-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, -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 -- -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 -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))) -- -Note that what `treemap` does is take some global, contextual + let rec tree_map (leaf_modifier : 'a -> 'b) (t : 'a tree) : 'b tree = + match t with + | Leaf i -> Leaf (leaf_modifier i) + | Node (l, r) -> Node (tree_map leaf_modifier l, + tree_map leaf_modifier r);; + +`tree_map` takes a tree and 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;; + tree_map 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 `tree_map` +code. However, because we've made what to do to each leaf a +parameter, we can decide to do something else to the leaves without +needing to rewrite `tree_map`. For instance, we can easily square +each leaf instead, by supplying the appropriate `int -> int` operation +in place of `double`: + + let square i = i * i;; + tree_map square t1;; + - : int tree = + Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) + +Note that what `tree_map` does is take some unchanging 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. +to each subpart of the computation. In other words, `tree_map` 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 -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 +In general, we're on a journey of making our `tree_map` function more and +more flexible. So the next step---combining the tree transformer with +a Reader monad---is to have the `tree_map` function return a (monadized) +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 -+ fun e -> . + _____|____ + | | + . . + __|___ __|___ + | | | | + e 2 e 3 e 5 . + __|___ + | | + e 7 e 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)`. - -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 '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 -- -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))));; -+tree`) into a reader monadic object of type `(int -> int) -> int +tree`: something that, when you apply it to an `int -> int` function +`e` returns an `int tree` in which each leaf `i` has been replaced +with `e i`. + +[Application note: this kind of reader object could provide a model +for Kaplan's characters. It turns an ordinary tree into one that +expects contextual information (here, the `e`) that can be +used to compute the content of indexicals embedded arbitrarily deeply +in the tree.] + +With our previous applications of the Reader monad, we always knew +which kind of environment to expect: either an assignment function, as +in the original calculator simulation; a world, as in the +intensionality monad; an individual, as in the Jacobson-inspired link +monad; etc. In the present case, we expect that our "environment" +will be some function of type `int -> int`. "Looking up" some `int` in +the environment will return us the `int` that comes out the other side +of that function. + + type 'a reader = (int -> int) -> 'a;; + 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 would be a simple matter to turn an *integer* into an `int reader`: + + let asker : int -> int reader = + fun (a : int) -> + fun (modifier : int -> int) -> modifier a;; + asker 2 (fun i -> i + i);; + - : int = 4 + +`asker a` is a monadic box that waits for an an environment (here, the argument `modifier`) and returns what that environment maps `a` to. + +How do we do the analagous transformation when our `int`s are scattered over the leaves of a tree? How do we turn an `int tree` into a reader? +A tree is not the kind of thing that we can apply a +function of type `int -> int` to. + +But we can do this: + + let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = + match t with + | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b)) + | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' -> + reader_bind (tree_monadize f r) (fun r' -> + reader_unit (Node (l', r'))));; 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, -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))) -+something of type `'a` into an `'b reader`---this is a function of the same type that you could bind an `'a reader` to, such as `asker` or `reader_unit`---and I'll show you how to +turn an `'a tree` into an `'b tree reader`. That is, if you show me how to do this: + + ------------ + 1 ---> | 1 | + ------------ + +then I'll give you back the ability to do this: + + ____________ + . | . | + __|___ ---> | __|___ | + | | | | | | + 1 2 | 1 2 | + ------------ + +And how will that boxed tree behave? Whatever actions you perform on it will be transmitted down to corresponding operations on its leaves. For instance, our `int reader` expects an `int -> int` environment. If supplying environment `e` to our `int reader` doubles the contained `int`: + + ------------ + 1 ---> | 1 | applied to e ~~> 2 + ------------ + +Then we can expect that supplying it to our `int tree reader` will double all the leaves: + + ____________ + . | . | . + __|___ ---> | __|___ | applied to e ~~> __|___ + | | | | | | | | + 1 2 | 1 2 | 2 4 + ------------ + +In more fanciful terms, the `tree_monadize` function builds plumbing that connects all of the leaves of a tree into one connected monadic network; it threads the +`'b reader` monad through the original tree's leaves. + + # tree_monadize asker t1 double;; + - : 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 -int2int_reader t1`) to a different `int->int` function---say, the +we apply the very same `int tree reader` (namely, `tree_monadize +asker 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))) -+ # tree_monadize asker t1 square;; + - : 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 + +For instance, we can use a State monad to count the number of leaves 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 a = fun s -> (a, s);; + let state_bind u f = fun s -> let (a, s') = u s in f a s';; -Gratifyingly, we can use the `treemonadizer` function without any +Gratifyingly, we can use the `tree_monadize` 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))));; -- -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 -- -Notice that we've counted each internal node twice---it's a good -exercise to adjust the code to count each node once. +`'b reader` with `'b state`, and substituting in the appropriate unit and bind: + + let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = + match t with + | Leaf a -> state_bind (f a) (fun b -> state_unit (Leaf b)) + | Node (l, r) -> state_bind (tree_monadize f l) (fun l' -> + state_bind (tree_monadize f r) (fun r' -> + state_unit (Node (l', r'))));; + +Then we can count the number of leaves in the tree: + + # let incrementer = fun a -> + fun s -> (a, s+1);; + + # tree_monadize incrementer t1 0;; + - : int tree * int = + (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5) + + . + ___|___ + | | + . . + ( _|__ _|__ , 5 ) + | | | | + 2 3 5 . + _|__ + | | + 7 11 + +Note that the value returned is a pair consisting of a tree and an +integer, 5, which represents the count of the leaves in the tree. + +Why does this work? Because the operation `incrementer` +takes an argument `a` and wraps it in an State monadic box that +increments the store and leaves behind a wrapped `a`. When we give that same operations to our +`tree_monadize` function, it then wraps an `int tree` in a box, one +that does the same store-incrementing for each of its leaves. + +We can use the state monad to annotate leaves with a number +corresponding to that leave's ordinal position. When we do so, we +reveal the order in which the monadic tree forces evaluation: + + # tree_monadize (fun a -> fun s -> ((a,s+1), s+1)) t1 0;; + - : int tree * int = + (Node + (Node (Leaf (2, 1), Leaf (3, 2)), + Node + (Leaf (5, 3), + Node (Leaf (7, 4), Leaf (11, 5)))), + 5) + +The key thing to notice is that instead of just wrapping `a` in the +monadic box, we wrap a pair of `a` and the current store. + +Reversing the annotation order requires reversing the order of the `state_bind` +operations. It's not obvious that this will type correctly, so think +it through: + + let rec tree_monadize_rev (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = + match t with + | Leaf a -> state_bind (f a) (fun b -> state_unit (Leaf b)) + | Node (l, r) -> state_bind (tree_monadize f r) (fun r' -> (* R first *) + state_bind (tree_monadize f l) (fun l'-> (* Then L *) + state_unit (Node (l', r'))));; + + # tree_monadize_rev (fun a -> fun s -> ((a,s+1), s+1)) t1 0;; + - : int tree * int = + (Node + (Node (Leaf (2, 5), Leaf (3, 4)), + Node + (Leaf (5, 3), + Node (Leaf (7, 2), Leaf (11, 1)))), + 5) + +Later, we will talk more about controlling the order in which nodes are visited. One more revealing example before getting down to business: replacing -`state` everywhere in `treemonadizer` with `list` gives us +`state` everywhere in `tree_monadize` with `list` lets us do: + + # let decider i = if i = 2 then [20; 21] else [i];; + # tree_monadize decider t1;; + - : int tree List_monad.m = + [ + Node (Node (Leaf 20, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))); + Node (Node (Leaf 21, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) + ] -

-# 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. +from some input to a result, this monadized tree gives us back a list of trees, +one for each choice of `int`s for its leaves. 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))));; -- -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] -- -We have found a way of collapsing a tree into a list of its leaves. - -The continuation monad is amazingly flexible; we can use it to +of leaves? + + type ('r,'a) continuation = ('a -> 'r) -> 'r;; + let continuation_unit a = fun k -> k a;; + let continuation_bind u f = fun k -> u (fun a -> f a k);; + + let rec tree_monadize (f : 'a -> ('r,'b) continuation) (t : 'a tree) : ('r,'b tree) continuation = + match t with + | Leaf a -> continuation_bind (f a) (fun b -> continuation_unit (Leaf b)) + | Node (l, r) -> continuation_bind (tree_monadize f l) (fun l' -> + continuation_bind (tree_monadize f r) (fun r' -> + continuation_unit (Node (l', r'))));; + +We use the Continuation monad described above, and insert the +`continuation` type in the appropriate place in the `tree_monadize` code. Then if we give the `tree_monadize` function an operation that converts `int`s into `'b`-wrapping Continuation monads, it will give us back a way to turn `int tree`s into corresponding `'b tree`-wrapping Continuation monads. + +So for example, we compute: + + # tree_monadize (fun a k -> a :: k ()) t1 (fun _ -> []);; + - : int list = [2; 3; 5; 7; 11] + +We have found a way of collapsing a tree into a list of its +leaves. Can you trace how this is working? Think first about what the +operation `fun a k -> a :: k a` does when you apply it to a +plain `int`, and the continuation `fun _ -> []`. Then given what we've +said about `tree_monadize`, what should we expect `tree_monadize (fun +a -> fun k -> a :: k a)` to do? + +Soon we'll return to the same-fringe problem. Since the +simple but inefficient way to solve it is to map each tree to a list +of its leaves, this transformation is on the path to a more efficient +solution. We'll just have to figure out how to postpone computing the +tail of the list until it's needed... + +The Continuation monad is amazingly flexible; we can use it to simulate some of the computations performed above. To see how, first -note that an interestingly uninteresting thing happens if we use the -continuation unit as our first argument to `treemonadizer`, and then +note that an interestingly uninteresting thing happens if we use +`continuation_unit` as our first argument to `tree_monadize`, 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))) -+ # tree_monadize continuation_unit t1 (fun t -> t);; + - : 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))) +interesting functions for the first argument of `tree_monadize`: -(* 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 tree reader: distributing a operation over the leaves *) + # tree_monadize (fun a -> fun k -> k (square a)) t1 (fun t -> t);; + - : int tree = + Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121))) -(* Counting leaves *) -# treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);; -- : int = 5 -+ (* Counting leaves *) + # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 0);; + - : int = 5 + +It's not immediately obvious to us how to simulate the List monadization of the tree using this technique. + +We could simulate the tree annotating example by setting the relevant +type to `(store -> 'result, 'a) continuation`. + +Andre Filinsky has proposed that the continuation monad is +able to simulate any other monad (Google for "mother of all monads"). + +If you want to see how to parameterize the definition of the `tree_monadize` function, so that you don't have to keep rewriting it for each new monad, see [this code](/code/tree_monadize.ml). + +The idea of using continuations to characterize natural language meaning +------------------------------------------------------------------------ + +We might a philosopher or a linguist be interested in continuations, +especially if efficiency of computation is usually not an issue? +Well, the application of continuations to the same-fringe problem +shows that continuations can manage order of evaluation in a +well-controlled manner. In a series of papers, one of us (Barker) and +Ken Shan have argued that a number of phenomena in natural langauge +semantics are sensitive to the order of evaluation. We can't +reproduce all of the intricate arguments here, but we can give a sense +of how the analyses use continuations to achieve an analysis of +natural language meaning. + +**Quantification and default quantifier scope construal**. + +We saw in the copy-string example ("abSd") and in the same-fringe example that +local properties of a structure (whether a character is `'S'` or not, which +integer occurs at some leaf position) can control global properties of +the computation (whether the preceeding string is copied or not, +whether the computation halts or proceeds). Local control of +surrounding context is a reasonable description of in-situ +quantification. + + (1) John saw everyone yesterday. + +This sentence means (roughly) + + forall x . yesterday(saw x) john + +That is, the quantifier *everyone* contributes a variable in the +direct object position, and a universal quantifier that takes scope +over the whole sentence. If we have a lexical meaning function like +the following: + + let lex (s:string) k = match s with + | "everyone" -> Node (Leaf "forall x", k "x") + | "someone" -> Node (Leaf "exists y", k "y") + | _ -> k s;; + +Then we can crudely approximate quantification as follows: + + # let sentence1 = Node (Leaf "John", + Node (Node (Leaf "saw", + Leaf "everyone"), + Leaf "yesterday"));; + + # tree_monadize lex sentence1 (fun x -> x);; + - : string tree = + Node + (Leaf "forall x", + Node (Leaf "John", Node (Node (Leaf "saw", Leaf "x"), Leaf "yesterday"))) + +In order to see the effects of evaluation order, +observe what happens when we combine two quantifiers in the same +sentence: + + # let sentence2 = Node (Leaf "everyone", Node (Leaf "saw", Leaf "someone"));; + # tree_monadize lex sentence2 (fun x -> x);; + - : string tree = + Node + (Leaf "forall x", + Node (Leaf "exists y", Node (Leaf "x", Node (Leaf "saw", Leaf "y")))) + +The universal takes scope over the existential. If, however, we +replace the usual `tree_monadizer` with `tree_monadizer_rev`, we get +inverse scope: + + # tree_monadize_rev lex sentence2 (fun x -> x);; + - : string tree = + Node + (Leaf "exists y", + Node (Leaf "forall x", Node (Leaf "x", Node (Leaf "saw", Leaf "y")))) + +There are many crucially important details about quantification that +are being simplified here, and the continuation treatment used here is not +scalable for a number of reasons. Nevertheless, it will serve to give +an idea of how continuations can provide insight into the behavior of +quantifiers. -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;; - -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));; -+The Tree monad +============== + +Of course, by now you may have realized that we are working with a new +monad, the binary, leaf-labeled Tree monad. Just as mere lists are in fact a monad, +so are trees. Here is the type constructor, unit, and bind: + + type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);; + let tree_unit (a: 'a) : 'a tree = Leaf a;; + let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree = + match u with + | 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`: - -\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5))) -

- . . - __|__ __|__ - | | | | - a1 . fa1 . - _|__ __|__ - | | | | - . a5 . fa5 - bind _|__ f = __|__ - | | | | - . a4 . fa4 - __|__ __|___ - | | | | - a2 a3 fa2 fa3 -+except that each leaf `a` has been replaced with the tree returned by `f a`: + + . . + __|__ __|__ + | | /\ | + a1 . f a1 . + _|__ __|__ + | | | /\ + . a5 . f a5 + bind _|__ f = __|__ + | | | /\ + . a4 . f a4 + __|__ __|___ + | | /\ /\ + a2 a3 f a2 f a3 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 (f a) 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 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 -built the exact same final tree. +build the exact same final tree. 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. + + +What's this have to do with tree\_monadize? +-------------------------------------------- + +Our different implementations of `tree_monadize` above were different *layerings* of the Tree monad with other monads (Reader, State, List, and Continuation). We'll explore that further here: [[Monad Transformers]].