Interfaces¶
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 Double
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 interfaces, which are similar to type classes in
Haskell or traits in Rust. To define an interface, we provide a collection of
overloadable functions. A simple example is the Show
interface, which is defined in the prelude and provides an interface for
converting values to String
:
interface Show a where
show : a -> String
This generates a function of the following type (which we call a
method of the Show
interface):
show : Show a => a -> String
We can read this as: “under the constraint that a
has an implementation
of Show
, take an input a
and return a String
.” An implementation
of an interface is defined by giving definitions of the methods of the interface.
For example, the
Show
implementation for Nat
could be defined as:
Show Nat where
show Z = "Z"
show (S k) = "s" ++ show k
Idris> show (S (S (S Z)))
"sssZ" : String
Only one implementation of an interface can be given for a type — implementations may
not overlap. Implementation declarations can themselves have constraints.
To help with resolution, the arguments of an implementation must be
constructors (either data or type constructors) or variables
(i.e. you cannot give an implementation for a function). For
example, to define a Show
implementation for vectors, we need to know
that there is a Show
implementation for the element type, because we are
going to use it to convert each element to a String
:
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
interface which provides methods for
comparing values for equality or inequality, with implementations for all of
the built-in types:
interface Eq a where
(==) : a -> a -> Bool
(/=) : a -> a -> Bool
To declare an implementation for a type, we have to give definitions of all
of the methods. For example, for an implementation of Eq
for Nat
:
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 interface declaration, in terms of the other method:
interface Eq a where
(==) : a -> a -> Bool
(/=) : a -> a -> Bool
x /= y = not (x == y)
x == y = not (x /= y)
A minimal complete implementation of Eq
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 Interfaces¶
Interfaces 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
interface which inherits methods from Eq
as well as
defining some of its own:
data Ordering = LT | EQ | GT
interface 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
interface 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
interface. 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, interfaces and implementations 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)
Note: Interfaces and mutual
blocks¶
Idris is strictly “define before use”, except in mutual
blocks.
In a mutual
block, Idris elaborates in two passes: types on the first
pass and definitions on the second. When the mutual block contains an
interface declaration, it elaborates the interface header but none of the
method types on the first pass, and elaborates the method types and any
default definitions on the second pass.
Functors and Applicatives¶
So far, we have seen single parameter interfaces, where the parameter
is of type Type
. In general, there can be any number of parameters
(even zero), 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
interface is defined in the
prelude:
interface 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
:
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 <*>
interface Functor f => Applicative (f : Type -> Type) where
pure : a -> f a
(<*>) : f (a -> b) -> f a -> f b
Monads and do
-notation¶
The Monad
interface 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:
interface 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
becomesv >>= (\x => e)
v; e
becomesv >>= (\_ => e)
let x = v; e
becomeslet x = v in e
IO
has an implementation of Monad
, defined using primitive functions.
We can also define an implementation for Maybe
, as follows:
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
pure (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.
*Interfaces> m_add (Just 20) (Just 22)
Just 42 : Maybe Int
*Interfaces> m_add (Just 20) Nothing
Nothing : Maybe Int
Sometimes we want to pattern match immediately on the result of a function
in do
notation. For example, let’s say we have a function readNumber
which reads a number from the console, returning a value of the form
Just x
if the number is valid, or Nothing
otherwise:
readNumber : IO (Maybe Nat)
readNumber = do
input <- getLine
if all isDigit (unpack input)
then pure (Just (cast input))
else pure Nothing
If we then use it to write a function to read two numbers, returning
Nothing
if neither are valid, then we would like to pattern match
on the result of readNumber
:
readNumbers : IO (Maybe (Nat, Nat))
readNumbers =
do x <- readNumber
case x of
Nothing => pure Nothing
Just x_ok => do y <- readNumber
case y of
Nothing => pure Nothing
Just y_ok => pure (Just (x_ok, y_ok))
If there’s a lot of error handling, this could get deeply nested very quickly!
So instead, we can combine the bind and the pattern match in one line. For example,
we could try pattern matching on values of the form Just x_ok
:
readNumbers : IO (Maybe (Nat, Nat))
readNumbers =
do Just x_ok <- readNumber
Just y_ok <- readNumber
pure (Just (x_ok, y_ok))
There is still a problem, however, because we’ve now omitted the case for
Nothing
so readNumbers
is no longer total! We can add the Nothing
case back as follows:
readNumbers : IO (Maybe (Nat, Nat))
readNumbers =
do Just x_ok <- readNumber | Nothing => pure Nothing
Just y_ok <- readNumber | Nothing => pure Nothing
pure (Just (x_ok, y_ok))
The effect of this version of readNumbers
is identical to the first (in
fact, it is syntactic sugar for it and directly translated back into that form).
The first part of each statement (Just x_ok <-
and Just y_ok <-
) gives
the preferred binding - if this matches, execution will continue with the rest
of the do
block. The second part gives the alternative bindings, of which
there may be more than one.
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 = pure (!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'
The list comprehension notation we saw in Section
More Expressions is more general, and applies to anything which
has an implementation of both Monad
and Alternative
:
interface 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; pure 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 give Maybe
an implementation of Applicative
as follows, where <*>
is defined in the same way as m_app
above (this is defined in the Idris library):
Applicative Maybe where
pure = Just
(Just f) <*> (Just a) = Just (f a)
_ <*> _ = Nothing
Using <*>
we can use this implementation 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 |]
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 String
s) 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 provide implementations of interfaces for it 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 give Eval
an implementation
of Applicative
. Before Eval
can have an implementation of
Applicative
it is necessary for Eval
to have an implementation of
Functor
:
Functor Eval where
map f (MkEval g) = MkEval (\e => map f (g e))
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 Implementations¶
It can be desirable to have multiple implementations of an interface for the same type, for example to provide alternative methods for sorting or printing values. To achieve this, implementations can be named as follows:
[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 implementation as normal, but with an explicit name,
myord
. The syntax compare @{myord}
gives an explicit implementation to
compare
, otherwise it would use the default implementation 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
implementation, then the named
implementation myord
as follows, at the Idris prompt:
*named_impl> show (sort testList)
"[sO, sssO, ssssO]" : String
*named_impl> show (sort @{myord} testList)
"[ssssO, sssO, sO]" : String
Sometimes, we also need access to a named parent implementation. For example,
the prelude defines the following Semigroup
interface:
interface Semigroup ty where
(<+>) : ty -> ty -> ty
Then it defines Monoid
, which extends Semigroup
with a “neutral”
value:
interface Semigroup ty => Monoid ty where
neutral : ty
We can define two different implementations of Semigroup
and
Monoid
for Nat
, one based on addition and one on multiplication:
[PlusNatSemi] Semigroup Nat where
(<+>) x y = x + y
[MultNatSemi] Semigroup Nat where
(<+>) x y = x * y
The neutral value for addition is 0
, but the neutral value for multiplication
is 1
. It’s important, therefore, that when we define implementations
of Monoid
they extend the correct Semigroup
implementation. We can
do this with a using
clause in the implementation as follows:
[PlusNatMonoid] Monoid Nat using PlusNatSemi where
neutral = 0
[MultNatMonoid] Monoid Nat using MultNatSemi where
neutral = 1
The using PlusNatSemi
clause indicates that PlusNatMonoid
should
extend PlusNatSemi
specifically.
Determining Parameters¶
When an interface has more than one parameter, it can help resolution if the parameters used to find an implementation are restricted. For example:
interface Monad m => MonadState s (m : Type -> Type) | m where
get : m s
put : s -> m ()
In this interface, only m
needs to be known to find an implementation of
this interface, and s
can then be determined from the implementation. This
is declared with the | m
after the interface declaration. We call m
a
determining parameter of the MonadState
interface, because it is the
parameter used to find an implementation.
[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 |