Tuples are also easy to de.
b.c) = (a. .b) data (a.d) .b. -. .b. .c.d) .ne playing this game: data (a.c.c) data (a. = (a.b.more pseudo-code Each declaration above de. .b) = (a.
with (...nes a tuple type of a particular length.) playing a role in both the expression syntax (as data constructor) and type-expression syntax (as type constructor). The vertical dots after the last declaration are intended to convey an in.
nite number of such declarations.more pseudo-code We can now see clearly what we described about lists earlier: [] is the empty list. they are recursive: data [a] = [] | a : [a] -. Lists are also easily handled. and : is the in. and more interestingly. re ecting the fact that tuples of all lengths are allowed in Haskell.
x .
[The way \:" is de.9 2. thus [1.2.4 Built-in Types Are Not Special list constructor. (: is right associative.) The type of [] is [a]. and the type of : is a->[a]->[a].3] must be equivalent to the list 1:2:3:[].
ned here is actually legal syntax|in.
and are distinguished from in.x constructors are permitted in data declarations.
x operators (for pattern-matching purposes) by the fact that they must begin with a \:" (a property trivially satis.
ed by \:").] At this point the reader should note carefully the dierences between tuples and lists. which the above de.
nitions make abundantly clear. In particular. note the recursive nature of the list type whose elements are homogeneous and of arbitrary length. and the non-recursive nature of a (particular) tuple type whose elements are heterogeneous and of .
4]. For [e1 . here is a concise de.xs ] This expression can intuitively be read as \the list of all f x such that x is drawn from xs. as in: [ (x.xs. : : : .xs is called a generator.4)].4.(1. and as with other functional languages.(2.en ). For example.(2.4).e2 . The typing rules for tuples and lists should now also be clear: For (e1 . : : : . if xs is [1. y <. Guards place constraints on the elements generated. The elements are selected as if the generators were \nested" from left to right (with the rightmost generator varying fastest).tn ).t2 . lists are pervasive in Haskell. Besides generators. Aside from the constructors for lists just discussed. Haskell provides an expression known as a list comprehension that is best explained by example: [ f x | x <. then the type of the tuple is (t1 . there is yet more syntactic sugar to aid in their creation. the result is [(1." The similarity to set notation is not a coincidence. of which more than one is allowed.2] and ys is [3.1 List Comprehensions and Arithmetic Sequences As with Lisp dialects. boolean expressions called guards are permitted. n 0.y) | x <. if ti is the type of ei .xed length. each ei must have the same type t.e2 . 2.3). The phrase x <. and the type of the list is [t]. n 2.ys ] This list comprehension forms the cartesian product of the two lists xs and ys. thus. : : : .3).en ].
7.8. which are best explained by a series of examples: [1.2..10] [1..9] [1. y>=x] To further support the use of lists.6. Haskell has special syntax for arithmetic sequences..xs.3.10] [1.9.3.xs. (in. ..] ) ) ) [1.7.7.10] [1.5.5..9.3.5.3. y<x ] [x] quicksort [y | y <.nition of everybody's favorite sorting algorithm: quicksort [] quicksort (x:xs) = = ++ ++ [] quicksort [y | y <.4.3.
and \in.nite sequence) More will be said about arithmetic sequences in Section 8.2.
.4.nite lists" in Section 3.
4.'l'.'e'. we note that the literal string "hello" is actually shorthand for the list of characters ['h'. Indeed.2 Strings As another example of syntactic sugar for built-in types. where String is a prede.'l'.10 3 FUNCTIONS 2. the type of "hello" is String.'o'].
ned type synonym (that we gave as an earlier example): type String = [Char] This means we can use prede.
First. one would expect functions to play a major role. and indeed they do. we look at several aspects of functions in Haskell.ned polymorphic list functions to operate on strings. In this section. consider this de. For example: "hello" ++ " world" 3 "hello world" ) Functions Since Haskell is a functional language.
i.nition of a function which adds its two arguments: add add x y :: Integer -> Integer -> Integer = x + y This is an example of a curried function. In other words. Integer->Integer->Integer. using add. we can de. since function application associates to the left. which is equivalent to Integer->(Integer->Integer).e. -> associates to the right.6 An application of add has the form add e1 e2 . and is equivalent to (add e1 ) e2 . Indeed. This is consistent with the type of add. applying add to one argument yields a new function which is then applied to the second argument..
x operator. and thus the right-hand side of the second equation parses as (f x) : (map f xs).] The map function is polymorphic and its type indicates clearly that its .
As an example of the use of map.3] ) [2. To get the eect of an uncurried function.3. as in: add (x. note also that the two a's must be instantiated with the same type (likewise for the b's). we can increment the elements in a list: map (add 1) [1.4] 6 The name curry derives from the person who popularized the idea: Haskell Curry.y) = x + y But then we see that this version of add is really just a function of one argument! .2.rst argument is a function. we could use a tuple.
11 3.1 Lambda Abstractions These examples demonstrate the .
3.rst-class nature of functions. which when used in this way are usually called higher-order functions.1 Lambda Abstractions Instead of using equations to de.
ne functions. we can also de.
In general. the equations: inc x add x y = x+1 = x+y are really shorthand for: inc add = \x -> x+1 = \x y -> x+y We will have more to say about such equivalences later. a function equivalent to inc could be written as \x -> x+1. then \x->exp has type t1 ->t2 . 3. given that x has type t1 and exp has type t2 . In fact.ne them \anonymously" via a lambda abstraction. For example. Similarly.2 In. the function add is equivalent to \x -> \y -> x+y. Nested lambda abstractions such as this may be written using the equivalent shorthand notation \x y -> x+y.
x Operators In.
and can also be de.x operators are really just functions.
ned using equations. For example. here is a de.
nition of a list concatenation operator: (++) [] ++ ys (x:xs) ++ ys :: [a] -> [a] -> [a] = ys = x : (xs++ys) [Lexically. in.
" as opposed to normal identi.x operators consist entirely of \symbols.
4). Haskell has no pre.ers which are alphanumeric (x2.
with the exception of minus (-).x operators. which is both in.
x and pre.
an important in.x.] As another example.
x operator on functions is that for function composition: (.2.) f . g :: (b->c) -> (a->b) -> (a->c) = \ x -> f (g x) 3.1 Sections Since in.
x operators are really just functions. In Haskell the partial application of an in. it makes sense to be able to partially apply them as well.
For example: (x+) (+y) (+) \y -> x+y \x -> x+y \x y -> x+y .x operator is called a section.
] The last form of section given above essentially coerces an in.12 3 FUNCTIONS [The parentheses are mandatory.
and is handy when passing an in.x operator into an equivalent functional value.
2.) given earlier. as in map (+) [1. We can now see that add de. It is also necessary when giving a function type signature.x operator as an argument to a function.3] (the reader should verify that this returns a list of functions!). as in the examples of (++) and (.
ned earlier is just (+). these de. and inc is just (+1)! Indeed.
nitions would do just .
ne: inc add = (+ 1) = (+) We can coerce an in.
but can we go the other way? Yes|we simply enclose an identi.x operator into a functional value.
x `add` y is the same as add x y. For example.er bound to a functional value in backquotes.7 Some functions read better this way. An example is the prede.
ned list membership predicate elem. the expression x `elem` xs can be read intuitively as \x is an element of xs." [There are some special rules regarding sections involving the pre.
x/in.
] At this point. the reader may be confused at having so many ways to de. see (x3.x operator -.5.4).x3.
in the treatment of in. and partly re ects the desire for consistency (for example.ne a function! The decision to provide these mechanisms partly re ects historical conventions.
3.x vs.2. regular functions).2 Fixity Declarations A .
xity declaration can be given for any in.
x operator or constructor (including those made from ordinary identi.
ers. This declaration speci. such as `elem`).
right-. normal application is assumed to have a precedence level of 10). or non-associativity. For example.es a precedence level from 0 to 9 (with 9 being the strongest. and left-. the .
Both of these specify right-associativity.xity declarations for ++ and . the . are: infixr 5 ++ infixr 9 .
Left associativity is speci. the other 9.rst with a precedence level of 5.
Also. and non-associativity by infix.ed via infixl. the .
xity of more than one operator may be speci.
ed with the same .
If no .xity declaration.
(See x5.xity declaration is given for a particular operator. it defaults to infixl 9.9 for a detailed de.
3 Functions are Non-strict Suppose bot is de.) 3.nition of the associativity rules.
i.ned by: 7 Note carefully that add is enclosed in backquotes.e. whereas `f` is an in. 'f' is a character. not apostrophes as used in the syntax of characters.
. most ASCII terminals distinguish these much better than the font used in this manuscript.x operator. Fortunately.
4 \In.13 3.
such as an end-of-. Abstractly. bot is a non-terminating expression. Such an error is not recoverable: programs will not continue past these errors.nite" Data Structures bot = bot In other words. such as 1/0. also have this value. Expressions that result in some kind of a run-time error. Errors encountered by the I/O system. we denote the value of a nonterminating expression as ? (read \bottom").
As a simple example. de. when applied to a nonterminating expression. For most programming languages. all functions are strict. In other words. consider const1. it also fails to terminate. (Such an I/O error is really not an error at all but rather an exception.) A function f is said to be strict if. f is strict i the value of f bot is ?. But this is not so in Haskell. Much more will be said about exceptions in Section 7. the constant 1 function. are recoverable and are handled in a dierent manner.le error.
An important example of this is a possibly in. and thus never gets caught in a nonterminating computation. or \by need". Computationally expensive values may be passed as arguments to functions without fear of them being computed if they are not needed. The main advantage is that they free the programmer from many concerns about evaluation order. For this reason. const1 (1/0) also evaluates properly to 1. For example. Since error and nonterminating values are semantically the same in Haskell. Operationally speaking. Non-strict functions are extremely useful in a variety of contexts. non-strict functions are also called \lazy functions".ned by: const1 x = 1 The value of const1 bot in Haskell is 1. the above argument also holds for errors. and are said to evaluate their arguments \lazily". since const1 does not \need" the value of its argument. it never attempts to evaluate it.
Another way of explaining non-strict functions is that Haskell computes using de.nite data structure.
Read a declaration such as v = 1/0 as `de.nitions rather than the assignments found in traditional languages.
Only if the value (de.ne v as 1/0' instead of `compute 1/0 and store the result in v'.
De. By itself. Programming using assignments requires careful attention to the ordering of the assignments: the meaning of the program depends on the order in which the assignments are executed. this declaration does not imply any computation.nition) of v is needed will the division by zero error occur.
nitions. are much simpler: they can be presented in any order without aecting the meaning of the program. in contrast.4 \In. 3.
This should not be surprising. is non-strict. since constructors are really just a special kind of function (the distinguishing feature being that they can be used in pattern matching). For example. the constructor for lists. (:). Non-strict constructors permit the de.nite" Data Structures One advantage of the non-strict nature of Haskell is that data constructors are non-strict. too..
From it we can construct an in.nite list of successive integers beginning with n.
^ is the in.nite list of squares: squares = map (^2) (numsfrom 0) (Note the use of a section.
eventually we expect to extract some .x exponentiation operator.) Of course.
and there are lots of prede.nite portion of the list for actual computation.
and others.ned functions in Haskell that do this sort of thing: take. The de. filter. takeWhile.
take removes the .".
4.9.16] The de.1.rst n elements from a list: take 5 squares ) [0.
thus saving space. For another example of the use of circularity. In most circumstances laziness has an important impact on eÆciency.nition of ones above is an example of a circular list. since an implementation can be expected to implement the list as a true circular structure. the Fibonacci sequence can be computed eÆciently as the following in..y) : zip xs ys = [] Note how fib.
nite list. is de.
For another application of in. as if it were \chasing its tail.ned in terms of itself. we can draw a picture of this computation as shown in Figure 1." Indeed.
. see Section 4.4.nite lists.
the actual de. Indeed. This is a somewhat odd function: From its type it looks as if it is returning a value of a polymorphic type about which it knows nothing.15 3. semantically that is exactly what value is always returned by error (recall that all errors have value ?).5 The Error Function Haskell has a built-in function called error whose type is String->a. since it never receives a value of that type as an argument! In fact.5 The Error Function 3. we can expect that a reasonable implementation will print the string argument to error for diagnostic purposes. Thus this function is useful when we wish to terminate a program when something has \gone wrong." For example. there is one value \shared" by all types: ?. However.
nition of head taken from the Standard Prelude is: head (x:xs) head [] 4 = x = error "head{PreludeList}: head []" Case Expressions and Pattern Matching Earlier we gave several examples of pattern matching in de.
8 Patterns are not \.ning functions|for example length and fringe. In this section we will look at the pattern-matching process in greater detail (x3.17).
" there is only a .rst-class.
both length and fringe de. We have already seen several examples of data constructor patterns.xed set of dierent kinds of patterns.
the former on the constructors of a \built-in" type (lists). the latter on a userde.ned earlier use such patterns.
Indeed.ned type (Tree). user-de. matching is permitted using the constructors of any type.
For example. 8 Pattern matching in Haskell is dierent from that found in logic programming languages such as Prolog. True) = False This example also demonstrates that nesting of patterns is permitted (to arbitrary depth).0). in particular. it can be viewed as \one-way" matching. 2. As a \side eect" of the successful match. (Int. formal parameters9 are also patterns|it's just that they never fail to match a value. strings. x4.4. Char.17. two of which we will introduce now (the other we will delay until Section 4. There are three other kinds of irrefutable patterns. etc. Float). in contrast to refutable patterns which may fail to match. (1. "hi".4). This includes tuples. here's a contrived function that matches against a tuple of \constants:" contrived :: ([a]. x3.ned or not. Bool) -> Bool contrived ([].2). numbers. 'b'. The pattern used in the contrived example above is refutable. characters. whereas Prolog allows \two-way" matching (via uni.3. the formal parameter is bound to the value it is being matched against. Patterns such as formal parameters that never fail to match are said to be irrefutable. For this reason patterns in any one equation are not allowed to have more than one occurrence of the same formal parameter (a property called linearity x3. String. Technically speaking.
. 9 The Report calls these variables. along with implicit backtracking in its evaluation mechanism.cation).
16 4 CASE EXPRESSIONS AND PATTERN MATCHING As-patterns. For example. a function that duplicates the . Sometimes it is convenient to name a pattern for use on the right-hand side of an equation.
) Note that x:xs appears both as a pattern on the left-hand side. the functions head and tail de. as-patterns always result in a successful match. although the sub-pattern (in this case x:xs) could. fail. To improve readability. and an expression on the right-hand side. For example. of course. which we can achieve using an as-pattern as follows:10 f s@(x:xs) = x:s Technically speaking. Wild-cards. we might prefer to write x:xs just once. Another common situation is matching against a value we really care nothing about.rst element in a list might be written as: f (x:xs) = x:x:xs (Recall that \:" associates to the right.
each binds nothing.2] is matched against [0. so the result is a failed match. then 1 fails to match 0. etc. but in contrast to a formal parameter. for this reason more than one is allowed in an equation. if [1. Each wild-card independently matches anything.1 Pattern-Matching Semantics So far we have discussed how individual patterns are matched. A successful match binds the formal parameters in the pattern. how some are refutable. some are irrefutable. succeed or diverge. and results in a run-time error. de. left-to-right.1 can be rewritten as: head (x:_) tail (_:xs) = x = xs in which we have \advertised" the fact that we don't care what a certain part of the input is. the value of the function application is ?. If all equations fail." Failure of a pattern anywhere in one equation results in failure of the whole equation. and the next equation is then tried. Pattern matching can either fail.ned in Section 2. For example. The matching process itself occurs \top-down. But what drives the overall process? In what order are the matches attempted? What if none succeeds? This section addresses these questions. 4. (Recall that bot.bot]. Divergence occurs when a value needed by the pattern contains an error (?).
2] is matched against [bot. as in this de.e. ?).) But if [1. is a variable bound to ?.ned earlier. then matching 1 against bot causes divergence (i. The other twist to this set of rules is that top-level patterns may also have a boolean guard.0].
as with patterns.nition of a function that forms an abstract version of a number's sign: sign x | x > 0 | x == 0 | x < 0 = 1 = 0 = -1 Note that a sequence of guards may be provided for the same pattern. and the . they are evaluated top-down.
rst that evaluates to True results in a successful match. 10 Another advantage to doing this is that a naive implementation might completely reconstruct x:xs rather than re-use the value being matched against. .
For example.2 An Example 4. consider this de.17 4.2 An Example The pattern-matching rules can have subtle eects on the meaning of functions.
n de.
ned" with respect to its second argument. whereas take1 is more de.
ned with respect to its .
It is diÆcult to say in this case which de.rst.
it may make a dierence. Just remember that in certain applications.nition is better. (The Standard Prelude includes a de.
In many circumstances we don't wish to de.3 Case Expressions Pattern matching provides a way to \dispatch control" based on structural properties of a value.nition corresponding to take.) 4.
ne a function every time we need to do this. but so far we have only shown how to do pattern matching in function de.
the meaning of pattern matching in function de. Haskell's case expression provides a way to solve this problem.nitions. Indeed.
nitions is speci.
ed in the Report in terms of case expressions. which are considered more primitive. In particular. a function de.
xk) of (p11 . is semantically equivalent to: f x1 x2 : : : xk = case (x1. : : : .nition of the form: f p11 : : : p1k = e1 ::: f pn1 : : : pnk = en where each pij is a pattern. : : : . pnk ) -> en where the xi are new identi. : : : . p1k ) -> e1 ::: (pn1 .
ers. (For a more general translation that includes guards.4.2.) For example. the de. see x4.
nition of take given earlier is equivalent to: .
x:xs) -> x : take (n-1) xs A point not made earlier is that._) -> [] (_. for type correctness.[]) -> [] (n. the types of the right-hand sides of a case expression or set of equations comprising a function de.ys) of (0.18 4 CASE EXPRESSIONS AND PATTERN MATCHING take m ys = case (m.
The pattern-matching rules for case expressions are the same as we have given for function de. more precisely. they must all share a common principal type.nition must all be the same.
In other words. and has the form ~pat. 4. Lazy patterns are irrefutable: matching a value v against ~pat always succeeds. and e2 and e3 must have the same (but otherwise arbitrary) type. regardless of pat.nitions. other than to note the convenience that case expressions oer. It is called a lazy pattern. Indeed. conditional expressions have the familiar form: if e1 then e2 else e3 which is really short-hand for: case e1 of True -> e2 False -> e3 From this expansion it should be clear that e1 must have type Bool. if-then-else when viewed as a function has type Bool->a->a->a. so there is really nothing new to learn here.4 Lazy Patterns There is one other kind of pattern allowed in Haskell. there's one use of a case expression that is so common that it has special syntax: the conditional expression. if an identi. In Haskell. Operationally speaking.
it will be bound to that portion of the value that would result if v were to successfully match pat.er in pat is later \used" on the right-hand-side. Lazy patterns are useful in contexts where in. and ? otherwise.
nite data structures are being de.
ned recursively. For example. in.
and in this context the in.nite lists are an excellent vehicle for writing simulation programs.
This situation is shown pictorially in Figure 2. and server replies to each request with some kind of response. Let us further assume that the structure of the server and client look something like this: client init (resp:resps) = init : client (next resp) resps server (req:reqs) = process req : server reqs .) Using streams to simulate the message sequences. (Note that client also takes an initial message as argument. Consider the simple case of simulating the interactions between a server process server and a client process client..
as used in the recursive setting of reqs and resps. returning an appropriate response. given a response from the server. Unfortunately.4 Lazy Patterns Figure 2: Client-Server Simulation where we assume that next is a function that. determines the next request. attempts a match on the response list before it has submitted its .19 4. this program has a serious problem: it will not produce any output! The problem is that client. and process is a function that processes a request from the client.
" One way to . the pattern matching is being done \too early.rst request! In other words.
x this is to rede.
ne client as follows: client init resps = init : client (next (head resps)) (tail resps) Although workable. in turn allowing the . allowing the initial request to be \submitted". A better solution is to use a lazy pattern: client init ~(resp:resps) = init : client (next resp) resps Because lazy patterns are irrefutable. the match will immediately succeed. this solution does not read as well as that given earlier.
As an example of this program in action. the engine is now \primed". if we de.rst response to be generated. and the recursion takes care of the rest.
9] As another example of the use of lazy patterns.1. consider the de.5.ne: init next resp process req then we see that: = 0 = resp = req+1 take 10 reqs ) [0.6.2.8.4.3.7.
nition of Fibonacci given earlier: fib = 1 : 1 : [ a+b | (a.b) <. both fib and tfib become bound within the scope of the declaration. [This kind of equation is called a pattern binding because it is a top-level equation in which the entire left-hand side is a pattern.e. i.zip fib (tail fib) ] We might try rewriting this using an as-pattern: fib@(1:tfib) = 1 : 1 : [ a+b | (a.] .zip fib tfib ] This version of fib has the (small) advantage of not using tail on the right-hand side. since it is available in \destructured" form on the left-hand side as tfib.b) <.
= . if only implicitly. and pattern bindings are treated as lazy patterns (i. consider: let y = a*b f x = (x+y)/y in f c + f d The set of bindings created by a let expression is mutually recursive. which only scopes over the expression which it encloses. we should be led to believe that this program will not generate any output. 4.. etc. and avoiding some anomalous situations which are beyond the scope of this tutorial. which requires a where clause: f x y | y>z | y==z | y<z where z = x*x = . The same properties and constraints on bindings in let expressions apply to those in where clauses. using the same reasoning as earlier. it does. and the reason is simple: in Haskell. function bindings. pattern bindings are assumed to have an implicit ~ in front of them. some kind of \block-structuring" form.e. however.. declarations. and pattern bindings. As a simple example. Note that this cannot be done with a let expression..6 Layout The reader may have been wondering how it is that Haskell programs avoid the use of semicolons. for the purpose of creating local bindings not seen elsewhere|i. Thus we see that lazy patterns play an important role in Haskell.... = . Sometimes it is convenient to scope bindings over several guarded equations. Curiously. The only kind of declarations permitted are type signatures. or some other kind of terminator. whereas a where clause is not|it is part of the syntax of function declarations and case expressions. but remember that a let expression is an expression. they carry an implicit ~). consider this let expression from the last section: let y = a*b f x = (x+y)/y in f c + f d .e. Where Clauses. Haskell's let expressions are useful whenever a nested set of bindings is required. In Haskell there are two ways to achieve this: Let Expressions. For example. A where clause is only allowed at the top level of a set of equations or case expression. re ecting the most common behavior expected of pattern bindings.20 4 CASE EXPRESSIONS AND PATTERN MATCHING Now.5 Lexical Scoping and Nested Forms It is often desirable to create a nested scope within an expression. to mark the end of equations. These two forms of nested scope seem very similar. 4.
let.11 Layout is actually shorthand for an explicit grouping mechanism. or of is what determines the starting column for the declarations in the where. this is a valid expression: let y = a*b. or case expression being written (the rule also applies to where used in the class and instance declarations to be introduced in Section 5). xB. (The do keyword.7. which deserves mention because it can be useful under certain circumstances. One way in which this explicit notation is useful is when more than one declaration is desired on a line. the next character following any of the keywords where. The let example above is equivalent to: let { y = a*b . The \termination" of a declaration happens when something appears at or to the left of the starting column associated with that binding form. but in practice. also uses layout). and its use is encouraged. z = a/b f x = (x+y)/z in f c + f d For another example of the expansion of layout into explicit delimiters. etc. the next line. thus enhancing readability. 5 Type Classes and Overloading There is one . It is easy to learn. for example. note that y and f begin in the same column. to be discussed later. f x = (x+y)/y } in f c + f d Note the explicit curly braces and semicolons. use of layout is rather intuitive. see x2. let. The use of layout greatly reduces the syntactic clutter associated with declaration lists.7. Thus we can begin the declarations on the same line as the keyword. Just remember two things: First. Second. The rules for layout are spelled out in detail in the Report (x2. just be sure that the starting column is further to the right than the starting column associated with the immediately surrounding clause (otherwise it would be ambiguous).3).
thus care must be taken when using an editor which may observe some other convention. There is another kind called ad hoc polymorphism. .nal feature of Haskell's type system that sets it apart from other programming languages. Here are some examples of ad hoc polymorphism: 11 Haskell observes the convention that tabs count as 8 blanks. better known as overloading. The kind of polymorphism that we have talked about so far is commonly called parametric polymorphism.
etc. 2. are often used to represent both .22 5 TYPE CLASSES AND OVERLOADING The literals 1.
xed and arbitrary precision integers. Numeric operators such as + are often de.
Note that these overloaded behaviors are dierent for each type (in fact the behavior is sometimes unde.ned to work on many dierent kinds of numbers. The equality operator (== in Haskell) usually works on numbers and many other (but not all) types.
type classes provide a structured way to control ad hoc polymorphism. or error). example: equality. for example. In Haskell. There are many types for which we would like equality de. Let's start with a simple. or overloading. really doesn't care what kind of elements are found in the leaves of a tree). whereas in parametric polymorphism the type truly does not matter (fringe.ned. but important.
For example. but some for which we would not.12 To highlight the issue. comparing the equality of functions is generally considered computationally intractable. consider this de. whereas we often want to compare two lists for equality.ned.
nition of the function elem which tests for membership in a list: x `elem` [] x `elem` (y:ys) = False = x==y || (x `elem` ys) [For the stylistic reason we discussed in Section 3.1. we have chosen to de.
ne elem in in.
x form. == and || are the in.
even though we just said that we don't expect == to be de. But this would imply that == has type a->a->Bool. respectively.x operators for equality and logical or. the type of elem \ought" to be: a->[a]->Bool.] Intuitively speaking.
ned for all types. Furthermore. as we have noted earlier. even if == were de.
ned on all types. Type classes conveniently solve both of these problems. They allow us to declare which types are instances of which class. and to provide de. In this sense. comparing two lists for equality is very dierent from comparing two integers. we expect == to be overloaded to carry on these various tasks. ==. of the appropriate type. and == is the single operation in the class.ned.
" (Note that == is only de.ned on it.
Contexts are placed at the front of type expressions.ned on pairs of objects of the same type. Pointer equality is not referentially transparent. for example. and thus does not sit well in a purely functional language. the eect of the above class declaration is to assign the following type to ==: (==) 12 :: (Eq a) => a -> a -> Bool The kind of equality we are referring to here is \value equality." and opposed to the \pointer equality" found. and is called a context. but rather it expresses a constraint on a type.) The constraint that a type a must be an instance of the class Eq is written Eq a. For example. with Java's ==. Thus Eq a is not a type expression. .
\For every type a that is an instance of the class Eq. This is the type that would be used for == in the elem example. This is just what we want|it expresses the fact that elem is not de. \For every type a that is an instance of the class Eq. and indeed the constraint imposed by the context propagates to the principal type for elem: elem :: (Eq a) => a -> [a] -> Bool This is read. == has type a->a->Bool". elem has type a->[a]->Bool".23 This should be read.
But how do we specify which types are instances of the class Eq. For example: instance Eq Integer where x == y = x `integerEq` y The de. and the actual behavior of == on each of those types? This is done with an instance declaration. just those for which we know how to compare elements for equality.ned on all types. So far so good.
just as for any other function de.nition of == is called a method..
we can now compare ." Given this declaration.nition of the method corresponding to the operation ==.
Recursive types such as Tree de. Similarly: instance Eq Float where x == y = x `floatEq` y allows us to compare oating point numbers using ==.xed precision integers for equality using ==.
n Haskell Report. contains a wealth of useful examples of type classes. The additional constraint is essentially saying that we can compare trees of a's for equality as long as we know how to compare a's for equality. Indeed. a static type error would result. a class Eq is de. especially the Prelude. If the context were omitted from the instance declaration.
ned that is slightly larger than the one de.
ned earlier: class Eq a where (==). If a method for a particular operation is omitted in an instance declaration. (/=) x /= y :: a -> a -> Bool = not (x == y) This is an example of a class with two operations. the other for inequality. then the default one de. in this case for the inequality operation /=. one for equality. It also demonstrates the use of a default method.
is used instead.ned in the class declaration. if it exists. the three instances of Eq de. For example.
ned earlier will work perfectly well with the above class declaration. yielding just the right de.
nition of inequality that we want: the logical negation of equality. .
For example.24 5 TYPE CLASSES AND OVERLOADING Haskell also supports a notion of class extension. we may wish to de.
Ord is a subclass of Eq). We say that Eq is a superclass of Ord (conversely. min :: a -> a -> a Note the context in the class declaration. but in addition has a set of comparison operations and minimum and maximum functions: class (Eq a) => Ord a where (<). (>) :: a -> a -> Bool max. (<=).ne a class Ord which inherits all of the operations in Eq. and any type which is an instance of Ord must also be an instance of Eq. (In the next Section we give a fuller de. (>=).
nition of Ord taken from the Prelude.) One bene.
since Ord \implies" Eq. rather than (Eq a. More importantly.t of such class inclusions is shorter contexts: a type expression for a function that uses operations from both the Eq and Ord classes can use the context (Ord a). For example. Ord a). the principal typing of quicksort de. methods for subclass operations can assume the existence of methods for superclass operations. the Ord declaration in the Standard Prelude contains this default method for (<): x < y = x <= y && x /= y As an example of the use of Ord.
1 is: quicksort :: (Ord a) => [a] -> [a] In other words. This typing for quicksort arises because of the use of the comparison operators < and >= in its de. quicksort only operates on lists of values of ordered types.ned in Section 2.4.
Class methods are treated as top level declarations in Haskell.. since classes may have more than one superclass. see x4. Class methods may have additional class constraints on any type variable except the one de. Contexts are also allowed in data declarations.2. They share the same namespace as ordinary variables.. a name cannot be used to denote both a class method and a variable or methods in dierent classes.1. the declaration class (Eq a.nition. Show a) => C a where . Haskell also permits multiple inheritance. For example. creates a class C which inherits operations from both Eq and Show.
the method m could not place any additional class constraints on type a. in this class: class C a where m :: Show b => a -> b the method m requires that type b is in class Show. So far. However. we have been using \. For example.ning the current class. These would instead have to be part of the context in the class declaration.
rst-order" types. But Tree by itself is a type constructor. For example. There are . and as such takes a type as an argument and returns a type as a result. the type constructor Tree has so far always been paired with an argument. as in Tree Integer (a tree containing Integer values) or Tree a (representing the family of trees containing a values).
The type T a b is parsed as (T a) b. lists. the types f -> g and (->) f g are the same. the type Tree Int Int should produce some sort of an error since the Tree type takes only a single argument. (->) is a type constructor. Similarly. Types such as tuples which use special syntax can be written in an alternative style which allows currying. Note that the type variable f is applied to other types in f a and f b. [Type applications are written in the same manner as function applications. the type system detects typing errors in expressions. the type constructors (as well as the data constructors) are (. Similarly.).] As we know. For functions. rather than Tree a. is an instance of Functor. So. An instance of Functor for type Tree would be: instance Functor Tree where fmap f (Leaf x) = Leaf (f x) fmap f (Branch t1 t2) = Branch (fmap f t1) (fmap f t2) This instance declaration declares that Tree. For tuples.25 no values in Haskell that have this type. and other data types. and so on. how does Haskell detect malformed type expressions? The answer is a second type system which ensures the correctness of types! Each type has an associated kind which ensures that the type is used correctly. (.). and here demonstrates the ability to describe generic \container" types. To begin. But what about errors due to malformed type expressions? The expression (+) 1 2 3 results in a type error since (+) takes only two arguments. Thus we would expect it to be bound to a type such as Tree which can be applied to an argument. the types [a] and [] a are the same. consider the following Functor class (taken from the Prelude): class Functor f where fmap :: (a -> b) -> f a -> f b The fmap function generalizes the map function used previously. but such \higher-order" types can be used in class declarations. Type expressions are classi.. This capability is quite useful. allowing functions such as fmap to work uniformly over arbitrary trees.
Members of the Functor class must all have the kind ! .1. the kind of v must be . See x4. since Integer has the kind . That is. The compiler infers kinds before doing type checking without any need for `kind declarations'. if the value v has type t . Kinds are simple enough that compilers should be able to provide descriptive error messages when kind con icts occur. Kinds stay in the background of a Haskell program except when an erroneous type signature leads to a kind error. a kinding error would result from an declaration such as instance Functor Integer where .ed into dierent kinds which take one of two possible forms: The symbol represents the kind of type associated with concrete data objects. then 1 ! 2 is the kind of types that take a type of kind 1 and return a type of kind 2 ...6 for more information about kinds. .1 and x4. Kinds do not appear directly in Haskell programs. the type Tree Int has the kind . If 1 and 2 are kinds. The type constructor Tree has the kind ! .
26 5 TYPE CLASSES AND OVERLOADING A Dierent Perspective. The . Before going on to further examples of the use of type classes. it is worth pointing out two other views of Haskell's type classes.
A particular object may be an instance of a class. it should be clear that types are not objects. In the following general statement about OOP. and will have a method corresponding to each operation. and in particular there is no notion of an object's or type's internal mutable state. yields a valid summary of Haskell's type class mechanism: \Classes capture common sets of operations. simply substituting type class for class." In contrast to OOP. forming notions of superclasses and subclasses.rst is by analogy with objectoriented programming (OOP). A default method may also be associated with an operation. and type for object. methods are not \looked up" at runtime but are simply passed as higher-order functions. In other words. and permitting inheritance of operations/methods.. We have shown how parametric polymorphism is useful in de. Classes may be arranged hierarchically.
Sometimes. that universal quanti. however.ning families of types by universally quantifying over all types.
The classes used by Haskell are similar to those used in other object-oriented languages such as C++ and Java. Comparison to Other Languages. there are some signi.e. Indeed. a type class). we can think of parametric polymorphism as a kind of overloading too! It's just that the overloading occurs implicitly over all types instead of a constrained set of types (i. However. Type classes can be seen as providing a structured way to do just this. such as those types whose elements can be compared for equality.cation is too broad|we wish to quantify over some smaller set of types.
cant dierences: Haskell separates the de.
nition of a type from the de.
nition of the methods associated with that type. A class in C++ or Java usually de.
In Haskell.nes both a data structure (the member variables) and the functions associated with the structure (the methods).. Haskell classes are roughly similar to a Java interface..
There is no access control (such as public or private class constituents) built into the Haskell class system. Instead. Again Here we examine some of the more advanced aspects of type declarations. 6. through the type system.1 The Newtype Declaration A common programming practice is to de.27 C++ and Java attach identifying information (such as a VTable) to the runtime representation of an object. such information is attached logically instead of physically to values. 6 Types. the module system must be used to hide or reveal components of a class. In Haskell.
This would not be possible if Natural were de. Natural. Natural would not be in Num.fromNatural y in if r < 0 then error "Unnatural subtraction" else toNatural r x * y = toNatural (fromNatural x * fromNatural y) Without this declaration. . whose only constructor contains a single Integer.y = let r = fromNatural x . natural numbers can be represented by the type Integer using the following declaration: newtype Natural = MakeNatural Integer This creates an entirely new type. Instances declared for the old type do not carry over to the new one. the whole purpose of this type is to introduce a dierent Num instance.ne a type whose representation is identical to an existing one but which has a separate identity in the type system. For example. In Haskell. Indeed. the newtype declaration creates a new type from an existing one.
. the data declaration incurs extra overhead in the representation of Natural values. All of this works using a data declaration instead of a newtype declaration. However. The use of newtype avoids the extra level of indirection (caused by laziness) that the data declaration would introduce.ned as a type synonym of Integer.
and type declarations.2. [Except for the keyword. the newtype declaration uses the same syntax as a data declaration with a single constructor containing a single .28 6 TYPES. data.3 of the report for a more discussion of the relation between newtype. AGAIN See section 4.
eld. This is appropriate since types de.
ned using newtype are nearly identical to those created by an ordinary data declaration.2 Field Labels The .] 6.
elds within a Haskell data type can be accessed either positionally or by name using .
Consider a data type for a two-dimensional point: data Point = Pt Float Float The two components of a Point are the .eld labels .
rst and second arguments to the constructor Pt. A function such as pointx pointx (Pt x _) :: Point -> Float = x may be used to refer to the .
but.rst component of a point in a more descriptive way. it becomes tedious to create such functions by hand. for large structures. Constructors in a data declaration may be declared with associated .
enclosed in braces.eld names . These .
This is an alternative way to de.eld names identify the components of constructor by name rather than by position.
pointy :: Float} This data type is identical to the earlier de.ne Point: data Point = Pt {pointx.
nition of Point. this declaration also de. The constructor Pt is the same in both cases. However.
nes two .
eld names. pointx and pointy. These .
The expression Pt {pointx=1. The use of ..eld names can be used as selector functions to extract a component from a structure. pointy=2} is identical to Pt 1 2. In this example.
eld names in the declaration of a data constructor does not preclude the positional style of .
eld access. pointy=2} and Pt 1 2 are allowed. both Pt {pointx=1. When constructing a value using .
some .eld names.
elds may be omitted. these absent .
elds are unde.
ned. Pattern matching using .
pointy = y}) = sqrt (x*x + y*y) .eld names uses a similar syntax for the constructor Pt: absPoint (Pt {pointx = x.
3 Strict Data Constructors 29 An update function uses .6.
eld values in an existing structure to .
This is not a destructive update: the update function merely creates a new copy of the object. . If p is a Point. then p {pointx=2} is a point with the same pointy as p but with pointx replaced by 2.ll in components of a new structure.
lling in the speci.
ed .
[The braces used in conjunction with .elds with new values.
However. the braces associated with .6).eld labels are somewhat special: Haskell syntax usually allows braces to be omitted using the layout rule (described in Section 4.
In a type with multiple constructors.eld names must be explicit. selection or update operations using .] Field names are not restricted to types with a single constructor (commonly called `record' types).
A .eld names may fail at runtime. Field labels share the top level namespace with ordinary variables and class methods. This is similar to the behavior of the head function when applied to an empty list.
However.eld name cannot be used in more than one data type in scope. within a data type. the same .
For example.eld name can be used in more than one of the constructors so long as it has the same typing in all cases. g :: Float} | C2 {f :: Int. in this data type data T = C1 {f :: Int. h :: Bool} the .
They make constructors with many components more manageable since . Thus if x is of type T.eld name f applies to both constructors in T. they are simply a convenient syntax for accessing the components of a data structure by name rather than by position. then x {f=5} will work for values created by either of the constructors in T. Field names does not change the basic nature of an algebraic data type.
elds can be added or removed without changing every reference to the constructor. For full details of .
This permits structures that contain elements which.1. each . see Section x4. would lead to an error or fail to terminate. if evaluated.3 Strict Data Constructors Data structures in Haskell are generally lazy : the components are not evaluated until needed. Lazy data structures enhance the expressiveness of Haskell and are an essential aspect of the Haskell programming style. Internally.eld labels and their semantics. 6.2.
eld of a lazy data object is wrapped up in a structure commonly referred to as a thunk that encapsulates the computation de.
ning the .
eld value. all components of a data structure are reduced to values before being placed in the structure. the tuple ('a'. The 'a' may be used without disturbing the other component of the tuple. There are a number of overheads associated with thunks: they take time to construct and evaluate. This thunk is not entered until the value is needed. they occupy space in the heap. thunks which contain errors (?) do not aect other elements of a data structure.?) is a perfectly legal Haskell value. For example. Most programming languages are strict instead of lazy: that is. and they cause the garbage collector to retain other structures needed for the evaluation of the thunk. To avoid these overheads. strictness ags in .
30 7 INPUT/OUTPUT data declarations allow speci.
c .
elds of a constructor to be evaluated immediately. selectively suppressing laziness. A .
There are a number of situations where it may be appropriate to use strictness ags: Structure components that are sure to be evaluated at some point during program execution. Structure components that are simple to evaluate and never cause errors. Types in which partially unde.eld marked by ! in a data declaration is evaluated when the structure is created instead of delayed in a thunk.
the complex number library de. For example.ned values are not meaningful.
nes the Complex type as: data RealFloat a => Complex a = !a :+ !a [note the in.
x de.
nition of the constructor :+.] This de.
nition marks the two components. of the complex number as being strict. This is a more compact representation of complex numbers but this comes at the expense of making a complex number with an unde. the real and imaginary parts.
totally unde. 1 :+ ? for example.ned component.
ned (?). As there is no real need for partially de.
it makes sense to use strictness ags to achieve a more eÆcient representation. It cannot be used in other type signatures or in any other type de. The strictness ag. can only appear in data declarations.ned complex numbers. !. Strictness ags may be used to address memory leaks: structures retained by the garbage collector but no longer necessary for computation.
2.nitions. See x4.1 for further details. They should be used with caution: laziness is one of the fundamental properties of Haskell and adding strictness ags may lead to hard to . although the same eect can be obtained using the seq or !$ functions. It is diÆcult to present exact guidelines for the use of strictness ags. There is no corresponding way to mark function arguments as being strict.
nd in.
writing .nite loops or have other unexpected consequences. 7 Input/Output The I/O system in Haskell is purely functional. Typical actions include reading and setting global variables. programs proceed via actions which examine and modify the current state of the world. In imperative languages. yet has all of the expressive power found in conventional programming languages.
reading input. Haskell's I/O system is built around a somewhat daunting mathematical foundation: the monad . and opening windows. Such actions are also a part of Haskell but are cleanly separated from the purely functional core of the language. However. monads are a conceptual structure into which I/O happens to . understanding of the underlying monad theory is not necessary to program using the I/O system. Rather.les.
It is no more necessary to understand monad theory to perform Haskell I/O than it is to understand group theory to do simple arithmetic.t. A detailed explanation of monads is found in Section 9. .
we will look more deeply into monads later. It's best to think of the I/O monad as simply an abstract data type.31 7.1 Basic I/O Operations The monadic operators that the I/O system is built upon are also used for other purposes. For now. we will avoid the term monad and concentrate on the use of the I/O system. Actions are de..
Instead of using this operator directly. or are a sequential composition of other actions. For example.' in other languages. to hide these sequencing operators under a syntax resembling more conventional languages. Actions which return no interesting values use the unit type. the return value is `tagged' with IO type. In the type system. as described in x3. 7.14. Actions are sequenced using an operator that has a rather cryptic name: >>= (or `bind'). a process similar to joining statements in sequential order using `. Thus the monad serves as the glue which binds together the actions in a program. a pattern bound to the result of an action using <-. A statement is either an action. The do notation can be trivially expanded to >>=. performs some action which returns a character. or a set of local de. The unit type is similar to void in other languages. when invoked. the putChar function: putChar :: Char -> IO () takes a character as an argument but returns nothing useful. (). The keyword do introduces a sequence of statements which are executed in order. the do notation. distinguishing actions from other values. The I/O monad contains primitives which build composite actions. For example. the type of the function getChar is: getChar :: IO Char The IO Char indicates that getChar.ned in system primitives.1 Basic I/O Operations Every I/O action returns a value. we choose some syntactic sugar.
The do notation uses layout in the same manner as let or where so we can omit braces and semicolons with proper indentation.nitions introduced using let.getChar putChar c The use of the name main is important: main is de. Here is a simple program to read and then print a character: main main :: IO () = do c <.
) This program performs two actions in sequence: .ned to be the entry point of a Haskell program (similar to the main function in C). we will have more to say about modules later. (The name main is special only in the module Main. usually IO (). and must have an IO type.
Unlike a let expression where variables are scoped over all de.rst it reads in a character. binding the result to the variable c. and then prints the character.
the variables de.nitions.
are only in scope in the following statements. .ned by <.
some unsafe functions available to get around this problem but these are better left to advanced programmers.2 Programming With Actions I/O actions are ordinary Haskell values: they may be passed to functions. how can we say x + print y in an expression so that y is printed out as the expression evaluates? The answer is that we can't! It is not possible to sneak into the imperative universe while in the midst of purely functional code. 7. such as the if. Any intervening construct. Debugging packages (like Trace) often make liberal use of these `forbidden functions' in an entirely safe manner. The last line of ready should read return (c == 'y'). placed in structures. The return function does just that: return :: a -> IO a The return function completes the set of sequencing primitives. We need to take this boolean and create an action that does nothing but return the boolean as its result.32 7 INPUT/OUTPUT There is still one missing piece. not an action.getLine return (c:l) Note the second do in the else clause. This fact is often quite distressing to programmers used to placing print statements liberally throughout their code during debugging. First. in fact. We are now ready to look at more complicated I/O functions. consider the ready function that reads a character and returns True if the character was a `y': ready ready :: IO Bool = do c <. but how do we return a value from a sequence of actions? For example. must use a new do to initiate further sequences of actions. and used as any other Haskell value. Each do introduces a single chain of statements. Any value `infected' by the imperative world must be tagged as such.getChar if c == '\n' then return "" else do l <. What about the other direction? Can we invoke some I/O actions within an ordinary expression? For example. Consider this list of actions: .Bad!!! This doesn't work because the second statement in the `do' is just a boolean value. There are. A function such as f :: Int -> Int -> Int absolutely cannot do any I/O since IO does not appear in the returned type. We can invoke actions and examine their results using do. the function getLine: getLine getLine :: IO String = do c <.getChar c == 'y' -. The return function admits an ordinary value such as a boolean to the realm of I/O actions.
3 Exception Handling todoList :: [IO ()] todoList = [putChar 'a'.33 7.getChar putChar c] This list doesn't actually invoke any actions|it simply holds them. To join these actions into a single action. a function such as sequence_ is needed: sequence_ :: [IO ()] -> IO () sequence_ [] = return () sequence_ (a:as) = do a sequence as This can be simpli. do putChar 'b' putChar 'c'. do c <.
This pattern of recursion is captured by the foldr function (see the Prelude for a de.ed by noting that do x.1).y is expanded to x >> y (see Section 9.
a better de.nition of foldr).
3 Exception Handling So far. 7. >>. one for each character in the string. mapping an imperative version of putChar over the string would be suÆcient to print it. What would happen if getChar encounters an end of . The return () used here is quite necessary { foldr needs a null action at the end of the chain of actions it creates (especially if there are no characters in the string!). The sequence_ function can be used to construct putStr from putChar: putStr putStr s :: String -> IO () = sequence_ (map putChar s) One of the dierences between Haskell and conventional imperative programming can be seen in putStr. An understanding of the operators upon which do is built is quite useful to the Haskell programmer. In an imperative language. These are usually generalized to arbitrary monads. the map function does not perform any action. we have avoided the issue of exceptions during I/O operations.nition of sequence_ is: sequence_ sequence_ :: [IO ()] -> IO () = foldr (>>) (return ()) The do notation is a useful tool but in this case the underlying monadic operator. In Haskell. Instead it creates a list of actions. is more appropriate. any function with a context including Monad m => works with the IO type. The Prelude and the libraries contains many functions which are useful for sequencing I/O actions. however. The folding operation in sequence_ uses the >> function to combine all of the individual actions into a single action.
le?13 To deal with exceptional conditions such as `.
can be caught and handled within the I/O monad. on the other hand.le not found' We use the term error for ?: a condition which cannot be recovered from such as non-termination or pattern match failure. 13 . Exceptions.
a handling mechanism is used.34 7 INPUT/OUTPUT within the I/O monad. No special syntax or semantics are used. similar in functionality to the one in standard ML. exception handling is part of the de.
nition of the I/O sequencing operations. IOError. Errors are encoded using a special data type. Predicates allow IOError values to be queried. the function isEOFError :: IOError -> Bool determines whether an error was caused by an end-of-. This type represents all possible exceptions that may occur within the I/O monad. This is an abstract type: no constructors for IOError are available to the user. For example.
By making IOError abstract. new sorts of errors may be added to the system without a noticeable change to the data type.le condition. The function isEOFError is de.
An exception handler has type IOError -> IO a. If an error occurs. its result is returned without invoking the handler. IO. For example. If only end-of-. and must be explicitly imported into a program. The catch function associates an exception handler with an action or set of actions: catch :: IO a -> (IOError -> IO a) -> IO a The arguments to catch are an action and a handler. If the action succeeds. it is passed to the handler as a value of type IOError and the action associated with the handler is then invoked.ned in a separate library. this version of getChar returns a newline when an error is encountered: getChar' getChar' :: IO Char = getChar `catch` (\e -> return '\n') This is rather crude since it treats all errors in the same manner.
we can rede. Using getChar'. and produce nested exception handlers.le is to be recognized. The type of ioError is ioError :: IOError -> IO a It is similar to return except that it transfers control to the exception handler instead of proceeding to the next I/O action.. Nested calls to catch are permitted.
ne getLine to demonstrate the use of nested handlers: getLine' getLine' where :: IO String = catch getLine'' (\err -> return ("Error: " ++ show err)) getLine'' = do c <.getChar' if c == '\n' then return "" else do l <.getLine' return (c:l) .
and Handles 35 The nested error handlers allow getChar' to catch end of .4 Files. Channels.7.
For convenience.le while any other error results in a string starting with "Error: " from getLine'. and Handles Aside from the I/O monad and the exception handling mechanism it provides. many of these functions are discussed in the Library Report instead of the main report. Opening a . Haskell provides a default exception handler at the topmost level of a program that prints out the exception and terminates the program. Many of these functions are in the IO library instead of the Prelude and thus must be explicitly imported to be in scope (modules and importing are discussed in Section 11). 7. I/O facilities in Haskell are for the most part quite similar to those in other languages.4 Files. Also. Channels..ned. The getChar function used previously can be de. stdout (standard output). which take a handle as an argument. including stdin (standard input). and stderr (standard error).
ned as: getChar = hGetChar stdin Haskell also allows the entire contents of a .
it may seem that getContents must immediately read an entire .le or channel to be returned as a single string: getContents :: Handle -> IO String Pragmatically.
e. resulting in poor space and time performance under certain conditions.le or channel. However. this is not the case. The key point is that getContents returns a \lazy" (i. non-strict) list of characters (recall that strings are just lists of characters in Haskell). An implementation can be expected to implement this demand-driven behavior by reading one character at a time from the . whose elements are read \by demand" just like any other list.
a Haskell program copies one .le as they are required by the computation. In this example.
le to another: .
getLine catch (openFile name mode) (\_ -> do putStrLn ("Cannot open "++ name ++ "\n") getAndOpenFile prompt mode) By using the lazy getContents function.hGetContents fromHandle hPutStr toHandle contents hClose toHandle putStr "Done.getAndOpenFile "Copy from: " ReadMode toHandle <. the entire contents of the .getAndOpenFile "Copy to: " WriteMode contents <." getAndOpenFile :: String -> IOMode -> IO Handle getAndOpenFile prompt mode = do putStr prompt name <.36 7 INPUT/OUTPUT main = do fromHandle <.
le need not be read into memory all at once. If hPutStr chooses to buer the output by writing the string in .
only one block of the input .xed sized blocks of characters.
The input .le needs to be in memory at once.
7.5 Haskell and Imperative Programming As a .le is closed implicitly when the last character has been read.
getChar if c == '\n' then return "" else do l <. yes. if c == `\n` then return "" else {l := getLine(). the getLine function: getLine = do c <. return c:l}} So. The imperative feel of the monadic code in a program does not detract from the functional aspect of Haskell.nal note. I/O programming raises an important issue: this style looks suspiciously like ordinary imperative programming. But there is one important dierence: There is no special semantics that the user needs to deal with. An experienced functional programmer should be able to minimize the imperative component of the program. and thus the I/O component of a program may appear similar to ordinary imperative code. in the end. For example. has Haskell simply re-invented the imperative wheel? In some sense. only using . In particular. equational reasoning in Haskell is not compromised. The I/O monad constitutes a small imperative sub-language inside Haskell.getLine return (c:l) bears a striking similarity to imperative code (not in any real language) : function getLine() { c := getChar().
imperative languages with functional subsets do not generally have any well-de. The monad cleanly separates the functional and imperative program components.37 the I/O monad for a minimal amount of top-level sequencing. In contrast.
8 Standard Haskell Classes In this section we introduce the prede.ned barrier between the purely functional and imperative worlds.
ned standard type classes in Haskell. We have simpli.
these are described in the Haskell Library Report. the Haskell report contains a more complete description. 8. some of the standard classes are part of the standard Haskell libraries. Also. The de.1 Equality and Ordered Classes The classes Eq and Ord have already been discussed.ed these classes somewhat by omitting some of the less interesting methods in these classes.
nition of Ord in the Prelude is somewhat more complex than the simpli.
In particular.ed version of Ord presented earlier. note the compare method: data Ordering compare = EQ | LT | GT :: Ord a => a -> a -> Ordering The compare method is suÆcient to de.
We can now see that arithmetic sequence expressions can be used to generate lists of any type that is an instance of Enum.] stands for enumFromThen 1 3 (see x3. for example. for instance.'z'] denotes the list of lower-case letters in alphabetical order.. so that. This includes not only most numeric types.. user-de.3.2 The Enumeration Class Class Enum has a set of operations that underlie the syntactic sugar of arithmetic sequences. 8.10 for the formal translation). the arithmetic sequence expression [1. Furthermore. ['a'. but also Char.ne all other methods (via defaults) in this class and is the best way to create Ord instances.
Violet] ) [Red. Violet] Note that such a sequence is arithmetic in the sense that the increment between values is constant.ned enumerated types like Color can easily be given Enum instance declarations. even though the values are not numbers. Green. Most types in Enum can be mapped onto .. Indigo. If so: [Red . Blue.
3 The Read and Show Classes The instances of class Show are those types that can be converted to character strings (typically for I/O). the fromEnum and toEnum convert between Int and a type in Enum. The simplest function in the class Show is show: . for these. The class Read provides operations for parsing character strings to obtain the values they may represent. 8.xed precision integers.
which results in "4". show takes any value of an appropriate type and returns its representation as a character string (list of characters). as in show (2+2).38 8 STANDARD HASKELL CLASSES show :: (Show a) => a -> String Naturally enough. This is .
all that concatenation gets to be a bit ineÆcient. but we typically need to produce more complex strings that may have the representations of many values in them. Speci.ne as far as it goes. as in "The sum of " ++ show x ++ " and " ++ show y ++ " is " ++ show (x+y) ++ "." and after a while.
let's consider a function to represent the binary trees of Section 2. showTree is potentially quadratic in the size of the tree.2. The second argument serves as a sort of string accumulator. To restore linear complexity.1 as a string. and show can now be de.cally. the function shows is provided: shows :: (Show a) => a -> String -> String shows takes a printable value and a string and returns that string with the value's representation concatenated at the front..
This is the default de.ned as shows with the null accumulator.
nition of show in the Show class de.
nition: show x = shows x "" We can use shows to de.
but the presentation of this function (and others like it) can be improved. we can avoid carrying accumulators around. ('|':) . Functions .). We can think of the typing as saying that showsTree maps a tree into a showing function. showsTree l . strings) to a function level. and also avoid amassing parentheses at the right end of long constructions. ('>':) Something more important than just tidying up the code has come about by this transformation: we have raised the presentation from an object level (in this case. by using functional composition: showsTree :: (Show a) => Tree a -> ShowS showsTree (Leaf x) = shows x showsTree (Branch l r) = ('<':) . let's create a type synonym: type ShowS = String -> String This is the type of a function that returns a string representation of something followed by an accumulator string. showsTree r . First. Second.ne a more eÆcient version of showTree.
which is a function that takes a string and returns a list of (a. and if there is more than one possible parse (an ambiguity).String)] Normally. If no parse was possible. The basic idea is a parser for a type a. The standard function reads is a parser for any instance of Read: reads :: (Read a) => ReadS a We can use this function to de. however. the resulting list contains more than one pair. a parser returns a singleton list. String) pairs [9]. let's turn to the inverse problem.8. the result is the empty list.3 The Read and Show Classes 39 like ('<' :) or ("a string" ++) are primitive showing functions. The Prelude provides a type synonym for such functions: type ReadS a = String -> [(a. Now that we can turn trees into strings. containing a value of type a that was read from the input string and the remaining string that follows what was parsed. and we build up more complex functions by function composition.
'|':t) <.reads s] Let's take a moment to examine this function de. (r.ne a parsing function for the string representation of binary trees produced by showsTree.t) <.readsTree s.readsTree t ] = [(Leaf x. '>':u) <. u) | (l. t) | (x. List comprehensions give us a convenient idiom for constructing such parsers:14 readsTree readsTree ('<':s) readsTree s :: (Read a) => ReadS (Tree a) = [(Branch l r.
There are two main cases to consider: If the .nition in detail.
otherwise.rst character of the string to be parsed is '<'. In the . we should have the representation of a branch. we have a leaf.
2. The tree r can be parsed from the beginning of t. The string remaining (following the representation of l) begins with '|'. Call the tail of this string t. subject to the following conditions: 1. calling the rest of the input string following the opening angle bracket s. and u is the tail. the .rst case. 4. 3. The string remaining from that parse begins with '>'. The tree l can be parsed from the beginning of the string s. any possible parse must be a tree Branch l r with remaining string u. Notice the expressive power we get from the combination of pattern matching with list comprehension: the form of a resulting parse is given by the main expression of the list comprehension.
rst two conditions above are expressed by the .
rst generator (\(l. '|':t) is drawn from the list of parses of s"). and the remaining conditions are expressed by the second generator. The second de..ning equation above just says that to parse the representation of a leaf.
g.String)] ) [(5)). " golden rings")] With this understanding.: (reads "5 golden rings") :: [(Integer. e. providing a reads that behaves as one would expect. "")] [] There are a couple of shortcomings in our de.
this lack of uniformity making the function de. allowing no white space before or between the elements of the tree representation.nition of readsTree. the other is that the way we parse our punctuation symbols is quite dierent from the way we parse leaf values and subtrees. One is that the parser is quite rigid.
We can address both of these problems by using the lexical analyzer provided by the Prelude: lex :: ReadS String lex normally returns a singleton list containing a pair of strings: the .nition harder to read.
v) (r. readsTree t. x) | ("<". This would allow us to use the generic overloaded functions from the Prelude to parse and display trees. we would automatically then be able to parse and display many other types containing trees as components. but also does not begin with a valid lexeme after any leading whitespace and comments. including comments. lex u. w) (">". As it turns out. The lexical rules are those of Haskell programs. x) ++ [(Leaf x. If the input string is empty or contains only whitespace and comments. lex returns []. which lex skips. t) (l. t) | (x. [Tree Integer].rst lexeme in the input string and the remainder of the input."")].reads s ] We may now wish to use readsTree and showsTree to declare (Read a) => Tree a an instance of Read and (Show a) => Tree a an instance of Show. along with whitespace. u) ("|". our tree parser now looks like this: readsTree readsTree s :: (Read a) => ReadS (Tree a) = [(Branch l r. if the input is not empty in this sense. used to properly parenthesize expressions containing in. lex w ] <. readsTree and showsTree are of almost the right types to be Show and Read methods The showsPrec and readsPrec methods are parameterized versions of shows and reads. Using the lexical analyzer. for example. lex returns [("". Moreover. t) <<<<<- lex s. readsTree v. The extra parameter is a precedence level.
For types such as Tree. The Show and Read instances for Tree are: instance Show a => Show (Tree a) where showsPrec _ x = showsTree x instance Read a => Read (Tree a) where readsPrec _ s = readsTree s . the precedence can be ignored.x constructors.
4 Derived Instances Alternatively.41 8. the Show instance could be de.
Note that the Show class de.ned in terms of showTree: instance Show a => Show (Tree a) where show t = showTree t This. will be less eÆcient than the ShowS version. however.
nes default methods for both showsPrec and show. allowing the user to de.
Since these defaults are mutually recursive. an instance declaration that de.ne either one of these in an instance declaration.
Other classes such as Num also have these \interlocking defaults".4 Derived Instances Recall the Eq instance for trees we presented in Section 5. where read is a specialization of reads: read :: (Read a) => String -> a This function fails if there is not a unique parse or if the input contains anything more than a representation of one value of type a (and possibly. then. Enum. 8.nes neither of these functions will loop when called. we don't need to go through this tedium every time we need equality operators for a new type. respectively. and two branches are equal i their left and right subtrees are equal. Instances of Ord. Ix. such a declaration is simple|and boring| to produce: we require that the element type in the leaves be an equality type. Read. show) (which should be the identity) to some trees. the Eq instance can be derived automatically from the data declaration if we so specify: data Tree a = Leaf a | Branch (Tree a) (Tree a) deriving Eq The deriving clause implicitly produces an Eq instance declaration just like the one in Section 5. comments and whitespace). two leaves are equal i they contain equal elements. and Show can also be generated by the deriving clause. We refer the interested reader to xD for details of the Read and Show classes. Any other two trees are unequal: instance (Eq a) => Eq (Tree a) where (Leaf x) == (Leaf y) = x == y (Branch l r) == (Branch l' r') = l == l' && r == r' _ == _ = False Fortunately. [More than one class name can be speci. We can test the Read and Show instances by applying (read .
] .
pseudo-code (Lists also have Show and Read instances. in particular. Ord) -. Recall that the built-in list type is semantically equivalent to an ordinary two-constructor type. "cat" < "catalog". for example.) The derived Eq and Ord instances for lists are the usual ones. In practice. Eq and Ord instances are almost always derived. this is the full declaration: data [a] = [] | a : [a] deriving (Eq. as lists of characters. rather than user-de. are ordered as determined by the underlying Char type. character strings. In fact. with an initial substring comparing less than a longer string.42 9 ABOUT MONADS the data declaration. and the arguments of a constructor are compared from left to right. which are not derived.
ned. we should provide our own de. In fact.
nitions of equality and ordering predicates only with some trepidation. for example. confusing readers of the program and confounding manual or automatic program transformations that rely on the (==) predicate's being an approximation to de. An intransitive (==) predicate. could be disastrous. being careful to maintain the expected algebraic properties of equivalence relations and total orders.
Wednesday . probably the most important example is that of an abstract data type in which dierent concrete values may represent the same abstract value. the ordering is that of the constructors in the data declaration.. have a Show instance but not a corresponding Read.] [Wednesday. Friday] [Monday. Friday] ) ) Derived Read (Show) instances are possible for all types whose component types also have Read (Show) instances. For example: data Day = Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday deriving (Enum) Here are some simple examples using the derived instances for this type: [Wednesday . Thursday. Some types. such as the function type (->). and here again. An enumerated type can have a derived Enum instance.. Nevertheless. it is sometimes necessary to provide Eq or Ord instances dierent from those that would be derived. Friday] [Monday.nitional equality.) The textual representation de. (Read and Show instances for most of the standard types are provided by the Prelude. Wednesday.
we obtain show [Monday . In this section we explore monadic programming in more detail. Monads are frequently encountered in Haskell: the IO system is constructed using a monad. .Tuesday.Wednesday]" About Monads Many newcomers to Haskell are puzzled by the concept of monads. a special syntax for monads has been provided (do expressions). For example. above.. and the standard libraries contain an entire module dedicated to monads. Wednesday] 9 ) "[Monday. if we add Show and Read to the deriving clause for type Day.ned by a derived Show instance is consistent with the appearance of constant Haskell expressions of the type in question.
There is no single way of explaining monads that works for everyone. 9.org.1 Monadic Classes This section is perhaps less \gentle" than the others. more explanations can be found at haskell.43 9. Another good introduction to practical programming using monads is Wadler's Monads for Functional Programming [10].1 Monadic Classes The Prelude contains a number of classes de. Here we address not only the language features that involve monads but also try to reveal the bigger picture: why monads are such an important tool and how they are used.
whilst the category theoretic terminology provides the names for the monadic classes and operations.ning monads are they are used in Haskell. A monad is constructed on top of a polymorphic type such as IO. it is not necessary to delve into abstract mathematics to get an intuitive understanding of how to use the monadic classes. These classes are based on the monad construct in category theory. The monad itself is de.
monads are governed by set of laws that should hold for the monadic operations. In the same sense. there is no guarantee of this: both == and /= are separate methods in the Eq class and there is no way to assure that == and =/ are related in this manner. and MonadPlus. In addition to IO. The monad laws give insight into the underlying structure of monads: by examining these laws. Functor. de. we hope to give a feel for how monads are used. This idea of laws is not unique to monads: Haskell includes other operations that are governed. already discussed in section 5. However. at least informally. by laws. but ought be obeyed by any instances of a monadic class. The Functor class. two other types in the Prelude are members of the monadic classes: lists ([]) and Maybe.ned by instance declarations associating the type with the some or all of the monadic classes. None of the monadic classes are derivable. Mathematically. x /= y and not (x == y) ought to be the same for any type of values being compared. Monad. For example. the monadic laws presented here are not enforced by Haskell.
These laws apply to fmap in the class Functor: fmap id = id fmap (f . fmap g These laws ensure that the container shape is unchanged by fmap and that the contents of the container are not re-arranged by the mapping operation. g) = fmap f . returning a container of the same shape. The map function applies an operation to the objects inside a container (polymorphic types can be thought of as containers for values of another type). The Monad class de.nes a single operation: fmap.
nes two basic operators: >>= (bind) and return. infixl 1 >>. >>= class Monad m where (>>=) :: (>>) :: return :: fail :: m >> k = m a -> m a -> a -> m String (a -> m b) -> m b m b -> m b a -> m a m >>= \_ -> k The bind operations. combine two monadic values while the return operation injects . >> and >>=.
The signature of >>= helps us to understand this operation: ma >>= \v -> mb combines a monadic value ma containing values of type a and a function which operates on a value v of type a. The >> function is used when the function does not need the value produced by the .44 9 ABOUT MONADS a value into the monad (container). The result is to combine ma and mb into a monadic value containing b. returning the monadic value mb.
in the IO monad. For example. of course. on the monad. passing the result of the .rst monadic operator. The precise meaning of binding depends. x >>= y performs two actions sequentially.
an action such as 'a' <. This. in the I/O monad. the empty list. The do syntax provides a simple shorthand for chains of monadic operations. e2 = do p <. The laws which govern >>= and return are: return a >>= k = k a m >>= return = m xs >>= return . We will see examples of this shortly. For example.. e2 = e1 >>= (\v -> case v of p -> e2.getChar will call fail if the character typed is not 'a'. The essential translation of do is captured in the following two rules: do e1 . The laws governing the mplus operator are as follows: m `mplus` mzero = m mzero `mplus` m = m The mplus operator is ordinary list concatenation in the list monad. pattern match failure calls the fail operation. .e1. terminates the program since in the I/O monad fail calls error. in turn. Thus the more complex translation is do p <.e1. The I/O monad has no zero element and is not a member of this class. e2 = e1 >> e2 e1 >>= \p -> e2 When the pattern in this second form of do is refutable. _ -> fail "s") where "s" is a string identifying the location of the do statement for possible use in an error message. lists and the Maybe type. the zero value is []. For the other built-in monads.rst into the second. This may raise an error (as in the IO monad) or return a \zero" (as in the list monad). these monadic operations can be understood in terms of passing zero or more values from one calculation to the next.
the signature of >>= becomes: (>>=) :: [a] -> (a -> [b]) -> [b] That is.45 9. The return function creates a singleton list. For lists. These operations should already be familiar: list comprehensions can easily be expressed using the monadic operations de. monadic binding involves joining together a set of calculations for each value in the list.2 Built-in Monads 9.2 Built-in Monads Given the monadic operations and the laws that govern them. what can we build? We have already examined the I/O monad in detail so we start with the two other built-in monads. given a list of a's and a function that maps an a onto a list of b's. binding applies this function to each of the a's in the input and returns all of the generated b's concatenated into a list. When used with lists.
3].3] >>= (\ x -> [1. y <.y) [1.[1.2.2.2.3] >>= (\y -> return (x/=y) >>= (\r -> case r of True -> return (x.return (x /= y) return (x.2.3] .2.[1.3] y <.[1.ned for lists. These following three expressions are all dierent syntax for the same thing: [(x. x /= y] do x <.[1.y) | x <.y) _ -> fail ""))) This de.2.3] True <.
nition depends on the de.
is generating a set of values which is passed on into the remainder of the monadic computation. once for each element of the list.y). Thus x <.31. For example.b]) "ab" "cd" mvLift2 (*) [1. You can think of it as transporting a function from outside the list monad."bc". In this sense. Essentially.[1. mvLift2 (+) [1. For example.2.3] invokes the remainder of the monadic computation three times.2.33] ["ac"."ad". (x. The monad de. returning a value for each possible combination of the two input arguments.nition of fail in this monad as the empty list. f.x y' <.13. The returned expression.4] [] ) ) ) [11. each <."bd"] [] This function is a specialized version of the LiftM2 function in the monad library.30] mvLift2 (\a b->[a. will be evaluated for all possible combinations of bindings that surround it. into the list monad in which computations take on multiple values.3] [10. this function: mvLift2 mvLift2 f x y :: (a -> b -> c) -> [a] -> [b] -> [c] = do x' <.23.21.20. the list monad can be thought of as describing functions of multi-valued arguments.y return (f x' y') turns an ordinary function of two arguments (f) into a function over multiple values (lists of arguments).
.ned for Maybe is similar to the list monad: the value Nothing serves as [] and Just x as [x].
That is.46 9 ABOUT MONADS 9. What they really provide is modularity. by de.3 Using Monads Explaining the monadic operators and their associated laws doesn't really show what monads are good for.
the state monad.ning an operation monadically. and then build a more complex monad with a similar de..
S) runSM s0 (SM c) = c s0 This example de.s)) -.run a computation in the SM monad runSM :: S -> SM a -> (a. f s)) -.S)) -. a state monad built around a state type S looks like this: data SM a = SM (S -> (a. Brie y.defines state propagation SM c1 >>= fc2 = SM (\s0 -> let (r.extracts the state from the monad readSM :: SM S readSM = SM (\s -> (s.The monadic type instance Monad SM where -.s1) = c1 s0 SM c2 = fc2 r in c2 s1) return k = SM (\s -> (k.nition.updates the state of the monad updateSM :: (S -> S) -> SM () -.s)) -.alters the state updateSM f = SM (\s -> (().
nes a new type. SM. a computation of type SM t de. to be a computation that implicitly carries a type S. That is.
The de.nes a value of type t while also interacting with (reading and writing) the state of type S.
nition of SM is simple: it consists of functions that take a state and produce two results: a returned value (of any type) and an updated state. This instance declaration de. The newtype declaration is often used here instead of data. We can't use a type synonym here: we need a type name like SM that can be used in instance declarations.
nes the `plumbing' of the monad: how to sequence two computations and the de.
nition of an empty computation. Sequencing (the >>= operator) de.
to the function that returns the second computation. Finally. c2. r. into c1. then passes the value coming out of this computation.nes a computation (denoted by the constructor SM) that passes an initial state. The de. s0. the state coming out of c1 is passed into c2 and the overall result is the result of c2.
we also need some monadic primitives. .nition of return is easier: return doesn't change the state at all. it only serves to bring a value into the monad. For example. While >>= and return are the basic monadic sequencing operations. A monadic primitive is simply an operation that uses the insides of the monad abstraction and taps into the `wheels and gears' that make the monad work. in the IO monad.
a change to the de.3 Using Monads 47 operators such as putChar are primitive since they deal with the inner workings of the IO monad. Similarly.9. Note that these depend on the inner structure of the monad . our state monad uses two primitives: readSM and updateSM.
nition of the SM type would require a change to these primitives. The de.
we need a function that runs computations in the monad. Finally.n). runSM. This takes an initial state and a computation and yields both the returned value of the computation and the .
Looking at the bigger picture. what we are trying to do is de.nal state.
we proceed on to a more complex example that includes the state monad.ne an overall computation as a series of steps (functions with type SM a). Rather than present any examples using this simple state monad. the use (or non-use) of the state is hidden: we don't invoke or sequence our computations dierently depending on whether or not they use S. However. sequenced using >>= and return. We de. These steps may interact with the state (via readSM or updateSM) or may ignore the state.
ne a small embedded language of resourceusing calculations. to build a library of operations and types speci. That is. we build a special purpose language implemented as a set of Haskell types and functions. Such languages use the basic tools of Haskell. functions and types.
when the resource is unavailable. If the resource is available. consider a computation that requires some sort of resource. In this example.cally tailored to a domain of interest. The de. We use the type R to denote a computation using resources controlled by our monad. computation proceeds. the computation suspends.
Left v) -> let R c2 = fc2 v in c2 r' (r'. Right pc1) -> (r'. Either a (R a))) Each computation is a function from available resources to remaining resources. (Left v))) The Resource type is used in the same manner as the state in the state monad. or a suspended computation. Right (pc1 >>= fc2))) return v = R (\r -> (r. of type R a. coupled with either a result. capturing the work done up to the point where resources were exhausted. The Monad instance for R is as follows: instance Monad R where R c1 >>= fc2 = R (\r -> case c1 r of (r'. This de. of type a.nition of R is as follows: data R a = R (Resource -> (Resource.
pc1. or a suspended computation.nition reads as follows: to combine two `resourceful' computations. v. The result will be either a value. which are used to determine the next computation (the call fc2 v). and remaining resources. pass the initial resources into c1. . and resources remaining at the point of suspension. c1 and fc2 (a function producing c2).
48 9 ABOUT MONADS The suspension must take the second computation into consideration: pc1 suspends only the .
The de.rst computation. so we must bind c2 to this to produce a suspension of the overall computation. c1.
This instance declaration de.nition of return leaves the resources unchanged while moving v into the monad.
nes the basic structure of the monad but does not determine how resources are used. We will demonstrate a very simple de. This monad could be used to control many types of resource or implement many dierent types of resource usage policies.
So far. Left v) else (r. If no steps are available. representing available computation steps: type Resource = Integer This function takes a step unless no steps are available: step step v :: a -> R a = c where c = R (\r -> if r /= 0 then (r-1.nition of resources as an example: we choose Resource to be an Integer. Right c)) The Left and Right constructors are part of the Either type. we have the tools to de. the step function suspends the current computation (this suspension is captured in c) and passes this suspended computation back into the monad. This function continues computation in R by returning v so long as there is at least one computational step resource available.
Finally. we need to address how computations in this monad are expressed. Consider an increment function in our monad: inc inc i :: R Integer -> R Integer = do iValue <.i step (iValue+1) This de.ne a sequence of \resourceful" computations (the monad) and we can express a form of resource usage using step.
This de.is necessary to pull the argument value out of the monad. The <. the type of iValue is Integer instead of R Integer.nes increment as a single step of computation.
nition isn't particularly satisfying. compared to the standard de. though.
Consider the de. Can we instead \dress up" existing operations like + so that they work in our monadic world? We'll start with a set of lifting functions. These bring existing functionality into the monad.nition of the increment function.
Using lift1. f. we add lift2: .nition of lift1 (this is slightly dierent from the liftM1 found in the Monad library): lift1 lift1 f :: (a -> b) -> (R a -> R b) = \ra1 -> do a1 <. and creates a function in R that executes the lifted function in a single step. First. inc becomes inc inc i :: R Integer -> R Integer = lift1 (i+1) This is better but still not ideal.ra1 step (f a1) This takes a function of a single argument.
Using lift2.3 Using Monads lift2 lift2 f :: (a -> b -> c) -> (R a -> R b -> R c) = \ra1 ra2 -> do a1 <.ra1 a2 <. this de. fromInteger The fromInteger function is applied implicitly to all integer constants in a Haskell program (see Section 10. .ra2 step (f a1 a2) Notice that this function explicitly sets the order of evaluation in the lifted function: the computation yielding a1 occurs before the computation for a2.49 9.3).
. We can now.nition allows integer constants to have the type R Integer.
To express interesting computations in R we will need a conditional.nally. The idea of providing new. we name the function ifR: ifR ifR tst thn els :: R Bool -> R a -> R a -> R a = do t <. Since we can't use if (it requires that the test be of type Bool instead of R Bool)..
Monads are particularly useful for encapsulating the semantics of these embedded languages in a clean and modular way. We're now ready to actually run some programs. This function runs a program in M given a maximum number of computation steps: .nitions for existing operations like + or if is an essential part of creating an embedded language in Haskell.
Left v) -> Just v _ -> Nothing We use the Maybe type to deal with the possibility of the computation not .50 9 ABOUT MONADS run run s (R p) :: Resource -> R a -> Maybe a = case (p s) of (_.
returning the value of the . we can add some more interesting functionality to this monad. Consider the following function: (|||) :: R a -> R a -> R a This runs two computations in parallel. We can now compute run 10 (fact 2) run 10 (fact 20) ) ) Just 2 Nothing Finally.nishing in the allotted number of steps.
One possible de.rst one to complete.
The de. Left v) -> (r+r'-1.r' must be 0 let R next = f c1' in next (r+r'-1)) This takes a step in c1. returning its value of c1 complete or. Left v) (r'. The oneStep function takes a single step in its argument. Right c1') -> -'. if c1 returns a suspended computation (c1'). either returning an evaluated value or passing the remainder of the computation into f.
If a .nition of oneStep is simple: it gives c1 a 1 as its resource argument.. (Our de. We can now evaluate expressions like run 100 (fact (-1) ||| (fact 3)) without looping since the two calculations are interleaved.
nition of fact loops for -1). For example. While this example is perhaps more advanced than others in this tutorial. it serves to illustrate the power of monads as a tool for de. Many variations are possible on this basic structure. allowing computations in M to interact with the outside world. We could also embed this monad inside the standard IO monad. we could extend the state to include a trace of the computation steps.
ning the basic semantics of a system. We also present this example as a model of a small Domain Speci.
c Language. something Haskell is particularly good at de.
see haskell. Of particular interest are Fran.org for many more examples. and Haskore. a language of reactive animations. Many other DSLs have been developed in Haskell. . a language of computer music.ning.
which in turn are based on Common Lisp [8].51 10 Numbers Haskell provides a rich collection of numeric types. are dynamically typed.) The standard types include . however. (Those languages. based on those of Scheme [7].
these include. ratios (rational numbers) formed from each integer type. is a subclass of Ord as well. (*) negate. The Num class provides several basic operations common to all numeric types. and single. abs :: (Num a) => a -> a -> a :: (Num a) => a -> a [negate is the function applied by Haskell's only pre. (-). We outline here the basic characteristics of the numeric type class structure and refer the reader to x6. We also note that Num is a subclass of Eq.and double-precision real and complex oating-point.xed. The subclass Real of Num. negation. and absolute value: (+). among others.and arbitrary-precision integers.1 Numeric Class Structure The numeric type classes (class Num and those that lie below it) account for many of the standard Haskell classes. addition. however. but not of Ord. this is because the order predicates do not apply to complex numbers. subtraction. multiplication.4 for details. 10.
-x*y is equivalent to negate (x*y). For example. minus.x operator. (Pre. we can't call it (-). because that is the subtraction function. so this name is provided instead.
x minus has the same syntactic precedence as in.
Integral b) => a -> b . ceiling: :: (Fractional a.)] Note that Num does not provide a division operator. with a range equivalent to at least 29-bit signed binary). this means that there is no attempt to provide Gaussian integers. A particular Haskell implementation might provide other integral types in addition to these. and a collection of functions that round to integral values by diering rules: properFraction truncate. two dierent kinds of division operators are provided in two non-overlapping subclasses of Num: The class Integral provides whole-number division and remainder operations. All other numeric types fall in the class Fractional. which provides the ordinary division operator (/). Integral b) => a -> (b. is lower than that of multiplication.x minus. The further subclass Floating contains trigonometric. floor. rather than of Num directly. machine integers. also known as \bignums") and Int (bounded. Note that Integral is a subclass of Real.a) :: (Fractional a. which decomposes a number into its whole and fractional parts. which. The RealFrac subclass of Fractional and Real provides a function properFraction. The standard instances of Integral are Integer (unbounded or mathematical integers. logarithmic. of course. round. and exponential functions.
52 10 NUMBERS The RealFloat subclass of Floating and RealFrac provides some specialized functions for eÆcient access to the components of a oating-point number. the exponent and signi.
cand. (Rational is a type synonym for Ratio Integer. which restricts the argument type.3 Numeric Coercions and Overloaded Literals The Standard Prelude and libraries provide several overloaded functions that serve as explicit coercions: . 10. ratios are not unique. Complex (found in the library Complex) is a type constructor that makes a complex type in class Floating from a RealFloat type: data (RealFloat a) => Complex a = !a :+ !a deriving (Eq. for instance.3. although the real part of x:+y is always x. respectively. we can use it in pattern matching: conjugate conjugate (x:+y) :: (RealFloat a) => Complex a -> Complex a = x :+ (-y) Similarly. Text) The ! symbols are strictness ags. Instead of pattern matching. Since :+ is a data constructor. denominator :: (Integral a) => Ratio a -> a Why the dierence? Complex numbers in cartesian form are unique|there are no nontrivial identities involving :+. the standard complex types are Complex Float and Complex Double. 10. rationals use the `%' function to form a ratio from two integers. Int. Float.) Ratio. The others are made from these by type constructors. but have a canonical (reduced) form that the implementation of the abstract data type must maintain. thus.2 Constructed Numbers Of the standard numeric types. is an abstract type constructor. On the other hand. the arguments are the cartesian real and imaginary parts. it is not necessarily the case. the type constructor Ratio (found in the Rational library) makes a rational type in class RealFrac from an instance of Integral. component extraction functions are provided: (%) :: (Integral a) => a -> a -> Ratio a numerator. however. that numerator (x%y) is equal to x. Integer. We can also see from the data declaration that a complex number is written x :+ y . Notice the context RealFloat a. and Double are primitive. The standard types Float and Double fall in class RealFloat. these were discussed in Section 6. Instead of a data constructor like :+.
7 has the type (Num a) => a. and 7.53 10. This means that we can use numeric literals in generic numeric functions.5 This rather indirect way of overloading numerals has the additional advantage that the method of interpreting a numeral as a number of a given type can be speci. Num b) => a -> b (RealFrac a. Thus. toInteger = fromRational . toRational Two of these are implicitly used to provide overloaded numeric literals: An integer numeral (without a decimal point) is actually equivalent to an application of fromInteger to the value of the numeral as an Integer. for example: halve halve x :: (Fractional a) => a -> a = x * 0.4 Default Numeric Types fromInteger fromRational toInteger toRational fromIntegral fromRealFrac :: :: :: :: :: :: (Num a) => Integer -> a (Fractional a) => Rational -> a (Integral a) => a -> Integer (RealFrac a) => a -> Rational (Integral a. Similarly. a oating numeral (with a decimal point) is regarded as an application of fromRational to the value of the numeral as a Rational. Fractional b) => a -> b fromIntegral fromRealFrac = fromInteger .3 has the type (Fractional a) => a.
For example. the Num instance of (RealFloat a) => Complex a contains this method: fromInteger x = fromInteger x :+ 0 This says that a Complex instance of fromInteger is de.ed in an Integral or Fractional instance declaration (since fromInteger and fromRational are operators of those classes. respectively).
even user-de. In this manner.ned to produce a complex number whose real part is supplied by an appropriate RealFloat instance of fromInteger.
quaternions) can make use of overloaded numerals. As another example. recall our .ned numeric types (say.
rst de.
the most general type of inc is (Num a) => a->a. since it is more speci.nition of inc from Section 2: inc inc n :: Integer -> Integer = n+1 Ignoring the type signature. however. The explicit type signature is legal.
c than the principal type (a more general type signature would cause a static error).4 Default Numeric Types Consider the following function de. and in this case would cause something like inc (1::Float) to be ill-typed. The type signature has the eect of restricting inc's type. 10.
and since 2 has the type (Num a) => a.5) The exponentiation function (^) (one of three dierent standard exponentiation operators with dierent typings. there . see x6.nition: rms rms x y :: (Floating a) => a -> a -> a = sqrt ((x^2 + y^2) * 0.5) has the type (Num a. the type of x^2 is (Num a. Integral b) => a -> b -> a.8. Integral b) => a. This is a problem.
54 11 MODULES is no way to resolve the overloading associated with the type variable b. but has otherwise vanished from the type expression. the programmer has speci. Essentially. since it is in the context.
ed that x should be squared. but has not speci.
ed whether it should be squared with an Int or an Integer value of two. we can . Of course.
above). and the . any Integral instance will do. if at least one of its classes is numeric and all of its classes are standard. When an ambiguous type variable is discovered (such as b. consisting of the keyword default followed by a parenthesized. however.x this: rms x y = sqrt ((x ^ (2::Integer) + y ^ (2::Integer)) * 0.. Because of the dierence between the numeric and general cases of the overloading ambiguity problem. Haskell provides a solution that is restricted to numbers: Each module may contain a default declaration. because there. In fact. whereas here. the default list is consulted.5) It's obvious that this sort of thing will soon grow tiresome. commaseparated list of numeric monotypes (types with no variables).
11 Modules A Haskell program consists of a collection of modules. Float) is in eect. if the default declaration default (Int. For example.4 for more details. Rational. which provides no defaults.rst type from the list that will satisfy the context of the type variable is used. but (Integer. the ambiguous exponent above will be resolved as type Int. The top level of a module contains any of the various declarations we have discussed: .3. Double) may also be appropriate.) The \default default" is (Integer. (See x4. A module in Haskell serves the dual purpose of controlling name-spaces and creating abstract data types. Double). Very cautious programmers may prefer default ().
function de. type signatures. data and type declarations. class and instance declarations.xity declarations.
nitions. and pattern bindings. Except for the fact that import declarations (to be described shortly) must appear .
Haskell's module design is relatively conservative: the name-space of modules is completely at. the declarations may appear in any order (the top-level scope is mutually recursive). and modules are in no way \.rst.
" Module names are alphanumeric and must begin with an uppercase letter. There is no formal connection between a Haskell module and the .rst-class.
In particular.le system that would (typically) support it. there is no connection between module names and .
and more than one module could conceivably reside in a single .le names.
le (one module may even span several .
les). a particular implementation will most likely adopt conventions that make the connection between modules and . Of course.
here's an example for a module whose name is Tree: . a module is really just one big declaration which begins with the keyword module. Technically speaking.les more stringent.
55 11.1 Quali.
and thus the declarations must all line up in the same column (typically the . [Because of the where keyword.ed Names module Tree ( Tree(Leaf.1.2.Branch). layout is active at the top level of a module. they were given as examples in Section 2. fringe ) where data Tree a = Leaf a | Branch (Tree a) (Tree a) fringe :: Tree a -> [a] fringe (Leaf x) = [x] fringe (Branch left right) = fringe left ++ fringe right The type Tree and the function fringe should be familiar.
The Tree module may now be imported into some other module: module Main (main) where import Tree ( Tree(Leaf. as in Tree(Leaf. The names in an export list need not be local to the exporting module..).1 Quali. (In the above example everything is explicitly exported. If the export list following the module keyword is omitted. Leaf. As short-hand. Exporting a subset of the constructors is also possible.Branch). fringe ) main = print (fringe (Branch (Leaf 1) (Leaf 2))) The various items being imported into and exported out of a module are called entities. Note the explicit import list in the import declaration. we could also write Tree(. 11.) Note that the name of a type and its constructors have be grouped together. all of the names bound at the top level of the module would be exported. Also note that the module name is the same as that of the type. any name in scope may be listed in an export list. and fringe.rst). omitting it would cause all entities exported from Tree to be imported.] This module explicitly exports Tree. Branch. this is allowed.Branch). so the eect would be the same.
What if two imported modules contain dierent entities with the same name? Haskell solves this problem using quali.ed Names There is an obvious problem with importing names directly into the namespace of module.
An import declaration may use the qualified keyword to cause the imported names to be pre.ed names .
These pre.xed by the name of the module imported.
[Quali.xes are followed by the `.' character without intervening whitespace.
A. x are quite dierent: the .x and A .ers are part of the lexical syntax. Thus.
rst is a quali.
ed name and the second a use of the in.
' function.] For example.x `. using the Tree module introduced above: .
.fringe (Branch (Leaf 1) (Leaf 2))) Some Haskell programmers prefer to use quali.56 11 MODULES module Fringe(fringe) where import Tree(Tree(.)) fringe :: Tree a -> [a] -.A different definition of fringe fringe (Leaf x) = [x] fringe (Branch x y) = fringe x module Main where import Tree ( Tree(Leaf. fringe ) import qualified Fringe ( fringe ) main = do print (fringe (Branch (Leaf 1) (Leaf 2))) print (Fringe.Branch).
Others prefer short names and only use quali.ers for all imported entities. making the source of each name explicit with every use.
Quali.ers when absolutely necessary.
The compiler knows whether entities from dierent modules are actually the same. For example. all operations on the ADT are done at an abstract level which does not depend on the representation.ers are used to resolve con icts between dierent entities which have the same name. 11.2 Abstract Data Types Aside from controlling namespaces. a suitable ADT for it might include the following operations: data Tree a leaf branch cell left. leaf. right isLeaf -:: :: :: :: :: just a -> Tree Tree Tree Tree the type name Tree a a -> Tree a -> Tree a a -> a a -> Tree a a -> Bool A module supporting this is: module TreeADT (Tree. For example. left. such name clashes are allowed: an entity can be imported by various routes without con ict. isLeaf) where data Tree a = Leaf a | Branch (Tree a) (Tree a) leaf branch cell (Leaf a) left (Branch l r) right (Branch l r) isLeaf (Leaf _) isLeaf _ = = = = = = = Leaf Branch a l r True False . the characteristic feature of an ADT is that the representation type is hidden. branch. modules provide the only way to build abstract data types (ADTs) in Haskell. right. although the Tree type is simple enough that we might not normally make it abstract. But what if the same entity is imported from more than one module? Fortunately. cell.
and the only way to build or take apart trees outside of the module is by using the various (abstract) operations. This is useful for explicitly excluding names that are used for other purposes without having to use quali. 11. Of course.3 More Features Here is a brief overview of some other aspects of the module system.3 More Features 57 Note in the export list that the type name Tree appears alone (i.11. Thus Leaf and Branch are not exported. An import declaration may selectively hide entities using a hiding clause in the import declaration. without its constructors). See the report for more details.e. the advantage of this information hiding is that at a later time we could change the representation type without aecting users of the type.
An import may contain an as clause to specify a dierent quali.ers for other imported names from the module.
This can be used to shorten quali.er than the name of the importing module.
ers from modules with long names or to easily adapt to a change in module name without changing all quali.
allowing the name length to be de. An explicit import of the Prelude overrides the implicit import of all Prelude names.ers. Thus. import Prelude hiding length will not import length from the Standard Prelude. Programs implicitly import the Prelude module.
Most of these are obvious|for instance. there cannot be more than one instance declaration for that combination of type and class anywhere in the program. or as ordinary variables. Class methods may be named either in the manner of data constructors. Other rules are not so obvious|for example. 12 Typing Pitfalls This short section give an intuitive description of a few common problems that novices run into using Haskell's type system. Every module exports all of its instance declarations and every import brings all instance declarations into scope. Although Haskell's module system is relatively conservative.ned dierently. The reader should read the Report for details (x5). for a given type and class. there are many rules concerning the import and export of values. it is illegal to import two dierent entities having the same name into the same scope. in parentheses following the class name. . Instance declarations are not explicitly named in import or export lists.
because identi.58 12 TYPING PITFALLS 12.1 Let-Bound Polymorphism Any language using the Hindley-Milner type system has what is called let-bound polymorphism.
. but length's result is an Int. In particular. The reason for this restriction is related to a subtle type ambiguity and is explained in full detail in the Report (x4. bound to a lambda abstraction whose principal type is a->a. this program is illegal: let f g = (g []. is used within f in two dierent ways: once with type [a]->[a].ers not bound using a let or where clause (or at the top level of a module) are limited with respect to their polymorphism.Wrong! (/) requires fractional arguments. A common numeric typing error is something like the following: average xs = sum xs / length xs -. More general numeric expressions sometimes cannot be quite so generic. g 'a') in f (\x->x) -. 12. The type mismatch must be corrected with an explicit coercion: average average xs :: (Fractional a) => [a] -> a = sum xs / fromIntegral (length xs) 12.5. a lambda-bound function (i.2 Numeric Overloading It is easy to forget at times that numerals are overloaded. and once with type Char->Char. one passed as argument to another function) cannot be instantiated in two dierent ways. and not implicitly coerced to the various numeric types.e.5).3 The Monomorphism Restriction The Haskell type system contains a restriction related to type classes that is not found in ordinary Hindley-Milner type systems: the monomorphism restriction.ill-typed expression because g. as in many other languages. A simpler explanation follows: The monomorphism restriction says that any identi. For example.
er bound by a pattern binding (which includes bindings to a single identi.
must be monomorphic. and having no explicit type signature.er). An identi.
er is monomorphic if is either not overloaded. or is overloaded but is used in at most one speci.
Violations of this restriction result in a static type error.c overloading and is not exported. A common violation of the restriction happens with functions de. Note that any type signature will do (as long it is type correct). The simplest way to avoid the problem is to provide an explicit type signature.
as in this de.ned in a higher-order manner.
nition of sum from the Standard Prelude: sum = foldl (+) 0 .
59 As is. We can . this would cause a static type error.
in order to assure eÆcient access to array elements.x the problem by adding the type signature: sum :: (Num a) => [a] -> a Also note that this problem would not have arisen if we had written: sum xs = foldl (+) 0 xs because the restriction only applies to pattern bindings. 13 Arrays Ideally. but pragmatically. arrays in a functional language would be regarded simply as functions from indices to values. which are isomorphic to . we need to be sure we can take advantage of the special properties of the domains of these functions.
therefore. either requiring a new copy of an array for each incremental rede.nite contiguous subsets of the integers. producing a new array that diers from the old one only at the given index. we have a function that produces an empty array of a given size and another that takes an array. Haskell. but as abstract data types with a subscript operation. In the incremental case. and a value. does not treat arrays as general functions with an application operation. an index. Two main approaches to functional arrays may be discerned: incremental and monolithic definition. Obviously. a naive implementation of such an array semantics would be intolerably ineÆcient.
nition. The monolithic approach. Any module using arrays must import the Array module. without reference to intermediate array values.1 Index types The Ix library de. Although Haskell has an incremental array update operator. 13. the main thrust of the array facility is monolithic. serious attempts at using this approach employ sophisticated static analysis and clever run-time devices to avoid excessive copying. or taking linear time for array lookup. Arrays are not part of the Standard Prelude|the standard library contains the array operators. on the other hand. constructs an array all at once. thus.
a) a -> Int inRange :: (a.a) -> a -> Bool Instance declarations are provided for Int. instances may be automatically derived for enumerated and tuple types. Char. Integer. Note that the . in addition. and tuples of Ix types up to length 5.nes a type class of array indices: class (Ord a) => Ix a where range :: (a. Bool. We regard the primitive types as vector indices. and tuples as indices of multidimensional rectangular arrays.a) -> [a] index :: (a.
these are typically the bounds (.rst argument of each of the operations of class Ix is a pair of indices.
9). such bounds would be written in a . zero-origin vector with Int indices would be (0.rst and last indices) of an array. For example. (In many other languages.1).100)).(100. while a 100 by 100 1-origin matrix might have the bounds ((1. the bounds of a 10-element.
60 13 ARRAYS form like 1:100. 1:100. but the present form .
4] [(0. (1.) Finally.0).1) ) 4 13.ts the type system better.2)) ) ) [0. (For a tuple type.0).2 Array Creation Haskell's monolithic array creation function forms an array from a pair of bounds and a list of index-value pairs (an association list): array :: (Ix a) => (a.(1.a) -> [(a. For example. this test is performed component-wise.2)) (1. (1. (0. for example. for example: index (1.4) range ((0.9) 2 1 ) index ((0. is a de.2. range (0.0). since each bound is of the same type as a general index.2)] The inRange predicate determines whether an index lies between a given pair of bounds. the index operation allows a particular element of an array to be addressed: given a bounds pair and an in-range index.1). (1. the operation yields the zero-origin ordinal of the index within the range.0).(1. in index order.1.b)] -> Array a b Here.) The range operation takes a bounds pair and produces the list of indices lying between those bounds.3.1). (0.2).
i*i) | i <.100) [(i.. Array subscripting is performed with the in.[1. this usage results in array expressions much like the array comprehensions of the language Id [6]. in fact.100]] This array expression is typical in using a list comprehension for the association list.nition of an array of the squares of numbers from 1 to 100: squares = array (1.
f i) | i <.100) We might generalize this example by parameterizing the bounds and the function to be applied to each index: mkArray mkArray f bnds :: (Ix a) => (a -> b) -> (a.a) -> Array a b = array bnds [(i. we could de.x operator !. and the bounds of an array can be extracted with the function bounds: squares!7 ) 49 bounds squares ) (1.range bnds] Thus.
ne squares as mkArray (\i -> i * i) (1.100). Many arrays are de.
n) ([(0. 1).n]]) .[2. for example. a!(i-2) + a!(i-1)) | i <. that is. (1. we have a function returning an array of Fibonacci numbers: fibs :: Int -> Array Int Int fibs n = a where a = array (0.. 1)] ++ [(i. Here.ned recursively. with the values of some elements depending on the values of others.
61 13. in which elements of the .3 Accumulation Another example of such a recurrence is the n by n wavefront matrix.
rst row and .
n)) ([((1.1). 1) | i <.n].n]] ++ [((i. a!(i. and north: wavefront wavefront n :: Int -> Array (Int.n]]) The wavefront matrix is so called because in a parallel implementation. the recurrence dictates that the computation can begin with the ..(n..j-1) + a!(i-1.j).1).j)) | i <.n]] ++ [((i.j-1) + a!(i-1.[1.[2..Int) Int = a where a = array ((1. northwest.rst column all have the value 1 and other elements are sums of their neighbors to the west..j).[2. 1) | j <.[2. j <.
It is important to note. traveling from northwest to southeast.rst row and column in parallel and proceed as a wedge-shaped wave. that no order of computation is speci. however.
ed by the association list. In each of our examples so far. we have given a unique association for each index of the array and only for the indices within the bounds of the array. we must do this in general for an array be fully de. and indeed.
but the value of the array at that index is then unde.ned. An association with an out-of-bounds index results in an error. if an index is missing or appears more than once. there is no immediate error. however.
ned.a) -> [Assoc a c] -> Array a b The .3 Accumulation We can relax the restriction that an index appear at most once in the association list by specifying how to combine multiple values associated with a single index. the result is called an accumulated array: accumArray :: (Ix a) -> (b -> c -> b) -> b -> (a. so that subscripting the array with such an index yields an error. 13.
and the initial value.9) . zero. a table of the number of occurrences of each value within the bounds: hist hist bnds is :: (Ix a. inRange bnds i] Suppose we have a collection of measurements on the interval [a .a) * s) s = 10 / (b . that is. this function takes a pair of bounds and a list of values (of an index type) and yields a histogram.a) -> [a] -> Array a b = accumArray (+) 0 bnds [(i. and we want to divide the interval into decades and count the number of measurements within each: decades decades a b :: (RealFrac a) => a -> a -> [a] -> Array Int Int = hist (0. 1) | i <. for example. and the remaining arguments are bounds and an association list.rst argument of accumArray is the accumulating function. as with the array function. Typically. the accumulating function is (+).a) . map decade where decade x = floor ((x . b ). Integral b) => (a.is. the second is an initial value (the same for each element of the array).
written as the in. Haskell also has an incremental array update function.62 13 ARRAYS 13.4 Incremental updates In addition to the monolithic array creation functions.
usually containing a proper subset of the indices of the array: (//) :: (Ix a) => Array a b -> [(a.x operator //. is written a // [(i. the simplest case. the indices in the association list must be unique for the values to be de. The reason for the square brackets is that the left argument of (//) is an association list. an array a with element i updated to v.b)] -> Array a b As with the array function. v)].
however.jHi]]) where ((iLo..j). here is a function to interchange two rows of a matrix: swapRows :: (Ix a.jHi]] ++ [((i'.jHi)) = bounds a 13. j))] ] where ((iLo..ned. ((i'.(iHi. Enum b) => a -> a -> Array (a. a!(i'.b) c swapRows i i' a = a // ([((i . a!(i . Never fear. a slight ineÆciency.5 An example: Matrix Multiplication We complete our introduction to Haskell arrays with the familiar example of matrix multiplication.j).(iHi.[jLo.jLo).[((i .jLo). it's like writing two loops where one will do in an imperative language. a!(i.j). Ix b..jHi].j)) | j <. we can perform the equivalent of a loop fusion optimization in Haskell: swapRows i i' a = a // [assoc | j <.j)). taking advantage of overloading to de. a!(i'.[jLo.j)) | j <.j). assoc <.jHi)) = bounds a The concatenation here of two separate list comprehensions over the same list of j indices is.b) c -> Array (a. For example.[jLo.
we get genericity over index types. Since only multiplication and addition on the element type of the matrices is involved. however.uj')) | otherwise = error "matMult: incompatible bounds" .lj').range (lj. Ix b.lj).k) * y!(k. the four row and column index types need not all be the same. Num d) => Array (a.ui') = ((li.ne a fairly general function. Additionally.range (li.j) | k <.(ui. we require that the left column indices and right row indices be of the same type.uj)) = bounds x ((li'.range (lj'.(ui'. that the bounds be equal: matMult :: (Ix a. Ix c. and in fact.uj') ] where ((li.c) d matMult x y = array resultBounds [((i.lj'). and moreover.b) d -> Array (b. if we are careful to apply only (!) and the operations of Ix to indices.uj)==(li'.uj)]) | i <.uj')) = bounds y resultBounds | (lj. j <. sum [x!(i. we get a function that multiplies matrices of any numeric type unless we try hard not to.c) d -> Array (a. For simplicity.ui).j).(ui.
13.5 An example: Matrix Multiplication 63 As an aside. we can also de.
ui).j)) | i <.uj)) = bounds x ((li'.k) * y!(k.range (lj'. j <.(ui'. sum' [x!(i.uj')) | otherwise = error "matMult: incompatible bounds" We can generalize further by making the function higher-order. Ix c) => ([f] -> g) -> (d -> e -> f) -> Array (a.ui).(ui.k) `star` y!(k.uj)==(li'.range (li.(ui.ne matMult using accumArray.b) d -> Array (b.lj').range (lj.c) e -> Array (a.uj')) = bounds y resultBounds | (lj.j).range (li.uj)) = bounds x ((li'.range (lj. j <.uj') k <.(ui'. resulting in a presentation that more closely resembles the usual formulation in an imperative language: matMult x y = accumArray (+) 0 resultBounds [((i.uj')) | otherwise = error "matMult: incompatible bounds" APL fans will recognize the usefulness of functions like the following: genMatMult maximum (-) genMatMult and (==) With the .ui') = ((li.lj').uj') ] where ((li.ui') = ((li.uj) ] where ((li. simply replacing sum and (*) by functional parameters: genMatMult :: (Ix a.range (lj'.(ui.lj).j).uj)==(li'. Ix b.uj)]) | i <.c) g genMatMult sum' star x y = array resultBounds [((i. x!(i.uj')) = bounds y resultBounds | (lj.j) | k <.lj).lj').(ui.lj').
and the (i . j ) is True if and only if the i -th row of the .rst of these. In the second case. the arguments are numeric matrices. j )-th element of the result is the maximum dierence between corresponding elements of the i -th row and j -th column of the inputs. the arguments are matrices of any equality type. and the result is a Boolean matrix in which element (i .
We could generalize still further by dropping the requirement that the . but merely appropriate for the function parameter star.rst argument and j -th column of the second are equal as vectors. Notice that the element types of genMatMult need not be the same.
clearly. two matrices could be considered conformable as long as the lengths of the columns of the .rst column index and second row index types be the same.
The reader may wish to derive this still more general version. (Hint: Use the index operation to determine the lengths.) .rst and the rows of the second are equal.
Here you will .org.64 14 REFERENCES The Next Stage A large collection of Haskell resources is available on the web at haskell.
Springer Verlag. Report on the Programming Language Haskell 98. pages 113{ 128. [6] R. Special thanks to Erik Meijer for his extensive comments on the new material added for version 1.Davie. SIGPLAN Notices. and David Rochberg at Yale University for their quick readings of earlier drafts of this manuscript. evolution. 1998. The Hugs system is both small and portable { it is an excellent vehicle for learning Haskell. Id (version 90. [10] P. Department of Computer Science Tech Report YALEU/DCS/RR-1106. Conception. 1984. Steele Jr. 21(3):359{411. Laboratory for Computer Science. Wadler. Sandra Loosemore. A Non-strict Purely Functional Language. Mass. . 1985. Cambridge University Press. Yale University. Monads for Functional Programming In Advanced Functional Programming . [8] G. Common Lisp: The Language. Bird. Department of Computer Science Tech Report YALEU/DCS/RR-1105. 1995. Hudak. [9] P..L. Yale University. Martin Odersky. Springer Verlag. and application of functional programming languages. Wadler.nd compilers.0) reference manual.4 of this tutorial. LNCS 925.). Amir Kishon. Introduction to Functional Programming using Haskell. 21(12):37{79. and Nick Carriero. Introduction to Functional Programming System Using Haskell. [3] P. Charles Consel. Massachusetts Institute of Technology. The revised3 report on the algorithmic language Scheme. Burlington. New York. In Proceedings of Conference on Functional Programming Languages and Computer Architecture. References [1] R. Feb 1999. [2] A. [4] Simon Peyton Jones (editor). 15 Acknowledgements Thanks to Patricia Fasel and Mark Mundt at Los Alamos. papers. Prentice Hall. [5] Simon Peyton Jones (editor) The Haskell 98 Library Report. demos. 201. September 1990. Haskell compilers or interpreters run on almost all hardware and operating systems.S. 1992. Technical report. LNCS Vol. and much valuable information about Haskell and functional programming. Digital Press. How to replace failure by a list of successes. Clinger (eds. ACM Computing Surveys. Feb 1999. 1989. Rees and W. Nikhil. December 1986. [7] J.
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