`α, β, γ, ...`

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 instantiated to,
we will have a "type box" of a certain sort that "contains" values 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 readily thought of as "containers" (lists, trees, and so on, with `α`s as their "elements"), with later examples we discuss it will be less natural to describe the boxed types that way. For example, where `R` is some fixed type, `R -> α` is a box type.
Also, for clarity: the *box type* is the type `α list` (or as we might just say, the `list` type operator); the *boxed type* is some specific instantiation of the free type variable `α`. We'll often write boxed types as a box containing what the free
type variable instantiates to. So if our box type is `α list`, and `α` instantiates to the specific type `int`, we would write:
__int__

for the type of a boxed `int`.
## 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 -> `__Q__

That is, they are functions from values 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 -> `__bool__

`int list -> `__int list__

In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type __Q__

is __bool__

).
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 type as:
__int__ -> __int list__

As semanticists, you are no doubt familiar with the debates between those who insist that propositions are sets of worlds and those who insist they are context change potentials. We hope to show you, in coming weeks, that propositions are (certain sorts of) Kleisli arrows. But this doesn't really compete with the other proposals; it is a generalization of them. Both of the other proposed structures can be construed as specific Kleisli arrows.
## A family of functions for each box type ##
We'll need a family of functions to help us work with box types. As will become clear, these have to be defined differently for each box type.
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.)
`map (/mæp/): (P -> Q) -> `__P__ -> __Q__

`map2 (/mæptu/): (P -> Q -> R) -> `__P__ -> __Q__ -> __R__

`mid (/εmaidεnt@tI/ aka unit, return, pure): P -> `__P__

`m$ or mapply (/εm@plai/): `__P -> Q__ -> __P__ -> __Q__

`<=< or mcomp : (Q -> `__R__) -> (P -> __Q__) -> (P -> __R__)

`>=> (flip mcomp, should we call it mpmoc?): (P -> `__Q__) -> (Q -> __R__) -> (P -> __R__)

`>>= or mbind : (`__Q__) -> (Q -> __R__) -> (__R__)

`=<< (flip mbind, should we call it mdnib?) (Q -> `__R__) -> (__Q__) -> (__R__)

`join: P -> `__P__

The menagerie 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: ```
k <=< j ≡
\a. (j a >>= k)
```

.
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:
`map (id : α -> α) = (id : `__α__ -> __α__)

`map (g ○ f) = (map g) ○ (map f)`

Essentially these say that `map` is a homomorphism from `(α -> β, ○, id)` to `(`__α__ -> __β__, ○', id')

, where `○'` and `id'` are `○` and `id` restricted to arguments of type _____

. That might be hard to digest because it's so abstract. Think of the following concrete example: if you take a `α list` (that's our __α__

), and apply `id` to each of its elements, that's the same as applying `id` to the list itself. That's the first law. And if you apply the composition of functions `g ○ f` to each of the list's elements, that's the same as first applying `f` to each of the elements, and then going through the elements of the resulting list and applying `g` to each of those elements. That's the second law. These laws obviously hold for our familiar notion of `map` in relation to lists.
* ***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 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.
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 =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:# 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 *)

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 =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:# safe_add (Some 12) (Some 4);; - : int option = Some 16 # safe_add (safe_div (Some 12) (Some 0)) (Some 4);; - : int option = None *)

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.)