-let t1 = Node ((Node ((Leaf 2), (Leaf 3))), - (Node ((Leaf 5),(Node ((Leaf 7), - (Leaf 11)))))) - - . - ___|___ - | | - . . -_|__ _|__ -| | | | -2 3 5 . - _|__ - | | - 7 11 -+ let t1 = Node ((Node ((Leaf 2), (Leaf 3))), + (Node ((Leaf 5),(Node ((Leaf 7), + (Leaf 11)))))) + . + ___|___ + | | + . . + _|__ _|__ + | | | | + 2 3 5 . + _|__ + | | + 7 11 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));; --`treemap` takes a function that transforms old leaves into new leaves, + 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));; + +`treemap` takes a function that transforms old leaves into new leaves, and maps that function over all the leaves in the tree, leaving the structure of the tree unchanged. For instance: -

-let double i = i + i;; -treemap double t1;; -- : int tree = -Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) - - . - ___|____ - | | - . . -_|__ __|__ -| | | | -4 6 10 . - __|___ - | | - 14 22 -+ let double i = i + i;; + treemap double t1;; + - : int tree = + Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) + + . + ___|____ + | | + . . + _|__ __|__ + | | | | + 4 6 10 . + __|___ + | | + 14 22 We could have built the doubling operation right into the `treemap` code. However, because what to do to each leaf is a parameter, we can @@ -82,17 +76,15 @@ 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;; -treemap square t1;; -- : int tree =ppp -Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -+ let square x = x * x;; + treemap square t1;; + - : int tree =ppp + Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) Note that what `treemap` does is take some global, contextual information---what to do to each leaf---and supplies that information to each subpart of the computation. In other words, `treemap` has the -behavior of a reader monad. Let's make that explicit. +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 @@ -101,21 +93,20 @@ 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 . + ____|____ + | | + . . + __|__ __|__ + | | | | + f2 f3 f5 . + __|___ + | | + f7 f11 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 +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)`. @@ -126,44 +117,37 @@ 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 '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 (x:'a): 'a reader = fun _ -> x;; + let reader_bind (u: 'a reader) (f:'a -> 'c reader):'c 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;; -int2int_reader 2 (fun i -> i + i);; -- : int = 4 -+ let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;; + 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. -

-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 -> - reader_bind (treemonadizer f r) (fun y -> - reader_unit (Node (x, y))));; -+ 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 -> + reader_bind (treemonadizer f r) (fun y -> + reader_unit (Node (x, y))));; This function says: give me a function `f` that knows how to turn -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, +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. -

-# treemonadizer int2int_reader t1 (fun i -> i + i);; -- : int tree = -Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) -+ # treemonadizer int2int_reader t1 (fun i -> i + i);; + - : int tree = + Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22))) Here, our environment is the doubling function (`fun i -> i + i`). If we apply the very same `int tree reader` (namely, `treemonadizer @@ -171,53 +155,46 @@ int2int_reader t1`) to a different `int->int` function---say, the squaring function, `fun i -> i * i`---we get an entirely different result: -

-# treemonadizer int2int_reader t1 (fun i -> i * i);; -- : int tree = -Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -+ # treemonadizer int2int_reader t1 (fun i -> i * i);; + - : 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 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);; -+ 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);; Gratifyingly, we can use the `treemonadizer` function without any modification whatsoever, except for replacing the (parametric) type `reader` with `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 -> - state_bind (treemonadizer f r) (fun y -> - state_unit (Node (x, y))));; -+ 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 -> + state_unit (Node (x, y))));; Then we can count the number of nodes in the tree: -

-# treemonadizer state_unit t1 0;; -- : int tree * int = -(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13) - - . - ___|___ - | | - . . -_|__ _|__ -| | | | -2 3 5 . - _|__ - | | - 7 11 -+ # treemonadizer state_unit t1 0;; + - : int tree * int = + (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13) + + . + ___|___ + | | + . . + _|__ _|__ + | | | | + 2 3 5 . + _|__ + | | + 7 11 Notice that we've counted each internal node twice---it's a good exercise to adjust the code to count each node once. @@ -225,41 +202,36 @@ 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 -

-# treemonadizer (fun x -> [ [x; square x] ]) t1;; -- : int list tree list = -[Node - (Node (Leaf [2; 4], Leaf [3; 9]), - Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))] -+ # treemonadizer (fun x -> [ [x; square x] ]) t1;; + - : int list tree list = + [Node + (Node (Leaf [2; 4], Leaf [3; 9]), + Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))] 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 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 -> - continuation_bind (treemonadizer f r) (fun y -> - continuation_unit (Node (x, y))));; -+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 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 -> + continuation_bind (treemonadizer f r) (fun y -> + continuation_unit (Node (x, y))));; 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 -> []);; -- : int list = [2; 3; 5; 7; 11] -+ # treemonadizer (fun a c -> a :: (c 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. @@ -269,37 +241,33 @@ 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);; -- : int tree = -Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) -+ # treemonadizer continuation_unit t1 (fun x -> x);; + - : int tree = + Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) 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);; -- : int tree = -Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) + (* Simulating the tree reader: distributing a operation over the leaves *) + # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);; + - : 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);; -- : int list tree = -Node - (Node (Leaf [2; 4], Leaf [3; 9]), - Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121]))) + (* Simulating the int list tree list *) + # treemonadizer (fun a c -> c [a; square a]) t1 (fun x -> x);; + - : 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);; -- : int = 5 -+ (* Counting leaves *) + # treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);; + - : int = 5 We could simulate the tree state example too, but it would require -generalizing the type of the continuation monad to +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 --------------------- @@ -307,13 +275,12 @@ The binary tree monad 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 = - match u with Leaf x -> f x - | Node (l, r) -> Node ((tree_bind l f), (tree_bind r f));; -+ 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 = + match u with + | Leaf x -> f x + | 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: @@ -326,60 +293,56 @@ resulting from `bind u f` is a tree with the same strucure as `u`, except that each leaf `a` has been replaced with `fa`: \tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5))) -

- . . - __|__ __|__ - | | | | - a1 . fa1 . - _|__ __|__ - | | | | - . a5 . fa5 - bind _|__ f = __|__ - | | | | - . a4 . fa4 - __|__ __|___ - | | | | - a2 a3 fa2 fa3 -+ + . . + __|__ __|__ + | | | | + a1 . fa1 . + _|__ __|__ + | | | | + . a5 . fa5 + bind _|__ f = __|__ + | | | | + . a4 . fa4 + __|__ __|___ + | | | | + a2 a3 fa2 fa3 Given this equivalence, the right identity law - Right identity: bind u unit = u + Right identity: bind u unit = u falls out once we realize that - bind (Leaf a) unit = unit a = Leaf a + bind (Leaf a) unit = unit a = Leaf a As for the associative law, - Associativity: bind (bind u f) g = bind u (\a. bind (fa) g) + Associativity: bind (bind u f) g = bind u (\a. bind (fa) g) 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))) )) -

- . - ____|____ - . . | | -bind __|__ f = __|_ = . . - | | | | __|__ __|__ - a1 a2 fa1 fa2 | | | | - a1 a1 a1 a1 -+ + . + ____|____ + . . | | + bind __|__ f = __|_ = . . + | | | | __|__ __|__ + a1 a2 fa1 fa2 | | | | + a1 a1 a1 a1 Now when we bind this tree to `g`, we get -

- . - ____|____ - | | - . . - __|__ __|__ - | | | | - ga1 ga1 ga1 ga1 -+ . + ____|____ + | | + . . + __|__ __|__ + | | | | + ga1 ga1 ga1 ga1 At this point, it should be easy to convince yourself that using the recipe on the right hand side of the associative law will @@ -390,6 +353,5 @@ So binary trees are a monad. Haskell combines this monad with the Option monad to provide a monad called a [SearchTree](http://hackage.haskell.org/packages/archive/tree-monad/0.2.1/doc/html/src/Control-Monad-SearchTree.html#SearchTree) -that is intended to -represent non-deterministic computations as a tree. +that is intended to represent non-deterministic computations as a tree.