From 26319cf2ffc188af7fc324143881d45fd7c322c8 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Wed, 1 Dec 2010 03:23:12 -0500 Subject: [PATCH] manip trees: tweaks Signed-off-by: Jim Pryor --- manipulating_trees_with_monads.mdwn | 31 ++++++++++++++++++++----------- 1 file changed, 20 insertions(+), 11 deletions(-) diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 20039c28..de8fc5fa 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -28,9 +28,9 @@ internal nodes?] 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)))))) + let t1 = Node (Node (Leaf 2, Leaf 3), + Node (Leaf 5, Node (Leaf 7, + Leaf 11))) . ___|___ | | @@ -47,8 +47,8 @@ Our first task will be to replace each leaf with its double: let rec treemap (newleaf : 'a -> 'b) (t : 'a tree) : 'b tree = match t with | Leaf x -> Leaf (newleaf x) - | Node (l, r) -> Node ((treemap newleaf l), - (treemap newleaf r));; + | Node (l, r) -> Node (treemap newleaf l, + treemap newleaf r);; `treemap` takes a function that transforms old leaves into new leaves, and maps that function over all the leaves in the tree, leaving the @@ -122,7 +122,7 @@ type `int -> int`. It's easy to figure out how to turn an `int` into an `int reader`: - let int2int_reader (x : 'a) : 'b reader = fun (op : 'a -> 'b) -> op x;; + let int2int_reader : 'a -> 'b reader = fun (a : 'a) -> fun (op : 'a -> 'b) -> op a;; int2int_reader 2 (fun i -> i + i);; - : int = 4 @@ -142,7 +142,7 @@ something of type `'a` into an `'b reader`, and I'll show you how to turn an `'a tree` into an `'a tree reader`. In more fanciful terms, the `treemonadizer` function builds plumbing that connects all of the leaves of a tree into one connected monadic network; it threads the -monad through the leaves. +`'b reader` monad through the leaves. # treemonadizer int2int_reader t1 (fun i -> i + i);; - : int tree = @@ -158,18 +158,18 @@ result: - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -Now that we have a tree transformer that accepts a monad as a +Now that we have a tree transformer that accepts a reader monad as a parameter, we can see what it would take to swap in a different monad. For instance, we can use a state monad to count the number of nodes in the tree. type 'a state = int -> 'a * int;; - let state_unit x i = (x, i+.5);; - let state_bind u f i = let (a, i') = u i in f a (i'+.5);; + let state_unit a = fun i -> (a, i);; + let state_bind u f = fun i -> let (a, i') = u i in f a (i' + 1);; Gratifyingly, we can use the `treemonadizer` function without any modification whatsoever, except for replacing the (parametric) type -`reader` with `state`: +`'b reader` with `'b state`, and substituting in the appropriate unit and bind: let rec treemonadizer (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = match t with @@ -198,6 +198,15 @@ Then we can count the number of nodes in the tree: Notice that we've counted each internal node twice---it's a good 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 -- 2.11.0