From: Chris Barker Date: Mon, 6 Dec 2010 12:21:31 +0000 (-0500) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=9fe62083953213cce34fc4458e36666902c5ee4b;hp=0638e3b38097e20ebf645fe7cb1ecec4f17aef7a edits --- diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 000772ad..2ec15d6a 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 @@ -248,6 +248,37 @@ 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' -> + state_bind (tree_monadize f l) (fun 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 @@ -327,6 +358,30 @@ 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 --------------------- diff --git a/monad_transformers.mdwn b/monad_transformers.mdwn index efe35b28..b72c035c 100644 --- a/monad_transformers.mdwn +++ b/monad_transformers.mdwn @@ -10,7 +10,12 @@ So far, we've defined monads as single-layered things. Though in the Groenendijk let bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = fun e -> (fun v -> f v e) (u e);; -We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v e) (u e)`, in order to factor out the parts where any Reader monad is being supplied as an argument to another function. Then if we want instead to add a Reader layer to some arbitrary other monad M, with its own M.unit and M.bind, here's how we do it: +We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v +e) (u e)`, in order to factor out the parts where any Reader monad is +being supplied as an argument to another function, as illustrated in +the `bind` function in the following example. Then if we want instead +to add a Reader layer to some arbitrary other monad M, with its own +M.unit and M.bind, here's how we do it: (* monadic operations for the ReaderT monadic transformer *) @@ -29,7 +34,11 @@ We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v e) (u e)`, let bind (u : ('a, M) readerT) (f : 'a -> ('b, M) readerT) : ('b, M) readerT = fun e -> M.bind (u e) (fun v -> f v e);; -Notice the key differences: where before we just returned `a`, now we instead return `M.unit a`. Where before we just supplied value `u e` of type `'a reader` as an argument to a function, now we instead `M.bind` the `'a reader` to that function. Notice also the differences in the types. +Notice the key differences: where before we just returned `a`, now we +instead return `M.unit a`. Where before we just supplied value `u e` +of type `'a reader` as an argument to a function, now we instead +`M.bind` the `'a reader` to that function. Notice also the differences +in the types. What is the relation between Reader and ReaderT? Well, suppose you started with the Identity monad: