X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=e739c997bada0d8b2c9f23fc07e4821e22574f63;hp=000772ad7df4d17122fa7beb7c36ffed88572cd0;hb=c6912ab54f5f042930b15e63b984480c1b98e6ca;hpb=0638e3b38097e20ebf645fe7cb1ecec4f17aef7a diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 000772ad..e739c997 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -22,7 +22,7 @@ the utility of replacing one monad with other. First, we'll be needing a lot of trees for the remainder of the course. Here again is a type constructor for leaf-labeled, binary trees: - type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree) + type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree);; [How would you adjust the type constructor to allow for labels on the internal nodes?] @@ -81,7 +81,7 @@ in place of `double`: let square i = i * i;; tree_map square t1;; - - : int tree =ppp + - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) Note that what `tree_map` does is take some unchanging contextual @@ -95,8 +95,6 @@ a Reader monad---is to have the `tree_map` function return a (monadized) tree that is ready to accept any `int -> int` function and produce the updated tree. -\tree (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11)))) - \f . _____|____ | | @@ -248,6 +246,37 @@ increments the state. When we give that same operations to our `tree_monadize` function, it then wraps an `int tree` in a box, one that does the same state-incrementing for each of its leaves. +We can use the state monad to replace leaves with a number +corresponding to that leave's ordinal position. When we do so, we +reveal the order in which the monadic tree forces evaluation: + + # tree_monadize (fun a -> fun s -> (s+1, s+1)) t1 0;; + - : int tree * int = + (Node (Node (Leaf 1, Leaf 2), Node (Leaf 3, Node (Leaf 4, Leaf 5))), 5) + +The key thing to notice is that instead of copying `a` into the +monadic box, we throw away the `a` and put a copy of the state in +instead. + +Reversing the order requires reversing the order of the state_bind +operations. It's not obvious that this will type correctly, so think +it through: + + let rec tree_monadize_rev (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = + match t with + | Leaf a -> state_bind (f a) (fun b -> state_unit (Leaf b)) + | Node (l, r) -> state_bind (tree_monadize f r) (fun r' -> (* R first *) + state_bind (tree_monadize f l) (fun l'-> (* Then L *) + state_unit (Node (l', r'))));; + + # tree_monadize_rev (fun a -> fun s -> (s+1, s+1)) t1 0;; + - : int tree * int = + (Node (Node (Leaf 5, Leaf 4), Node (Leaf 3, Node (Leaf 2, Leaf 1))), 5) + +We will need below to depend on controlling the order in which nodes +are visited when we use the continuation monad to solve the +same-fringe problem. + One more revealing example before getting down to business: replacing `state` everywhere in `tree_monadize` with `list` gives us @@ -259,7 +288,7 @@ One more revealing example before getting down to business: replacing Unlike the previous cases, instead of turning a tree into a function from some input to a result, this transformer replaces each `int` with -a list of `int`'s. We might also have done this with a Reader monad, though then our environments would need to be of type `int -> int list`. Experiment with what happens if you supply the `tree_monadize` based on the List monad an operation like `fun -> [ i; [2*i; 3*i] ]`. Use small trees for your experiment. +a list of `int`'s. We might also have done this with a Reader monad, though then our environments would need to be of type `int -> int list`. Experiment with what happens if you supply the `tree_monadize` based on the List monad an operation like `fun i -> [2*i; 3*i]`. Use small trees for your experiment. [Why is the argument to `tree_monadize` `int -> int list list` instead of `int -> int list`? Well, as usual, the List monad bind operation @@ -327,6 +356,30 @@ generalizing the type of the Continuation monad to If you want to see how to parameterize the definition of the `tree_monadize` function, so that you don't have to keep rewriting it for each new monad, see [this code](/code/tree_monadize.ml). +Using continuations to solve the same fringe problem +---------------------------------------------------- + +We've seen two solutions to the same fringe problem so far. +The simplest is to map each tree to a list of its leaves, then compare +the lists. But if the fringes differ in an early position, we've +wasted our time visiting the rest of the tree. + +The second solution was to use tree zippers and mutable state to +simulate coroutines. We would unzip the first tree until we found the +next leaf, then store the zipper structure in the mutable variable +while we turned our attention to the other tree. Because we stop as +soon as we find the first mismatched leaf, this solution does not have +the flaw just mentioned of the solution that maps both trees to a list +of leaves before beginning comparison. + +Since zippers are just continuations reified, we expect that the +solution in terms of zippers can be reworked using continuations, and +this is indeed the case. To make this work in the most convenient +way, we need to use the fully general type for continuations just mentioned. + +tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);; + + The Binary Tree monad --------------------- @@ -352,8 +405,6 @@ 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 `f a`: -\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5))) - . . __|__ __|__ | | | | @@ -383,9 +434,6 @@ As for the associative law, we'll give an example that will show how an inductive proof would proceed. Let `f a = Node (Leaf a, Leaf a)`. Then -\tree (. (. (. (. (a1) (a2))))) -\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))))) - . ____|____ . . | |