A Gentle Introduction to Haskell, Version 98
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In this section we introduce the predefined standard type classes in Haskell. We have simplified these classes somewhat by omitting some of the less interesting methods in these classes; the Haskell report contains a more complete description. Also, some of the standard classes are part of the standard Haskell libraries; these are described in the Haskell Library Report.
The classes Eq and Ord have already been discussed. The
definition of Ord in the Prelude is somewhat more complex than the
simplified version of Ord presented earlier. In particular, note the
compare method:
data Ordering = EQ | LT | GT
compare :: Ord a => a -> a -> Ordering
The compare method is sufficient to define all other
methods (via defaults) in this class and is the best way to create
Ord instances.
Class Enum has a set of operations that underlie the syntactic sugar of arithmetic sequences; for example, the arithmetic sequence expression [1,3..] stands for enumFromThen 1 3 (see §3.10 for the formal translation). We can now see that arithmetic sequence expressions can be used to generate lists of any type that is an instance of Enum. This includes not only most numeric types, but also Char, so that, for instance, ['a'..'z'] denotes the list of lower-case letters in alphabetical order. Furthermore, user-defined enumerated types like Color can easily be given Enum instance declarations. If so:
[Red .. Violet] => [Red, Green, Blue, Indigo, Violet]
Note that such a sequence is arithmetic in the sense that the increment between values is constant, even though the values are not numbers. Most types in Enum can be mapped onto fixed precision integers; for these, the fromEnum and toEnum convert between Int and a type in Enum.
The instances of class Show are those types that can be converted
to character strings (typically for I/O). The class Read
provides operations for parsing character strings to obtain the values
they may represent. The simplest function in the class Show is
show:
show :: (Show a) => a -> String
Naturally enough, show takes any value of an appropriate type and
returns its representation as a character string (list of characters),
as in show (2+2), which results in "4". This is fine as far as
it goes, but we typically need to produce more complex strings
that may have the representations of many values in them, as in
"The sum of " ++ show x ++ " and " ++ show y ++ " is " ++ show (x+y) ++ "."
and after a while, all that concatenation gets to be a bit
inefficient. Specifically, let's consider a function to represent
the binary trees of Section 2.2.1 as a string,
with suitable markings to show the nesting of subtrees and the
separation of left and right branches (provided the element type is
representable as a string):
showTree :: (Show a) => Tree a -> String
showTree (Leaf x) = show x
showTree (Branch l r) = "<" ++ showTree l ++ "|" ++ showTree r ++ ">"
Because (++) has time complexity linear in the length of its
left argument, showTree is potentially quadratic in the size of the
tree.
To restore linear complexity, 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. The second argument serves as a sort of string
accumulator, and show can now be defined as shows with
the null accumulator. This is the default definition of
show in the Show class definition:
show x = shows x ""
We can use shows to define a more efficient version of showTree,
which also has a string accumulator argument:
showsTree :: (Show a) => Tree a -> String -> String
showsTree (Leaf x) s = shows x s
showsTree (Branch l r) s= '<' : showsTree l ('|' : showsTree r ('>' : s))
This solves our efficiency problem (showsTree has linear complexity),
but the presentation of this function (and others like it) can be
improved. First, 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.
Second, we can avoid carrying accumulators around, and also avoid
amassing parentheses at the right end of long constructions, by using
functional composition:
showsTree :: (Show a) => Tree a -> ShowS
showsTree (Leaf x) = shows x
showsTree (Branch l r) = ('<':) . showsTree l . ('|':) . showsTree r . ('>':)
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, strings) to a function level.
We can think of the typing as saying that showsTree maps a tree
into a showing function. Functions like ('<' :) or
("a string" ++) are primitive showing functions, and we build up
more complex functions by function composition.
Now that we can turn trees into strings, let's turn to the inverse
problem. The basic idea is a parser for a type a, which
is a function that takes a string and returns a list of (a, String)
pairs [9]. The Prelude provides
a type synonym for such functions:
type ReadS a = String -> [(a,String)]
Normally, a parser returns a singleton list, containing a value
of type a that was read from the input string and the remaining
string that follows what was parsed. If no parse was possible, however,
the result is the empty list, and if there is more than one possible
parse (an ambiguity), the resulting list contains more than one pair.
The standard function reads is a parser for any instance of Read:
reads :: (Read a) => ReadS a
We can use this function to define a parsing function for the string
representation of binary trees produced by showsTree. List comprehensions
give us a convenient idiom for constructing such parsers: (An
even more elegant approach to parsing uses monads and parser
combinators. These are part of a standard parsing library distributed
with most Haskell systems.)
readsTree :: (Read a) => ReadS (Tree a)
readsTree ('<':s) = [(Branch l r, u) | (l, '|':t) <- readsTree s,
(r, '>':u) <- readsTree t ]
readsTree s = [(Leaf x, t) | (x,t) <- reads s]
Let's take a moment to examine this function definition in detail.
There are two main cases to consider: If the first character of the
string to be parsed is '<', we should have the representation of
a branch; otherwise, we have a leaf. In the first case, calling the
rest of the input string following the opening angle bracket s,
any possible parse must be a tree Branch l r with remaining string u,
subject to the following conditions:
The second defining equation above just says that to parse the representation of a leaf, we parse a representation of the element type of the tree and apply the constructor Leaf to the value thus obtained.
We'll accept on faith for the moment that there is a Read (and Show) instance of Integer (among many other types), providing a reads that behaves as one would expect, e.g.:
(reads "5 golden rings") :: [(Integer,String)] => [(5, " golden rings")]
With this understanding, the reader should verify the following evaluations:
readsTree "<1|<2|3>>" | => | [(Branch (Leaf 1) (Branch (Leaf 2) (Leaf 3)), "")] |
readsTree "<1|2" | => | [] |
There are a couple of shortcomings in our definition of readsTree.
One is that the parser is quite rigid, allowing no white space before
or between the elements of the tree representation; the other is that
the way we parse our punctuation symbols is quite different from the
way we parse leaf values and subtrees, this lack of uniformity making
the function definition harder to read. 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 first lexeme in the input string and the remainder
of the input. The lexical rules are those of Haskell programs,
including comments, which lex skips, along with whitespace.
If the input string is empty or contains only whitespace and comments,
lex returns [("","")]; if the input is not empty in this sense,
but also does not begin with a valid lexeme after any leading whitespace
and comments, lex returns [].
Using the lexical analyzer, our tree parser now looks like this:
readsTree :: (Read a) => ReadS (Tree a)
readsTree s = [(Branch l r, x) | ("<", t) <- lex s,
(l, u) <- readsTree t,
("|", v) <- lex u,
(r, w) <- readsTree v,
(">", x) <- lex w ]
++
[(Leaf x, t) | (x, t) <- 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. This would allow us to
use the generic overloaded functions from the Prelude to parse and
display trees. Moreover, we would automatically then be able to parse
and display many other types containing trees as components, for
example, [Tree Integer]. As it turns out, 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. The extra parameter is a precedence level, used to properly
parenthesize expressions containing infix constructors. For types
such as Tree, the precedence can be ignored. 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
Alternatively, the Show instance could be defined in terms of
showTree:
instance Show a => Show (Tree a) where
show t = showTree t
This, however, will be less efficient than the ShowS version. Note
that the Show class defines default methods for both showsPrec and
show, allowing the user to define either one of these in an instance
declaration. Since these defaults are mutually recursive, an instance
declaration that defines neither of these functions will loop when
called. Other classes such as Num also have these "interlocking
defaults".
We refer the interested reader to §D for details of the Read and Show classes.
We can test the Read and Show instances by applying (read . show)
(which should be the identity) to some trees, 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, comments and whitespace).
Recall the Eq instance for trees we presented in Section
5; such a declaration is
simple---and boring---to produce: we require that the
element type in the leaves be an equality type; then, two leaves are
equal iff they contain equal elements, and two branches are equal iff
their left and right subtrees are equal, respectively. 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, we don't need to go through this tedium every time we
need equality operators for a new type; 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. Instances of Ord,
Enum, Ix, Read, and Show can also be generated by the
deriving clause.
[More than one class name can be specified, in which case the list
of names must be parenthesized and the names separated by commas.]
The derived Ord instance for Tree is slightly more complicated than
the Eq instance:
instance (Ord a) => Ord (Tree a) where
(Leaf _) <= (Branch _) = True
(Leaf x) <= (Leaf y) = x <= y
(Branch _) <= (Leaf _) = False
(Branch l r) <= (Branch l' r') = l == l' && r <= r' || l <= l'
This specifies a lexicographic order: Constructors are ordered
by the order of their appearance in the data declaration, and the
arguments of a constructor are compared from left to right. Recall
that the built-in list type is semantically equivalent to an ordinary
two-constructor type. In fact, this is the full declaration:
data [a] = [] | a : [a] deriving (Eq, Ord) -- pseudo-code
(Lists also have Show and Read instances, which are not derived.)
The derived Eq and Ord instances for lists are the usual ones; in
particular, character strings, as lists of characters, are ordered as
determined by the underlying Char type, with an initial substring
comparing less than a longer string; for example, "cat" < "catalog".
In practice, Eq and Ord instances are almost always derived, rather than user-defined. In fact, we should provide our own definitions of equality and ordering predicates only with some trepidation, being careful to maintain the expected algebraic properties of equivalence relations and total orders. An intransitive (==) predicate, for example, could be disastrous, confusing readers of the program and confounding manual or automatic program transformations that rely on the (==) predicate's being an approximation to definitional equality. Nevertheless, it is sometimes necessary to provide Eq or Ord instances different from those that would be derived; probably the most important example is that of an abstract data type in which different concrete values may represent the same abstract value.
An enumerated type can have a derived Enum instance, and here again,
the ordering is that of the constructors in the data declaration.
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 .. Friday] | => | [Wednesday, Thursday, Friday] |
[Monday, Wednesday ..] | => | [Monday, Wednesday, Friday] |
Derived Read (Show) instances are possible for all types whose component types also have Read (Show) instances. (Read and Show instances for most of the standard types are provided by the Prelude. Some types, such as the function type (->), have a Show instance but not a corresponding Read.) The textual representation defined by a derived Show instance is consistent with the appearance of constant Haskell expressions of the type in question. For example, if we add Show and Read to the deriving clause for type Day, above, we obtain
show [Monday .. Wednesday] => "[Monday,Tuesday,Wednesday]"