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 type as:
+Note that the left-hand schema `P` is permitted to itself be a boxed
+type. That is, if `α list` is our box type, and `P` is the boxed type
+`int list`, we can write the boxed type that has `P` as its left-hand
+side as
<code><u>int</u> -> <u>int list</u></code>
+If it's clear that we're uniformly talking about the same box type (in
+this example, `α list`), we can equivalently write
+
+<code><u>int</u> -> <span class="box2">int</span></code>
+
Here are some examples of values of these Kleisli arrow types, where the box type is `α list`, and the Kleisli arrow types are <code>int -> <u>int</u></code> (that is, `int -> int list`) or <code>int -> <u>bool</u></code>:
<pre>\x. [x]
<code>mid (/εmaidεnt@tI/): P -> <u>P</u></code>
-> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. In the class presentation Jim called it `𝟭`. This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`.
+> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. In the class presentation Jim called it `𝟭`; 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 α` and by the singleton function for the box type `List α`.
<code>m$ or mapply (/εm@plai/): <u>P -> Q</u> -> <u>P</u> -> <u>Q</u></code>
-> In Haskell, this is called `Control.Monad.ap` or `Control.Applicative.<*>`. In the class presentation Jim called it `●`.
+> We'll use `m$` as an infix operator, reminiscent of `$` which is just ordinary function application (also expressed by mere juxtaposition). In the class presentation Jim called `m$` `●`. In Haskell, it's called `Control.Monad.ap` or `Control.Applicative.<*>`.
<code><=< or mcomp : (Q -> <u>R</u>) -> (P -> <u>Q</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>
-> In Haskell, this is `Control.Monad.>=>`.
+> 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.
<code>>>= or mbind : (<u>Q</u>) -> (Q -> <u>R</u>) -> (<u>R</u>)</code>
> In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
+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".
-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.
-
-Haskell's name "bind" for `>>=` is not well chosen from our perspective, but this is too deeply entrenched by now. We've at least preprended an `m` to the front of it.
+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 `η` --- having nothing to do with `η`-reduction in the Lambda Calculus.)
-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 `η` --- having nothing to do with `η`-reduction in the Lambda Calculus.) 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>.
+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>. 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
Moreover, with `map2` in hand, `map3`, `map4`, ... `mapN` are easily definable.) These
have to obey the following MapN Laws:
- TODO LAWS
+ 1. <code>mid (id : P->P) : <u>P</u> -> <u>P</u></code> 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$`ing of `gs` to xs: `((mid ○) m$ 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`. In examples we'll be working with at first, order _never_ matters; but down the road, sometimes it will. This Law states a class of cases where it's guaranteed not to.
+ 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: TODO
* ***Monad*** (or "Composables") A MapNable box type is a *Monad* if there
You could just as well express the Monad laws using `>=>`:
- l >=> (k >=> j) == (l >=> k) >-> j
+ l >=> (k >=> j) == (l >=> k) >=> j
k >=> mid == k
mid >=> k == k
> <pre>map f ○ mid == mid ○ f<br>map f ○ join == join ○ map (map f)</pre>
> The Monad Laws then take the form:
> <pre>join ○ (map join) == join ○ join<br>join ○ mid == id == join ○ map mid</pre>
- > The first of these says that if you have a triply-boxed 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
instead mapped a second box around the elements of the original (with `map mid`, leaving the original box on the outside), and then merged them.<p>
+ > The first of these says that if you have a triply-boxed 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.<p>
> The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type <code><u>α</u></code>:
> <pre>μ ○ M(μ) == μ ○ μ<br>μ ○ η == id == μ ○ M(η)</pre></small>
+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`).
+
+<!--
+Some global transformations that we work with in semantics, like Veltman's test functions, can't directly be expressed in terms of the primitive Monad operations? For example, there's no `j` such that `xs >>= j == mzero` if `xs` anywhere contains the value `1`.
+-->
+
## 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:
<pre>
-f : α -> β; g and h have types of the same format (note that α and β are permitted to be, but needn't be, boxed types)
-j : α -> <u>β</u>; k and l have types of the same format
-u : <u>α</u>; v and xs and ys have types of the same format
+f : α -> β; g and h have types of the same form
+ also sometimes these will have types of the form α -> β -> γ
+ note that α and β are permitted to be, but needn't be, boxed types
+j : α -> <u>β</u>; k and l have types of the same form
+u : <u>α</u>; v and xs and ys have types of the same form
+
w : <span class="box2">α</span>
</pre>
* <code>mzero</code> is a value of type <code><u>α</u></code> that is exemplified by `Nothing` for the box type `Maybe α` and by `[]` for the box type `List α`. 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. mid (const id) m$ u m$ v`. It works like this: `u >> v == u >>= const v`. This is often useful, and won't in general be identical to just `v`. For example, using the box type `List α`, `[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.*>`.
+* 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 α`, `[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.*>`.
-* Haskell has a correlative notion `Control.Applicative.<*`, definable as `\u v. mid const m$ 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 it anymore. Maybe in the future they'll call `<*` `<<`.
+* 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.
* <code>mapconst</code> is definable as `map ○ 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 ()`.