X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=d0897efd40ef34360d051c96a349accffb6fdca5;hp=de8fc5fa6c7c4e6c0c9e4ec3ce3571c076cf9375;hb=12ba49b7826c64a85032e1640db29d4c947347f9;hpb=26319cf2ffc188af7fc324143881d45fd7c322c8 diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index de8fc5fa..d0897efd 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))) @@ -92,6 +92,7 @@ a reader monad---is to have the treemap 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 . _____|____ @@ -107,7 +108,7 @@ 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 +133,7 @@ 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 +165,8 @@ 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 +174,9 @@ 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 +211,7 @@ 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 +221,21 @@ 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 +244,7 @@ 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 +255,7 @@ 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))) @@ -257,25 +263,25 @@ 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 k -> k (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 k -> k [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 k -> 1 + k a) t1 (fun i -> 0);; - : int = 5 We could simulate the tree state example too, but it would require generalizing the type of the continuation monad to - type ('a -> 'b -> 'c) continuation = ('a -> 'b) -> 'c;; + type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;; The binary tree monad --------------------- @@ -284,37 +290,37 @@ 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: - Left identity: bind (unit a) f = bind (Leaf a) f = fa + Left identity: bind (unit a) f = bind (Leaf a) f = f a To check the other two laws, we need to make the following observation: it is easy to prove based on `tree_bind` by a simple 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 `fa`: +except that each leaf `a` has been replaced with `f a`: -\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5))) +\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5))) . . __|__ __|__ | | | | - a1 . fa1 . + a1 . f a1 . _|__ __|__ | | | | - . a5 . fa5 + . a5 . f a5 bind _|__ f = __|__ | | | | - . a4 . fa4 + . a4 . f a4 __|__ __|___ | | | | - a2 a3 fa2 fa3 + a2 a3 f a2 f a3 Given this equivalence, the right identity law @@ -331,26 +337,26 @@ 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))) )) +\tree (. (. (. (. (a1) (a2))))) +\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))))) . ____|____ . . | | bind __|__ f = __|_ = . . | | | | __|__ __|__ - a1 a2 fa1 fa2 | | | | + a1 a2 f a1 f a2 | | | | a1 a1 a1 a1 Now when we bind this tree to `g`, we get - . - ____|____ - | | - . . - __|__ __|__ - | | | | - ga1 ga1 ga1 ga1 + . + _____|______ + | | + . . + __|__ __|__ + | | | | + g a1 g a1 g a1 g a1 At this point, it should be easy to convince yourself that using the recipe on the right hand side of the associative law will