X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek7_introducing_monads.mdwn;h=0e9ab56f31b6f42592fe0dcd20801f7da6f9b892;hp=d73d4cbb956fab595ad67968815bf78b613a1663;hb=5b304e0cbe18ba3bc9b372e3ca8faf3edaa4e8e5;hpb=caf7c722676ab5c359ca8ab78f999445453158e6
diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn
index d73d4cbb..0e9ab56f 100644
--- a/topics/week7_introducing_monads.mdwn
+++ b/topics/week7_introducing_monads.mdwn
@@ -327,6 +327,10 @@ Identity is a monad. Here is a demonstration that the laws hold:
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:
@@ -358,8 +362,51 @@ Contrast that to `m$` (`mapply`), which operates not on two *box-producing funct
let xs = [7, 5] in
mapply js xs ==> [49, 25, 14, 10]
+These implementations of `<=<` and `m$` for lists use the "crossing" strategy for pairing up multiple lists, as opposed to the "zipping" strategy. Nothing forces that choice; you could also define `m$` using the "zipping" strategy instead. (But then you wouldn't be able to build a corresponding Monad; see below.) Haskell talks of the List Monad in the first case, and the ZipList Applicative in the second case.
+
+Sticking with the "crossing" strategy, here's how to motivate our implementation of `<=<`. Recall that we have on the one hand, an operation `filter` for lists, that applies a predicate to each element of the list, and returns a list containing only those elements which satisfied the predicate. But the elements it does retain, it retains unaltered. On the other hand, we have the operation `map` for lists, that is capable of _changing_ the list elements in the result. But it doesn't have the power to throw list elements away; elements in the source map one-to-one to elements in the result. In many cases, we want something in between `filter` and `map`. We want to be able to ignore or discard some list elements, and possibly also to change the list elements we keep. One way of doing this is to have a function `optmap`, defined like this:
+
+ let rec optmap (f : Î± -> Î² option) (xs : Î± list) : Î² list =
+ match xs with
+ | [] -> []
+ | x' :: xs' ->
+ (match f x' with
+ | None -> optmap f xs'
+ | Some b -> b :: optmap f xs')
+
+Then we retain only those `Î±`s for which `f` returns `Some b`; when `f` returns `None`, we just leave out any corresponding element in the result.
+
+That can be helpful, but it only enables us to have _zero or one_ elements in the result corresponding to a given element in the source list. What if we sometimes want more? Well, here is a more general approach:
+
+ let rec catmap (k : Î± -> Î² list) (xs : Î± list) : Î² list =
+ match xs with
+ | [] -> []
+ | x' :: xs' -> List.append (k x') (catmap f xs')
+
+Now we can have as many elements in the result for a given `Î±` as `k` cares to return. Another way to write `catmap k xs` is as `List.concat (map k xs)`. And this is just the definition of `mbind` or `>>=` for the List Monad. The definition of `mcomp` or `<=<`, that we gave above, differs only in that it's the way to compose two functions `j` and `k`, that you'd want to `catmap`, rather than the way to `catmap` one of those functions over a value that's already a list.
+
+This example is a good intuitive basis for thinking about the notions of `mbind` and `mcomp` more generally. Thus `mbind` for the option/Maybe type takes an option value, applies `k` to its element (if there is one), and returns the resulting option value. `mbind` for a tree with `Î±`-labeled leaves would apply `k` to each of the leaves, and return a tree containing arbitrarily large subtrees in place of all its former leaves, depending on what `k` returned.
+
++[3, 2, 0, 1] >>=_{Î± list} (\a -> dup a a) ==> [3, 3, 3, 2, 2, 1]
+
+ Some a >>=_{Î± option} (\a -> Some 0) ==> Some 0
+ None >>=_{Î± option} (\a -> Some 0) ==> None
+
+ . _____
+ / \ . / \
+ . 3 >>=_{(Î±,unit) tree} (\a -> / \ ) ==> _/_ .
+ / \ a a / \ / \
+1 2 . . 3 3
+ / \ / \
+ 1 1 2 2
+

+
+
+Though as we warned before, only some of the Monads we'll be working with are naturally thought of "containers"; so in other cases the similarity of their `mbind` operations to what we have here will be more abstract.
+
-The question came up in class of when box types might fail to be Mappable, or Mappables might fail to be MapNables, or MapNables might fail to be Monads.
+The question came up in class of **when box types might fail to be Mappable, or Mappables might fail to be MapNables, or MapNables might fail to be Monads**.
For the first failure, we noted that it's easy to define a `map` operation for the box type `R -> Î±`, for a fixed type `R`. You `map` a function of type `P -> Q` over a value of the boxed type __P__

, that is a value of type `R -> P`, by just returning a function that takes some `R` as input, first supplies it to your `R -> P` value, and then supplies the result to your `map`ped function of type `P -> Q`. (We will be working with this Mappable extensively; in fact it's not just a Mappable but more specifically a Monad.)