X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=inline;f=zipper-lists-continuations.mdwn;h=8858a8930699f29335825e27600ff8c2f6bbb253;hb=3f6acf2fe4a06afd2da601e360c86c906e1b7182;hp=2e84616ae05081cb6e7b9ccdced8487f641ef3e5;hpb=47bc0c16fec3c91dc1635ac6f42fe457e58fc41c;p=lambda.git diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index 2e84616a..8858a893 100644 --- a/zipper-lists-continuations.mdwn +++ b/zipper-lists-continuations.mdwn @@ -73,7 +73,7 @@ The **State Monad** is similar. Once we've decided to use the following type co Then our unit is naturally: - let s_unit (x : 'a) : ('a state) = fun (s : store) -> (x, s) + let s_unit (a : 'a) : ('a state) = fun (s : store) -> (a, s) And we can reason our way to the bind function in a way similar to the reasoning given above. First, we need to apply `f` to the contents of the `u` box: @@ -100,7 +100,7 @@ Our other familiar monad is the **List Monad**, which we were told looks like this: type 'a list = ['a];; - l_unit (x : 'a) = [x];; + l_unit (a : 'a) = [a];; l_bind u f = List.concat (List.map f u);; Recall that `List.map` take a function and a list and returns the @@ -167,7 +167,7 @@ general than an ordinary OCaml list, but we'll see how to map them into OCaml lists soon. We don't need to fully grasp the role of the `'b`'s in order to proceed to build a monad: - l'_unit (x : 'a) : ('a, 'b) list = fun x -> fun f z -> f x z + l'_unit (a : 'a) : ('a, 'b) list = fun a -> fun f z -> f a z No problem. Arriving at bind is a little more complicated, but exactly the same principles apply, you just have to be careful and @@ -299,7 +299,7 @@ generalized quantifier `fun pred -> pred j` of type `(e -> t) -> t`. Let's write a general function that will map individuals into their corresponding generalized quantifier: - gqize (x : e) = fun (p : e -> t) -> p x + gqize (a : e) = fun (p : e -> t) -> p a This function wraps up an individual in a fancy box. That is to say, we are in the presence of a monad. The type constructor, the unit and @@ -308,16 +308,16 @@ belabor the construction of the bind function, the derivation is similar to the List monad just given: type 'a continuation = ('a -> 'b) -> 'b - c_unit (x : 'a) = fun (p : 'a -> 'b) -> p x + c_unit (a : 'a) = fun (p : 'a -> 'b) -> p a c_bind (u : ('a -> 'b) -> 'b) (f : 'a -> ('c -> 'd) -> 'd) : ('c -> 'd) -> 'd = - fun (k : 'a -> 'b) -> u (fun (x : 'a) -> f x k) + fun (k : 'a -> 'b) -> u (fun (a : 'a) -> f a k) How similar is it to the List monad? Let's examine the type constructor and the terms from the list monad derived above: type ('a, 'b) list' = ('a -> 'b -> 'b) -> 'b -> 'b - l'_unit x = fun f -> f x - l'_bind u f = fun k -> u (fun x -> f x k) + l'_unit a = fun f -> f a + l'_bind u f = fun k -> u (fun a -> f a k) (We performed a sneaky but valid eta reduction in the unit term.) @@ -341,3 +341,304 @@ side, and non-determinism on the list monad side. Refunctionalizing zippers ------------------------- +Manipulating trees with monads +------------------------------ + +This thread 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 +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 during the remainder of the +course. Here's a type constructor for binary trees: + + type 'a tree = Leaf of 'a | Node of ('a tree * 'a tree) + +These are trees in which the internal nodes do not have labels. [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 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 +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. + +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 +updated tree. + +\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11)))) +
+\f . + ____|____ + | | + . . +__|__ __|__ +| | | | +f2 f3 f5 . + __|___ + | | + f7 f11 ++ +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))));; ++ +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))) ++ +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 +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))) ++ +Now that we have a tree transducer that accepts a 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 +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);; ++ +Gratifyingly, we can use the `treemonadizer` 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. + +One more revealing example before getting down to business: replacing +`state` everywhere in `treemonadizer` with `list` gives us + +
+# 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. + +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 +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 +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))) ++ +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))) + +(* 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]))) + +(* Counting leaves *) +# treemonadizer (fun a c -> 1 + c a) t1 (fun x -> 0);; +- : int = 5 ++ +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;; +