X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek7_introducing_monads.mdwn;h=8b3428c30102ce66c37c2e05dcf7cef50a0f93fe;hp=840521a045b171c9f11dab164eec5ba2e8070c0a;hb=0e24e1b63b8c5a7a1ffbe6ada981d0d24b6d6914;hpb=f932e2ef9505b6b863a7957be94b526c5af9d99f diff --git a/topics/week7_introducing_monads.mdwn b/topics/week7_introducing_monads.mdwn index 840521a0..8b3428c3 100644 --- a/topics/week7_introducing_monads.mdwn +++ b/topics/week7_introducing_monads.mdwn @@ -1,5 +1,4 @@ - The [[tradition in the functional programming @@ -383,7 +382,7 @@ That can be helpful, but it only enables us to have _zero or one_ elements in th | [] -> [] | x' :: xs' -> List.append (k x') (catmap f xs') -Now we can have as many elements in the result for a given `α` 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. +Now we can have as many elements in the result for a given `α` as `k` cares to return. Another way to write `catmap k xs` is as (Haskell) `concat (map k cs)` or (OCaml) `List.flatten (List.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 `α`-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. @@ -392,6 +391,7 @@ This example is a good intuitive basis for thinking about the notions of `mbind` Some a >>=α option (\a -> Some 0) ==> Some 0 None >>=α option (\a -> Some 0) ==> None + Some a >>=α option (\a -> None ) ==> None . / \ @@ -438,6 +438,8 @@ As we mentioned above, the notions of Monads have their origin in Category Theor [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. +[5](http://www.stephendiehl.com/posts/monads.html) + Here are some papers that introduced Monads into functional programming: