X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=de8fc5fa6c7c4e6c0c9e4ec3ce3571c076cf9375;hp=62c94652cb90dbcda52bc66471d451079043d4be;hb=26319cf2ffc188af7fc324143881d45fd7c322c8;hpb=9b648adf10491e437a56379a2714b55b629a3db6 diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 62c94652..de8fc5fa 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -28,14 +28,14 @@ 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))) . ___|___ | | . . - _|__ _|__ + _|_ _|__ | | | | 2 3 5 . _|__ @@ -44,11 +44,11 @@ We'll be using trees where the nodes are integers, e.g., Our first task will be to replace each leaf with its double: - let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) = + 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 @@ -87,51 +87,50 @@ to each subpart of the computation. In other words, `treemap` has the behavior of a reader monad. Let's make that explicit. In general, we're on a journey of making our treemap function more and -more flexible. So the next step---combining the tree transducer with +more flexible. So the next step---combining the tree transformer with 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 +tree that is ready to accept any `int -> int` function and produce the updated tree. -\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11)))) - \f . - ____|____ - | | - . . - __|__ __|__ - | | | | - f2 f3 f5 . - __|___ - | | - f7 f11 + \f . + _____|____ + | | + . . + __|___ __|___ + | | | | + f 2 f 3 f 5 . + __|___ + | | + 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 returns an `int -tree` in which each leaf `x` has been replaced with `(f x)`. +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`. 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 this situation, it will be enough for now to expect that our reader will expect a function of -type `int->int`. +type `int -> int`. - type 'a reader = (int->int) -> 'a;; (* mnemonic: e for environment *) - let reader_unit (x:'a): 'a reader = fun _ -> x;; - let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c reader = fun e -> f (u e) e;; + type 'a reader = (int -> int) -> 'a;; (* mnemonic: e for environment *) + let reader_unit (a : 'a) : 'a reader = fun _ -> a;; + let reader_bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader = fun e -> f (u e) e;; 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 But what do we do when the integers are scattered over the leaves of a tree? A binary tree is not the kind of thing that we can apply a -function of type `int->int` to. +function of type `int -> int` to. - let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader = + 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')) | Node (l, r) -> reader_bind (treemonadizer f l) (fun x -> @@ -143,7 +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 = @@ -151,7 +150,7 @@ monad through the leaves. Here, our environment is the doubling function (`fun i -> i + i`). If we apply the very same `int tree reader` (namely, `treemonadizer -int2int_reader t1`) to a different `int->int` function---say, the +int2int_reader t1`) to a different `int -> int` function---say, the squaring function, `fun i -> i * i`---we get an entirely different result: @@ -159,20 +158,20 @@ result: - : int tree = Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -Now that we have a tree transducer 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 = + 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 -> @@ -199,6 +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 @@ -219,7 +227,7 @@ of leaves? let continuation_unit x c = c x;; let continuation_bind u f c = u (fun a -> f a c);; - let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation = + 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')) | Node (l, r) -> continuation_bind (treemonadizer f l) (fun x -> @@ -276,8 +284,8 @@ 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 rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree = + let tree_unit (x: 'a) = Leaf x;; + let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = match u with | Leaf x -> f x | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));;