X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_manipulating_trees_with_monads.mdwn;h=0000000000000000000000000000000000000000;hp=1a4a1dd1bba534ade11e010bd6394611a65771da;hb=f6c0df2dd622cb0cb5552fc1d940cd2a73b44792;hpb=5c25429aa8ed3c65ab1cb67a244a3b21919f079c diff --git a/topics/_manipulating_trees_with_monads.mdwn b/topics/_manipulating_trees_with_monads.mdwn deleted file mode 100644 index 1a4a1dd1..00000000 --- a/topics/_manipulating_trees_with_monads.mdwn +++ /dev/null @@ -1,465 +0,0 @@ -[[!toc]] - -Manipulating trees with monads ------------------------------- - -This topic develops an idea based on a suggestion of Ken Shan's. -We'll build a series of functions that operate on trees, doing various -things, including updating leaves with a Reader monad, counting nodes -with a State monad, copying the tree with a List monad, and converting -a tree into a list of leaves with a Continuation monad. It will turn -out that the continuation monad can simulate the behavior of each of -the other monads. - -From an engineering standpoint, we'll build a tree machine that -deals in monads. We can modify the behavior of the system by swapping -one monad for another. We've already seen how adding a monad can add -a layer of funtionality without disturbing the underlying system, for -instance, in the way that the Reader monad allowed us to add a layer -of intensionality to an extensional grammar. But we have not yet seen -the utility of replacing one monad with other. - -First, we'll be needing a lot of trees for the remainder of the -course. Here again is a type constructor for leaf-labeled, binary trees: - - type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree);; - -[How would you adjust the type constructor to allow for labels on the -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))) - . - ___|___ - | | - . . - _|_ _|__ - | | | | - 2 3 5 . - _|__ - | | - 7 11 - -Our first task will be to replace each leaf with its double: - - 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, `tree_map` has the -behavior of a Reader monad. Let's make that explicit. - -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. - - 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 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`---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, `tree_monadize -asker t1`) to a different `int -> int` function---say, the -squaring function, `fun i -> i * i`---we get an entirely different -result: - - # 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 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 leaves in -the tree. - - 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 `tree_monadize` function without any -modification whatsoever, except for replacing the (parametric) type -`'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 `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))) - ] - - -Unlike the previous cases, instead of turning a tree into a function -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 ('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 -`continuation_unit` as our first argument to `tree_monadize`, and then -apply the result to the identity function: - - # 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 `tree_monadize`: - - (* 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 *) - # 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 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 = 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 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 - -falls out once we realize that - - bind (Leaf a) unit = unit a = Leaf a - -As for the associative law, - - 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 - - . - ____|____ - . . | | - bind __|__ f = __|_ = . . - | | | | __|__ __|__ - a1 a2 f a1 f a2 | | | | - a1 a1 a1 a1 - -Now when we bind this tree to `g`, we get - - . - _____|______ - | | - . . - __|__ __|__ - | | | | - 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 -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. - - -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]]. -