X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=0d9e33df3425822ec12e1b4b8aab11fa2594475c;hp=e7cccecd03afbd648474de6ddee6bd8a6cc8c16e;hb=a1a38a5cfaede25b7e6299cac3275e1ccfd9b2db;hpb=d0a9dde6d449c9973b29704d7326ef978cba6da6 diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index e7cccecd..0d9e33df 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. - -From an engineering standpoint, we'll build a tree transformer that +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 +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))) . ___|___ | | @@ -50,9 +52,9 @@ Our first task will be to replace each leaf with its double: | Node (l, r) -> Node (tree_map leaf_modifier l, tree_map leaf_modifier r);; -`tree_map` 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: +`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;; @@ -71,61 +73,76 @@ 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 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. -\tree (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11)))) - - \f . - _____|____ - | | - . . - __|___ __|___ - | | | | - f 2 f 3 f 5 . - __|___ - | | - f 7 f 11 + 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 `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, 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 *) +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;; + 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 int_readerize : int -> int reader = fun (a : int) -> fun (modifier : int -> int) -> modifier a;; - int_readerize 2 (fun i -> i + i);; + let asker : int -> int reader = + fun (a : int) -> + fun (modifier : int -> int) -> modifier a;; + asker 2 (fun i -> i + i);; - : int = 4 -But 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? +`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. @@ -139,7 +156,7 @@ But we can do this: 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---and I'll show you how to +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: ------------ @@ -173,24 +190,24 @@ Then we can expect that supplying it to our `int tree reader` will double all th 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;; + # 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 -int_readerize t1`) to a different `int -> int` function---say, the +asker t1`) to a different `int -> int` function---say, the squaring function, `fun i -> i * i`---we get an entirely different result: - # tree_monadize int_readerize t1 square;; + # 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 +For instance, we can use a State monad to count the number of leaves in the tree. type 'a state = int -> 'a * int;; @@ -210,67 +227,123 @@ modification whatsoever, except for replacing the (parametric) type Then we can count the number of leaves in the tree: - # tree_monadize (fun a -> fun s -> (a, s+1)) t1 0;; + # 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) - . - ___|___ - | | - . . - _|__ _|__ - | | | | - 2 3 5 . - _|__ - | | - 7 11 + . + ___|___ + | | + . . + ( _|__ _|__ , 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) -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. +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. -One more revealing example before getting down to business: replacing -`state` everywhere in `tree_monadize` with `list` gives us +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) - # tree_monadize (fun i -> [ [i; square i] ]) t1;; - - : int list tree list = - [Node - (Node (Leaf [2; 4], Leaf [3; 9]), - Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))] +Later, we will talk more about controlling the order in which nodes are visited. -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. 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. +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 ('a, 'r) continuation = ('a -> 'r) -> 'r;; + 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 -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation = + 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. +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 -> fun k -> a :: k a) t1 (fun t -> []);; + # 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 -> 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? +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 +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 @@ -288,37 +361,114 @@ interesting functions for the first argument of `tree_monadize`: - : 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 -> 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 -> 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 +It's not immediately obvious to us how to simulate the List monadization of the tree using this technique. - type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;; +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")))) -The binary tree monad ---------------------- +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. -Of course, by now you may have realized that we have discovered a new -monad, the binary tree monad: + +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) = 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: @@ -328,22 +478,20 @@ 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 `f a`: - -\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5))) +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 @@ -361,9 +509,6 @@ As for the associative law, 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))))) - . ____|____ . . | | @@ -384,7 +529,7 @@ Now when we bind this tree to `g`, we get 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. @@ -393,3 +538,9 @@ 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]]. +