X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=2ec15d6a6c9a8ee37ec39304a5f4d1ee75befffc;hp=c92065c41bec611a3625d6b58ac2671408646b37;hb=9fe62083953213cce34fc4458e36666902c5ee4b;hpb=f6950034eb1c228badf3364375595032a56e3afb diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index c92065c4..2ec15d6a 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -3,24 +3,26 @@ Manipulating trees with monads ------------------------------ -This topic develops an idea based on a detailed suggestion of Ken -Shan's. We'll build a series of functions that operate on trees, -doing various things, including replacing leaves, counting nodes, and -converting a tree to a list of leaves. The end result will be an -application for continuations. +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, replacing leaves 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 transformer 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 +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) + 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?] @@ -30,7 +32,7 @@ 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))) + Leaf 11))) . ___|___ | | @@ -71,24 +73,25 @@ structure of the tree unchanged. For instance: 14 22 We could have built the doubling operation right into the `tree_map` -code. However, because we've left what to do to each leaf as 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`: +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 =ppp + - : 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 global, contextual +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. +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) +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. @@ -106,16 +109,25 @@ updated tree. 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 `f` returns an `int -tree` in which each leaf `i` has been replaced with `f i`. - -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 the present case, it will be -enough to expect that our "environment" will be some function of type -`int -> int`. +tree`) into a reader monadic 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 `i` has been replaced +with `f 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 `λ f`) 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;; (* mnemonic: e for environment *) let reader_unit (a : 'a) : 'a reader = fun _ -> a;; @@ -135,16 +147,44 @@ But we can do this: let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader = match t with - | Leaf i -> reader_bind (f i) (fun i' -> reader_unit (Leaf i')) - | Node (l, r) -> reader_bind (tree_monadize f l) (fun x -> - reader_bind (tree_monadize f r) (fun y -> - reader_unit (Node (x, y))));; + | 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 `'b tree reader`. 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 +something of type `'a` into an `'b reader`---this is a function of the same type that you could bind an `'a reader` to---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 int_readerize t1 double;; @@ -161,10 +201,10 @@ result: - : 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 +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 +For instance, we can use a State monad to count the number of leaves in the tree. type 'a state = int -> 'a * int;; @@ -177,14 +217,14 @@ modification whatsoever, except for replacing the (parametric) type let rec tree_monadize (f : 'a -> 'b state) (t : 'a tree) : 'b tree state = match t with - | Leaf i -> state_bind (f i) (fun i' -> state_unit (Leaf i')) - | Node (l, r) -> state_bind (tree_monadize f l) (fun x -> - state_bind (tree_monadize f r) (fun y -> - state_unit (Node (x, y))));; + | 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: - # tree_monadize (fun a s -> (a, s+1)) t1 0;; + # tree_monadize (fun a -> fun s -> (a, s+1)) t1 0;; - : int tree * int = (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5) @@ -192,13 +232,52 @@ Then we can count the number of leaves in the tree: ___|___ | | . . - _|__ _|__ + _|__ _|__ , 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 `fun a -> fun s -> (a, s+1)` +takes an `int` and wraps it in an `int state` monadic box that +increments the state. 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 state-incrementing for each of its leaves. + +We can use the state monad to replace 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 -> (s+1, s+1)) t1 0;; + - : int tree * int = + (Node (Node (Leaf 1, Leaf 2), Node (Leaf 3, Node (Leaf 4, Leaf 5))), 5) + +The key thing to notice is that instead of copying `a` into the +monadic box, we throw away the `a` and put a copy of the state in +instead. + +Reversing the 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' -> + state_bind (tree_monadize f l) (fun l' -> + state_unit (Node (l', r'))));; + + # tree_monadize_rev (fun a -> fun s -> (s+1, s+1)) t1 0;; + - : int tree * int = + (Node (Node (Leaf 5, Leaf 4), Node (Leaf 3, Node (Leaf 2, Leaf 1))), 5) + +We will need below to depend on controlling the order in which nodes +are visited when we use the continuation monad to solve the +same-fringe problem. One more revealing example before getting down to business: replacing `state` everywhere in `tree_monadize` with `list` gives us @@ -211,12 +290,13 @@ One more revealing example before getting down to business: replacing 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. - - +a list of `int`'s. We might also have done this with a Reader monad, though then our environments would need to be of type `int -> int list`. Experiment with what happens if you supply the `tree_monadize` based on the List monad an operation like `fun -> [ i; [2*i; 3*i] ]`. Use small trees for your experiment. +[Why is the argument to `tree_monadize` `int -> int list list` instead +of `int -> int list`? Well, as usual, the List monad bind operation +will erase the outer list box, so if we want to replace the leaves +with lists, we have to nest the replacement lists inside a disposable +box.] Now for the main point. What if we wanted to convert a tree to a list of leaves? @@ -227,27 +307,28 @@ of leaves? let rec tree_monadize (f : 'a -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = match t with - | Leaf i -> continuation_bind (f i) (fun i' -> continuation_unit (Leaf i')) - | Node (l, r) -> continuation_bind (tree_monadize f l) (fun x -> - continuation_bind (tree_monadize f r) (fun y -> - continuation_unit (Node (x, y))));; + | 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. -We use the continuation monad described above, and insert the -`continuation` type in the appropriate place in the `tree_monadize` code. -We then compute: +So for example, we compute: - # tree_monadize (fun a k -> a :: (k a)) t1 (fun t -> []);; + # tree_monadize (fun a -> fun k -> a :: k 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. +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 -> fun 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? -The continuation monad is amazingly flexible; we can use it to +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 i -> i);; + # tree_monadize continuation_unit t1 (fun t -> t);; - : int tree = Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))) @@ -255,41 +336,66 @@ 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 k -> k (square a)) t1 (fun i -> i);; + # 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))) (* Simulating the int list tree list *) - # tree_monadize (fun a k -> k [a; square a]) t1 (fun i -> i);; + # tree_monadize (fun a -> fun k -> k [a; square a]) t1 (fun t -> t);; - : int list tree = Node (Node (Leaf [2; 4], Leaf [3; 9]), Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121]))) (* Counting leaves *) - # tree_monadize (fun a k -> 1 + k a) t1 (fun i -> 0);; + # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 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;; 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). +Using continuations to solve the same fringe problem +---------------------------------------------------- -The binary tree monad +We've seen two solutions to the same fringe problem so far. +The simplest is to map each tree to a list of its leaves, then compare +the lists. But if the fringes differ in an early position, we've +wasted our time visiting the rest of the tree. + +The second solution was to use tree zippers and mutable state to +simulate coroutines. We would unzip the first tree until we found the +next leaf, then store the zipper structure in the mutable variable +while we turned our attention to the other tree. Because we stop as +soon as we find the first mismatched leaf, this solution does not have +the flaw just mentioned of the solution that maps both trees to a list +of leaves before beginning comparison. + +Since zippers are just continuations reified, we expect that the +solution in terms of zippers can be reworked using continuations, and +this is indeed the case. To make this work in the most convenient +way, we need to use the fully general type for continuations just mentioned. + +tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);; + + + +The Binary Tree monad --------------------- Of course, by now you may have realized that we have discovered a new -monad, the binary tree monad: +monad, the Binary 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) = Leaf a;; + 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));; + | 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: @@ -364,3 +470,29 @@ 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\_mondadize? +-------------------------------------------- + +So we've defined a Tree monad: + + 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);; + +What's this have to do with the `tree_monadize` functions we defined earlier? + + 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'))));; + +... and so on for different monads? + +The answer is that each of those `tree_monadize` functions is adding a Tree monad *layer* to a pre-existing Reader (and so on) monad. We discuss that further here: [[Monad Transformers]]. +