Type Classes

We often want to define functions which work across several different data types. For example, we would like arithmetic operators to work on Int, Integer and Float at the very least. We would like == to work on the majority of data types. We would like to be able to display different types in a uniform way.

To achieve this, we use a feature which has proved to be effective in Haskell, namely type classes. To define a type class, we provide a collection of overloaded operations which describe the interface for instances of that class. A simple example is the Show type class, which is defined in the prelude and provides an interface for converting values to String:

class Show a where
    show : a -> String

This generates a function of the following type (which we call a method of the Show class):

show : Show a => a -> String

We can read this as: “under the constraint that a is an instance of Show, take an input a and return a String.” An instance of a class is defined with an instance declaration, which provides implementations of the function for a specific type. For example, the Show instance for Nat could be defined as:

instance Show Nat where
    show Z = "Z"
    show (S k) = "s" ++ show k

Idris> show (S (S (S Z)))
"sssZ" : String

Only one instance of a class can be given for a type — instances may not overlap. Instance declarations can themselves have constraints. To help with resolution, the arguments of an instance must be constructors (either data or type constructors), variables or constants (i.e. you cannot give an instance for a function). For example, to define a Show instance for vectors, we need to know that there is a Show instance for the element type, because we are going to use it to convert each element to a String:

instance Show a => Show (Vect n a) where
    show xs = "[" ++ show' xs ++ "]" where
        show' : Vect n a -> String
        show' Nil        = ""
        show' (x :: Nil) = show x
        show' (x :: xs)  = show x ++ ", " ++ show' xs

Default Definitions

The library defines an Eq class which provides an interface for comparing values for equality or inequality, with instances for all of the built-in types:

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

To declare an instance of a type, we have to give definitions of all of the methods. For example, for an instance of Eq for Nat:

instance Eq Nat where
    Z     == Z     = True
    (S x) == (S y) = x == y
    Z     == (S y) = False
    (S x) == Z     = False

    x /= y = not (x == y)

It is hard to imagine many cases where the /= method will be anything other than the negation of the result of applying the == method. It is therefore convenient to give a default definition for each method in the class declaration, in terms of the other method:

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

    x /= y = not (x == y)
    x == y = not (x /= y)

A minimal complete definition of an Eq instance requires either == or /= to be defined, but does not require both. If a method definition is missing, and there is a default definition for it, then the default is used instead.

Extending Classes

Classes can also be extended. A logical next step from an equality relation Eq is to define an ordering relation Ord. We can define an Ord class which inherits methods from Eq as well as defining some of its own:

data Ordering = LT | EQ | GT

class Eq a => Ord a where
    compare : a -> a -> Ordering

    (<) : a -> a -> Bool
    (>) : a -> a -> Bool
    (<=) : a -> a -> Bool
    (>=) : a -> a -> Bool
    max : a -> a -> a
    min : a -> a -> a

The Ord class allows us to compare two values and determine their ordering. Only the compare method is required; every other method has a default definition. Using this we can write functions such as sort, a function which sorts a list into increasing order, provided that the element type of the list is in the Ord class. We give the constraints on the type variables left of the fat arrow =>, and the function type to the right of the fat arrow:

sort : Ord a => List a -> List a

Functions, classes and instances can have multiple constraints. Multiple constraints are written in brackets in a comma separated list, for example:

sortAndShow : (Ord a, Show a) => List a -> String
sortAndShow xs = show (sort xs)

Functors and Applicatives

So far, we have seen single parameter type classes, where the parameter is of type Type. In general, there can be any number (greater than 0) of parameters, and the parameters can have any type. If the type of the parameter is not Type, we need to give an explicit type declaration. For example, the Functor class is defined in the library:

class Functor (f : Type -> Type) where
    map : (m : a -> b) -> f a -> f b

A functor allows a function to be applied across a structure, for example to apply a function to every element in a List:

instance Functor List where
  map f []      = []
  map f (x::xs) = f x :: map f xs

Idris> map (*2) [1..10]
[2, 4, 6, 8, 10, 12, 14, 16, 18, 20] : List Integer

Having defined Functor, we can define Applicative which abstracts the notion of function application:

infixl 2 <*>

class Functor f => Applicative (f : Type -> Type) where
    pure  : a -> f a
    (<*>) : f (a -> b) -> f a -> f b

Monads and do-notation

The Monad class allows us to encapsulate binding and computation, and is the basis of do-notation introduced in Section “do” notation. It extends Applicative as defined above, and is defined as follows:

class Applicative m => Monad (m : Type -> Type) where
    (>>=)  : m a -> (a -> m b) -> m b

Inside a do block, the following syntactic transformations are applied:

• x <- v; e becomes v >>= (\x => e)
• v; e becomes v >>= (\_ => e)
• let x = v; e becomes let x = v in e

IO is an instance of Monad, defined using primitive functions. We can also define an instance for Maybe, as follows:

instance Monad Maybe where
    Nothing  >>= k = Nothing
    (Just x) >>= k = k x

Using this we can, for example, define a function which adds two Maybe Int, using the monad to encapsulate the error handling:

m_add : Maybe Int -> Maybe Int -> Maybe Int
m_add x y = do x' <- x -- Extract value from x
               y' <- y -- Extract value from y
               return (x' + y') -- Add them

This function will extract the values from x and y, if they are both available, or return Nothing if one or both are not (“fail fast”). Managing the Nothing cases is achieved by the >>= operator, hidden by the do notation.

*classes> m_add (Just 20) (Just 22)
Just 42 : Maybe Int
*classes> m_add (Just 20) Nothing
Nothing : Maybe Int

!-notation

In many cases, using do-notation can make programs unnecessarily verbose, particularly in cases such as m_add above where the value bound is used once, immediately. In these cases, we can use a shorthand version, as follows:

m_add : Maybe Int -> Maybe Int -> Maybe Int
m_add x y = return (!x + !y)

The notation !expr means that the expression expr should be evaluated and then implicitly bound. Conceptually, we can think of ! as being a prefix function with the following type:

(!) : m a -> a

Note, however, that it is not really a function, merely syntax! In practice, a subexpression !expr will lift expr as high as possible within its current scope, bind it to a fresh name x, and replace !expr with x. Expressions are lifted depth first, left to right. In practice, !-notation allows us to program in a more direct style, while still giving a notational clue as to which expressions are monadic.

For example, the expression:

let y = 42 in f !(g !(print y) !x)

is lifted to:

let y = 42 in do y' <- print y
                 x' <- x
                 g' <- g y' x'
                 f g'

Monad comprehensions

The list comprehension notation we saw in Section More Expressions is more general, and applies to anything which is an instance of both Monad and Alternative:

class Applicative f => Alternative (f : Type -> Type) where
    empty : f a
    (<|>) : f a -> f a -> f a

In general, a comprehension takes the form [ exp | qual1, qual2, …, qualn ] where quali can be one of:

• A generator x <- e
• A guard, which is an expression of type Bool
• A let binding let x = e

To translate a comprehension [exp | qual1, qual2, …, qualn], first any qualifier qual which is a guard is translated to guard qual, using the following function:

guard : Alternative f => Bool -> f ()

Then the comprehension is converted to do notation:

do { qual1; qual2; ...; qualn; return exp; }

Using monad comprehensions, an alternative definition for m_add would be:

m_add : Maybe Int -> Maybe Int -> Maybe Int
m_add x y = [ x' + y' | x' <- x, y' <- y ]

Idiom brackets

While do notation gives an alternative meaning to sequencing, idioms give an alternative meaning to application. The notation and larger example in this section is inspired by Conor McBride and Ross Paterson’s paper “Applicative Programming with Effects” [1].

First, let us revisit m_add above. All it is really doing is applying an operator to two values extracted from Maybe Int. We could abstract out the application:

m_app : Maybe (a -> b) -> Maybe a -> Maybe b
m_app (Just f) (Just a) = Just (f a)
m_app _        _        = Nothing

Using this, we can write an alternative m_add which uses this alternative notion of function application, with explicit calls to m_app:

m_add' : Maybe Int -> Maybe Int -> Maybe Int
m_add' x y = m_app (m_app (Just (+)) x) y

Rather than having to insert m_app everywhere there is an application, we can use idiom brackets to do the job for us. To do this, we can make Maybe an instance of Applicative as follows, where <*> is defined in the same way as m_app above (this is defined in the Idris library):

instance Applicative Maybe where
    pure = Just

    (Just f) <*> (Just a) = Just (f a)
    _        <*> _        = Nothing

Using <*> we can use this instance as follows, where a function application [| f a1 …an |] is translated into pure f <*> a1 <*> … <*> an:

m_add' : Maybe Int -> Maybe Int -> Maybe Int
m_add' x y = [| x + y |]

An error-handling interpreter

Idiom notation is commonly useful when defining evaluators. McBride and Paterson describe such an evaluator [1], for a language similar to the following:

data Expr = Var String      -- variables
          | Val Int         -- values
          | Add Expr Expr   -- addition

Evaluation will take place relative to a context mapping variables (represented as Strings) to Int values, and can possibly fail. We define a data type Eval to wrap an evaluator:

data Eval : Type -> Type where
     MkEval : (List (String, Int) -> Maybe a) -> Eval a

Wrapping the evaluator in a data type means we will be able to make it an instance of a type class later. We begin by defining a function to retrieve values from the context during evaluation:

fetch : String -> Eval Int
fetch x = MkEval (\e => fetchVal e) where
    fetchVal : List (String, Int) -> Maybe Int
    fetchVal [] = Nothing
    fetchVal ((v, val) :: xs) = if (x == v)
                                  then (Just val)
                                  else (fetchVal xs)

When defining an evaluator for the language, we will be applying functions in the context of an Eval, so it is natural to make Eval an instance of Applicative. Before Eval can be an instance of Applicative it is necessary to make Eval an instance of Functor:

instance Functor Eval where
    map f (MkEval g) = MkEval (\e => map f (g e))

instance Applicative Eval where
    pure x = MkEval (\e => Just x)

    (<*>) (MkEval f) (MkEval g) = MkEval (\x => app (f x) (g x)) where
        app : Maybe (a -> b) -> Maybe a -> Maybe b
        app (Just fx) (Just gx) = Just (fx gx)
        app _         _         = Nothing

Evaluating an expression can now make use of the idiomatic application to handle errors:

eval : Expr -> Eval Int
eval (Var x)   = fetch x
eval (Val x)   = [| x |]
eval (Add x y) = [| eval x + eval y |]

runEval : List (String, Int) -> Expr -> Maybe Int
runEval env e = case eval e of
    MkEval envFn => envFn env

Named Instances

It can be desirable to have multiple instances of a type class, for example to provide alternative methods for sorting or printing values. To achieve this, instances can be named as follows:

instance [myord] Ord Nat where
   compare Z (S n)     = GT
   compare (S n) Z     = LT
   compare Z Z         = EQ
   compare (S x) (S y) = compare @{myord} x y

This declares an instance as normal, but with an explicit name, myord. The syntax compare @{myord} gives an explicit instance to compare, otherwise it would use the default instance for Nat. We can use this, for example, to sort a list of Nat in reverse. Given the following list:

testList : List Nat
testList = [3,4,1]

We can sort it using the default Ord instance, then the named instance myord as follows, at the Idris prompt:

*named_instance> show (sort testList)
"[sO, sssO, ssssO]" : String
*named_instance> show (sort @{myord} testList)
"[ssssO, sssO, sO]" : String

Determining Parameters

When a class has more than one parameter, it can help resolution if the parameters used to resolve the type class are restricted. For example:

class Monad m => MonadState s (m : Type -> Type) | m where
  get : m s
  put : s -> m ()

In this class, only m needs to be known to resolve this class, and s can then be determined from the instance. This is declared with the | m after the class declaration. We call m a determining parameter of the MonadState class, because it is the parameter used to resolve an instance.

[1]	(1, 2) Conor McBride and Ross Paterson. 2008. Applicative programming with effects. J. Funct. Program. 18, 1 (January 2008), 1-13. DOI=10.1017/S0956796807006326 http://dx.doi.org/10.1017/S0956796807006326