From cc6aefeaf1088b59459dcfcbab5f4a209ae99853 Mon Sep 17 00:00:00 2001 From: Chris Barker Date: Tue, 30 Nov 2010 16:37:26 -0500 Subject: [PATCH] edits --- manipulating_trees_with_monads.mdwn | 7 +- week11.mdwn | 393 ------------------------------------ 2 files changed, 3 insertions(+), 397 deletions(-) diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn index 1be499b0..d4b0bd87 100644 --- a/manipulating_trees_with_monads.mdwn +++ b/manipulating_trees_with_monads.mdwn @@ -1,10 +1,9 @@ [[!toc]] - Manipulating trees with monads ------------------------------ -This thread develops an idea based on a detailed suggestion of Ken +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 @@ -12,11 +11,11 @@ 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 +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.) +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: diff --git a/week11.mdwn b/week11.mdwn index d0ecb704..52f80b67 100644 --- a/week11.mdwn +++ b/week11.mdwn @@ -511,399 +511,6 @@ So now, guess what would be the result of doing the following: -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 -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;; - -The binary tree monad ---------------------- - -Of course, by now you may have realized that we have discovered a new -monad, the binary tree monad: - -
-type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
-let tree_unit (x:'a) = Leaf x;;
-let rec tree_bind (u:'a tree) (f:'a -> 'b tree):'b tree = 
-  match u with Leaf x -> f x 
-             | 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 = fa - -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 `fa`: - -\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5))) -
-                .                         .       
-              __|__                     __|__   
-              |   |                     |   |   
-              a1  .                    fa1  .   
-                 _|__                     __|__ 
-                 |  |                     |   | 
-                 .  a5                    .  fa5
-   bind         _|__       f   =        __|__   
-                |  |                    |   |   
-                .  a4                   .  fa4  
-              __|__                   __|___   
-              |   |                   |    |   
-              a2  a3                 fa2  fa3         
-
- -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 (fa) g) - -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))) )) -
-                                           .
-                                       ____|____
-          .               .            |       |
-bind    __|__   f  =    __|_    =      .       .
-        |   |           |   |        __|__   __|__
-        a1  a2         fa1 fa2       |   |   |   |
-                                     a1  a1  a1  a1  
-
- -Now when we bind this tree to `g`, we get - -
-           .
-       ____|____
-       |       |
-       .       .
-     __|__   __|__
-     |   |   |   |
-    ga1 ga1 ga1 ga1  
-
- -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. - -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. - ##[[List Monad as Continuation Monad]]## ##[[Manipulating Trees with Monads]]## -- 2.11.0