From: Chris Barker Date: Sat, 4 Dec 2010 16:35:38 +0000 (-0500) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=0c843ed2f741748d662a8eaea08c5e95061486fe;hp=-c edits --- 0c843ed2f741748d662a8eaea08c5e95061486fe diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index ba990d10..52b8508c 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -3,11 +3,13 @@ Manipulating trees with monads ------------------------------ -This topic develops an idea based on a detailed suggestion of Ken -Shan's. We'll build a series of functions that operate on trees, -doing various things, including replacing leaves, counting nodes, and -converting a tree to a list of leaves. The end result will be an -application for continuations. +This topic develops an idea based on a suggestion of Ken Shan's. +We'll build a series of functions that operate on trees, doing various +things, including updating leaves with a Reader monad, counting nodes +with a State monad, replacing leaves with a List monad, and converting +a tree into a list of leaves with a Continuation monad. It will turn +out that the continuation monad can simulate the behavior of each of +the other monads. From an engineering standpoint, we'll build a tree transformer that deals in monads. We can modify the behavior of the system by swapping @@ -30,7 +32,7 @@ We'll be using trees where the nodes are integers, e.g., let t1 = Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, - Leaf 11))) + Leaf 11))) . ___|___ | | @@ -71,10 +73,11 @@ structure of the tree unchanged. For instance: 14 22 We could have built the doubling operation right into the `tree_map` -code. However, because we've left what to do to each leaf as a parameter, we can -decide to do something else to the leaves without needing to rewrite -`tree_map`. For instance, we can easily square each leaf instead by -supplying the appropriate `int -> int` operation in place of `double`: +code. However, because we've made what to do to each leaf a +parameter, we can decide to do something else to the leaves without +needing to rewrite `tree_map`. For instance, we can easily square +each leaf instead by supplying the appropriate `int -> int` operation +in place of `double`: let square i = i * i;; tree_map square t1;; @@ -106,14 +109,25 @@ updated tree. f 7 f 11 That is, we want to transform the ordinary tree `t1` (of type `int -tree`) into a reader object of type `(int -> int) -> int tree`: something -that, when you apply it to an `int -> int` function `f` returns an `int -tree` in which each leaf `i` has been replaced with `f i`. - -With previous readers, we always knew which kind of environment to -expect: either an assignment function (the original calculator -simulation), a world (the intensionality monad), an integer (the -Jacobson-inspired link monad), etc. In the present case, we expect that our "environment" will be some function of type `int -> int`. "Looking up" some `int` in the environment will return us the `int` that comes out the other side of that function. +tree`) into a reader monadic object of type `(int -> int) -> int +tree`: something that, when you apply it to an `int -> int` function +`f` returns an `int tree` in which each leaf `i` has been replaced +with `f i`. + +[Application note: this kind of reader object could provide a model +for Kaplan's characters. It turns an ordinary tree into one that +expects contextual information (here, the `λ f`) that can be +used to compute the content of indexicals embedded arbitrarily deeply +in the tree.] + +With our previous applications of the Reader monad, we always knew +which kind of environment to expect: either an assignment function, as +in the original calculator simulation; a world, as in the +intensionality monad; an individual, as in the Jacobson-inspired link +monad; etc. In the present case, we expect that our "environment" +will be some function of type `int -> int`. "Looking up" some `int` in +the environment will return us the `int` that comes out the other side +of that function. type 'a reader = (int -> int) -> 'a;; (* mnemonic: e for environment *) let reader_unit (a : 'a) : 'a reader = fun _ -> a;; @@ -218,14 +232,21 @@ Then we can count the number of leaves in the tree: ___|___ | | . . - _|__ _|__ + _|__ _|__ , 5 | | | | 2 3 5 . _|__ | | 7 11 -Why does this work? Because the operation `fun a -> fun s -> (a, s+1)` takes an `int` and wraps it in an `int state` monadic box that 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. +Note that the value returned is a pair consisting of a tree and an +integer, 5, which represents the count of the leaves in the tree. + +Why does this work? Because the operation `fun a -> fun s -> (a, s+1)` +takes an `int` and wraps it in an `int state` monadic box that +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. One more revealing example before getting down to business: replacing `state` everywhere in `tree_monadize` with `list` gives us @@ -240,11 +261,11 @@ 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. - - - +[Why is the argument to tree_monadize `int -> int list list` instead +of `int -> int list`? Well, as usual, the List monad bind operation +will erase the outer list box, so if we want to replace the leaves +with lists, we have to nest the replacement lists inside a disposable +box.] Now for the main point. What if we wanted to convert a tree to a list of leaves? @@ -311,7 +332,8 @@ The Binary Tree monad --------------------- Of course, by now you may have realized that we have discovered a new -monad, the Binary Tree monad: +monad, the Binary Tree monad. Just as mere lists are in fact a monad, +so are trees. Here is the type constructor, unit, and bind: type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; let tree_unit (a: 'a) : 'a tree = Leaf a;;