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Type classes

CIS 194 Week 5
25 September 2014

Before we get down to the business of the day, we need a little header information to get us going:

{-# LANGUAGE FlexibleInstances #-}

That’s a so-called language pragma. GHC includes many features which are not part of the standardized Haskell language. To enable these features, we use language pragmas. There are lots of these language pragmas available — we’ll see only a few over the course of the semester.

import Data.Char  ( isUpper, toUpper )
import Data.Maybe ( mapMaybe )
import Text.Read  ( readMaybe )

Putting the fun in function

We have seen now several cases of using a functional programming style. Here, we will look at several functions using a very functional style to help you get acclimated to this mode of programming.

Functional combinators

First, we need a few more combinators to get us going:

(.) :: (b -> c) -> (a -> b) -> a -> c
(.) f g x = f (g x)

The (.) operator, part of the Haskell Prelude, is just function composition. Say we want to take every element of a list and add 1 and then multiply by 4. Here is a good way to do it:

add1Mul4 :: [Int] -> [Int]
add1Mul4 x = map ((*4) . (+1)) x

While we’re at it, we should also show the ($) operator, which has a trivial-looking definition:

($) :: (a -> b) -> a -> b
f $ x = f x

Why have such a thing? Because ($) is parsed as an operator, and this is useful for avoiding parentheses. For example, if we wish to negate the number of even numbers in a list, we could say

negateNumEvens1 :: [Int] -> Int
negateNumEvens1 x = negate (length (filter even x))

or

negateNumEvens2 :: [Int] -> Int
negateNumEvens2 x = negate $ length $ filter even x

No more parentheses!

Point-free programming

Both examples above just apply a sequence of operations to a parameter. The parameter itself – x – isn’t important at all. So, let’s leave it out:

add1Mul4' :: [Int] -> [Int]
add1Mul4' = map ((*4) . (+1))

negateNumEvens3 :: [Int] -> Int
negateNumEvens3 = negate . length . filter even

Point-free notation is where the (.) operator really comes in: it’s the way of combining functions when you want to perform one after another.

Lambda

It is sometimes necessary to create an anonymous function, or lambda expression. This is best explained by example. Say we want to duplicate every string in a list:

duplicate1 :: [String] -> [String]
duplicate1 = map dup
  where dup x = x ++ x

It’s a tiny bit silly to name dup. Instead, we can make an anonymous function:

duplicate2 :: [String] -> [String]
duplicate2 = map (\x -> x ++ x)

The backslash binds the variables after it in the expression that follows the ->. For anything but the shortest examples, it’s better to use a named helper function, though.

Functional examples

Say we want to count the total number of uppercase letters in a list of strings. Here is a good way to do it:

numUppercase :: [String] -> Int
numUppercase = length . filter isUpper . concat

It may look like the implementation ends up building and then discarding large lists. But, Haskell style includes functions like these so much, GHC knows how to optimize these really well. The internal lists never get built!

Now, say we want to make all characters in a list of strings to be uppercase. Here we go:

listToUpper :: [String] -> [String]
listToUpper = map (map toUpper)

Say we want to find all the digits in a string and add them together. Here is one way:

sumStringDigits :: String -> Int
sumStringDigits = sum . mapMaybe read_digit
  where read_digit :: Char -> Maybe Int
        read_digit = readMaybe . listify

        listify :: a -> [a]
        listify x = [x]   -- also written (:[])  [[Think about why...]]

Further reading

Previous versions of CIS194 have focused more on the functional style of programming. In light of the fact that other Penn courses (specifically, CIS 120) do this quite well, I am choosing this lighter presentation. But, the old lecture notes are great. Find them here.

Type classes

Though you never knew it, there are two different forms of polymorphism. The polymorphism we have seen so far is parametric polymorphism, which we can also call universal polymorphism (though that isn’t a standard term). A function like length :: [a] -> Int works for any type a. But, sometimes we don’t want to be universal. Sometimes, we want a function to work for several types, but not every type.

A great example of this is (+). We want to be able to add Ints and Integers and Doubles, but not Maybe Chars. This sort of polymorphism – where multiple types are allowed, but not every type – is called ad-hoc polymorphism. Haskell uses type classes to implement ad-hoc polymorphism.

A Haskell type class defines a set of operations. We can then choose several types that support those operations via class instances. (Note: These are not the same as object-oriented classes and instances!) Intuitively, type classes correspond to sets of types which have certain operations defined for them.

As an example, let’s look in detail at the Eq type class.

class Eq a where
  (==) :: a -> a -> Bool
  (/=) :: a -> a -> Bool

We can read this as follows: Eq is declared to be a type class with a single (type) parameter, a. Any type a which wants to be an instance of Eq must define two functions, (==) and (/=), with the indicated type signatures. For example, to make Int an instance of Eq we would have to define (==) :: Int -> Int -> Bool and (/=) :: Int -> Int -> Bool. (Of course, there’s no need, since the standard Prelude already defines an Int instance of Eq for us.)

Let’s look at the type of (==) again:

(==) :: Eq a => a -> a -> Bool

The Eq a that comes before the => is a type class constraint. We can read this as saying that for any type a, as long as a is an instance of Eq, (==) can take two values of type a and return a Bool. It is a type error to call the function (==) on some type which is not an instance of Eq. If a normal polymorphic type is a promise that the function will work for whatever type the caller chooses, a type class polymorphic function is a restricted promise that the function will work for any type the caller chooses, as long as the chosen type is an instance of the required type class(es).

The important thing to note is that when (==) (or any type class method) is used, the compiler uses type inference to figure out which implementation of (==) should be chosen, based on the inferred types of its arguments. In other words, it is something like using an overloaded method in a language like Java.

To get a better handle on how this works in practice, let’s make our own type and declare an instance of Eq for it.

data Foo = F Int | G Char

instance Eq Foo where
  (F i1) == (F i2) = i1 == i2
  (G c1) == (G c2) = c1 == c2
  _ == _ = False

  foo1 /= foo2 = not (foo1 == foo2)

It’s a bit annoying that we have to define both (==) and (/=). In fact, type classes can give default implementations of methods in terms of other methods, which should be used whenever an instance does not override the default definition with its own. So we could imagine declaring Eq like this:

class Eq a where
  (==) :: a -> a -> Bool
  (/=) :: a -> a -> Bool
  x /= y = not (x == y)

Now anyone declaring an instance of Eq only has to specify an implementation of (==), and they will get (/=) for free. But if for some reason they want to override the default implementation of (/=) with their own, they can do that as well.

In fact, the Eq class is actually declared like this:

class Eq a where
  (==), (/=) :: a -> a -> Bool
  x == y = not (x /= y)
  x /= y = not (x == y)

This means that when we make an instance of Eq, we can define either (==) or (/=), whichever is more convenient; the other one will be automatically defined in terms of the one we specify. (However, we have to be careful: if we don’t specify either one, we get infinite recursion!)

As it turns out, Eq (along with a few other standard type classes) is special: GHC is able to automatically generate instances of Eq for us. Like so:

data Foo' = F' Int | G' Char
  deriving (Eq, Ord, Show)

This tells GHC to automatically derive instances of the Eq, Ord, and Show type classes for our data type Foo. This deriving mechanism is baked into Haskell – you can’t make your own class and tell GHC how to derive instances. The full list of derivable classes is Eq, Ord, Enum, Ix, Bounded, Show, and Read. Not each of this is applicable to each datatype, though. GHC does provide extensions that allow other classes to be derived; see the GHC manual for details.

Type classes and Java interfaces

Type classes are quite similar to Java interfaces. Both define a set of types/classes which implement a specified list of operations. However, there are a couple of important ways in which type classes are more general than Java interfaces:

1. When a Java class is defined, any interfaces it implements must be declared. Type class instances, on the other hand, are declared separately from the declaration of the corresponding types, and can even be put in a separate module.

2. The types that can be specified for type class methods are more general and flexible than the signatures that can be given for Java interface methods, especially when multi-parameter type classes enter the picture. For example, consider a hypothetical type class

   class Blerg a b where
     blerg :: a -> b -> Bool

   Using blerg amounts to doing multiple dispatch: which implementation of blerg the compiler should choose depends on both the types a and b. There is no easy way to do this in Java.

   Haskell type classes can also easily handle binary (or ternary, or …) methods, as in

   class Num a where
     (+) :: a -> a -> a
     ...

   There is no nice way to do this in Java: for one thing, one of the two arguments would have to be the “privileged” one which is actually getting the (+) method invoked on it, and this asymmetry is awkward. Furthermore, because of Java’s subtyping, getting two arguments of a certain interface type does not guarantee that they are actually the same type, which makes implementing binary operators such as (+) awkward (usually requiring some runtime type checks).

Standard type classes

Here are some other standard type classes you should know about:

• Ord is for types whose elements can be totally ordered, that is, where any two elements can be compared to see which is less than the other. It provides comparison operations like (<) and (<=), and also the compare function.

• Num is for “numeric” types, which support things like addition, subtraction, and multipication. One very important thing to note is that integer literals are actually type class polymorphic:

  Prelude> :t 5
  5 :: Num a => a
  

  This means that literals like 5 can be used as Ints, Integers, Doubles, or any other type which is an instance of Num (Rational, Complex Double, or even a type you define…)

• Show defines the method show, which is used to convert values into Strings.

• Read is the dual of Show.

• Integral represents whole number types such as Int and Integer.

A type class example

As an example of making our own type class, consider the following:

class Listable a where
  toList :: a -> [Int]

We can think of Listable as the class of things which can be converted to a list of Ints. Look at the type of toList:

toList :: Listable a => a -> [Int]

Let’s make some instances for Listable. First, an Int can be converted to an [Int] just by creating a singleton list, and Bool can be converted similarly, say, by translating True to 1 and False to 0:

instance Listable Int where
  -- toList :: Int -> [Int]
  toList x = [x]

instance Listable Bool where
  toList True  = [1]
  toList False = [0]

We don’t need to do any work to convert a list of Int to a list of Int:

instance Listable [Int] where
  toList = id

Finally, here’s a binary tree type which we can convert to a list by flattening:

data Tree a = Empty | Node a (Tree a) (Tree a)

instance Listable (Tree Int) where
  toList Empty        = []
  toList (Node x l r) = toList l ++ [x] ++ toList r

If we implement other functions in terms of toList, they also get a Listable constraint. For example:

-- to compute sumL, first convert to a list of Ints, then sum
sumL x = sum (toList x)

ghci informs us that type type of sumL is

sumL :: Listable a => a -> Int

which makes sense: sumL will work only for types which are instances of Listable, since it uses toList. What about this one?

foo x y = sum (toList x) == sum (toList y) || x < y

ghci informs us that the type of foo is

foo :: (Listable a, Ord a) => a -> a -> Bool

That is, foo works over types which are instances of both Listable and Ord, since it uses both toList and comparison on the arguments.

As a final, and more complex, example, consider this instance:

instance (Listable a, Listable b) => Listable (a,b) where
  toList (x,y) = toList x ++ toList y

Notice how we can put type class constraints on an instance as well as on a function type. This says that a pair type (a,b) is an instance of Listable as long as a and b both are. Then we get to use toList on values of types a and b in our definition of toList for a pair. Note that this definition is not recursive! The version of toList that we are defining is calling other versions of toList, not itself.

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