Preamble
Introduction
Getting Started
Language Basics (Solutions)
Type Basics (Solutions)
IO (Solutions)
Modules (Solutions)
Recursion
Complexity

As you've probably ascertained by this point, the type system is integral to Haskell. While this chapter is called "Advanced Types", you will probably find it to be more general than that and it must not be skipped simply because you're not interested in the type system.

## Type Synonyms

Type synonyms exist in Haskell simply for convenience: their removal would not make Haskell any less powerful.

Consider the case when you are constantly dealing with lists of three-dimensional points. For instance, you might have a function with type [(Double,Double,Double)] -> Double -> [(Double,Double,Double)]. Since you are a good software engineer, you want to place type signatures on all your top-level functions. However, typing [(Double,Double,Double)] all the time gets very tedious. To get around this, you can define a type synonym:

type List3D = [(Double,Double,Double)]

Now, the type signature for your functions may be written List3D -> Double -> List3D.

We should note that type synonyms cannot be self-referential. That is, you cannot have:

This is because this is an "infinite type." Since Haskell removes type synonyms very early on, any instance of BadType will be replaced by Int -> BadType, which will result in an infinite loop.

To make a recursive type, one would use newtype

newtype GoodType =  MakeGoodType (Int -> GoodType)

Type synonyms can also be parameterized. For instance, you might want to be able to change the types of the points in the list of 3D points. For this, you could define:

type List3D a = [(a,a,a)]

Then your references to [(Double,Double,Double)] would become List3D Double.

## Newtypes

Consider the problem in which you need to have a type which is very much like Int, but its ordering is defined differently. Perhaps you wish to order Ints first by even numbers then by odd numbers (that is, all odd numbers are greater than any even number and within the odd/even subsets, ordering is standard).

Unfortunately, you cannot define a new instance of Ord for Int because then Haskell won't know which one to use. What you want is to define a type which is isomorphic to Int.

Note

"Isomorphic" is a common term in mathematics which basically means "structurally identical." For instance, in graph theory, if you have two graphs which are identical except they have different labels on the nodes, they are isomorphic. In our context, two types are isomorphic if they have the same underlying structure.

One way to do this would be to define a new datatype:

data MyInt = MyInt Int

We could then write appropriate code for this datatype. The problem (and this is very subtle) is that this type is not truly isomorphic to Int: it has one more value. When we think of the type Int, we usually think that it takes all values of integers, but it really has one more value: $\bot$ (pronounced "bottom"), which is used to represent erroneous or undefined computations. Thus, MyInt has not only values MyInt 0, MyInt 1 and so on, but also MyInt $\bot$. However, since datatypes can themselves be undefined, it has an additional value: $\bot$ which differs from MyInt $\bot$ and this makes the types non-isomorphic. (See the section on Bottom for more information on bottom.)

Disregarding that subtlety, there may be efficiency issues with this representation: now, instead of simply storing an integer, we have to store a pointer to an integer and have to follow that pointer whenever we need the value of a MyInt.

To get around these problems, Haskell has a newtype construction. A newtype is a cross between a datatype and a type synonym: it has a constructor like a datatype, but it can have only one constructor and this constructor can have only one argument. For instance, we can define:

newtype MyInt = MyInt Int

But we cannot define any of:

Of course, the fact that we cannot define Bad2 as above is not a big issue: we simply use type instead:

type Good2 = Good2 Int Double

or (almost equivalently) declare a newtype alias to the existing tuple type:

newtype Good2 = Good2 (Int,Double)

o Now, suppose we've defined MyInt as a newtype:

instance Ord MyInt where
MyInt i < MyInt j
| odd i  && odd  j = i < j
| even i && even j = i < j
| even i           = True
| otherwise        = False
where odd x = (x `mod` 2) == 1
even  = not . odd

Like datatype, we can still derive classes like Show and Eq over newtypes (in fact, I'm implicitly assuming we have derived Eq over MyInt -- where is my assumption in the above code?).

Moreover, in recent versions of GHC (see the section on Ghc), on newtypes, you are allowed to derive any class of which the base type (in this case, Int) is an instance. For example, we could derive Num on MyInt to provide arithmetic functions over it.

Pattern matching over newtypes is exactly as in datatypes. We can write constructor and destructor functions for MyInt as follows:

mkMyInt i = MyInt i
unMyInt (MyInt i) = i

## Datatypes

We've already seen datatypes used in a variety of contexts. This section concludes some of the discussion and introduces some of the common datatypes in Haskell. It also provides a more theoretical underpinning to what datatypes actually are.

### Strict Fields

One of the great things about Haskell is that computation is performed lazily. However, sometimes this leads to inefficiencies. One way around this problem is to use datatypes with strict fields. Before we talk about the solution, let's spend some time to get a bit more comfortable with how bottom works in to the picture (for more theory, see the section on Bottom).

Suppose we've defined the unit datatype (this one of the simplest datatypes you can define):

data Unit = Unit

This datatype has exactly one constructor, Unit, which takes no arguments. In a strict language like ML, there would be exactly one value of type Unit: namely, Unit. This is not quite so in Haskell. In fact, there are two values of type Unit. One of them is Unit. The other is bottom (written $\bot$).

You can think of bottom as representing a computation which won't halt. For instance, suppose we define the value:

foo = foo

This is perfectly valid Haskell code and simply says that when you want to evaluate foo, all you need to do is evaluate foo. Clearly this is an "infinite loop."

What is the type of foo? Simply a. We cannot say anything more about it than that. The fact that foo has type a in fact tells us that it must be an infinite loop (or some other such strange value). However, since foo has type a and thus can have any type, it can also have type Unit. We could write, for instance:

foo :: Unit
foo = foo

Thus, we have found a second value with type Unit. In fact, we have found all values of type Unit. Any other non-terminating function or error-producing function will have exactly the same effect as foo (though Haskell provides some more utility with the function error).

This means, for instance, that there are actually four values with type Maybe Unit. They are: $\bot$, Nothing, Just $\bot$ and Just Unit. However, it could be the fact that you, as a programmer, know that you will never come across the third of these. Namely, you want the argument to Just to be strict. This means that if the argument to Just is bottom, then the entire structure becomes bottom. You use an exclamation point to specify a constructor as strict. We can define a strict version of Maybe as:

data SMaybe a = SNothing | SJust !a

There are now only three values of SMaybe. We can see the difference by writing the following program:

module Main where

import System

data SMaybe a = SNothing | SJust !a  deriving Show

main = do
[cmd] <- getArgs
case cmd of
"a" -> printJust   undefined
"b" -> printJust   Nothing
"c" -> printJust  (Just undefined)
"d" -> printJust  (Just ())

"e" -> printSJust  undefined
"f" -> printSJust  SNothing
"g" -> printSJust (SJust undefined)
"h" -> printSJust (SJust ())

printJust :: Maybe () -> IO ()
printJust Nothing = putStrLn "Nothing"
printJust (Just x) = do putStr "Just "; print x

printSJust :: SMaybe () -> IO ()
printSJust SNothing = putStrLn "Nothing"
printSJust (SJust x) = do putStr "Just "; print x

Here, depending on what command line argument is passed, we will do something different. The outputs for the various options are:

Example:

% ./strict a
Fail: Prelude.undefined

% ./strict b
Nothing

% ./strict c
Just
Fail: Prelude.undefined

% ./strict d
Just ()

% ./strict e
Fail: Prelude.undefined

% ./strict f
Nothing

% ./strict g
Fail: Prelude.undefined

% ./strict h
Just ()

The thing worth noting here is the difference between cases "c" and "g". In the "c" case, the Just is printed, because this is printed before the undefined value is evaluated. However, in the "g" case, since the constructor is strict, as soon as you match the SJust, you also match the value. In this case, the value is undefined, so the whole thing fails before it gets a chance to do anything.

## Classes

We have already encountered type classes a few times, but only in the context of previously existing type classes. This section is about how to define your own. We will begin the discussion by talking about Pong and then move on to a useful generalization of computations.

### Pong

The discussion here will be motivated by the construction of the game Pong (see the appendix on Pong for the full code). In Pong, there are three things drawn on the screen: the two paddles and the ball. While the paddles and the ball are different in a few respects, they share many commonalities, such as position, velocity, acceleration, color, shape, and so on. We can express these commonalities by defining a class for Pong entities, which we call Entity. We make such a definition as follows:

class Entity a where
getPosition :: a -> (Int,Int)
getVelocity :: a -> (Int,Int)
getAcceleration :: a -> (Int,Int)
getColor :: a -> Color
getShape :: a -> Shape

This code defines a typeclass Entity. This class has five methods: getPosition, getVelocity, getAcceleration, getColor and getShape with the corresponding types.

The first line here uses the keyword class to introduce a new typeclass. We can read this typeclass definition as "There is a typeclass 'Entity'; a type 'a' is an instance of Entity if it provides the following five functions: ...". To see how we can write an instance of this class, let us define a player (paddle) datatype:

playerNumber :: Int }

Given this data declaration, we can define Paddle to be an instance of Entity:

getShape = Rectangle 5 . paddleHeight

The actual Haskell types of the class functions all have included the context Entity a =>. For example, getPosition has type Entity{a => a -> (Int,Int)}. However, it will turn out that many of our routines will need entities to also be instances of Eq. We can therefore choose to make Entity a subclass of Eq: namely, you can only be an instance of Entity if you are already an instance of Eq. To do this, we change the first line of the class declaration to:

class Eq a => Entity a where

Now, in order to define Paddles to be instances of Entity we will first need them to be instances of Eq -- we can do this by deriving the class.

### Computations

Let's think back to our original motivation for defining the Maybe datatype from the section on Datatypes-maybe. We wanted to be able to express that functions (i.e., computations) can fail.

Let us consider the case of performing search on a graph. Allow us to take a small aside to set up a small graph library:

data Graph v e = Graph [(Int,v)] [(Int,Int,e)]

The Graph datatype takes two type arguments which correspond to vertex and edge labels. The first argument to the Graph constructor is a list (set) of vertices; the second is the list (set) of edges. We will assume these lists are always sorted and that each vertex has a unique id and that there is at most one edge between any two vertices.

Suppose we want to search for a path between two vertices. Perhaps there is no path between those vertices. To represent this, we will use the Maybe datatype. If it succeeds, it will return the list of vertices traversed. Our search function could be written (naively) as follows:

search :: Graph v e -> Int -> Int -> Maybe [Int]
search g@(Graph vl el) src dst
| src == dst = Just [src]
| otherwise  = search' el
where search' [] = Nothing
search' ((u,v,_):es)
| src == u  =
case search g v dst of
Just p  -> Just (u:p)
Nothing -> search' es
| otherwise = search' es

This algorithm works as follows (try to read along): to search in a graph g from src to dst, first we check to see if these are equal. If they are, we have found our way and just return the trivial solution. Otherwise, we want to traverse the edge-list. If we're traversing the edge-list and it is empty, we've failed, so we return Nothing. Otherwise, we're looking at an edge from u to v. If u is our source, then we consider this step and recursively search the graph from v to dst. If this fails, we try the rest of the edges; if this succeeds, we put our current position before the path found and return. If u is not our source, this edge is useless and we continue traversing the edge-list.

This algorithm is terrible: namely, if the graph contains cycles, it can loop indefinitely. Nevertheless, it is sufficient for now. Be sure you understand it well: things only get more complicated.

Now, there are cases where the Maybe datatype is not sufficient: perhaps we wish to include an error message together with the failure. We could define a datatype to express this as:

data Failable a = Success a | Fail String

Now, failures come with a failure string to express what went wrong. We can rewrite our search function to use this datatype:

search2 :: Graph v e -> Int -> Int -> Failable [Int]
search2 g@(Graph vl el) src dst
| src == dst = Success [src]
| otherwise  = search' el
where search' [] = Fail "No path"
search' ((u,v,_):es)
| src == u  =
case search2 g v dst of
Success p -> Success (u:p)
_         -> search' es
| otherwise = search' es

This code is a straightforward translation of the above.

There is another option for this computation: perhaps we want not just one path, but all possible paths. We can express this as a function which returns a list of lists of vertices. The basic idea is the same:

search3 :: Graph v e -> Int -> Int -> [[Int]]
search3 g@(Graph vl el) src dst
| src == dst = [[src]]
| otherwise  = search' el
where search' [] = []
search' ((u,v,_):es)
| src == u  =
map (u:) (search3 g v dst) ++
search' es
| otherwise = search' es

The code here has gotten a little shorter, thanks to the standard prelude map function, though it is essentially the same.

We may ask ourselves what all of these have in common and try to gobble up those commonalities in a class. In essence, we need some way of representing success and some way of representing failure. Furthermore, we need a way to combine two successes (in the first two cases, the first success is chosen; in the third, they are strung together). Finally, we need to be able to augment a previous success (if there was one) with some new value. We can fit this all into a class as follows:

class Computation c where
success :: a -> c a
failure :: String -> c a
augment :: c a -> (a -> c b) -> c b
combine :: c a -> c a -> c a

In this class declaration, we're saying that c is an instance of the class Computation if it provides four functions: success, failure, augment and combine. The success function takes a value of type a and returns it wrapped up in c, representing a successful computation. The failure function takes a String and returns a computation representing a failure. The combine function takes two previous computations and produces a new one which is the combination of both. The augment function is a bit more complex.

The augment function takes some previously given computation (namely, c a) and a function which takes the value of that computation (the a) and returns a b and produces a b inside of that computation. Note that in our current situation, giving augment the type c a -> (a -> a) -> c a would have been sufficient, since a is always [Int], but we make it more general this time just for generality.

How augment works is probably best shown by example. We can define Maybe, Failable and [] to be instances of Computation as:

instance Computation Maybe where
success = Just
failure = const Nothing
augment (Just x) f = f x
augment Nothing  _ = Nothing
combine Nothing y = y
combine x _ = x

Here, success is represented with Just and failure ignores its argument and returns Nothing. The combine function takes the first success we found and ignores the rest. The function augment checks to see if we succeeded before (and thus had a Just something) and, if we did, applies f to it. If we failed before (and thus had a Nothing), we ignore the function and return Nothing.

instance Computation Failable where
success = Success
failure = Fail
augment (Success x) f = f x
augment (Fail s) _ = Fail s
combine (Fail _) y = y
combine x _ = x

These definitions are obvious. Finally:

instance Computation [] where
success a = [a]
failure = const []
augment l f = concat (map f l)
combine = (++)

Here, the value of a successful computation is a singleton list containing that value. Failure is represented with the empty list and to combine previous successes we simply catenate them. Finally, augmenting a computation amounts to mapping the function across the list of previous computations and concatentate them. we apply the function to each element in the list and then concatenate the results.

Using these computations, we can express all of the above versions of search as:

searchAll g@(Graph vl el) src dst
| src == dst = success [src]
| otherwise  = search' el
where search' [] = failure "no path"
search' ((u,v,_):es)
| src == u  = (searchAll g v dst `augment`
(success . (u:)))
`combine` search' es
| otherwise = search' es

In this, we see the uses of all the functions from the class Computation.

If you've understood this discussion of computations, you are in a very good position as you have understood the concept of monads, probably the most difficult concept in Haskell. In fact, the Computation class is almost exactly the Monad class, except that success is called return, failure is called fail and augment is called >>= (read "bind"). The combine function isn't actually required by monads, but is found in the MonadPlus class for reasons which will become obvious later.

If you didn't understand everything here, read through it again and then wait for the proper discussion of monads in the chapter Monads.

## Instances

We have already seen how to declare instances of some simple classes; allow us to consider some more advanced classes here. There is a Functor class defined in the Functor module.

Note

The name "functor", like "monad" comes from category theory. There, a functor is like a function, but instead of mapping elements to elements, it maps structures to structures.

The definition of the functor class is:

class Functor f where
fmap :: (a -> b) -> f a -> f b

The type definition for fmap (not to mention its name) is very similar to the function map over lists. In fact, fmap is essentially a generalization of map to arbitrary structures (and, of course, lists are already instances of Functor). However, we can also define other structures to be instances of functors. Consider the following datatype for binary trees:

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

We can immediately identify that the BinTree type essentially "raises" a type a into trees of that type. There is a naturally associated functor which goes along with this raising. We can write the instance:

instance Functor BinTree where
fmap f (Leaf a) = Leaf (f a)
fmap f (Branch left right) =
Branch (fmap f left) (fmap f right)

Now, we've seen how to make something like BinTree an instance of Eq by using the deriving keyword, but here we will do it by hand. We want to make BinTree as instances of Eq but obviously we cannot do this unless a is itself an instance of Eq. We can specify this dependence in the instance declaration:

instance Eq a => Eq (BinTree a) where
Leaf a == Leaf b = a == b
Branch l r == Branch l' r' = l == l' && r == r'
_ == _ = False

The first line of this can be read "if a is an instance of Eq, then BinTree a is also an instance of Eq". We then provide the definitions. If we did not include the Eq a => part, the compiler would complain because we're trying to use the == function on as in the second line.

The "Eq a =>" part of the definition is called the "context." We should note that there are some restrictions on what can appear in the context and what can appear in the declaration. For instance, we're not allowed to have instance declarations that don't contain type constructors on the right hand side. To see why, consider the following declarations:

class MyEq a where
myeq :: a -> a -> Bool

instance Eq a => MyEq a where
myeq = (==)

As it stands, there doesn't seem to be anything wrong with this definition. However, if elsewhere in a program we had the definition:

instance MyEq a => Eq a where
(==) = myeq

In this case, if we're trying to establish if some type is an instance of Eq, we could reduce it to trying to find out if that type is an instance of MyEq, which we could in turn reduce to trying to find out if that type is an instance of Eq, and so on. The compiler protects itself against this by refusing the first instance declaration.

This is commonly known as the closed-world assumption. That is, we're assuming, when we write a definition like the first one, that there won't be any declarations like the second. However, this assumption is invalid because there's nothing to prevent the second declaration (or some equally evil declaration). The closed world assumption can also bite you in cases like:

class OnlyInts a where
foo :: a -> a -> Bool

instance OnlyInts Int where
foo = (==)

bar :: OnlyInts a => a -> Bool
bar = foo 5

We've again made the closed-world assumption: we've assumed that the only instance of OnlyInts is Int, but there's no reason another instance couldn't be defined elsewhere, ruining our definition of bar.

## Kinds

Let us take a moment and think about what types are available in Haskell. We have simple types, like Int, Char, Double and so on. We then have type constructors like Maybe which take a type (like Char) and produce a new type, Maybe Char. Similarly, the type constructor [] (lists) takes a type (like Int) and produces [Int]. We have more complex things like -> (function arrow) which takes two types (say Int and Bool) and produces a new type Int -> Bool.

In a sense, these types themselves have type. Types like Int have some sort of basic type. Types like Maybe have a type which takes something of basic type and returns something of basic type. And so forth.

Talking about the types of types becomes unwieldy and highly ambiguous, so we call the types of types "kinds." What we have been calling "basic types" have kind "*". Something of kind * is something which can have an actual value. There is also a single kind constructor, -> with which we can build more complex kinds.

Consider Maybe. This takes something of kind * and produces something of kind *. Thus, the kind of Maybe is * -> *. Recall the definition of Pair from the section on Datatypes-pairs:

data Pair a b = Pair a b

Here, Pair is a type constructor which takes two arguments, each of kind * and produces a type of kind *. Thus, the kind of Pair is * -> (* -> *). However, we again assume associativity so we just write * -> * -> *.

Let us make a slightly strange datatype definition:

data Strange c a b =
MkStrange (c a) (c b)

Before we analyze the kind of Strange, let's think about what it does. It is essentially a pairing constructor, though it doesn't pair actual elements, but elements within another constructor. For instance, think of c as Maybe. Then MkStrange pairs Maybes of the two types a and b. However, c need not be Maybe but could instead by [], or many other things.

What do we know about c, though? We know that it must have kind * -> *. This is because we have c a on the right hand side. The type variables a and b each have kind * as before. Thus, the kind of Strange is (* -> *) -> * -> * -> *. That is, it takes a constructor (c) of kind * -> * together with two types of kind * and produces something of kind *.

A question may arise regarding how we know a has kind * and not some other kind k. In fact, the inferred kind for Strange is (k -> *) -> k -> k -> *. However, this requires polymorphism on the kind level, which is too complex, so we make a default assumption that k = *.

Note

There are extensions to GHC which allow you to specify the kind of constructors directly. For instance, if you wanted a different kind, you could write this explicitly:

data Strange (c :: (* -> *) -> *) a b = MkStrange (c a) (c b)

to give a different kind to Strange.

The notation of kinds suggests that we can perform partial application, as we can for functions. And, in fact, we can. For instance, we could have:

type MaybePair = Strange Maybe

The kind of MaybePair is, not surprisingly, * -> * -> *.

We should note here that all of the following definitions are acceptable:

type MaybePair1     = Strange Maybe
type MaybePair2 a   = Strange Maybe a
type MaybePair3 a b = Strange Maybe a b

These all appear to be the same, but they are in fact not identical as far as Haskell's type system is concerned. The following are all valid type definitions using the above:

type MaybePair1a = MaybePair1
type MaybePair1b = MaybePair1 Int
type MaybePair1c = MaybePair1 Int Double

type MaybePair2b = MaybePair2 Int
type MaybePair2c = MaybePair2 Int Double

type MaybePair3c = MaybePair3 Int Double

But the following are not valid:

type MaybePair2a = MaybePair2

type MaybePair3a = MaybePair3
type MaybePair3b = MaybePair3 Int

This is because while it is possible to partially apply type constructors on datatypes, it is not possible on type synonyms. For instance, the reason MaybePair2a is invalid is because MaybePair2 is defined as a type synonym with one argument and we have given it none. The same applies for the invalid MaybePair3 definitions.

what is it?