As mentioned at the top of the page, in Category Theory presentations of monads they usually talk about "endofunctors", which are mappings from a Category to itself. In the uses they make of this notion, the endofunctors combine the role of a box type _
and of the `map` that goes together with it.
* ***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:
TODO 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
The name "bind" is not well chosen from our perspective, but this is too deeply entrenched by now.
## Examples ##
To take a trivial (but, as we will see, still useful) example,
consider the Identity box type: `α`. So if `α` is type `bool`,
then a boxed `α` is ... a `bool`. That is, α = α
.
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))
The Identity monad is favored by 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 k j a = concat (map k (j a)) = List.flatten (List.map k (j a))
= foldr (\b ks -> (k b) ++ ks) [] (j a) = List.fold_right (fun b ks -> List.append (k b) ks) [] (j a)
= [c | b <- j a, c <- k b]
In the first two definitions of `mcomp`, we give the definition first in Haskell and then in the equivalent OCaml. The three different definitions of `mcomp` (one for each line) are all equivalent, and it is easy to show that they obey the Monad Laws. (You will do this in the homework.)
In words, `mcomp k j a` feeds the `a` (which has type `α`) to `j`, which returns a list of `β`s;
each `β` in that list is fed to `k`, which returns a list of `γ`s. The
final result is the concatenation of those lists of `γ`s.
For example:
let j a = [a*a, a+a] in
let k b = [b, b+1] in
mcomp k j 7 ==> [49, 50, 14, 15]
`j 7` produced `[49, 14]`, which after being fed through `k` 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 js = [(\a->a*a),(\a->a+a)] in
let xs = [7, 5] in
mapply js 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 figure out 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.
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;;
~~~~~~~~~~~~~
Error: This expression has type int option
but an expression was expected of type int
*)
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:
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 =
# 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
*)
Calling the function now involves some extra verbosity, but it gives us what we need: now we can try to divide by anything we
want, without fear that we're going to trigger system errors.
I prefer to line up the `match` alternatives by using OCaml's
built-in tuple type:
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);;
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 already triggered a
presupposition failure:
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 =
# safe_add (Some 12) (Some 4);;
- : int option = Some 16
# safe_add (safe_div (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
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, using the monadic machinery we introduced above.
As we said, there needs to be different `>>=`, `map2` and so on operations for each
monad or box type we'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 monadic operators are 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 `>>=` and `map2` operations by hand:
let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
match u with
| None -> None
| Some x -> j x;;
let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
let safe_add3 = map2 (+);; (* that was easy *)
let safe_div3 (u: int option) (v: int option) =
u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
Haskell has an even more user-friendly notation for defining `safe_div3`, namely:
safe_div3 :: Maybe Int -> Maybe Int -> Maybe Int
safe_div3 u v = do {x <- u;
y <- v;
if 0 == y then Nothing else Just (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
*)
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` values or real numbers---those details are hidden inside of the
`bind` function.
Note also that our definition of `safe_div3` recovers some of the simplicity of
the original `safe_div`, without the complexity introduced by `safe_div2`. We now
add exactly what extra is needed to track the no-division-by-zero presupposition. Here, too, we don't
need to keep track of what other presuppositions may have already failed
for whatever reason on our inputs.
(Linguistics 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/Maybe monad.)