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Introducing Monads
==================

The [[tradition in the functional programming
literature|https://wiki.haskell.org/Monad_tutorials_timeline]] is to
introduce monads using a metaphor: monads are spacesuits, monads are
monsters, monads are burritos. These metaphors can be helpful, and they
can be unhelpful. There's a backlash about the metaphors that tells people
to instead just look at the formal definition. We'll give that to you below, but it's
sometimes sloganized as
[A monad is just a monoid in the category of endofunctors, what's the problem?](http://stackoverflow.com/questions/3870088).
Without some intuitive guidance, this can also be unhelpful. We'll try to find a good balance. 

The closest we will come to metaphorical talk is to suggest that
monadic types place objects inside of *boxes*, and that monads wrap
and unwrap boxes to expose or enclose the objects inside of them. In
any case, our emphasis will be on starting with the abstract structure
of monads, followed by instances of monads from the philosophical and
linguistics literature.

## Box types: type expressions with one free type variable

Recall that we've been using lower-case Greek letters
<code>&alpha;, &beta;, &gamma;, ...</code> as type variables. We'll
use `P`, `Q`, `R`, and `S` as schematic metavariables over type expressions, that may or may not contain unbound
type variables. For instance, we might have

    P_1 ≡ int
    P_2 ≡ α -> α
    P_3 ≡ ∀α. α -> α
    P_4 ≡ ∀α. α -> β 

etc.

A *box type* will be a type expression that contains exactly one free
type variable. (You could extend this to expressions with more free variables; then you'd have
to specify which one of them the box is capturing. But let's keep it simple.) Some examples (using OCaml's type conventions):

    α option
    α list
    (α, R) tree    (assuming R contains no free type variables)
    (α, α) tree

The idea is that whatever type the free type variable α might be,
the boxed type will be a box that "contains" an object of type `α`.
For instance, if `α list` is our box type, and `α` is the type
`int`, then in this context, `int list` is the type of a boxed integer.

Warning: although our initial motivating examples are naturally thought of as "containers" (lists, trees, and so on, with `α`s as their "elments"), with later examples we discuss it will less intuitive to describe the box types that way. For example, where `R` is some fixed type, `R -> α` is a box type.

The *box type* is the type `α list` (or as we might just say, `list`); the *boxed type* is some specific instantiantion of the free type variable `α`. We'll often write boxed types as a box containing the instance of the free
type variable. So if our box type is `α list`, and `α` is instantiated with the specific type `int`, we would write:

<u>int</u>

for the type of a boxed `int`. (We'll fool with the markup to make this a genuine box later; for now it will just display as underlined.)



## Kleisli arrows

A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Don't worry about why they're called that, or if you like go read some Category Theory. All we need to know is that these are functions whose type has the form:

P -> <u>Q</u>

That is, they are functions from objects of one type `P` to a boxed type `Q`, for some choice of type expressions `P` and `Q`.
For instance, the following are Kleisli arrows:

int -> <u>bool</u>

int list -> <u>int list</u>

In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <u>Q</u> is <u>bool</u>).

Note that the left-hand schema `P` is permitted to itself be a boxed type. That is, where
if `α list` is our box type, we can write the second arrow as

<u>int</u> -> <u>Q</u>

We'll need a number of classes of functions to help us maneuver in the
presence of box types. We will want to define a different instance of
each of these for whichever box type we're dealing with. (This will
become clearly shortly.)

Here are the types of our crucial functions, together with our pronunciation, and some other names the functions go by. (Usually the type doesn't fix their behavior, which will be discussed below.)

<code>map (/mæp/): (P -> Q) -> <u>P</u> -> <u>Q</u></code>

<code>map2 (/mæptu/): (P -> Q -> R) -> <u>P</u> -> <u>Q</u> -> <u>R</u></code>

<code>mid (/εmaidεnt@tI/ aka unit, return, pure): P -> <u>P</u></code>

<code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>

<code>&lt;=&lt; or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>

<code>&gt;=&gt; or mpmoc (m-flipcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>

<code>&gt;&gt;= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>

<code>=&lt;&lt;mdnib (or m-flipbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>

<code>join: <u>2<u>P</u></u> -> <u>P</u></code> 

The managerie isn't quite as bewildering as you might suppose. Many of these will
be interdefinable. For example, here is how `mcomp` and `mbind` are related: <code>k <=< j ≡
\a. (j a >>= k)</code>.

In most cases of interest, instances of these systems of functions will provide
certain useful guarantees.

*   ***Mappable*** (in Haskelese, "Functors") At the most general level, box types are *Mappable*
if there is a `map` function defined for that box type with the type given above. This
has to obey the following Map Laws:

    LAWS

*   ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
       if there are in addition `map2`, `mid`, and `mapply`.  (Given either
       of `map2` and `mapply`, you can define the other, and also `map`.
       Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
have to obey the following MapN Laws:


* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
       is in addition an associative `mcomp` having `mid` as its left and
       right identity. That is, the following Monad Laws must hold:

        mcomp (mcomp j k) l (that is, (j <=< k) <=< l) = mcomp j (mcomp k l)
        mcomp mid k (that is, mid <=< k) = k
        mcomp k mid (that is, k <=< mid) = k

If you have any of `mcomp`, `mpmoc`, `mbind`, or `join`, you can use them to define the others.
Also, with these functions you can define `m$` and `map2` from *MapNables*. So all you really need
are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.

Here are some interdefinitions: TODO. Names in Haskell TODO.

## Examples

To take a trivial (but, as we will see, still useful) example,
consider the identity box type Id: `α`. So if `α` is type `bool`,
then a boxed `α` is ... a `bool`. In terms of the box analogy, the
Identity box type is a completely invisible box. With the following
definitions

    mid ≡ \p. p
    mcomp ≡ \f g x.f (g x)

Identity is a monad.  Here is a demonstration that the laws hold:

    mcomp mid k == (\fgx.f(gx)) (\p.p) k
                ~~> \x.(\p.p)(kx)
                ~~> \x.kx
                ~~> k
    mcomp k mid == (\fgx.f(gx)) k (\p.p)
                ~~> \x.k((\p.p)x)
                ~~> \x.kx
                ~~> k
    mcomp (mcomp j k) l == mcomp ((\fgx.f(gx)) j k) l
                       ~~> mcomp (\x.j(kx)) l
                        == (\fgx.f(gx)) (\x.j(kx)) l
                       ~~> \x.(\x.j(kx))(lx)
                       ~~> \x.j(k(lx))
    mcomp j (mcomp k l) == mcomp j ((\fgx.f(gx)) k l)
                       ~~> mcomp j (\x.k(lx))
                        == (\fgx.f(gx)) j (\x.k(lx))
                       ~~> \x.j((\x.k(lx)) x)
                       ~~> \x.j(k(lx))

Id is the favorite monad of mimes.

To take a slightly less trivial (and even more useful) example,
consider the box type `α list`, with the following operations:

    mid: α -> [α]
    mid a = [a]
 
    mcomp: (β -> [γ]) -> (α -> [β]) -> (α -> [γ])
    mcomp f g a = concat (map f (g a))
                = foldr (\b -> \gs -> (f b) ++ gs) [] (g a) 
                = [c | b <- g a, c <- f b]

These three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises).

In words, `mcomp f g a` feeds the `a` (which has type `α`) to `g`, which returns a list of `β`s;
each `β` in that list is fed to `f`, which returns a list of `γ`s. The
final result is the concatenation of those lists of `γ`s.

For example: 

    let f b = [b, b+1] in
    let g a = [a*a, a+a] in
    mcomp f g 7 ==> [49, 50, 14, 15]

`g 7` produced `[49, 14]`, which after being fed through `f` gave us `[49, 50, 14, 15]`.

Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:

    let gs = [(\a->a*a),(\a->a+a)] in
    let xs = [7, 5] in
    mapply gs xs ==> [49, 25, 14, 10]


As we illustrated in class, there are clear patterns shared between lists and option types and trees, so perhaps you can see why people want to identify the general structures. But it probably isn't obvious yet why it would be useful to do so. To a large extent, this will only emerge over the next few classes. But we'll begin to demonstrate the usefulness of these patterns by talking through a simple example, that uses the Monadic functions of the Option/Maybe box type.


Safe division
-------------

Integer division presupposes that its second argument
(the divisor) is not zero, upon pain of presupposition failure.
Here's what my OCaml interpreter says:

    # 12/0;;
    Exception: Division_by_zero.

Say we want to explicitly allow for the possibility that
division will return something other than a number.
To do that, we'll use OCaml's `option` type, which works like this:

    # type 'a option = None | Some of 'a;;
    # None;;
    - : 'a option = None
    # Some 3;;
    - : int option = Some 3

So if a division is normal, we return some number, but if the divisor is
zero, we return `None`. As a mnemonic aid, we'll prepend a `safe_` to the start of our new divide function.

<pre>
let safe_div (x:int) (y:int) =
  match y with
    | 0 -> None
    | _ -> Some (x / y);;

(*
val safe_div : int -> int -> int option = fun
# safe_div 12 2;;
- : int option = Some 6
# safe_div 12 0;;
- : int option = None
# safe_div (safe_div 12 2) 3;;
# safe_div (safe_div 12 2) 3;;
            ~~~~~~~~~~~~~
Error: This expression has type int option
       but an expression was expected of type int
*)
</pre>

This starts off well: dividing 12 by 2, no problem; dividing 12 by 0,
just the behavior we were hoping for. But we want to be able to use
the output of the safe-division function as input for further division
operations. So we have to jack up the types of the inputs:

<pre>
let safe_div2 (u:int option) (v:int option) =
  match u with
          None -> None
        | Some x -> (match v with
                                  Some 0 -> None
                                | Some y -> Some (x / y));;

(*
val safe_div2 : int option -> int option -> int option = <fun>
# safe_div2 (Some 12) (Some 2);;
- : int option = Some 6
# safe_div2 (Some 12) (Some 0);;
- : int option = None
# safe_div2 (safe_div2 (Some 12) (Some 0)) (Some 3);;
- : int option = None
*)
</pre>

Beautiful, just what we need: now we can try to divide by anything we
want, without fear that we're going to trigger any system errors.

I prefer to line up the `match` alternatives by using OCaml's
built-in tuple type:

<pre>
let safe_div2 (u:int option) (v:int option) =
  match (u, v) with
    | (None, _) -> None
    | (_, None) -> None
    | (_, Some 0) -> None
    | (Some x, Some y) -> Some (x / y);;
</pre>

So far so good. But what if we want to combine division with
other arithmetic operations? We need to make those other operations
aware of the possibility that one of their arguments has triggered a
presupposition failure:

<pre>
let safe_add (u:int option) (v:int option) =
  match (u, v) with
    | (None, _) -> None
    | (_, None) -> None
    | (Some x, Some y) -> Some (x + y);;

(*
val safe_add : int option -> int option -> int option = <fun>
# safe_add (Some 12) (Some 4);;
- : int option = Some 16
# safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
</pre>

This works, but is somewhat disappointing: the `safe_add` operation
doesn't trigger any presupposition of its own, so it is a shame that
it needs to be adjusted because someone else might make trouble.

But we can automate the adjustment. The standard way in OCaml,
Haskell, and other functional programming languages, is to use the monadic
`bind` operator, `>>=`. (The name "bind" is not well chosen from our
perspective, but this is too deeply entrenched by now.) As mentioned above,
there needs to be a different `>>=` operator for each Monad or box type you're working with.
Haskell finesses this by "overloading" the single symbol `>>=`; you can just input that
symbol and it will calculate from the context of the surrounding type constraints what
monad you must have meant. In OCaml, the `>>=` or `bind` operator is not pre-defined, but we will
give you a library that has definitions for all the standard monads, as in Haskell.
For now, though, we will define our `bind` operation by hand:

<pre>
let bind (u: int option) (f: int -> (int option)) =
  match u with
    |  None -> None
    | Some x -> f x;;

let safe_add3 (u: int option) (v: int option) =
  bind u (fun x -> bind v (fun y -> Some (x + y)));;

(* This is really just `map2 (+)`, using the `map2` operation that corresponds to
   definition of `bind`. *)

let safe_div3 (u: int option) (v: int option) =
  bind u (fun x -> bind v (fun y -> if 0 = y then None else Some (x / y)));;

(* This goes back to some of the simplicity of the original safe_div, without the complexity
   introduced by safe_div2. *)
</pre>

The above definitions look even simpler if you focus on the fact that `safe_add3` can be written as simply `map2 (+)`, and that `safe_div3` could be written as `u >>= fun x -> v >>= fun y -> if 0 = y then None else Some (x / y)`. Haskell has an even more user-friendly notation for this, namely:

    safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
    safe_div3 u v = do {x <- u;
                        y <- v;
                        if 0 == y then Nothing else return (x `div` y)}

Let's see our new functions in action:

(*
# safe_div3 (safe_div3 (Some 12) (Some 2)) (Some 3);;
- : int option = Some 2
#  safe_div3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
- : int option = None
# safe_add3 (safe_div3 (Some 12) (Some 0)) (Some 3);;
- : int option = None
*)
</pre>

Compare the new definitions of `safe_add3` and `safe_div3` closely: the definition
for `safe_add3` shows what it looks like to equip an ordinary operation to
survive in dangerous presupposition-filled world. Note that the new
definition of `safe_add3` does not need to test whether its arguments are
None objects or real numbers---those details are hidden inside of the
`bind` function.

The definition of `safe_div3` shows exactly what extra needs to be said in
order to trigger the no-division-by-zero presupposition. Here, too, we don't
need to keep track of what presuppositions may have already failed
for whatever reason on our inputs.

(Linguitics note: Dividing by zero is supposed to feel like a kind of
presupposition failure. If we wanted to adapt this approach to
building a simple account of presupposition projection, we would have
to do several things. First, we would have to make use of the
polymorphism of the `option` type. In the arithmetic example, we only
made use of `int option`s, but when we're composing natural language
expression meanings, we'll need to use types like `N option`, `Det option`,
`VP option`, and so on. But that works automatically, because we can use
any type for the `'a` in `'a option`. Ultimately, we'd want to have a
theory of accommodation, and a theory of the situations in which
material within the sentence can satisfy presuppositions for other
material that otherwise would trigger a presupposition violation; but,
not surprisingly, these refinements will require some more
sophisticated techniques than the super-simple Option monad.)