int
@@ -75,7 +75,7 @@ A lot of what we'll be doing concerns types that are called *Kleisli arrows*. Do
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:
+For instance, the following are Kleisli arrow types:
int -> bool
@@ -83,12 +83,25 @@ For instance, the following are Kleisli arrows:
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:
+Note that either of the schemas `P` or `Q` are permitted to themselves be boxed
+types. That is, 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.
+And also what the rhs there is a boxing of is itself a boxed type (with the same kind of box):, so we can write it as:
+
+int -> int
+
+We have to be careful though not to to unthinkingly equivocate between different kinds of boxes.
+
+Here are some examples of values of these Kleisli arrow types, where the box type is `α 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 ## @@ -99,32 +112,41 @@ Here are the types of our crucial functions, together with our pronunciation, an
map (/mæp/): (P -> Q) -> P -> Q
+> In Haskell, this is the function `fmap` from the `Prelude` and `Data.Functor`; also called `<$>` in `Data.Functor` and `Control.Applicative`, and also called `Control.Applicative.liftA` and `Control.Monad.liftM`.
+
map2 (/mæptu/): (P -> Q -> R) -> P -> Q -> R
-mid (/εmaidεnt@tI/ aka unit, return, pure): P -> P
+> In Haskell, this is called `Control.Applicative.liftA2` and `Control.Monad.liftM2`.
+
+mid (/εmaidεnt@tI/): P -> P
+
+> In Haskell, this is called `Control.Monad.return` and `Control.Applicative.pure`. In other theoretical contexts it is sometimes called `unit` or `η`. In the class presentation Jim called it `ð`; but now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".) This notion is exemplified by `Just` for the box type `Maybe α` and by the singleton function for the box type `List α`.
m$ or mapply (/εm@plai/): P -> Q -> P -> Q
+> We'll use `m$` as a left-associative infix operator, reminiscent of (the right-associative) `$` which is just ordinary function application (also expressed by mere left-associative 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)
+> 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)
=<< (flip mbind, should we call it mdnib?) (Q -> R) -> (Q) -> (R)
-join: P -> P
-
+join: P -> P
-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.
+> In Haskell, this is `Control.Monad.join`. In other theoretical contexts it is sometimes called `μ`.
-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 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".
-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` `ð`. But now we've decided that `mid` is better. (Think of it as "m" plus "identity", not as the start of "midway".)
+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.)
-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).
+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
@@ -151,8 +173,28 @@ 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
-
+ 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 (the right-hand operand of `m$`) are an `mid`'d value, the order of `m$`ing doesn't matter: `fs m$ (mid x) = mid ($x) m$ fs`. (Though 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 _left_-hand operand of `m$` is a `mid`'d value, the order of `m$`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
@@ -164,7 +206,7 @@ has to obey the following Map Laws:
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
@@ -182,25 +224,78 @@ has to obey the following Map Laws:
> 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 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.
join â mid == id == join â map mid
+ > 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.
> The Category Theorist would state these Laws like this, where `M` is the endofunctor that takes us from type `α` to type α:
- >
μ â M(μ) == μ â μ+ >
μ â η = id == μ â M(η)
μ â M(μ) == μ â μ+ + +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 be 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), on the other hand, 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: + +
μ â η == id == μ â M(η)
+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 -> β; k and l have types of the same form
+u : α; v and xs and ys have types of the same form
+
+w : α
+
+
+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 α 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. 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. 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.
+* 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 ()`.
-Here are some interdefinitions: TODO
-Names in Haskell: TODO
## 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.
+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 â¡ \p. p
- mcomp â¡ \f g x.f (g x)
+ mid â¡ \p. p, that is, our familiar combinator I
+ mcomp â¡ \f g x. f (g x), that is, ordinary function composition (â) (aka the B combinator)
Identity is a monad. Here is a demonstration that the laws hold:
@@ -250,7 +345,7 @@ For example:
`j 7` produced `[49, 14]`, which after being fed through `k` gave us `[49, 50, 14, 15]`.
-Contrast that to `m$` (`mapply`, which operates not on two *box-producing functions*, but instead on two *values of a boxed type*, one containing functions to be applied to the values in the other box, via some predefined scheme. Thus:
+Contrast that to `m$` (`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