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Type Classes

> module Classes where

> import Test.HUnit ( (~:), (~?=), runTestTT, Test(TestList) )
> import Data.Char ( Char )
> import Text.Read ( Read )
> import qualified Data.List as List
> import Prelude hiding (lookup)

Our first qualified type

Question: What is the type of (+)?

We've most often used (+) to add Ints, as in:

> fancySeven :: Int
> fancySeven = 3 + 4

So you might guess that the type of (+) is:

(+) :: Int -> Int -> Int

But if you think a little harder, you may remember we've also used (+) to add Floats and Doubles, as in:

> fancyEight :: Float
> fancyEight = 3.2 + 4.8

So it must also be the case that:

(+) :: Float -> Float -> Float

At this point, you might guess that (+) has the type

(+) :: a -> a -> a

since it seems to work for many different types. But this type would be too general: it doesn't make much sense to add a Bool to a Bool or an Char -> Char to an Char -> Char.

We need a type in the middle: (+) should work on any kind of numbers, but not on other things. If we look up the actual type, we find this:

(+) :: Num a => a -> a -> a

What's going on here? What's that fancy => thing?

In this type, Num a is a "type class constraint". The type says that (+) should work at any type a, so long as a is a member of the Num type class.

Num is one of many type classes in Haskell's standard library. Types like Int and Double are members of this class because they support operations like (+) and (-).

Type classes are Haskell's solution for overloading, the ability to define functions with the same name but different operations.

For example, the operation of (+) for Ints is very different than that for Doubles. The compiler generates different machine instructions!

This is the key difference between overloaded functions, like (+), and (parametrically)-polymorphic functions, like list length. The length function behaves the same, no matter whether it is working with a list of Ints or a list of Doubles. However, the behavior of (+) really does depend on the type of argument that it is working with.

Eq

Let's consider another function we've been using quite a bit:

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

Again, this makes sense: We've used equality at many different types, but it doesn't work at every type: there is no obvious way to check for equality on functions, for example.

Furthermore, this type also says that we can only test arguments with the same type for equality. We will get a compile-time type error if we try to compare a Bool and an [Char], for example.

> -- >>> True == "True"

Let's peek at the definition of the Eq type class:

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

This declares Eq to be a type class with a single parameter, a. To show that some type is a member of the class, we must provide definitions of (==) and (/=) for the type. We do this with an "instance" declaration.

For example, consider the following type:

> data PrimaryColor = Red | Green | Blue

We can tell Haskell that PrimaryColors can be compared for equality like this:

> instance Eq PrimaryColor where
>     (==) :: PrimaryColor -> PrimaryColor -> Bool
>     Red   == Red   = True
>     Blue  == Blue  = True
>     Green == Green = True
>     _     == _     = False

>     (/=) :: PrimaryColor -> PrimaryColor -> Bool
>     Red   /= Red   = False
>     Blue  /= Blue  = False
>     Green /= Green = False
>     _     /= _     = True

This code means that we are making the overloaded (==) and (/=) operators available for the PrimaryColor type.

If we ask for information about this type, we can see that it is now a member of the Eq type class.

> -- >>> :info PrimaryColor

With this instance declaration we can use (==) and (/=) on PrimaryColors! Try it out!

> -- >>> Red == Red

It might seem annoying, though, that we had to provide both (==) and (/=)...

Fortunately, we don't. Type classes are allowed to provide "default definitions" for member functions. For example, the full definition of Eq from the Prelude is:

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

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

So to define Eq for a new type, we only actually have to provide one of (==) and (/=). Haskell can figure out the other for us. But, we better provide at least one definition! Otherwise, we'll get an infinite loop for both.

Let's do another example. We'd like to define Eq for the type Tree that we saw last time. But we have a bit of a problem: to check if trees are equal, we'll need to know if the data in each pair of Branchs is equal. Put another way, we'll only be able to compare two Tree as if a is an instance of Eq.

> data Tree a = Empty | Branch a (Tree a) (Tree a)

No worries, Haskell lets us put type class constraints on our instance declarations. See if you can finish this instance for trees. (No cheating by using 'deriving' like we saw in Datatypes!)

> instance Eq a => Eq (Tree a) where
>     (==) :: Tree a -> Tree a -> Bool
>     t1 == t2 = undefined

This code tells Haskell how to compare Tree as for equality as long as it already knows how to compare as.

Let's try it out:

> tree1, tree2 :: Tree Int
> tree1 = Branch 3 (Branch 2 Empty Empty) (Branch 1 Empty Empty)
> tree2 = Branch 3 Empty Empty

either inline

> -- >>> tree1 == tree1

> -- >>> tree1 == tree2 

> -- >>> tree1 == Empty

or as unit tests.

> testTreeEq :: Test
> testTreeEq = TestList [ "tree1 == tree1"  ~: tree1 == tree1  ~?= True,
>                         "tree1 /= tree2"  ~: tree1 == tree2  ~?= False,
>                         "tree1 /= Empty"  ~: tree1 == Empty  ~?= False ]

More qualified types

We can now explain the types of a few functions that we glossed over before. Type class constraints don't just appear on the functions defined as members of a type class; they can appear anywhere we want to use such a function. For example the standard library function lookup can be used to find a member of an "association list" pairing keys with their associated values. Let's peek at its implementation:

> lookup :: Eq a => a -> [(a,b)] -> Maybe b
> lookup _ []          = Nothing
> lookup a ((a',b):ps) = if a == a' then Just b
>                                   else lookup a ps

The idea is that lookup checks to see if the given value appears as the first element of any pair in the list. To implement lookup, we need to use the (==) function to check if we've reached the right pair. So, the type of lookup records that there must be an Eq instance for a; otherwise, the compiler wouldn't have an implementation of (==) for this type.

What about a function that uses lookup, what is its type? Note how the equality constraint propagates to the type of this function.

> lookupDefault :: Eq a => a -> [(a, p)] -> p -> p
> lookupDefault x xs def = case lookup x xs of
>                              Just y  -> y
>                              Nothing -> def

Other overloaded operations?

Quickly, without looking at the rest of the lecture notes, brainstorm as many overloaded operations as you can. What overloaded functions have you seen in other languages? What about from your mathematics classes?

FILL IN EXAMPLES HERE

Overloading is sometimes called ad hoc polymorphism, for good reason. Allowing unrelated functions to have the same name can lead to messy, hard-to-understand code and unpredictable behavior.

Type classes are Haskell's technology to make ad hoc polymorphism less ad hoc. How do they encourage more principled design?

• First, the type class itself means that the types of overloaded functions must follow a common pattern. For example, the type of (+) states that any instance must take two arguments of the same type.

  (+) :: Num a => a -> a -> a

  Haskell won't allow you to overload (+) to work with a String and an Int at the same time, for example.

• Second, type classes usually come with laws, or specific properties that all instances of the type class are expected to adhere to. For example, all instances of (==) should satisfy reflexivity, symmetry and transitivity. Likewise, all instances of (+) should be commutative and associative.

  Any time you see a type class, you should ask yourself "what are the laws" that should hold for instances of this class?

  Note that there is no way for the Haskell language to enforce these laws when instances are defined; the type checker doesn't know about them. So it is up to programmers to check that they hold (informally) for their instances.

Deriving

We've now written a couple Eq instances ourselves, and you might guess that most of our future Eq instances will have a very similar structure. They will recurse down datatypes, making sure the two terms use the same constructors and that any subterms are equal. And you'd be right!

To avoid this kind of boilerplate, Haskell provides a nifty mechanism called deriving. When you declare a new datatype, you may ask Haskell to automatically derive an Eq instance of this form. For example, if we wanted equality for the Shape type we saw last time, we could have written:

> data Point = Point Double Double
>      deriving (Eq)

> data Shape = Circle Point Float
>            | Rectangle Point Point
>      deriving (Eq)

The 'deriving' keywords instructs the compiler to automatically create an instance of the Eq type class for the datatype. Haskell can derive an Eq instance as long as it already has one available for any data that appears as arguments to constructors. Since it already knows how to compare Doubles, this one works.

It won't always work, though, consider this datatype, which can contain functions on Ints.

> data IntFunctions = OneArg (Int -> Int)
>                   | TwoArg (Int -> Int -> Int)

There are no Eq instances for functions (this is a classic example of an undecidable problem!). So, if we added deriving (Eq) to this type, we'd get an error. Try it out!

Of course, not every type class supports this "deriving" mechanism. GHC can derive instances of a handful of classes for us (we'll see a few more today), but for most standard library type classes and any classes you define, you must write the instances out yourself.

Show and Read

Time for a couple more type classes from the standard library.

Though we haven't talked about it explicitly, we've been using Haskell's printing functions quite a bit. Every time we've run code in ghci and looked at the output, Haskell had to figure out how to convert the data to a String. A few times we've even explicitly used a function called show, which converts a value to a String.

Here is the type of show:

show :: Show a => a -> String

Aha, another type class! This says that the function show converts an a to a String, as long as a is a member of the Show class. Let's look at the full definition of this class:

class Show a where
   show      :: a   -> String

   showsPrec :: Int -> a -> ShowS
   showList  :: [a] -> ShowS

To define an instance of Show, you must implement either show or showsPrec. We've already discussed show, which is a bit like Java's toString. The second function, showsPrec, takes an extra Int argument which can be used to indicate the "precedence" of the printing context - this is useful in some situations to determine, for example, whether parentheses are needed in the output. Its return type, ShowS is used for more efficient printing. For now, you don't need to worry about these details. The third function, showList, exists so that users can specify a special way of printing lists of values, if desired for a given type. Usually, though, the default instance works fine.

By convention, instances of Show should produce valid Haskell expressions (i.e., expressions that can be read by the Haskell parser).

In the other direction, there is a type class called Read. The most primitive function in this class is

read :: Read a => String -> a

Notice that the type variable a doesn't appear in any arguments to this function. In general, to use read, you must make sure the type of the output is clear from context or provide it specifically. For example, if you try this in ghci

ghci> read "3"

you will get an error: ghci doesn't know whether to interpret the string "3" as an Int or a Float or even a Bool. You can help it, though

ghci> read "3" :: Int
3

What if you can't read?

ghci> read "3" :: Bool
*** Exception: Prelude.read: no parse

This exception is irritating, as there isn't much you can do to recover from it. Therefore, I like to use the GHC-specific version of read, from the library Text.Read called

readMaybe :: Read a => String -> Maybe a

This (non-partial) version provides a way to recover from a misparse; so is much more useful.

You can see the details of the Read type class in the standard library and in Text.Read. As one might expect, parsing values is a little more complicated than printing them. One important thing to remember, though, is that the read and show functions should be inverses. So, for example

read (show 3) :: Int

should return 3, and

show (read "3" :: Int)

should return "3".

This should work for all instances of show and read. For any string x that is readable as a value of type A, i.e. (read x is not an error), it should be the case that

show (read x :: A) == x

and, if there is an equality instance for A, we should have:

read (show a) == a

Both Show and Read are derivable:

> data SadColors = Black | Brown | Grey
>     deriving (Eq, Show, Read)

Notice that if we type a value into ghci and the corresponding type doesn't have a Show instance, we get an error saying ghci doesn't know how to display the value:

ghci> Empty

  <interactive>:1:1:
      No instance for (Show (Tree a0))
        arising from a use of `print'
      Possible fix: add an instance declaration for (Show (Tree a0))
      In a stmt of an interactive GHCi command: print it


ghci> \x -> (x,x)

  <interactive>:1:1:
      No instance for (Show (t0 -> (t0, t0)))
        arising from a use of `print'
      Possible fix:
        add an instance declaration for (Show (t0 -> (t0, t0)))
      In a stmt of an interactive GHCi command: print it

Type classes vs. OOP

At this point, many of you are probably thinking that type classes are a lot like Java's classes or interfaces. You're right! Both provide a way to describe functions that can be implemented for many types.

There are some important differences, though:

• In Java, when you define a new class, you must specify all the interfaces it implements right away. Haskell lets you add a new instance declaration at any time.

  This is very convenient: we often define new type classes and want to be able to add instances for types that are already around. We wouldn't want to have to change the standard library just to write a new type class and give it an instance for Int!

• Type classes are better integrated in the standard library than Java interfaces. In particular, every object in Java has equals and toString methods. This leads to some silliness, since not every type of data can sensibly be checked for equality or printed effectively. The result is that calling equals on objects that don't actually implement it may result in a run-time error or a nonsensical result.

  By contrast, Haskell will warn us at compile time if we try to use (==) on a term that doesn't support it. It's all tracked in the types!

• Haskell supports multiple inheritance and multi-parameter type classes.

  In Haskell, classes may require that types be members of an arbitrary number of other classes. For example, you might write a class for "Serializable" data that can be written to a file and demand that Serializable types implement both Read and Show:

  class (Read a, Show a) => Serializable a where
      toFile :: a -> ByteString
      ...

  Also, type classes in Haskell may have multiple type arguments. Often it's useful to think of such classes as describing a relationship between types. For example, we can define a class:

  class Convertible a b where
      convert :: a -> b

  Instances of Convertible a b show how to convert from one type to another. For example, we can convert from Chars to Ints using Haskell's built in ord function, which gets the ASCII code of a character:

  instance Convertible Char Int where
      convert = ord

  Or we can convert from Trees to Lists using an inorder traversal:

  instance Convertible (Tree a) [a] where
      convert = infixOrder

  Java doesn't have analogues for these features.

Ord

Let's talk about another type class from the standard library. This one is for comparisons. It is used in the type of (<):

(<) :: Ord a => a -> a -> Bool

Ord is a type class for things that can be ordered. Here is its definition:

class Eq a => Ord a where
    compare              :: a -> a -> Ordering
    (<), (<=), (>), (>=) :: a -> a -> Bool
    max, min             :: a -> a -> a

Notice that to be a member of the Ord class, a type must also have an implementation of Eq.

Most of these methods we've seen before, so let's talk about the one we haven't:

compare :: Ord a => a -> a -> Ordering

This uses a new type from the standard library:

data Ordering = LT | EQ | GT

So compare takes two terms and tells us whether the first is less than, equal to, or greater than than the second. Most built in types already have Ord instances (try some examples in ghci).

What properties should instances of Ord satisfy? Write some below:

Ord is derivable, like Eq, Show and Read. However, note that because of the superclass constraint, we must derive Eq at the same time as Ord.

> data MyThree = One | Two | Three deriving (Eq, Ord)

When you derive Ord, Haskell will order the constructors in the order that they appear in your source file. So, in this case, One < Two < Three.

> -- >>> One <= Two

> -- >>> Three <= Two

Alternatively, if you're writing your own Ord instance, you only need to provide compare or (<=); there are default definitions of the rest. (Don't forget to make an instance of the Eq class first.)

The Ord type class shows up all over the standard library. For example, Data.List has a function which sorts lists:

sort :: Ord a => [a] -> [a]

As you'd expect, we need to know an ordering on as in order to sort lists of them! But if this ordering exists, sort can use it.

> sorted :: [ MyThree ]
> sorted = List.sort [ Two, One, Three ]

If there is no ordering on a's you'll get a type error from the compiler.

> -- >>> List.sort [\x -> 1, \x -> 2]
> -- No instance for (Show (p0 -> Integer))
> --   arising from a use of ‘evalPrint’
> --   (maybe you haven't applied a function to enough arguments?)

Overloading and Syntax

Type classes have been a part of Haskell since the beginning of the language design. Because of that it is integrated into the language syntax in somewhat subtle ways.

For example, it is not just functions that are overloaded. What is the type of 3?

ghci> :type 3
3 :: Num a => a

Literal integers, such as 3 or 552 are overloaded in Haskell.

How does this work? The answer lies in the Num type class. There is a lot more in Num besides (+). Let's take a look!

ghci> :i Num

class Num a where
    (+) :: a -> a -> a
    (-) :: a -> a -> a
    (*) :: a -> a -> a
    negate :: a -> a
    abs :: a -> a
    signum :: a -> a
    fromInteger :: Integer -> a
    -- Defined in 'GHC.Num'

The parser converts a literal number to an Integer, but then the Num type class can convert that syntax to any numeric type.

This syntax is convenient because it allows all numeric types to use the same syntax for constants.

ghc> 1 :: Double

and

ghc> 1 :: Integer

It also allows expressions like this to work, even though (+) requires both arguments to have the same type.

ghc> 1 + 2.0

What happens in this expression? The (+) must have arguments that are the same type, and 2.0 is a Double, so the type system knows that the Integer 1 must first be converted to a Double before it can be added to 2.0.

Enum and Bounded

Previously, we observed that we could use the [a..b] list syntax on both Ints and Chars. For example:

> tenToThirty :: [Int]
> tenToThirty = [10..30]

> abcde :: [Char]
> abcde = ['a'..'e']

But obviously this syntax can't work on every type. Indeed, it works on only the ones that implement the Enum type class! This class describes sequentially ordered types -- i.e., those that can be enumerated.

class Enum a  where
    succ, pred           :: a -> a

    toEnum               :: Int -> a
    fromEnum             :: a -> Int

    -- These are used in haskell's translation of [n..] and [n..m]
    enumFrom            :: a -> [a]
    enumFromThen        :: a -> a -> [a]
    enumFromTo          :: a -> a -> [a]
    enumFromThenTo      :: a -> a -> a -> [a]

OK, one more basic type class. Recall that Int isn't arbitrary precision: it represents an actual machine-sized number in your computer. Of course, this varies from machine to machine (64-bit computers have a lot more Ints than 32-bit ones). And Haskell supports a number of other bounded datatypes too -- Char, Word, etc.

It would sure be nice if there were a uniform way to find out how big these things are on a given computer...

Enter Bounded!

class Bounded a where
    minBound, maxBound     :: a

So to find the biggest Int on my machine, we can write:

> biggestInt :: Int
> biggestInt = maxBound

Of course, if we tried to write

biggestInteger :: Integer
biggestInteger = maxBound

we would get a type error, since Integers are unbounded as long as they can be stored in memory. (Haskell uses a data structure to represent as many digits as we need.) Again the compiler protects us from basic mistakes like this.

Many standard library types support Enum and Bounded. They are also both derivable - but only for datatypes whose constructors don't take any arguments.

For example, if we have a datatype for the days of the week

> data Day = Sunday | Monday | Tuesday | Wednesday | Thursday | Friday | Saturday
>    deriving (Eq, Ord, Enum, Bounded, Read, Show)

Then we can see the first day

> -- >>> minBound :: Day

and last day

> -- >>> maxBound :: Day

as well as enumerate a list of all of them.

> -- >>> daysOfWeek

> daysOfWeek :: [Day]
> daysOfWeek = [minBound ..]

Functor

Now it's time for everybody's favorite type class...

Recall the map function on lists

map :: (a -> b) -> [a] -> [b]

that takes a function and applies it to every element of a list, creating a new list with the results. We also saw that the same pattern can be used for Trees:

treeMap :: (a -> b) -> Tree a -> Tree b

If you think a little, you'll realize that map makes sense for pretty much any data structure that holds a single type of values. It would be nice if we could factor this pattern out into a class, to keep track of the types that support map.

Behold, Functor:

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

instance Eq a => Eq (Tree a) where
    ...

instance Functor Tree where
    ...

Functor is a little different than the other classes we've seen so far. It's a "constructor" class, because the types it works on are constructors like Tree, Maybe and [] -- ones that take another type as their argument. Notice how the f in the class declaration is applied to other types.

The standard library defines:

instance Functor [] where
    fmap :: (a -> b) -> [a] -> [b]
    fmap = map

We can define a Functor instance for our own trees:

> instance Functor Tree where
>     fmap :: (a -> b) -> Tree a -> Tree b
>     fmap = treeMap where
>         treeMap _f Empty = Empty
>         treeMap f (Branch x l r) = Branch (f x) (treeMap f l) (treeMap f r)

The standard library also defines Functor instances for a number of other types. For example, Maybe is a Functor:

instance Functor Maybe where
    fmap :: (a -> b) -> Maybe a -> Maybe b
    fmap f (Just x) = Just (f x)
    fmap f Nothing  = Nothing

Furthermore, the standard library also defines the infix operator <$> as another synonym for fmap, riffing on the visual similarity to the $ operator. This overloaded operator shows up frequently in idiomatic Haskell code when applying a function to all of the elements in some data structure. For example:

 > Data.Char.toUpper <$> "abcd"
 "ABCD"
 > Data.Char.toUpper <$> Just 'a'
 Just 'A'
 > Data.Char.toUpper <$> Branch 'a' Empty Empty
 Branch 'A' Empty Empty

Functor is very useful, and you'll see many more examples of it in the weeks to come.

See if you can define a Functor instance for this type:

> data Two a = MkTwo a a deriving (Eq, Show, Read, Ord)

> instance Functor Two where
>     fmap = undefined

In the meantime, think about what laws instances of this class should satisfy. (We'll come back to this.)

INTERLUDE: Kinds

How are Tree and Two different than Int and Bool? Why does the functor instance for lists say Functor [] ?

The answer is that types themselves have types. To keep things straight, we refer to the types of types as "kinds".

For example, the kind of normal types, like Int and Bool is Type (This kind is also written as *, but it is still pronounced "type".) The kind of parameterized types, like Tree and Two is * -> * or Type -> Type. In other words, we can think of Tree like a function that needs a normal type argument (like Int) to produce a normal type (i.e. Tree Int). We also use the words type constructor for Tree and Two, because they are not normal types, but construct them.

If we are ever confused, we can ask GHC to tell us the kinds of types.

> -- >>> :k Tree

> -- >>> :k Int

> -- >>> :k Bool

> -- >>> :k Tree Int

> -- >>> :k Two

> -- >>> :k Two Int

For lists, the type constructor is written [] when it is by itself. If it is applied to a normal type, we write it as [Int]. We can also write this application in prefix notation, i.e. as [] Int (which means the same thing as [Int]). The [Int] notation is special syntax for the application of the list constructor; this behavior is not available for user-defined types.

Knowing the kinds of types helps us to figure out what type class instances make sense. The valid instances of the Functor and Monad type classes all have kind Type -> Type. So we could never write Functor Int or Monad Bool. On the other hand, the valid instances of the Show type class must have kind Type. So this tells you that to show a list, the instance needs to read Show (Tree a) instead of Show Tree.

Are there other kinds besides Type and Type -> Type? What if we had a datatype parameterized by two different types? For example, the Haskell standard library includes the following definition:

data Either a b = Left a | Right b

What does GHC say is the kind of Either?

> -- >>> :k Either

Monad

Last, the most famous of all Haskell type classes: The warm fuzzy thing called 'Monad'.

We saw an example of the IO monad with code like this:

> main :: IO ()
> main = do
>     putStrLn "This is the Classes lecture. What is your name?"
>     inpStr <- getLine
>     putStrLn $ "Welcome to Haskell, " ++ inpStr ++ "!"
>     putStrLn "Now running tests."
>     _ <- runTestTT testTreeEq
>     return ()

This code works because IO is an instance of the Monad type class. We'll see more instances of this class in the next few lectures. Don't try to understand it all at once! We'll start with just looking at what's going on at a syntactic level.

class  Monad m  where

    -- | Sequentially compose two actions, passing any value produced
    -- by the first as an argument to the second.
    (>>=)       :: m a -> (a -> m b) -> m b

    -- | Inject a value into the monadic type.
    return      :: a -> m a

    -- | Sequentially compose two actions, discarding any value produced
    -- by the first, like sequencing operators (such as the semicolon)
    -- in imperative languages.
    (>>)        :: m a -> m b -> m b
    m >> k      = m >>= \_ -> k         -- default definition

You can see the use of return in the last line of the main function above. In fact, it must be the last line in any computation because it doesn't compose multiple actions together.

> nop :: IO ()
> nop = do
>     return ()

We've also been using (>>=), but only behind the scenes. You've missed it because of another feature of Haskell---the "do" syntax for composing sequences of actions.

For example

 ex :: IO Int
 ex = do
   y <- doSomething
   return (y + 3)

is really shorthand for an application of the infix >>= operator.

 ex = doSomething >>= ( \y -> return (y + 3))

The >>= operator is called "bind" in Haskell.

If the action that we do, doesn't return a result, then we can combine it with the next action with the (>>) operation. For example,

  ex2 :: IO Int
  ex2 = do
   doSomethingElse
   return 5

is shorthand for

  ex2 = doSomethingElse >> return 5

Finally we can everything together and mix it up. So that code like this

  main :: IO ()
  main = do
      x <- doSomething
      doSomethingElse
      y <- andSoOn
      return ()

is really shorthand for this:

  doSomething >>= ( \x ->
    doSomethingElse >>
      (andSoOn >>= ( \y ->
        return () )))

So everytime that you see do there is some monad involved (though, as we'll see later, not necessarily IO!).

More information

• For more details, "Classes, Jim, but not as we know them," lecture from OPLSS13.