From: Jim Pryor Date: Wed, 1 Dec 2010 08:37:18 +0000 (-0500) Subject: manip trees: tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=bdb385fa72c726c61102aa4b0ef2914025c371ee;ds=sidebyside manip trees: tweaks Signed-off-by: Jim Pryor --- diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index de8fc5fa..379ead54 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -46,7 +46,7 @@ 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) + | Leaf i -> Leaf (newleaf i) | Node (l, r) -> Node (treemap newleaf l, treemap newleaf r);; @@ -76,7 +76,7 @@ decide to do something else to the leaves without needing to rewrite `treemap`. For instance, we can easily square each leaf instead by supplying the appropriate `int -> int` operation in place of `double`: - let square x = x * x;; + let square i = i * i;; treemap square t1;; - : int tree =ppp Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) @@ -107,7 +107,7 @@ updated tree. 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 `x` has been replaced with `f x`. +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 @@ -132,7 +132,7 @@ function of type `int -> int` to. let rec treemonadizer (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = match t with - | Leaf x -> reader_bind (f x) (fun x' -> reader_unit (Leaf x')) + | Leaf i -> reader_bind (f i) (fun i' -> reader_unit (Leaf i')) | Node (l, r) -> reader_bind (treemonadizer f l) (fun x -> reader_bind (treemonadizer f r) (fun y -> reader_unit (Node (x, y))));; @@ -164,8 +164,8 @@ 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 a = fun i -> (a, i);; - let state_bind u f = fun i -> let (a, i') = u i in f a (i' + 1);; + let state_unit a = fun s -> (a, s);; + let state_bind_and_count u f = fun s -> let (a, s') = u s in f a (s' + 1);; Gratifyingly, we can use the `treemonadizer` function without any modification whatsoever, except for replacing the (parametric) type @@ -173,9 +173,9 @@ modification whatsoever, except for replacing the (parametric) type let rec treemonadizer (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = match t with - | Leaf x -> state_bind (f x) (fun x' -> state_unit (Leaf x')) - | Node (l, r) -> state_bind (treemonadizer f l) (fun x -> - state_bind (treemonadizer f r) (fun y -> + | Leaf i -> state_bind_and_count (f i) (fun i' -> state_unit (Leaf i')) + | Node (l, r) -> state_bind_and_count (treemonadizer f l) (fun x -> + state_bind_and_count (treemonadizer f r) (fun y -> state_unit (Node (x, y))));; Then we can count the number of nodes in the tree: @@ -210,7 +210,7 @@ But I assume Chris means here, adjust the code so that no corrections of this so 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 i -> [ [i; square i] ]) t1;; - : int list tree list = [Node (Node (Leaf [2; 4], Leaf [3; 9]), @@ -220,16 +220,21 @@ 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. + + + Now for the main point. What if we wanted to convert a tree to a list of leaves? type ('a, 'r) continuation = ('a -> 'r) -> 'r;; - let continuation_unit x c = c x;; - let continuation_bind u f c = u (fun a -> f a c);; + let continuation_unit a = fun k -> k a;; + let continuation_bind u f = fun k -> u (fun a -> f a k);; let rec treemonadizer (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = match t with - | Leaf x -> continuation_bind (f x) (fun x' -> continuation_unit (Leaf x')) + | Leaf i -> continuation_bind (f i) (fun i' -> continuation_unit (Leaf i')) | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x -> continuation_bind (treemonadizer f r) (fun y -> continuation_unit (Node (x, y))));; @@ -238,7 +243,7 @@ We use the continuation monad described above, and insert the `continuation` type in the appropriate place in the `treemonadizer` code. We then compute: - # treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);; + # treemonadizer (fun a k -> a :: (k a)) t1 (fun t -> []);; - : int list = [2; 3; 5; 7; 11] We have found a way of collapsing a tree into a list of its leaves. @@ -249,7 +254,7 @@ note that an interestingly uninteresting thing happens if we use the continuation unit as our first argument to `treemonadizer`, and then apply the result to the identity function: - # treemonadizer continuation_unit t1 (fun x -> x);; + # treemonadizer continuation_unit t1 (fun i -> i);; - : int tree = Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) @@ -257,19 +262,19 @@ That is, nothing happens. But we can begin to substitute more interesting functions for the first argument of `treemonadizer`: (* Simulating the tree reader: distributing a operation over the leaves *) - # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);; + # treemonadizer (fun a c -> c (square a)) t1 (fun i -> i);; - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) (* Simulating the int list tree list *) - # treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);; + # treemonadizer (fun a c -> c [a; square a]) t1 (fun i -> i);; - : int list tree = Node (Node (Leaf [2; 4], Leaf [3; 9]), Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121]))) (* Counting leaves *) - # treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);; + # treemonadizer (fun a c -> 1 + c a) t1 (fun i -> 0);; - : int = 5 We could simulate the tree state example too, but it would require @@ -284,10 +289,10 @@ Of course, by now you may have realized that we have discovered a new monad, the binary tree monad: type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; - let tree_unit (x: 'a) = Leaf x;; + let tree_unit (a: 'a) = Leaf a;; let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = match u with - | Leaf x -> f x + | Leaf a -> f a | 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: