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.
+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 values inside of *boxes*, and that monads wrap
of monads, followed by instances of monads from the philosophical and
linguistics literature.
+> <small>After you've read this once and are coming back to re-read it to try to digest the details further, the "endofunctors" that slogan is talking about are a combination of our boxes and their associated maps. Their "monoidal" character is captured in the Monad Laws, where a "monoid"---don't confuse with a mon*ad*---is a simpler algebraic notion, meaning a universe with some associative operation that has an identity. For advanced study, here are some further links on the relation between monads as we're working with them and monads as they appear in category theory:
+[1](http://en.wikipedia.org/wiki/Outline_of_category_theory)
+[2](http://lambda1.jimpryor.net/advanced_topics/monads_in_category_theory/)
+[3](http://en.wikibooks.org/wiki/Haskell/Category_theory)
+[4](https://wiki.haskell.org/Category_theory), where you should follow the further links discussing Functors, Natural Transformations, and Monads.</small>
+
+
## Box types: type expressions with one free type variable ##
Recall that we've been using lower-case Greek letters
(α, α) tree
The idea is that whatever type the free type variable `α` might be instantiated to,
-we will be a "type box" of a certain sort that "contains" values of type `α`. For instance,
+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.
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:
-<u>int</u>
+<code><u>int</u></code>
-for the type of a boxed `int`. (We'll fool with the markup to make this show a genuine box later; for now it will just display as underlined.)
+for the type of a boxed `int`.
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>
+<code>P -> <u>Q</u></code>
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 -> <u>bool</u>
+<code>int -> <u>bool</u></code>
-int list -> <u>int list</u>
+<code>int list -> <u>int list</u></code>
In the first, `P` has become `int` and `Q` has become `bool`. (The boxed type <code><u>Q</u></code> is <code><u>bool</u></code>).
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>
+<code><u>int</u> -> <u>int list</u></code>
-<<<<<<< HEAD
-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 clear shortly.)
-=======
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.
->>>>>>> ecff6bbae7c00556584b51913b934bdade0cff40
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>=<< or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
-<code>join: <span style="border-bottom: 3px double;">P</span> -> <u>P</u></code>
-
-
-Test1: <span style="border-bottom: 3px double"><d>P</d></span>
-
-Test2: <span style="text-decoration: overline"><u>P</u></span>
-
-Test3: <span style="font-style: italic;">XX</span>
-
-Test4: <span class="double">YY</span>
+<code>join: <span class="box2">P</span> -> <u>P</u></code>
The menagerie isn't quite as bewildering as you might suppose. Many of these will
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
+ TODO 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:
+ have to obey the following MapN Laws:
+
+ TODO LAWS
* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
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.
+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 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
+consider the Identity box type: `α`. So if `α` is type `bool`,
+then a boxed `α` is ... a `bool`. That is, <code><u>α</u> = α</code>.
+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.
+ 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:
= 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).
+The last three definitions of `mcomp` are all equivalent, and it is easy to see that they obey the monad laws (see exercises TODO).
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
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.
+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 ##
# 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,
+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));;
+ | 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>
*)
</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.
+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:
<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);;
+ | (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
+aware of the possibility that one of their arguments has already triggered a
presupposition failure:
<pre>
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.
+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 `>>=` or `bind` operator is not pre-defined, but we will
+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 `bind` operation by hand:
+For now, though, we will define our `>>=` and `map2` operations by hand:
<pre>
-let bind (u: int option) (f: int -> (int option)) =
+let (>>=) (u : 'a option) (j : 'a -> 'b option) : 'b option =
match u with
- | None -> None
- | Some x -> f x;;
+ | None -> None
+ | Some x -> j x;;
-let safe_add3 (u: int option) (v: int option) =
- bind u (fun x -> bind v (fun y -> Some (x + y)));;
+let map2 (f : 'a -> 'b -> 'c) (u : 'a option) (v : 'b option) : 'c option =
+ u >>= (fun x -> v >>= (fun y -> Some (f x y)));;
-(* This is really just `map2 (+)`, using the `map2` operation that corresponds to
- definition of `bind`. *)
+let safe_add3 = map2 (+);; (* that was easy *)
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. *)
+ u >>= (fun x -> v >>= (fun y -> if 0 = y then None else Some (x / y)));;
</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:
+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 return (x `div` y)}
+ if 0 == y then Nothing else Just (x `div` y)}
Let's see our new functions in action:
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
+`None` values 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
+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
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.)
+sophisticated techniques than the super-simple Option/Maybe monad.)