Q
is bool
).
Note that the lefthand schema `P` is permitted to itself be a boxed type. That is, where
if `Î± list` is our box type, we can write the second arrow as
+Note that the lefthand schema `P` is permitted to itself be a boxed type. That is, where if `Î± list` is our box type, we can write the second type as:
int > Q
+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.
+Here are some examples of values of these Kleisli arrow types, where the box type is `Î± list`, and the Kleisli arrow types are int > int
(that is, `int > int list`) or int > bool
:
+
+\x. [x] +\x. [odd? x, odd? x] +\x. prime_factors_of x +\x. [0, 0, 0]+ +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 arrow types. ## A family of functions for each box type ## @@ 91,35 +105,60 @@ Here are the types of our crucial functions, together with our pronunciation, an
map (/mÃ¦p/): (P > Q) > P > Q
+> In Haskell, this is the function `fmap` from the `Prelude` and `Data.Functor`; also called `<$>` in `Data.Functor` and `Control.Applicative`, and also called `Control.Applicative.liftA` and `Control.Monad.liftM`.
+
map2 (/mÃ¦ptu/): (P > Q > R) > P > Q > R
mid (/ÎµmaidÎµnt@tI/ aka unit, return, pure): P > P
+> In Haskell, this is called `Control.Applicative.liftA2` and `Control.Monad.liftM2`.
+
+mid (/ÎµmaidÎµnt@tI/): P > P
+
+> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `Î·`. In the class presentation Jim called it `ð`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".) This notion is exemplified by `Just` for the box type `Maybe Î±` and by the singleton function for the box type `List Î±`.
m$ or mapply (/Îµm@plai/): P > Q > P > Q
+> We'll use `m$` as a leftassociative infix operator, reminiscent of (the rightassociative) `$` which is just ordinary function application (also expressed by mere leftassociative juxtaposition). In the class presentation Jim called `m$` `â`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`.
+
<=< or mcomp : (Q > R) > (P > Q) > (P > R)
>=> or mpmoc (flip mcomp): (P > Q) > (Q > R) > (P > R)
+> In Haskell, this is `Control.Monad.<=<`.
+
+>=> (flip mcomp, should we call it mpmoc?): (P > Q) > (Q > R) > (P > R)
+
+> In Haskell, this is `Control.Monad.>=>`. 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.
>>= or mbind : (Q) > (Q > R) > (R)
=<< or mdnib (flip mbind) (Q) > (Q > R) > (R)
+=<< (flip mbind, should we call it mdnib?) (Q > R) > (Q) > (R)
+
+join: P > P
join: P > P
+> In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `Î¼`.
+Haskell uses the symbol `>>=` but calls it "bind". This is not well chosen from the perspective of formal semantics, but it's too deeply entrenched to change. We've at least preprended an "m" to the front of "bind".
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 â¡
\a. (j a >>= k)
.
+Haskell's names "return" and "pure" for `mid` are even less well chosen, and we think it will be clearer in our discussion to use a different name. (Also, in other theoretical contexts this notion goes by other names, anyway, like `unit` or `Î·`  having nothing to do with `Î·`reduction in the Lambda Calculus.)
In most cases of interest, instances of these systems of functions will provide
certain useful guarantees.
+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 â¡ \a. (j a >>= k)
. We'll state some other interdefinitions below.
+
+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 `â`. 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
+ map (id : Î± > Î±) == (id : Î± > Î±)
+ map (g â f) == (map g) â (map f)
+
+ Essentially these say that `map` is a homomorphism from the algebra of `(universe Î± > Î², operation â, elsment id)` to that of (Î± > Î², â', id')
, where `â'` and `id'` are `â` 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 `Î± list` (that's our Î±
), 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 â 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.
+
+ > 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
@@ 127,32 +166,124 @@ has to obey the following Map Laws:
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
+ 1. mid (id : P>P) : P > P
is a left identity for `m$`, that is: `(mid id) m$ xs = xs`
+ 2. `mid (f a) = (mid f) m$ (mid a)`
+ 3. The `map2`ing of composition onto boxes `fs` and `gs` of functions, when `m$`'d to a box `xs` of arguments == the `m$`ing of `fs` to the `m$`ing of `gs` to xs: `(mid (â) m$ fs m$ gs) m$ xs = fs m$ (gs m$ xs)`.
+ 4. When the arguments are `mid`'d, the order of `m$`ing doesn't matter: `fs m$ (mid x) = mid ($x) m$ fs`. (Note that it's `mid ($x)`, or `mid (\f. f x)` that gets `m$`d onto `fs`, not the original `mid x`.) Here's an example where the order *does* matter: `[succ,pred] m$ [1,2] == [2,3,0,1]`, but `[($1),($2)] m$ [succ,pred] == [2,0,3,1]`. This Law states a class of cases where the order is guaranteed not to matter.
+ 5. A consequence of the laws already stated is that when the functions are `mid`'d, the order of `m$`ing doesn't matter either: `mid f m$ xs == map (flip ($)) xs m$ mid f`.
+
+
+
+* ***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
+ 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
+
+ You could just as well express the Monad laws using `>=>`:
+
+ l >=> (k >=> j) == (l >=> k) >=> j
+ k >=> mid == k
+ mid >=> k == 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 with Monads, all you really need to get the whole system of functions are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
+
+ In practice, you will often work with `>>=`. In the Haskell manuals, they express the Monad Laws using `>>=` instead of the composition operators. This looks similar, but doesn't have the same symmetry:
+
+ u >>= (\a > k a >>= j) == (u >>= k) >>= j
+ u >>= mid == u
+ mid a >>= k == k a
+
+ Also, Haskell calls `mid` `return` or `pure`, but we've stuck to our terminology in this context.
+
+ > In Category Theory discussion, the Monad Laws are instead expressed in terms of `join` (which they call `Î¼`) and `mid` (which they call `Î·`). These are assumed to be "natural transformations" for their box type, which means that they satisfy these equations with that box type's `map`:
+ > map f â mid == mid â f+ > The Monad Laws then take the form: + >
map f â join == join â map (map f)
join â (map join) == join â join+ > The first of these says that if you have a triplyboxed type, and you first merge the inner two boxes (with `map join`), and then merge the resulting box with the outermost box, that's the same as if you had first merged the outer two boxes, and then merged the resulting box with the innermost box. The second law says that if you take a box type and wrap a second box around it (with `mid`) and then merge them, that's the same as if you had done nothing, or if you had instead wrapped a second box around each element of the original (with `map mid`, leaving the original box on the outside), and then merged them.
join â mid == id == join â map mid
+ > The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `Î±` to type Î±
:
+ >
Î¼ â M(Î¼) == Î¼ â Î¼+ + +As hinted in last week's homework and explained in class, the operations available in a Mappable system exactly preserve the "structure" of the boxed type they're operating on, and moreover are only sensitive to what content is in the corresponding original position. If you say `map f [1,2,3]`, then what ends up in the first position of the result depends only on how `f` and `1` combine. + +For MapNable operations, on the other hand, the structure of the result may instead by a complex function of the structure of the original arguments. But only of their structure, not of their contents. And if you say `map2 f [10,20] [1,2,3]`, what ends up in the first position of the result depends only on how `f` and `10` and `1` combine. + +With `map`, you can supply an `f` such that `map f [3,2,0,1] == [[3,3,3],[2,2],[],[1]]`. But you can't transform `[3,2,0,1]` to `[3,3,3,2,2,1]`, and you can't do that with MapNable operations, either. That would involve the structure of the result (here, the length of the list) being sensitive to the content, and not merely the structure, of the original. + +For Monads (Composables), you can perform more radical transformations of that sort. For example, `join (map (\x. dup x x) [3,2,0,1])` would give us `[3,3,3,2,2,1]` (for a suitable definition of `dup`). + + + + +## Interdefinitions and Subsidiary notions## + +We said above that various of these box type operations can be defined in terms of others. Here is a list of various ways in which they're related. We try to stick to the consistent typing conventions that: + +
Î¼ â Î· == id == Î¼ â M(Î·)
+f : Î± > Î²; g and h have types of the same form
+ also sometimes these will have types of the form Î± > Î² > Î³
+ note that Î± and Î² are permitted to be, but needn't be, boxed types
+j : Î± > Î²; k and l have types of the same form
+u : Î±; v and xs and ys have types of the same form
+
+w : Î±
+
+
+But we may sometimes slip.
+
+Here are some ways the different notions are related:
+
++j >=> k â¡= \a. (j a >>= k) +u >>= k == (id >=> k) u; or ((\(). u) >=> k) () +u >>= k == join (map k u) +join w == w >>= id +map2 f xs ys == xs >>= (\x. ys >>= (\y. mid (f x y))) +map2 f xs ys == (map f xs) m$ ys, using m$ as an infix operator +fs m$ xs == fs >>= (\f. map f xs) +m$ == map2 id +map f xs == mid f m$ xs +map f u == u >>= mid â f ++ + +Here are some other monadic notion that you may sometimes encounter: + +*
mzero
is a value of type Î±
that is exemplified by `Nothing` for the box type `Maybe Î±` and by `[]` for the box type `List Î±`. It has the behavior that `anything m$ mzero == mzero == mzero m$ anything == mzero >>= anything`. In Haskell, this notion is called `Control.Applicative.empty` or `Control.Monad.mzero`.
+
+* Haskell has a notion `>>` definable as `\u v. map (const id) u m$ v`, or as `u >> v == u >>= const v`. This is often useful, and `u >> v` won't in general be identical to just `v`. For example, using the box type `List Î±`, `[1,2,3] >> [4,5] == [4,5,4,5,4,5]`. But in the special case of `mzero`, it is a consequence of what we said above that `anything >> mzero == mzero`. Haskell also calls `>>` `Control.Applicative.*>`.
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$` and `map2` from *MapNables*. So all you really need
are a definition of `mid`, on the one hand, and one of `mcomp`, `mbind`, or `join`, on the other.
+* Haskell has a correlative notion `Control.Applicative.<*`, definable as `\u v. map const u m$ v`. For example, `[1,2,3] <* [4,5] == [1,1,2,2,3,3]`. You might expect Haskell to call `<*` `<<`, but they don't. They used to use `<<` for `flip (>>)` instead, but now they seem not to use `<<` anymore.
Here are some interdefinitions: TODO
+* mapconst
is definable as `map â const`. For example `mapconst 4 [1,2,3] == [4,4,4]`. Haskell calls `mapconst` `<$` in `Data.Functor` and `Control.Applicative`. They also use `$>` for `flip mapconst`, and `Control.Monad.void` for `mapconst ()`.
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: `Î±`. So if `Î±` is type `bool`,
then a boxed `Î±` is ... a `bool`. That is, Î± = Î±
.
+then a boxed `Î±` is ... a `bool`. That is, Î± == Î±
.
In terms of the box analogy, the Identity box type is a completely invisible box. With the following
definitions:
@@ 180,41 +311,41 @@ Identity is a monad. Here is a demonstration that the laws hold:
~~> \x.j((\x.k(lx)) x)
~~> \x.j(k(lx))
The Identity Monad is favored by mimes.
+The Identity monad is favored by mimes.
To take a slightly less trivial (and even more useful) example,
consider the box type `Î± list`, with the following operations:
 mid: Î± > [Î±]
+ mid : Î± > [Î±]
mid a = [a]
 mcomp: (Î² > [Î³]) > (Î± > [Î²]) > (Î± > [Î³])
 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 : (Î² > [Î³]) > (Î± > [Î²]) > (Î± > [Î³])
+ 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 `Î±`) to `g`, which returns a list of `Î²`s;
each `Î²` in that list is fed to `f`, which returns a list of `Î³`s. The
+In words, `mcomp k j a` feeds the `a` (which has type `Î±`) to `j`, which returns a list of `Î²`s;
+each `Î²` in that list is fed to `k`, which returns a list of `Î³`s. The
final result is the concatenation of those lists of `Î³`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 *boxproducing 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 ##
@@ 325,7 +456,7 @@ 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.
+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 predefined, but we will