X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2Fweek14_manipulating_trees_with_monads.mdwn;h=1a4a1dd1bba534ade11e010bd6394611a65771da;hp=0000000000000000000000000000000000000000;hb=f6c0df2dd622cb0cb5552fc1d940cd2a73b44792;hpb=5c25429aa8ed3c65ab1cb67a244a3b21919f079c diff --git a/topics/week14_manipulating_trees_with_monads.mdwn b/topics/week14_manipulating_trees_with_monads.mdwn new file mode 100644 index 00000000..1a4a1dd1 --- /dev/null +++ b/topics/week14_manipulating_trees_with_monads.mdwn @@ -0,0 +1,465 @@ +[[!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]]. +