The Gentle Introduction To Haskell
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2  Values, Types, and Other Goodies

Because Haskell is a purely functional language, all computations are done via the evaluation of expressions (syntactic terms) to yield values (abstract entities that we regard as answers). Every value has an associated type. (Intuitively, we can think of types as sets of values.) Examples of expressions include atomic values such as the integer 5, the character 'a', and the function \x -> x+1, as well as structured values such as the list [1,2,3] and the pair ('b',4).

Just as expressions denote values, type expressions are syntactic terms that denote type values (or just types). Examples of type expressions include the atomic types Int (fixed-precision integers), Char (characters), Int->Int (functions mapping Int to Int), as well as the structured types [Int] (homogeneous lists of integers) and (Char,Int) (character, integer pairs).

All Haskell values are "first-class"---they may be passed as arguments to functions, returned as results, placed in data structures, etc. Haskell types, on the other hand, are not first-class. Types in a sense describe values, and the association of a value with its type is called a typing. Using the examples of values and types above, we write typings as follows:

                          5  :: Int
                         'a' :: Char
                         inc :: Int -> Int
                     [1,2,3] :: [Int]
                     ('b',4) :: (Char,Int)

The "::" can be read "has type."

Functions in Haskell are normally defined by a series of equations. For example, the function inc can be defined by the single equation:

inc n          = n+1

An equation is an example of a declaration. Another kind of declaration is a type signature declaration (§4.4.1), with which we can declare an explicit typing for inc:

inc            :: Int -> Int

We will have much more to say about function definitions in Section 3.

For pedagogical purposes, when we wish to indicate that an expression e1 evaluates, or "reduces," to another expression or value e2, we will write:

e1 => e2

For example, note that:

inc (inc 3) => 5

Haskell's static type system defines the formal relationship between types and values (§4.1.3). The static type system ensures that Haskell programs are type safe; that is, that the programmer has not mismatched types in some way. For example, we cannot generally add together two characters, so the expression 'a'+'b' is ill-typed. The main advantage of statically typed languages is well-known: All type errors are detected at compile-time. Not all errors are caught by the type system; an expression such as 1/0 is typable but its evaluation will result in an error at execution time. Still, the type system finds many program errors at compile time, aids the user in reasoning about programs, and also permits a compiler to generate more efficient code (for example, no run-time type tags or tests are required).

The type system also ensures that user-supplied type signatures are correct. In fact, Haskell's type system is powerful enough to allow us to avoid writing any type signatures at all, (With a few exceptions to be described later.) in which case we say that the type system infers the correct types for us. Nevertheless, judicious placement of type signatures is a good idea, as we did for inc, since type signatures are a very effective form of documentation and help bring programming errors to light.

[The reader will note that we have capitalized identifiers that denote specific types, such as Int and Char, but not identifiers that denote values, such as inc. This is not just a convention: it is enforced by Haskell's lexical syntax. In fact, the case of the other characters matters, too: foo, fOo, and fOO are all distinct identifiers.]

2.1  Polymorphic Types

Haskell also incorporates polymorphic types---types that are universally quantified in some way over all types. Polymorphic type expressions essentially describe families of types. For example, (forall a)[a] is the family of types consisting of, for every type a, the type of lists of a. Lists of integers (e.g. [1,2,3]), lists of characters (['a','b','c']), even lists of lists of integers, etc., are all members of this family. (Note, however, that [2,'b'] is not a valid example, since there is no single type that contains both 2 and 'b'.)

[Identifiers such as  a  above are called type variables, and are uncapitalized to distinguish them from specific types such as Int. Furthermore, since Haskell has only universally quantified types, there is no need to explicitly write out the symbol for universal quantification, and thus we simply write [a] in the example above. In other words, all type variables are implicitly universally quantified.]

Lists are a commonly used data structure in functional languages, and are a good vehicle for explaining the principles of polymorphism. The list [1,2,3] in Haskell is actually shorthand for the list 1:(2:(3:[])), where [] is the empty list and : is the infix operator that adds its first argument to the front of its second argument (a list). (: and [] are like Lisp's cons and nil, respectively.) Since : is right associative, we can also write this list as 1:2:3:[].

As an example of a user-defined function that operates on lists, consider the problem of counting the number of elements in a list:

length                  :: [a] -> Int
length []               =  0
length (x:xs)           =  1 + length xs

This definition is almost self-explanatory. We can read the equations as saying: "The length of the empty list is 0, and the length of a list whose first element is x and remainder is xs is 1 plus the length of xs." (Note the naming convention used here; xs is the plural of x, and should be read that way.)

Although intuitive, this example highlights an important aspect of Haskell that is yet to be explained: pattern matching. The left-hand sides of the equations contain patterns such as [] and x:xs. In a function application these patterns are matched against actual parameters in a fairly intuitive way ([] only matches the empty list, and x:xs will successfully match any list with at least one element, binding x to the first element and xs to the rest of the list). If the match succeeds, the right-hand side is evaluated and returned as the result of the application. If it fails, the next equation is tried, and if all equations fail, an error results.

Defining functions by pattern matching is quite common in Haskell, and the user should become familiar with the various kinds of patterns that are allowed; we will return to this issue in Section 4.

The length function is also an example of a polymorphic function. It can be applied to a list containing elements of any type, for example [Int], [Char], or [[Int]].

length [1,2,3] => 3
length ['a','b','c'] => 3
length [[1],[2],[3]] => 3

Here are two other useful polymorphic functions on lists that will be used later. Function head returns the first element of a list, function tail returns all but the first.

head                    :: [a] -> a
head (x:xs)             =  x

tail                    :: [a] -> [a]
tail (x:xs)             =  xs

Unlike length, these functions are not defined for all possible values of their argument. A runtime error occurs when these functions are applied to an empty list.

With polymorphic types, we find that some types are in a sense strictly more general than others in the sense that the set of values they define is larger. For example, the type [a] is more general than [Char]. In other words, the latter type can be derived from the former by a suitable substitution for a. With regard to this generalization ordering, Haskell's type system possesses two important properties: First, every well-typed expression is guaranteed to have a unique principal type (explained below), and second, the principal type can be inferred automatically (§4.1.3). In comparison to a monomorphically typed language such as C, the reader will find that polymorphism improves expressiveness, and type inference lessens the burden of types on the programmer.

An expression's or function's principal type is the least general type that, intuitively, "contains all instances of the expression." For example, the principal type of head is [a]->a; the types [b]->a, a->a, or even a are too general, whereas something like [Int]->Int is too specific. 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, ML, Miranda, ("Miranda" is a trademark of Research Software, Ltd.) and several other (mostly functional) languages.

2.2  User-Defined Types

We can define our own types in Haskell using a data declaration, which we introduce via a series of examples (§4.2.1).

An important predefined type in Haskell is that of truth values:

data Bool               = False | True

The type being defined here is Bool, and it has exactly two values: True and False. Type Bool is an example of a (nullary) type constructor, and True and False are (also nullary) data constructors (or just constructors, for short).

Similarly, we might wish to define a color type:

data Color              = Red | Green | Blue | Indigo | Violet

Both Bool and Color are examples of enumerated types, since they consist of a finite 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, a type like Point is often called a tuple type, since it is essentially just a cartesian product (in this case binary) of other types. (Tuples are somewhat like records in other languages.) In contrast, multi-constructor types, such as Bool and Color, are called (disjoint) union or sum types.

More importantly, however, Point is an example of a polymorphic type: for any type t, it defines the type of cartesian points that use t as the coordinate type. The Point type can now be seen clearly as a unary type constructor, since from the type t it constructs a new type Point t. (In the same sense, using the list example given earlier, [] is also a type constructor. Given any type t we can "apply" [] to yield a new type [t]. The Haskell syntax allows [] t to be written as [t]. Similarly, -> is a type constructor: given two types t and u, t->u is the type of functions mapping elements of type t to elements of type u.)

Note that the type of the binary data constructor Pt is a -> a -> Point a, and thus the following typings are valid:

Pt  2.0  3.0            :: Point Float
Pt  'a'  'b'            :: Point Char
Pt True False           :: Point Bool

On the other hand, an expression such as Pt 'a' 1 is ill-typed because 'a' and 1 are of different types.

It is important to distinguish between applying a data constructor to yield a value, and applying a type constructor to yield a type; the former happens at run-time and is how we compute things in Haskell, whereas the latter happens at compile-time and is part of the type system's process of ensuring type safety.

[Type constructors such as Point and data constructors such as Pt are in separate namespaces. This allows the same name to be used for both a type constructor and data constructor, as in the following:

data Point a = Point a a

While this may seem a little confusing at first, it serves to make the link between a type and its data constructor more obvious. ]

2.2.1  Recursive Types

Types can also be recursive, as in the type of binary trees:

data Tree a             = Leaf a | Branch (Tree a) (Tree a) 

Here we have defined a polymorphic binary tree type whose elements are either leaf nodes containing a value of type a, or internal nodes ("branches") containing (recursively) two sub-trees.

When reading data declarations such as this, remember again that Tree is a type constructor, whereas Branch and Leaf are data constructors. Aside from establishing a connection between these constructors, the above declaration is essentially defining the following types for Branch and Leaf:

Branch                  :: Tree a -> Tree a -> Tree a
Leaf                    :: a -> Tree a

With this example we have defined a type sufficiently rich to allow defining some interesting (recursive) functions that use it. For example, suppose we wish to define a function fringe that returns a list of all the elements in the leaves of a tree from left to right. It's usually helpful to write down the type of new functions first; in this case we see that the type should be Tree a -> [a]. That is, fringe is a polymorphic function that, for any type a, maps trees of a into lists of a. A suitable definition follows:

fringe                     :: Tree a -> [a]
fringe (Leaf x)            =  [x]
fringe (Branch left right) =  fringe left ++ fringe right

Where ++ is the infix operator that concatenates two lists (its full definition will be given in Section 8.5). As with the length example given earlier, the fringe function is defined using pattern matching, except that here we see patterns involving user-defined constructors: Leaf and Branch. [Note that the formal parameters are easily identified as the ones beginning with lower-case letters.]

2.3  Type Synonyms

For convenience, Haskell provides a way to define type synonyms; i.e. names for commonly used types. Type synonyms are created using a type declaration (§4.2.2). Here are several examples:

type String             = [Char]
type Person             = (Name,Address)
type Name               = String
data Address            = None | Addr String

Type synonyms do not define new types, but simply give new names for existing types. For example, the type Person -> Name is precisely equivalent to (String,Address) -> String. The new names are often shorter than the types they are synonymous with, but this is not the only purpose of type synonyms: they can also improve readability of programs by being more mnemonic; indeed, the above examples highlight this. We can even give new names to polymorphic types:

type AssocList a b              = [(a,b)]

This is the type of "association lists" which associate values of type a with those of type b.

2.4  Built-in Types Are Not Special

Earlier we introduced several "built-in" types such as lists, tuples, integers, and characters. We have also shown how new user-defined types can be defined. Aside from special syntax, are the built-in types in any way more special than the user-defined ones? The answer is no. The special syntax is for convenience and for consistency with historical convention, but has no semantic consequence.

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 defining them. For example, the Char type might be written as:

data Char       = 'a' | 'b' | 'c' | ...         -- This is not valid
                | 'A' | 'B' | 'C' | ...         -- Haskell code!
                | '1' | '2' | '3' | ...

These constructor names are not syntactically valid; to fix them we would have to write something like:

data Char       = Ca | Cb | Cc | ...
                | CA | CB | CC | ...
                | C1 | C2 | C3 | ...

Even though these constructors are more concise, they are quite unconventional for representing characters.

In any case, writing "pseudo-Haskell" code in this way helps us to see through the special syntax. We see now that Char is just an enumerated type consisting of a large number of nullary constructors. Thinking of Char in this way makes it clear why, for example, we can pattern-match against characters in function definitions; i.e., 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; the characters -- and all subsequent characters to the end of the line are ignored. Haskell also permits nested comments which have the form {-...-} and can appear anywhere (§2.2).]

Similarly, we could define Int and Integer by:

data Int     = -65532 | ... | -1 | 0 | 1 | ... | 65532  -- more pseudo-code
data Integer =       ... -2 | -1 | 0 | 1 | 2 ...

where -65532 and 65532, say, are the maximum and minimum fixed precision integers for a given implementation. Int is a much larger enumeration than Char, but it's still finite! In contrast, the pseudo-code for Integer (the type of arbitrary precision integers) is intended to convey an infinite enumeration.

Tuples are also easy to define playing this game:

data (a,b)              = (a,b)                         -- more pseudo-code
data (a,b,c)            = (a,b,c)
data (a,b,c,d)          = (a,b,c,d)
 .                         .
 .                         .
 .                         .

Each declaration above defines a tuple type of a particular length, with (...) 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 infinite number of such declarations, reflecting the fact that tuples of all lengths are allowed in Haskell.

Lists are also easily handled, and more interestingly, they are recursive:

 data [a]               = [] | a : [a]                  -- more pseudo-code

We can now see clearly what we described about lists earlier: [] is the empty list, and : is the infix list constructor; thus [1,2,3] must be equivalent to the list 1:2:3:[]. (: is right associative.) The type of [] is [a], and the type of : is a->[a]->[a].

[The way ":" is defined here is actually legal syntax---infix constructors are permitted in data declarations, and are distinguished from infix operators (for pattern-matching purposes) by the fact that they must begin with a ":" (a property trivially satisfied by ":").]

At this point the reader should note carefully the differences between tuples and lists, which the above definitions 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 fixed length. The typing rules for tuples and lists should now also be clear:

For (e1,e2,...,en), n>=2, if ti is the type of ei, then the type of the tuple is (t1,t2,...,tn).

For [e1,e2,...,en], n>=0, each ei must have the same type t, and the type of the list is [t].

2.4.1  List Comprehensions and Arithmetic Sequences

As with Lisp dialects, lists are pervasive in Haskell, and as with other functional languages, there is yet more syntactic sugar to aid in their creation. 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 <- xs ]

This expression can intuitively be read as "the list of all f x such that x is drawn from xs." The similarity to set notation is not a coincidence. The phrase x <- xs is called a generator, of which more than one is allowed, as in:

[ (x,y) | x <- xs, y <- ys ]

This list comprehension forms the cartesian product of the two lists xs and ys. The elements are selected as if the generators were "nested" from left to right (with the rightmost generator varying fastest); thus, if xs is [1,2] and ys is [3,4], the result is [(1,3),(1,4),(2,3),(2,4)].

Besides generators, boolean expressions called guards are permitted. Guards place constraints on the elements generated. For example, here is a concise definition of everybody's favorite sorting algorithm:

quicksort  []           =  []
quicksort (x:xs)        =  quicksort [y | y <- xs, y<x ]
                        ++ [x]
                        ++ quicksort [y | y <- xs, y>=x]

To further support the use of lists, Haskell has special syntax for arithmetic sequences, which are best explained by a series of examples:

[1..10] => [1,2,3,4,5,6,7,8,9,10]
[1,3..10] => [1,3,5,7,9]
[1,3..] => [1,3,5,7,9, ... (infinite sequence)

More will be said about arithmetic sequences in Section 8.2, and "infinite lists" in Section 3.4.

2.4.2  Strings

As another example of syntactic sugar for built-in types, we note that the literal string "hello" is actually shorthand for the list of characters ['h','e','l','l','o']. Indeed, the type of "hello" is String, where String is a predefined type synonym (that we gave as an earlier example):

type String             = [Char]

This means we can use predefined polymorphic list functions to operate on strings. For example:

"hello" ++ " world" => "hello world"

The Gentle Introduction To Haskell
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