From: Chris Barker Date: Sun, 28 Nov 2010 03:26:49 +0000 (-0500) Subject: edit X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=db89aab6e40647f64f6f51ed6281c0cbce361550 edit --- diff --git a/zipper-lists-continuations.mdwn b/zipper-lists-continuations.mdwn index a826eed8..fdfc2a55 100644 --- a/zipper-lists-continuations.mdwn +++ b/zipper-lists-continuations.mdwn @@ -303,3 +303,304 @@ 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 +excerice 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;; +