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
+if `α list` is our box type, we can write the second type as:
-<u>int</u> -> <u>Q</u>
+<code><u>int</u> -> <u>int list</u></code>
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.
<code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</u>) -> (P -> <u>R</u>)</code>
-<code>>=> or mpmoc (flip mcomp): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
+<code>>=> (flip mcomp, should we call it mpmoc?): (P -> <u>Q</u>) -> (Q -> <u>R</u>) -> (P -> <u>R</u>)</code>
<code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
-<code>=<< or mdnib (flip mbind) (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
+<code>=<< (flip mbind, should we call it mdnib?) (Q -> <u>R</u>) -> (<u>Q</u>) -> (<u>R</u>)</code>
+
+<code>join: <span class="box2">P</span> -> <u>P</u></code>
+
-<code>join: <span class="double">P</span> -> <u>P</u></code>
+In the class handout, we gave the types for `>=>` twice, and once was correct but the other was a typo. The above is the correct typing.
+Also, in the handout we called `mid` `𝟭`. But now we've decided that `mid`
+is better. (Think of it as "m" plus "identity", not as the start of "midway".)
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: <code>k <=< j ≡
\a. (j a >>= k)</code>.
-In most cases of interest, instances of these systems of functions will provide
-certain useful guarantees.
+We will move freely back and forth between using `>=>` and using `<=<` (aka `mcomp`), which
+is just `>=>` with its arguments flipped. `<=<` has the virtue that it corresponds more
+closely to the ordinary mathematical symbol `○`. But `>=>` has the virtue
+that its types flow more naturally from left to right.
+
+These functions come together in several systems, and have to be defined in a way that coheres with the other functions in the system:
* ***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:
- TODO LAWS
+ <code>map (id : α -> α) = (id : <u>α</u> -> <u>α</u>)</code>
+ <code>map (g ○ f) = (map g) ○ (map f)</code>
+
+ Essentially these say that `map` is a homomorphism from the algebra of `(universe α -> β, operation ○, elsment id)` to that of <code>(<u>α</u> -> <u>β</u>, ○', id')</code>, where `○'` and `id'` are `○` and `id` restricted to arguments of type <code><u>_</u></code>. 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 <code><u>α</u></code>), 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.
+
+ > <small>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 <code><u>_</u></code> and of the `map` that goes together with it.</small>
+
* ***MapNable*** (in Haskelese, "Applicatives") A Mappable box type is *MapNable*
if there are in addition `map2`, `mid`, and `mapply`. (Given either
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`. In terms of the box analogy, the
-Identity box type is a completely invisible box. With the following
+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
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.
+ 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 : 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]
+ 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]
-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 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 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
+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 f b = [b, b+1] in
- let g a = [a*a, a+a] in
- mcomp f g 7 ==> [49, 50, 14, 15]
+ let j a = [a*a, a+a] in
+ let k b = [b, b+1] in
+ mcomp k j 7 ==> [49, 50, 14, 15]
-`g 7` produced `[49, 14]`, which after being fed through `f` gave us `[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 gs = [(\a->a*a),(\a->a+a)] in
+ let js = [(\a->a*a),(\a->a+a)] in
let xs = [7, 5] in
- mapply gs xs ==> [49, 25, 14, 10]
+ 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.
+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 ##
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.
+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
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.
Note also that our definition of `safe_div3` recovers some of the simplicity 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.)