X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=zipper-lists-continuations.mdwn;h=d16a24e803351fc7af2dd3054ec48c07edc10402;hp=4aef298a75b11286c14b73b48c081df76f993da9;hb=fd01533abbf1c7e00f28285431e925298e728f42;hpb=fc2cec698d708b9baab616cbedfbd4481c4057cd diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index 4aef298a..d16a24e8 100644 --- a/zipper-lists-continuations.mdwn +++ b/zipper-lists-continuations.mdwn @@ -561,13 +561,13 @@ _|__ _|__ Notice that we've counted each internal node twice---it's a good -excerice to adjust the code to count each node once. +exercise to adjust the code to count each node once. One more revealing example before getting down to business: replacing `state` everywhere in `treemonadizer` with `list` gives us
-# treemonadizer (fun x -> [[x; square x]]) t1;; +# treemonadizer (fun x -> [ [x; square x] ]) t1;; - : int list tree list = [Node (Node (Leaf [2; 4], Leaf [3; 9]), @@ -642,3 +642,94 @@ generalizing the type of the continuation monad to type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;; +The tree monad +-------------- + +Of course, by now you may have realized that we have discovered a new +monad, the tree monad: + ++type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; +let tree_unit (x:'a) = Leaf x;; +let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree = + match u with Leaf x -> f x + | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));; ++ +For once, let's check the Monad laws. The left identity law is easy: + + Left identity: bind (unit a) f = bind (Leaf a) f = fa + +To check the other two laws, we need to make the following +observation: it is easy to prove based on tree_bind by a simple +induction on the structure of the first argument that the tree +resulting from `bind u f` is a tree with the same strucure as `u`, +except that each leaf `a` has been replaced with `fa`: + +\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5))) ++ . . + __|__ __|__ + | | | | + a1 . fa1 . + _|__ __|__ + | | | | + . a5 . fa5 + bind _|__ f = __|__ + | | | | + . a4 . fa4 + __|__ __|___ + | | | | + a2 a3 fa2 fa3 ++ +Given this equivalence, the right identity law + + Right identity: bind u unit = u + +falls out once we realize that + + bind (Leaf a) unit = unit a = Leaf a + +As for the associative law, + + Associativity: bind (bind u f) g = bind u (\a. bind (fa) g) + +we'll give an example that will show how an inductive proof would +have to proceed. Let `f a = Node (Leaf a, Leaf a)`. Then + +\tree (. (. (. (. (a1)(a2))))) +\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))) )) ++ . + ____|____ + . . | | +bind __|__ f = __|_ = . . + | | | | __|__ __|__ + a1 a2 fa1 fa2 | | | | + a1 a1 a1 a1 ++ +Now when we bind this tree to `g`, we get + ++ . + ____|____ + | | + . . + __|__ __|__ + | | | | + ga1 ga1 ga1 ga1 ++ +At this point, it should be easy to convince yourself that +using the recipe on the right hand side of the associative law will +built the exact same final tree. + +So binary trees are a monad. + +Haskell combines this monad with the Option monad to provide a monad +called a +[SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) +that is intended to +represent non-deterministic computations as a tree.