<!-- (1, 7) ... explain why not (1, 8) -->
-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.,
-
-
-<pre>
-let t1 = Node ((Node ((Leaf 2), (Leaf 3))),
- (Node ((Leaf 5),(Node ((Leaf 7),
- (Leaf 11))))))
-
- .
- ___|___
- | |
- . .
-_|__ _|__
-| | | |
-2 3 5 .
- _|__
- | |
- 7 11
-</pre>
-
-Our first task will be to replace each leaf with its double:
-
-<pre>
-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));;
-</pre>
-`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:
-
-<pre>
-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
-</pre>
-
-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`:
-
-<pre>
-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)))
-</pre>
-
-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))))
-<pre>
-\f .
- ____|____
- | |
- . .
-__|__ __|__
-| | | |
-f2 f3 f5 .
- __|___
- | |
- f7 f11
-</pre>
-
-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`.
-
-<pre>
-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;;
-</pre>
-
-It's easy to figure out how to turn an `int` into an `int reader`:
-
-<pre>
-let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
-int2int_reader 2 (fun i -> i + i);;
-- : int = 4
-</pre>
-
-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.
-
-<pre>
-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))));;
-</pre>
-
-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.
-
-<pre>
-# treemonadizer int2int_reader t1 (fun i -> i + i);;
-- : int tree =
-Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
-</pre>
-
-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:
-
-<pre>
-# treemonadizer int2int_reader t1 (fun i -> i * i);;
-- : int tree =
-Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
-</pre>
-
-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.
-
-<pre>
-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);;
-</pre>
-
-Gratifyingly, we can use the `treemonadizer` function without any
-modification whatsoever, except for replacing the (parametric) type
-`reader` with `state`:
-
-<pre>
-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))));;
-</pre>
-
-Then we can count the number of nodes in the tree:
-
-<pre>
-# 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
-</pre>
-
-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
-
-<pre>
-# 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])))]
-</pre>
-
-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?
-
-<pre>
-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))));;
-</pre>
-
-We use the continuation monad described above, and insert the
-`continuation` type in the appropriate place in the `treemonadizer` code.
-We then compute:
-
-<pre>
-# treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
-- : int list = [2; 3; 5; 7; 11]
-</pre>
-
-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:
-
-<pre>
-# treemonadizer continuation_unit t1 (fun x -> x);;
-- : int tree =
-Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
-</pre>
-
-That is, nothing happens. But we can begin to substitute more
-interesting functions for the first argument of `treemonadizer`:
-
-<pre>
-(* 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
-</pre>
-
-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:
-
-<pre>
-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));;
-</pre>
-
-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)))
-<pre>
- . .
- __|__ __|__
- | | | |
- a1 . fa1 .
- _|__ __|__
- | | | |
- . a5 . fa5
- bind _|__ f = __|__
- | | | |
- . a4 . fa4
- __|__ __|___
- | | | |
- a2 a3 fa2 fa3
-</pre>
-
-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))) ))
-<pre>
- .
- ____|____
- . . | |
-bind __|__ f = __|_ = . .
- | | | | __|__ __|__
- a1 a2 fa1 fa2 | | | |
- a1 a1 a1 a1
-</pre>
-
-Now when we bind this tree to `g`, we get
-
-<pre>
- .
- ____|____
- | |
- . .
- __|__ __|__
- | | | |
- ga1 ga1 ga1 ga1
-</pre>
-
-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]]##