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.
+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, replacing leaves 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 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
+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?]
let t1 = Node (Node (Leaf 2, Leaf 3),
Node (Leaf 5, Node (Leaf 7,
- Leaf 11)))
+ Leaf 11)))
.
___|___
| |
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.
f 7 f 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.
+tree`) into a reader monadic 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`.
+
+[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 `λ f`) 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;; (* mnemonic: e for environment *)
let reader_unit (a : 'a) : 'a reader = fun _ -> a;;
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;;
___|___
| |
. .
- _|__ _|__
+ _|__ _|__ , 5
| | | |
2 3 5 .
_|__
| |
7 11
-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.
+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 `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.
+
+We can use the state monad to replace 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 -> (s+1, s+1)) t1 0;;
+ - : int tree * int =
+ (Node (Node (Leaf 1, Leaf 2), Node (Leaf 3, Node (Leaf 4, Leaf 5))), 5)
+
+The key thing to notice is that instead of copying `a` into the
+monadic box, we throw away the `a` and put a copy of the state in
+instead.
+
+Reversing the 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' ->
+ state_bind (tree_monadize f l) (fun l' ->
+ state_unit (Node (l', r'))));;
+
+ # tree_monadize_rev (fun a -> fun s -> (s+1, s+1)) t1 0;;
+ - : int tree * int =
+ (Node (Node (Leaf 5, Leaf 4), Node (Leaf 3, Node (Leaf 2, Leaf 1))), 5)
+
+We will need below to depend on controlling the order in which nodes
+are visited when we use the continuation monad to solve the
+same-fringe problem.
One more revealing example before getting down to business: replacing
`state` everywhere in `tree_monadize` with `list` gives us
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.
-
-
-<!--
-FIXME: We don't make it clear why the fun has to be int -> int list list, instead of int -> int list
--->
+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.
+[Why is the argument to `tree_monadize` `int -> int list list` instead
+of `int -> int list`? Well, as usual, the List monad bind operation
+will erase the outer list box, so if we want to replace the leaves
+with lists, we have to nest the replacement lists inside a disposable
+box.]
Now for the main point. What if we wanted to convert a tree to a list
of leaves?
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:
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?
-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
- : int = 5
We could simulate the tree state example too, but it would require
-generalizing the type of the continuation monad to
+generalizing the type of the Continuation monad to
type ('a, 'b, 'c) continuation = ('a -> 'b) -> 'c;;
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).
+Using continuations to solve the same fringe problem
+----------------------------------------------------
+
+We've seen two solutions to the same fringe problem so far.
+The simplest is to map each tree to a list of its leaves, then compare
+the lists. But if the fringes differ in an early position, we've
+wasted our time visiting the rest of the tree.
+
+The second solution was to use tree zippers and mutable state to
+simulate coroutines. We would unzip the first tree until we found the
+next leaf, then store the zipper structure in the mutable variable
+while we turned our attention to the other tree. Because we stop as
+soon as we find the first mismatched leaf, this solution does not have
+the flaw just mentioned of the solution that maps both trees to a list
+of leaves before beginning comparison.
+
+Since zippers are just continuations reified, we expect that the
+solution in terms of zippers can be reworked using continuations, and
+this is indeed the case. To make this work in the most convenient
+way, we need to use the fully general type for continuations just mentioned.
+
+tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);;
+
-The binary tree monad
+
+The Binary Tree monad
---------------------
Of course, by now you may have realized that we have discovered a new
-monad, the binary tree monad:
+monad, the Binary 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:
[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\_mondadize?
+--------------------------------------------
+
+So we've defined a Tree monad:
+
+ type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
+ 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);;
+
+What's this have to do with the `tree_monadize` functions we defined earlier?
+
+ let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
+ match t with
+ | Leaf a -> reader_bind (f a) (fun b -> reader_unit (Leaf b))
+ | Node (l, r) -> reader_bind (tree_monadize f l) (fun l' ->
+ reader_bind (tree_monadize f r) (fun r' ->
+ reader_unit (Node (l', r'))));;
+
+... and so on for different monads?
+
+The answer is that each of those `tree_monadize` functions is adding a Tree monad *layer* to a pre-existing Reader (and so on) monad. We discuss that further here: [[Monad Transformers]].
+