Although lists and tuples are useful, we’ll often want to construct new data types of our own. This allows us to add structure to the values in our programs. Instead of using an anonymous tuple, we can give a collection of related values a name and a distinct type. Defining our own types also improves the type safety of our code: Haskell will not allow us to accidentally mix values of two types that are structurally similar but have different names.
For motivation, we’ll consider a few kinds of data that a small online bookstore might need to manage. We won’t make any attempt at complete or realistic data definitions, but at least we’re tying them to the real world.
We define a new data type using the data keyword.
data BookInfo = Book Int String [String]
deriving (Show)The BookInfo after the data keyword is the name of our new
type. We call BookInfo a type constructor. Once we have
defined a type, we will use its type constructor to refer to it.
As we’ve already mentioned, a type name, and hence a type
constructor, must start with a capital letter.
The Book that follows is the name of the value constructor
(sometimes called a data constructor). We use this to create a
value of the BookInfo type. A value constructor’s name must also
start with a capital letter.
After Book, the Int, String, and [String] that follow are
the components of the type. A component serves the same purpose
in Haskell as a field in a structure or class would in another
language: it’s a “slot” where we keep a value. (We’ll often refer
to components as fields.)
In this example, the Int represents a book’s identifier (e.g. in
a stock database), String its title, and [String] the names of
its authors.
To make the link to a concept we’ve already seen, the BookInfo
Type contains the same components as a 3-tuple of type
(Int, String, [String]), but it has a distinct type. We can’t
accidentally (or deliberately) use one in a context where the
other is expected. For instance, a bookstore is also likely to
carry magazines.
data MagazineInfo = Magazine Int String [String]
deriving (Show)Even though this MagazineInfo type has the same structure as our
BookInfo type, Haskell treats the types as distinct because
their type and value constructors have different names.
We can create a new value of type BookInfo by treating Book as
a function, and applying it with arguments of types Int,
String, and [String].
myInfo = Book 9780135072455 "Algebra of Programming"
["Richard Bird", "Oege de Moor"]Once we have defined a type, we can experiment with it in ghci.
We begin by using the :load command to load our source file.
ghci> :load BookStore
[1 of 1] Compiling Main ( BookStore.hs, interpreted )
Ok, one module loaded.
Remember the myInfo variable we defined in our source file? Here
it is.
ghci> myInfo
Book 9780135072455 "Algebra of Programming" ["Richard Bird","Oege de Moor"]
ghci> :type myInfo
myInfo :: BookInfo
We can construct new values interactively in ghci, too.
ghci> Book 0 "The Book of Imaginary Beings" ["Jorge Luis Borges"]
Book 0 "The Book of Imaginary Beings" ["Jorge Luis Borges"]
The ghci command :type lets us see what the type of an
expression is.
ghci> :type Book 1 "Cosmicomics" ["Italo Calvino"]
Book 1 "Cosmicomics" ["Italo Calvino"] :: BookInfo
ghci> cities = Book 173 "Use of Weapons" ["Iain M. Banks"]
To find out more about a type, we can use some of ghci’s
browsing capabilities. The :info command gets ghci to tell us
everything it knows about a name.
ghci> :info BookInfo
data BookInfo = Book Int String [String]
-- Defined at BookStore.hs:2:1
instance [safe] Show BookInfo -- Defined at BookStore.hs:3:27
We can also find out why we use Book to construct a new value
of type BookStore.
ghci> :type Book
Book :: Int -> String -> [String] -> BookInfo
We can treat a value constructor as just another function, one that happens to create and return a new value of the type we desire.
When we introduced the type BookStore, we deliberately chose to
give the type constructor BookStore a different name from the
value constructor Book, purely to make it obvious which was
which.
However, in Haskell, the names of types and values are independent of each other. We only use a type constructor (i.e. the type’s name) in a type declaration or a type signature. We only use a value constructor in actual code. Because these uses are distinct, there is no ambiguity if we give a type constructor and a value constructor the same name. If we are writing a type signature, we must be referring to a type constructor. If we are writing an expression, we must be using the value constructor.
-- We will introduce the CustomerID type shortly.
data BookReview = BookReview BookInfo CustomerID StringThis definition says that the type named BookReview has a value
constructor that is also named BookReview.
Not only is it legal for a value constructor to have the same name as its type constructor, it’s normal: you’ll see this all the time in regular Haskell code.
We can introduce a synonym for an existing type at any time, to
give a type a more descriptive name. For example, the String in
our BookReview type doesn’t tell us what the string is for, but
we can clarify this.
type CustomerID = Int
type ReviewBody = String
data BetterReview = BetterReview BookInfo CustomerID ReviewBodyThe type keyword introduces a type synonym. The new name is on
the left of the =, with the existing name on the right. The two
names identify the same type, so type synonyms are purely for
making code more readable.
We can also use a type synonym to create a shorter name for a verbose type.
type BookRecord = (BookInfo, BookReview)This states that we can use BookRecord as a synonym for the
tuple (BookInfo, BookReview). A type synonym only creates a new
name that refers to an existing type[fn:1]. We still use the same
value constructors to create a value of the type.
The familiar Bool is the simplest common example of a category
of type called an algebraic data type. An algebraic data type
can have more than one value constructor.
data Bool = False | TrueThe Bool type has two value constructors, True and False.
Each value constructor is separated in the definition by a |
character, which we can read as “or”: we can construct a Bool
that has the value True, or the value False. When a type has
more than one value constructor, they are usually referred to as
alternatives or cases. We can use any one of the alternatives
to create a value of that type.
Each of an algebraic data type’s value constructors can take zero or more arguments. As an example, here’s one way we might represent billing information.
type CardHolder = String
type CardNumber = String
type Address = [String]
data BillingInfo = CreditCard CardNumber CardHolder Address
| CashOnDelivery
| Invoice CustomerID
deriving (Show)Here, we’re saying that we support three ways to bill our
customers. If they want to pay by credit card, they must supply a
card number, the holder’s name, and the holder’s billing address
as arguments to the CreditCard value constructor. Alternatively,
they can pay the person who delivers their shipment. Since we
don’t need to store any extra information about this, we specify
no arguments for the CashOnDelivery constructor. Finally, we can
send an invoice to the specified customer, in which case we need
their CustomerID as an argument to the Invoice constructor.
When we use a value constructor to create a value of type
BillingInfo, we must supply the arguments that it requires.
ghci> :type CreditCard
CreditCard :: CardNumber -> CardHolder -> Address -> BillingInfo
ghci> CreditCard "2901650221064486" "Thomas Gradgrind" ["Dickens", "England"]
CreditCard "2901650221064486" "Thomas Gradgrind" ["Dickens","England"]
ghci> :type it
it :: BillingInfo
ghci> Invoice
<interactive>:1:1: error:
• No instance for (Show (CustomerID -> BillingInfo))
arising from a use of ‘print’
(maybe you haven't applied a function to enough arguments?)
• In a stmt of an interactive GHCi command: print it
ghci> :type it
it :: BillingInfo
The No instance error message arose because we did not supply an
argument to the Invoice constructor. As a result, we were trying
to print the Invoice constructor itself. That constructor
requires an argument and returns a value, so it is a function. We
cannot print functions in Haskell, which is ultimately why the
interpreter complained.
There is some overlap between tuples and user-defined algebraic
data types. If we wanted to, we could represent our BookInfo
type from earlier as an (Int, String, [String]) tuple.
ghci> Book 2 "The Wealth of Networks" ["Yochai Benkler"]
Book 2 "The Wealth of Networks" ["Yochai Benkler"]
ghci> (2, "The Wealth of Networks", ["Yochai Benkler"])
(2,"The Wealth of Networks",["Yochai Benkler"])
Algebraic data types allow us to distinguish between otherwise identical pieces of information. Two tuples with elements of the same type are structurally identical, so they have the same type.
a = ("Porpoise", "Grey")
b = ("Table", "Oak")Since they have different names, two algebraic data types have distinct types, even if they are otherwise structurally equivalent.
data Cetacean = Cetacean String String
data Furniture = Furniture String String
c = Cetacean "Porpoise" "Grey"
d = Furniture "Table" "Oak"This lets us bring the type system to bear in writing programs with fewer bugs. With the tuples we defined above, we could conceivably pass a description of a whale to a function expecting a chair, and the type system could not help us. With the algebraic data types, there is no such possibility of confusion.
Here is a more subtle example. Consider the following representations of a two-dimensional vector.
-- x and y coordinates or lengths.
data Cartesian2D = Cartesian2D Double Double
deriving (Eq, Show)
-- Angle and distance (magnitude).
data Polar2D = Polar2D Double Double
deriving (Eq, Show)The cartesian and polar forms use the same types for their two
elements. However, the meanings of the elements are different.
Because Cartesian2D and Polar2D are distinct types, the type
system will not let us accidentally use a Cartesian2D value
where a Polar2D is expected, or vice versa.
ghci> Cartesian2D (sqrt 2) (sqrt 2) == Polar2D (pi / 4) 2
<interactive>:2:34: error:
• Couldn't match expected type ‘Cartesian2D’
with actual type ‘Polar2D’
• In the second argument of ‘(==)’, namely ‘Polar2D (pi / 4) 2’
In the expression:
Cartesian2D (sqrt 2) (sqrt 2) == Polar2D (pi / 4) 2
In an equation for ‘it’:
it = Cartesian2D (sqrt 2) (sqrt 2) == Polar2D (pi / 4) 2
The (==) operator requires its arguments to have the same type.
If we used tuples to represent these values, we could quickly land ourselves in hot water by mixing the two representations inappropriately.
ghci> (1, 2) == (1, 2)
True
The type system can’t rescue us here: as far as it’s concerned,
we’re comparing two (Double, Double) pairs, which is a perfectly
valid thing to do. Indeed, we cannot tell by inspection which of
these values is supposed to be polar or cartesian, but (1,2)
has a different meaning in each representation.
There is no hard and fast rule for deciding when it’s better to
use a tuple or a distinct data type, but here’s a rule of thumb to
follow. If you’re using compound values widely in your code (as
almost all non-trivial programs do), adding data declarations
will benefit you in both type safety and readability. For smaller,
localised uses, a tuple is usually fine.
Algebraic data types provide a single powerful way to describe data types. Other languages often need several different features to achieve the same degree of expressiveness. Here are some analogues from C and C++, which might make it clearer what we can do with algebraic data types, and how they relate to concepts that might be more familiar.
With just one constructor, an algebraic data type is similar to a
tuple: it groups related values together into a compound value. It
corresponds to a struct in C or C++, and its components
correspond to the fields of a struct. Here’s a C equivalent of
the BookInfo type that we defined earlier.
struct book_info {
int id;
char *name;
char **authors;
};The main difference between the two is that the fields in the Haskell type are anonymous and positional.
data BookInfo = Book Int String [String]
deriving (Show)By positional, we mean that the section number is in the first field of the Haskell type, and the title is in the second. We refer to them by location, not by name.
In the section called “Pattern matching”
the fields of a BookStore value. In
the section called “Record syntax”
syntax for defining data types that looks a little more C-like.
Algebraic data types also serve where we’d use an enum in C or
C++, to represent a range of symbolic values. Such algebraic data
types are sometimes referred to as enumeration types. Here’s an
example from C.
enum roygbiv {
red,
orange,
yellow,
green,
blue,
indigo,
violet,
};And here’s a Haskell equivalent.
data Roygbiv = Red
| Orange
| Yellow
| Green
| Blue
| Indigo
| Violet
deriving (Eq, Show)We can try these out in ghci.
ghci> :type Yellow
Yellow :: Roygbiv
ghci> :type Red
Red :: Roygbiv
ghci> Red == Yellow
False
ghci> Green == Green
True
In C, the elements of an enum are integers. We can use an
integer in a context where an enum is expected, and vice versa:
a C compiler will automatically convert values between the two
types. This can be a source of nasty bugs. In Haskell, this kind
of problem does not occur. For example, we cannot use a Roygbiv
value where an Int is expected.
ghci> take 3 "foobar"
"foo"
ghci> take Red "foobar"
<interactive>:3:6: error:
• Couldn't match expected type ‘Int’ with actual type ‘Roygbiv’
• In the first argument of ‘take’, namely ‘Red’
In the expression: take Red "foobar"
In an equation for ‘it’: it = take Red "foobar"
If an algebraic data type has multiple alternatives, we can think
of it as similar to a union in C or C++. A big difference
between the two is that a union doesn’t tell us which alternative
is actually present; we have to explicitly and manually track
which alternative we’re using, usually in another field of an
enclosing struct. This means that unions can be sources of nasty
bugs, where our notion of which alternative we should be using is
incorrect.
enum shape_type {
shape_circle,
shape_poly,
};
struct circle {
struct vector centre;
float radius;
};
struct poly {
size_t num_vertices;
struct vector *vertices;
};
struct shape
{
enum shape_type type;
union {
struct circle circle;
struct poly poly;
} shape;
};In the example above, the union can contain valid data for
either a struct circle or a struct poly. We have to use the
enum shape_type by hand to indicate which kind of value is
currently stored in the union.
The Haskell version of this code is both dramatically shorter and safer than the C equivalent.
type Vector = (Double, Double)
data Shape = Circle Vector Double
| Poly [Vector]If we create a Shape value using the Circle constructor, the
fact that we created a Circle is stored. When we later use a
Circle, we can’t accidentally treat it as a Square. We will
see why in the section called “Pattern matching”
Now that we’ve seen how to construct values with algebraic data types, let’s discuss how we work with these values. If we have a value of some type, there are two things we would like to be able to do.
- If the type has more than one value constructor, we need to be able to tell which value constructor was used to create the value.
- If the value constructor has data components, we need to be able to extract those values.
Haskell has a simple, but tremendously useful, pattern matching facility that lets us do both of these things.
A pattern lets us look inside a value and bind variables to the
data it contains. Here’s an example of pattern matching in action
on a Bool value: we’re going to reproduce the not function.
myNot True = False
myNot False = TrueIt might seem that we have two functions named myNot here, but
Haskell lets us define a function as a series of equations:
these two clauses are defining the behavior of the same function
for different patterns of input. On each line, the patterns are
the items following the function name, up until the = sign.
To understand how pattern matching works, let’s step through an
example, say myNot False.
When we apply myNot, the Haskell runtime checks the value we
supply against the value constructor in the first pattern. This
does not match, so it tries against the second pattern. That match
succeeds, so it uses the right hand side of that equation as the
result of the function application.
Here is a slightly more extended example. This function adds together the elements of a list.
sumList (x:xs) = x + sumList xs
sumList [] = 0Let us step through the evaluation of sumList [1,2]. The list
notation [1,2] is shorthand for the expression (1 : (2 : [])).
We begin by trying to match the pattern in the first equation of
the definition of sumList. In the (x : xs) pattern, the ”:”
is the familiar list constructor, (:). We are now using it to
match against a value, not to construct one. The value
(1 : (2 : [])) was constructed with (:), so the constructor in
the value matches the constructor in the pattern. We say that the
pattern matches, or that the match succeeds.
The variables x and xs are now “bound to” the constructor’s
arguments, so x is given the value 1, and xs the value
2 : [].
The expression we are now evaluating is 1 + sumList (2 : []). We
must now recursively apply sumList to the value 2 : []. Once
again, this was constructed using (:), so the match succeeds. In
our recursive application of sumList, x is now bound to 2,
and xs to [].
We are now evaluating 1 + (2 + sumList []). In this recursive
application of sumList, the value we are matching against is
[]. The value’s constructor does not match the constructor in
the first pattern, so we skip this equation. Instead, we “fall
through” to the next pattern, which matches. The right hand side
of this equation is thus chosen as the result of this application.
The result of sumList [1,2] is thus 1 + (2 + (0)), or 3.
As a final note, there already exists a standard function, sum,
that performs this sum-of-a-list for us. Our sumList is purely
for illustration.
Let’s step back and take a look at the relationship between constructing a value and pattern matching on it.
We apply a value constructor to build a value. The expression
Book 9 "Close Calls" ["John Long"] applies the Book
constructor to the values 9, "Close Calls", and
["John Long"] to produce a new value of type BookInfo.
When we pattern match against the Book constructor, we reverse
the construction process. First of all, we check to see if the
value was created using that constructor. If it was, we inspect it
to obtain the individual values that we originally supplied to the
constructor when we created the value.
Let’s consider what happens if we match the pattern
(Book id name authors) against our example expression.
- The match will succeed, because the constructor in the value matches the one in our pattern.
- The variable
idwill be bound to9. - The variable
namewill be bound to"Close Calls". - The variable
authorswill be bound to["John Long"].
Because pattern matching acts as the inverse of construction, it’s sometimes referred to as /de/construction.
The syntax for pattern matching on a tuple is similar to the syntax for constructing a tuple. Here’s a function that returns the last element of a 3-tuple.
third (a, b, c) = cThere’s no limit on how “deep” within a value a pattern can look. This definition looks both inside a tuple and inside a list within that tuple.
complicated (True, a, x:xs, 5) = (a, xs)We can try this out interactively.
ghci> :load Tuple.hs
[1 of 1] Compiling Main ( Tuple.hs, interpreted )
Ok, one module loaded.
ghci> complicated (True, 1, [1,2,3], 5)
(1,[2,3])
Wherever a literal value is present in a pattern (True and 5
in the tuple pattern above), that value must match exactly for the
pattern match to succeed. If every pattern within a series of
equations fails to match, we get a runtime error.
ghci> complicated (False, 1, [1,2,3], 5)
*** Exception: Tuple.hs:10:0-39: Non-exhaustive patterns in function complicated
For an explanation of this error message, skip forward a little, to the section called “Exhaustive patterns and wild cards”
We can pattern match on an algebraic data type using its value
constructors. Recall the BookInfo type we defined earlier: we
can extract the values from a BookInfo as follows.
bookID (Book id title authors) = id
bookTitle (Book id title authors) = title
bookAuthors (Book id title authors) = authorsLet’s see it in action.
ghci> bookID (Book 3 "Probability Theory" ["E.T.H. Jaynes"])
3
ghci> bookTitle (Book 3 "Probability Theory" ["E.T.H. Jaynes"])
"Probability Theory"
ghci> bookAuthors (Book 3 "Probability Theory" ["E.T.H. Jaynes"])
["E.T.H. Jaynes"]
The compiler can infer the types of the accessor functions based on the constructor we’re using in our pattern.
ghci> :type bookID
bookID :: BookInfo -> Int
ghci> :type bookTitle
bookTitle :: BookInfo -> String
ghci> :type bookAuthors
bookAuthors :: BookInfo -> [String]
If we use a literal value in a pattern, the corresponding part of
the value we’re matching against must contain an identical value.
For instance, the pattern (3 : xs) first of all checks that a
value is a non-empty list, by matching against the (:)
constructor. It also ensures that the head of the list has the
exact value 3. If both of these conditions hold, the tail of the
list will be bound to the variable xs.
As you read functions that match on lists, you’ll frequently find
that the names of the variables inside a pattern resemble
(x : xs) or (d : ds). This is a popular naming convention. The
idea is that the name xs has an ”s” on the end of its name as
if it’s the “plural” of x, because x contains the head of the
list, and xs the remaining elements.
We can indicate that we don’t care what is present in part of a
pattern. The notation for this is the underscore character ”_”,
which we call a wild card. We use it as follows.
nicerID (Book id _ _ ) = id
nicerTitle (Book _ title _ ) = title
nicerAuthors (Book _ _ authors) = authorsHere, we have tidier versions of the accessor functions we introduced earlier. Now, there’s no question about which element we’re using in each function.
In a pattern, a wild card acts similarly to a variable, but it doesn’t bind a new variable. As the examples above indicate, we can use more than one wild card in a single pattern.
Another advantage of wild cards is that a Haskell compiler can warn us if we introduce a variable name in a pattern, but do not use it in a function’s body. Defining a variable, but forgetting to use it, can often indicate the presence of a bug, so this is a helpful feature. If we use a wild card instead of a variable that we do not intend to use, the compiler won’t complain.
When writing a series of patterns, it’s important to cover all of
a type’s constructors. For example, if we’re inspecting a list, we
should have one equation that matches the non-empty constructor
(:), and one that matches the empty-list constructor [].
Let’s see what happens if we fail to cover all the cases. Here, we
deliberately omit a check for the [] constructor.
badExample (x:xs) = x + badExample xsIf we apply this to a value that it cannot match, we’ll get an error at runtime: our software has a bug!
ghci> badExample []
*** Exception: BadPattern.hs:4:0-36: Non-exhaustive patterns in function badExample
In this example, no equation in the function’s definition matches
the value [].
If we need to provide a default behavior in cases where we don’t care about specific constructors, we can use a wild card pattern.
goodExample (x:xs) = x + goodExample xs
goodExample _ = 0The wild card above will match the [] constructor, so applying
this function does not lead to a crash.
ghci> goodExample []
0
ghci> goodExample [1,2]
3
Writing accessor functions for each of a data type’s components can be repetitive and tedious.
nicerID (Book id _ _ ) = id
nicerTitle (Book _ title _ ) = title
nicerAuthors (Book _ _ authors) = authorsWe call this kind of code boilerplate: necessary, but bulky and irksome. Haskell programmers don’t like boilerplate. Fortunately, the language addresses this particular boilerplate problem: we can define a data type, and accessors for each of its components, simultaneously. (The positions of the commas here is a matter of preference. If you like, put them at the end of a line instead of the beginning.)
data Customer = Customer {
customerID :: CustomerID
, customerName :: String
, customerAddress :: Address
} deriving (Show)This is almost exactly identical in meaning to the following, more familiar form.
data Customer = Customer Int String [String]
deriving (Show)
customerID :: Customer -> Int
customerID (Customer id _ _) = id
customerName :: Customer -> String
customerName (Customer _ name _) = name
customerAddress :: Customer -> [String]
customerAddress (Customer _ _ address) = addressFor each of the fields that we name in our type definition, Haskell creates an accessor function of that name.
ghci> :type customerID
customerID :: Customer -> CustomerID
We can still use the usual application syntax to create a value of this type.
customer1 = Customer 271828 "J.R. Hacker"
["255 Syntax Ct",
"Milpitas, CA 95134",
"USA"]Record syntax adds a more verbose notation for creating a value. This can sometimes make code more readable.
customer2 = Customer {
customerID = 271828
, customerAddress = ["1048576 Disk Drive",
"Milpitas, CA 95134",
"USA"]
, customerName = "Jane Q. Citizen"
}If we use this form, we can vary the order in which we list fields. Here, we have moved the name and address fields from their positions in the declaration of the type.
When we define a type using record syntax, it also changes the way the type’s values are printed.
ghci> customer1
Customer {customerID = 271828, customerName = "J.R. Hacker", customerAddress = ["255 Syntax Ct","Milpitas, CA 95134","USA"]}
For comparison, let’s look at a BookInfo value; we defined this
type without record syntax.
ghci> cities
Book 173 "Use of Weapons" ["Iain M. Banks"]
The accessor functions that we get “for free” when we use record syntax really are normal Haskell functions.
ghci> :type customerName
customerName :: Customer -> String
ghci> customerName customer1
"J.R. Hacker"
The standard System.Time module makes good use of record
syntax. Here’s a type defined in that module:
data CalendarTime = CalendarTime {
ctYear :: Int,
ctMonth :: Month,
ctDay, ctHour, ctMin, ctSec :: Int,
ctPicosec :: Integer,
ctWDay :: Day,
ctYDay :: Int,
ctTZName :: String,
ctTZ :: Int,
ctIsDST :: Bool
}In the absence of record syntax, it would be painful to extract specific fields from a type like this. The notation makes it easier to work with large structures.
We’ve repeatedly mentioned that the list type is polymorphic: the
elements of a list can be of any type. We can also add
polymorphism to our own types. To do this, we introduce type
variables into a type declaration. The prelude defines a type
named Maybe: we can use this to represent a value that could be
either present or missing, e.g. a field in a database row that
could be null.
data Maybe a = Just a
| NothingHere, the variable a is not a regular variable: it’s a type
variable. It indicates that the Maybe type takes another type as
its parameter. This lets us use Maybe on values of any type.
someBool = Just True
someString = Just "something"As usual, we can experiment with this type in ghci.
ghci> Just 1.5
Just 1.5
ghci> Nothing
Nothing
ghci> :type Just "invisible bike"
Just "invisible bike" :: Maybe [Char]
Maybe is a polymorphic, or generic, type. We give the Maybe type
constructor a parameter to create a specific type, such as
Maybe Int or Maybe [Bool]. As we might expect, these types are
distinct.
We can nest uses of parameterised types inside each other, but when we do, we may need to use parentheses to tell the Haskell compiler how to parse our expression.
wrapped = Just (Just "wrapped")To once again extend an analogy to more familiar languages, parameterised types bear some resemblance to templates in C++, and to generics in Java. Just be aware that this is a shallow analogy. Templates and generics were added to their respective languages long after the languages were initially defined, and have an awkward feel. Haskell’s parameterised types are simpler and easier to use, as the language was designed with them from the beginning.
The familiar list type is recursive: it’s defined in terms of
itself. To understand this, let’s create our own list-like type.
We’ll use Cons in place of the (:) constructor, and Nil in
place of [].
-- file ListADT.hs
data List a = Cons a (List a)
| Nil
deriving (Show)Because List a appears on both the left and the right of the =
sign, the type’s definition refers to itself. If we want to use
the Cons constructor to create a new value, we must supply one
value of type a, and another of type List a. Let’s see where
this leads us in practice.
The simplest value of type List a that we can create is Nil.
Save the type definition in a file, then load it into ghci.
ghci> Nil
Nil
Because Nil has a List type, we can use it as a parameter to
Cons.
ghci> Cons 0 Nil
Cons 0 Nil
And because Cons 0 Nil has the type List a, we can use this as
a parameter to Cons.
ghci> Cons 1 it
Cons 1 (Cons 0 Nil)
ghci> Cons 2 it
Cons 2 (Cons 1 (Cons 0 Nil))
ghci> Cons 3 it
Cons 3 (Cons 2 (Cons 1 (Cons 0 Nil)))
We could continue in this fashion indefinitely, creating ever
longer Cons chains, each with a single Nil at the end.
By inspection, this clearly substitutes a Cons for every (:),
and a Nil for each []. This covers both of the built-in list
type’s constructors. The two types are isomorphic; they have the
same shape.
ghci> fromList "durian"
Cons 'd' (Cons 'u' (Cons 'r' (Cons 'i' (Cons 'a' (Cons 'n' Nil)))))
ghci> fromList [Just True, Nothing, Just False]
Cons (Just True) (Cons Nothing (Cons (Just False) Nil))
For a third example of what a recursive type is, here is a definition of a binary tree type.
data Tree a = Node a (Tree a) (Tree a)
| Empty
deriving (Show)A binary tree is either a node with two children, which are themselves binary trees, or an empty value.
This time, let’s search for insight by comparing our definition with one from a more familiar language. Here’s a similar class definition in Java.
class Tree<A>
{
A value;
Tree<A> left;
Tree<A> right;
public Tree(A v, Tree<A> l, Tree<A> r)
{
value = v;
left = l;
right = r;
}
}The one significant difference is that Java lets us use the
special value null anywhere to indicate “nothing”, so we can use
null to indicate that a node is missing a left or right child.
Here’s a small function that constructs a tree with two leaves (a
leaf, by convention, has no children).
class Example
{
static Tree<String> simpleTree()
{
return new Tree<String>(
"parent",
new Tree<String>("left leaf", null, null),
new Tree<String>("right leaf", null, null));
}
}In Haskell, we don’t have an equivalent of null. We could use
the Maybe type to provide a similar effect, but that bloats the
pattern matching. Instead, we’ve decided to use a no-argument
Empty constructor. Where the Java example provides null to the
Tree constructor, we supply Empty in Haskell.
simpleTree = Node "parent" (Node "left child" Empty Empty)
(Node "right child" Empty Empty)- Write the converse of
fromListfor the List type: a function that takes a List a and generates a[a]. - Define a tree type that has only one constructor, like our Java
example. Instead of the
Emptyconstructor, use theMaybetype to refer to a node’s children.
Haskell provides a standard function, error :: String -> a, that
we can call when something has gone terribly wrong in our code. We
give it a string parameter, which is the error message to display.
Its type signature looks peculiar: how can it produce a value of
any type a given only a string?
It has a result type of a so that we can call it anywhere and it
will always have the right type. However, it does not return a
value like a normal function: instead, it immediately aborts
evaluation, and prints the error message we give it.
The mySecond function returns the second element of its input
list, but fails if its input list isn’t long enough.
mySecond :: [a] -> a
mySecond xs = if null (tail xs)
then error "list too short"
else head (tail xs)As usual, we can see how this works in practice in ghci.
ghci> mySecond "xi"
'i'
ghci> mySecond [2]
*** Exception: list too short
ghci> head (mySecond [[9]])
*** Exception: list too short
Notice the third case above, where we try to use the result of the
call to mySecond as the argument to another function. Evaluation
still terminates and drops us back to the ghci prompt. This is
the major weakness of using error: it doesn’t let our caller
distinguish between a recoverable error and a problem so severe
that it really should terminate our program.
As we have already seen, a pattern matching failure causes a similar unrecoverable error.
ghci> mySecond []
*** Exception: Prelude.tail: empty list
We can use the Maybe type to represent the possibility of an
error.
If we want to indicate that an operation has failed, we can use
the Nothing constructor. Otherwise, we wrap our value with the
Just constructor.
Let’s see how our mySecond function changes if we return a
Maybe value instead of calling error.
safeSecond :: [a] -> Maybe a
safeSecond [] = Nothing
safeSecond xs = if null (tail xs)
then Nothing
else Just (head (tail xs))If the list we’re passed is too short, we return Nothing to our
caller. This lets them decide what to do, where a call to error
would force a crash.
ghci> safeSecond []
Nothing
ghci> safeSecond [1]
Nothing
ghci> safeSecond [1,2]
Just 2
ghci> safeSecond [1,2,3]
Just 2
To return to an earlier topic, we can further improve the readability of this function with pattern matching.
tidySecond :: [a] -> Maybe a
tidySecond (_:x:_) = Just x
tidySecond _ = NothingThe first pattern only matches if the list is at least two
elements long (it contains two list constructors), and it binds
the variable x to the list’s second element. The second pattern
is matched if the first fails.
Within the body of a function, we can introduce new local
variables whenever we need them, using a let expression. Here is
a simple function that determines whether we should lend some
money to a customer. We meet a money reserve of at least 100, we
return our new balance after subtracting the amount we have
loaned.
lend amount balance = let reserve = 100
newBalance = balance - amount
in if balance < reserve
then Nothing
else Just newBalanceThe keywords to look out for here are let, which starts a block
of variable declarations, and in, which ends it. Each line
introduces a new variable. The name is on the left of the =, and
the expression to which it is bound is on the right.
We can use the names of a variable in a let block both within
the block of declarations and in the expression that follows the
in keyword.
In general, we’ll refer to the places within our code where we can use a name as the name’s scope. If we can use a name, it’s in scope, otherwise it’s out of scope. If a name is visible throughout a source file, we say it’s at the top level.
We can “nest” multiple let blocks inside each other in an
expression.
foo = let a = 1
in let b = 2
in a + bIt’s perfectly legal, but not exactly wise, to repeat a variable
name in a nested let expression.
bar = let x = 1
in ((let x = "foo" in x), x)Here, the inner x is hiding, or shadowing, the outer x. It
has the same name, but a different type and value.
ghci> bar
("foo",1)
We can also shadow a function’s parameters, leading to even stranger results. What is the type of this function?
quux a = let a = "foo"
in a ++ "eek!"Because the function’s argument a is never used in the body of
the function, due to being shadowed by the let-bound a, the
argument can have any type at all.
ghci> :type quux
quux :: t -> [Char]
We can use another mechanism to introduce local variables: the
where clause. The definitions in a where clause apply to the
code that precedes it. Here’s a similar function to lend,
using where instead of let.
lend2 amount balance = if amount < reserve * 0.5
then Just newBalance
else Nothing
where reserve = 100
newBalance = balance - amountWhile a where clause may initially seem weird, it offers a
wonderful aid to readability. It lets us direct our reader’s focus
to the important details of an expression, with the supporting
definitions following afterwards. After a while, you may find
yourself missing where clauses in languages that lack them.
As with let expressions, white space is significant in where
clauses. We will talk more about the layout rules shortly, in
the section called “The offside rule and white space in an expression”
You’ll have noticed that Haskell’s syntax for defining a variable
looks very similar to its syntax for defining a function. This
symmetry is preserved in let and where blocks: we can define
local functions just as easily as local variables.
pluralise :: String -> [Int] -> [String]
pluralise word counts = map plural counts
where plural 0 = "no " ++ word ++ "s"
plural 1 = "one " ++ word
plural n = show n ++ " " ++ word ++ "s"We have defined a local function, plural, that consists of
several equations. Local functions can freely use variables from
the scopes that enclose them: here, we use word from the
definition of the outer function pluralise. In the definition of
pluralise, the map function (which we’ll be revisiting in the
next chapter) applies the local function plural to every element
of the counts list.
We can also define variables, as well as functions, at the top level of a source file.
itemName = "Weighted Companion Cube"In our definitions of lend and lend2, the left margin of our
text wandered around quite a bit. This was not an accident:
normally in Haskell white space has meaning: it uses the code
layout as defined in the report as a cue to parse it. This is
sometimes called the offside rule.
At the beginning of a source file, the first top level declaration or definition can start in any column, and the Haskell compiler or interpreter remembers that indentation level. Every subsequent top level declaration must have the same indentation.
Here’s an illustration of the top level indentation rule. Our
first file, GoodIndent.hs, is well behaved.
-- This is the leftmost column.
-- It's fine for top-level declarations to start in any column...
firstGoodIndentation = 1
-- ...provided all subsequent declarations do, too!
secondGoodIndentation = 2Our second, BadIndent.hs, doesn’t play by the rules.
-- This is the leftmost column.
-- Our first declaration is in column 4.
firstBadIndentation = 1
-- Our second is left of the first, which is illegal!
secondBadIndentation = 2Here’s what happens when we try to load the two files into ghci.
ghci> :load GoodIndent.hs
[1 of 1] Compiling Main ( GoodIndent.hs, interpreted )
Ok, one module loaded.
ghci> :load BadIndent.hs
[1 of 1] Compiling Main ( BadIndent.hs, interpreted )
BadIndent.hs:8:3: error:
parse error on input ‘secondBadIndentation’
|
8 | secondBadIndentation = 2
| ^^^^^^^^^^^^^^^^^^^^
Failed, no modules loaded.
An empty following line is treated as a continuation of the current item, as is a following line indented further to the right.
The rules for let expressions and where clauses are similar.
After a let or where keyword, the Haskell compiler or
interpreter remembers the indentation of the next token it sees.
If the line that follows is empty, or its indentation is further
to the right, it is considered to continue the previous line. If
the indentation is the same as the start of the preceding item,
this is treated as beginning a new item in the same block.
foo = let firstDefinition = blah blah
-- a comment-only line is treated as empty
continuation blah
-- we reduce the indentation, so this is a new definition
secondDefinition = yada yada
continuation yada
in whateverHere are nested uses of let and where.
bar = let b = 2
c = True
in let a = b
in (a, c)The name a is only visible within the inner let expression.
It’s not visible in the outer let. If we try to use the name a
there, we’ll get a compilation error. The indentation gives both
us and the compiler a visual cue as to what is currently in scope.
foo = x
where x = y
where y = 2Similarly, the scope of the first where clause is the definition
of foo, but the scope of the second is just the first where
clause.
The indentation we use for the let and where clauses makes our
intentions easy to figure out.
The default in Haskell code is to indent using spaces. ghc and
ghci will warn you if you indent with tabs unless you disable it
with the -Wno-tabs flag. The reason for this is that it is
easier to align expressions using spaces.
If you like tabs you can use them as long as you correctly align
expressions. For the compiler a tab equals to eight spaces and
uses this amount to determine if an indented expression is
correctly aligned. In the next example you’ll see the b aligned
with the a only if the tab width equals eight in whichever app
you are using to read this text.
x = let a = 1
b = 2
in a + bWe need the amount of characters before the a to be equal to
eight or the b won’t be aligned and won’t compile. Totally
impracticall. It is better to break the expressions as below.
x = let
a = 1
b = 2
in a + bAs long as you correctly align expressions you can even mix spaces and tabs.
We can use explicit structuring instead of layout to indicate what
we mean. To do so, we start a block of equations with an opening
curly brace; separate each item with a semicolon; and finish the
block with a closing curly brace. The following two uses of let
have the same meanings.
bar = let a = 1
b = 2
c = 3
in a + b + c
foo = let { a = 1; b = 2;
c = 3 }
in a + b + cWhen we use explicit structuring, the normal layout rules don’t
apply, which is why we can get away with unusual indentation in
the second let expression.
We can use explicit structuring anywhere that we’d normally use
layout. It’s valid for where clauses, and even top-level
declarations. Just remember that although the facility exists,
explicit structuring is hardly ever actually used in Haskell
programs.
Function definitions are not the only place where we can use
pattern matching. The case construct lets us match patterns
within an expression. Here’s what it looks like. This function
(defined for us in Data.Maybe) unwraps a Maybe value, using a
default if the value is Nothing.
fromMaybe defval wrapped =
case wrapped of
Nothing -> defval
Just value -> valueThe case keyword is followed by an arbitrary expression: the
pattern match is performed against the result of this expression.
The of keyword signifies the end of the expression and the
beginning of the block of patterns and expressions.
Each item in the block consists of a pattern, followed by an arrow
->, followed by an expression to evaluate if that pattern
matches. These expressions must all have the same type. The result
of the case expression is the result of the expression
associated with the first pattern to match. Matches are attempted
from top to bottom.
To express “here’s the expression to evaluate if none of the other
patterns match”, we just use the wild card pattern _ as the last
in our list of patterns. If a pattern match fails, we will get the
same kind of runtime error as we saw earlier.
There are a few ways in which new Haskell programmers can misunderstand or misuse patterns. Here are some attempts at pattern matching gone awry. Depending on what you expect one of these examples to do, it might contain a surprise.
data Fruit = Apple | Orange
apple = "apple"
orange = "orange"
whichFruit :: String -> Fruit
whichFruit f = case f of
apple -> Apple
orange -> OrangeA naive glance suggests that this code is trying to check the
value f to see whether it matches the value apple or orange.
It is easier to spot the mistake if we rewrite the code in an equational style.
equational apple = Apple
equational orange = OrangeNow can you see the problem? Here, it is more obvious apple does
not refer to the top level value named apple: it is a local
pattern variable.
Here’s a corrected version of this function.
betterFruit f = case f of
"apple" -> Apple
"orange" -> OrangeWe fixed the problem by matching against the literal values
"apple" and "orange".
What if we want to compare the values stored in two nodes of type Tree, and return one of them if they’re equal? Here’s an attempt.
bad_nodesAreSame (Node a _ _) (Node a _ _) = Just a
bad_nodesAreSame _ _ = NothingA name can only appear once in a set of pattern bindings. We cannot place a variable in multiple positions to express the notion “this value and that should be identical”. Instead, we’ll solve this problem using guards, another invaluable Haskell feature.
Pattern matching limites us to performing fixed tests of a value’s shape. Although this is useful, we will often want to make a more expressive check before evaluating a function’s body. Haskell provides a feature, guards, that give us this ability. We’ll introduce the idea with a modification of the function we wrote to compare two nodes of a tree.
nodesAreSame (Node a _ _) (Node b _ _)
| a == b = Just a
nodesAreSame _ _ = NothingIn this example, we use pattern matching to ensure that we are looking at values of the right shape, and a guard to compare pieces of them.
A pattern can be followed by zero or more guards, each an
expression of type Bool. A guard is introduced by a | symbol.
This is followed by the guard expression, then an = symbol (or
-> if we’re in a case expression), then the body to use if the
guard expression evaluates to True. If a pattern matches, each
guard associated with that pattern is evaluated, in the order in
which they are written. If a guard succeeds, the body affiliated
with it is used as the result of the function. If no guard
succeeds, pattern matching moves on to the next pattern.
When a guard expression is evaluated, all of the variables mentioned in the pattern with which it is associated are bound and can be used.
Here is a reworked version of our lend function that uses
guards.
lend3 amount balance
| amount <= 0 = Nothing
| amount > reserve * 0.5 = Nothing
| otherwise = Just newBalance
where reserve = 100
newBalance = balance - amountThe special-looking guard expression otherwise is simply a
variable bound to the value True, to aid readability.
We can use guards anywhere that we can use patterns. Writing a
function as a series of equations using pattern matching and
guards can make it much clearer. Remember the myDrop function we
defined in the section called “Conditional evaluation”
myDrop n xs = if n <= 0 || null xs
then xs
else myDrop (n - 1) (tail xs)Here is a reformulation that uses patterns and guards.
niceDrop n xs | n <= 0 = xs
niceDrop _ [] = []
niceDrop n (_:xs) = niceDrop (n - 1) xsThis change in style lets us enumerate up front the cases in
which we expect a function to behave differently. If we bury the
decisions inside a function as if expressions, the code becomes
harder to read.
- Write a function that computes the number of elements in a
list. To test it, ensure that it gives the same answers as the
standard
lengthfunction. - Add a type signature for your function to your source file. To
test it, load the source file into
ghciagain. - Write a function that computes the mean of a list, i.e. the sum
of all elements in the list divided by its length. (You may
need to use the
fromIntegralfunction to convert the length of the list from an integer into a floating point number). - Turn a list into a palindrome, i.e. it should read the same
both backwards and forwards. For example, given the list
[1,2,3], your function should return[1,2,3,3,2,1]. - Write a function that determines whether its input list is a palindrome.
- Create a function that sorts a list of lists based on the
length of each sublist. (You may want to look at the
sortByfunction from theData.Listmodule.) - Define a function that joins a list of lists together using a separator value:
intersperse :: a -> [[a]] -> [a]The separator should appear between elements of the list, but should not follow the last element. Your function should behave as follows.
ghci> :load Intersperse
[1 of 1] Compiling Main ( Intersperse.hs, interpreted )
Ok, modules loaded: Main.
ghci> intersperse ',' []
""
ghci> intersperse ',' ["foo"]
"foo"
ghci> intersperse ',' ["foo","bar","baz","quux"]
"foo,bar,baz,quux"
- Using the binary tree type that we defined earlier in this
chapter, write a function that will determine the height of the
tree. The height is the largest number of hops from the root to
an
Empty. For example, the treeEmptyhas height zero;Node "x" Empty Emptyhas height one;Node "x" Empty (Node "y" Empty Empty)has height two; and so on. - Consider three two-dimensional points a, b, and c. If we
look at the angle formed by the line segment from a to b
and the line segment from b to c, it either turns left,
turns right, or forms a straight line. Define a
Directiondata type that lets you represent these possibilities. - Write a function that calculates the turn made by three 2D points and returns a Direction.
- Define a function that takes a list of 2D points and computes
the direction of each successive triple. Given a list of
points
[a,b,c,d,e], it should begin by computing the turn made by[a,b,c], then the turn made by[b,c,d], then[c,d,e]. Your function should return a list ofDirection. - Using the code from the preceding three exercises, implement Graham’s scan algorithm for the convex hull of a set of 2D points. You can find good description of what a convex hull is, and how the Graham scan algorithm should work, on Wikipedia.
[fn:1] If you are familiar with C or C++, it is analogous to a
typedef.