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.
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;;
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.
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).
-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. So far, we've defined monads as single-layered things. Though in the Groenendijk, Stokhoff, and Veltmann homework, we had to figure out how to combine Reader, State, and Set monads in an ad-hoc way. In practice, one often wants to combine the abilities of several monads. Corresponding to each monad like Reader, there's a corresponding ReaderT **monad transformer**. That takes an existing monad M and adds a Reader monad layer to it. The way these are defined parallels the way the single-layer versions are defined. For example, here's the Reader monad:
+
+ (* monadic operations for the Reader monad *)
+
+ type 'a reader =
+ env -> 'a;;
+ let unit (a : 'a) : 'a reader =
+ fun e -> a;;
+ let bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader =
+ fun e -> (fun v -> f v e) (u e);;
+
+We've just beta-expanded the familiar `f (u e) e` into `(fun v -> f v e) (u e)`, in order to factor out the parts where any Reader monad is being supplied as an argument to another function. Then if we want instead to add a Reader layer to some arbitrary other monad M, with its own M.unit and M.bind, here's how we do it:
+
+ (* monadic operations for the ReaderT monadic transformer *)
+
+ (* We're not giving valid OCaml code, but rather something
+ * that's conceptually easier to digest.
+ * How you really need to write this in OCaml is more circuitous...
+ * see http://lambda.jimpryor.net/code/tree_monadize.ml for some details. *)
+
+ type ('a, M) readerT =
+ env -> 'a M;;
+ (* this is just an 'a M reader; but don't rely on that pattern to generalize *)
+
+ let unit (a : 'a) : ('a, M) readerT =
+ fun e -> M.unit a;;
+
+ let bind (u : ('a, M) readerT) (f : 'a -> ('b, M) readerT) : ('b, M) readerT =
+ fun e -> M.bind (u e) (fun v -> f v e);;
+
+Notice the key differences: where before we just returned `a`, now we instead return `M.unit a`. Where before we just supplied value `u e` of type `'a reader` as an argument to a function, now we instead `M.bind` the `'a reader` to that function. Notice also the differences in the types.
+
+What is the relation between Reader and ReaderT? Well, suppose you started with the Identity monad:
+
+ type 'a identity = 'a;;
+ let unit (a : 'a) : 'a = a;;
+ let bind (u : 'a) (f : 'a -> 'b) : 'b = f u;;
+
+and you used the ReaderT transformer to add a Reader monad layer to the Identity monad. What do you suppose you would get?
+
+The relations between the State monad and the StateT monadic transformer are parallel:
+
+ (* monadic operations for the State monad *)
+
+ type 'a state =
+ store -> ('a * store);;
+
+ let unit (a : 'a) : 'a state =
+ fun s -> (a, s);;
+
+ let bind (u : 'a state) (f : 'a -> 'b state) : 'b state =
+ fun s -> (fun (a, s') -> f a s') (u s);;
+
+We've used `(fun (a, s') -> f a s') (u s)` instead of the more familiar `let (a, s') = u s in f a s'` in order to factor out the part where a value of type `'a state` is supplied as an argument to a function. Now StateT will be:
+
+ (* monadic operations for the StateT monadic transformer *)
+
+ type ('a, M) stateT =
+ store -> ('a * store) M;;
+ (* notice this is not an 'a M state *)
+
+ let unit (a : 'a) : ('a, M) stateT =
+ fun s -> M.unit (a, s);;
+
+ let bind (u : ('a, M) stateT) (f : 'a -> ('b, M) stateT) : ('b, M) stateT =
+ fun s -> M.bind (u s) (fun (a, s') -> f a s');;
+
+Do you see the pattern? Where ordinarily we'd return an `'a` value, now we instead return an `'a M` value. Where ordinarily we'd supply a `'a state` value as an argument to a function, now we instead `M.bind` it to that function.
+
+Okay, now let's do the same thing for our Tree monad.
+
+ (* monadic operations for the Tree monad *)
+
+ type 'a tree =
+ Leaf of 'a | Node of ('a tree) * ('a tree);;
+
+ let unit (a: 'a) : 'a tree =
+ Leaf a;;
+
+ let rec bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
+ match u with
+ | Leaf a -> f a;;
+ | Node (l, r) -> (fun l' r' -> Node (l', r')) (bind l f) (bind r f);;
+
+ (* monadic operations for the TreeT monadic transformer *)
+ (* NOTE THIS IS NOT YET WORKING --- STILL REFINING *)
+
+ type ('a, M) treeT =
+ 'a tree M;;
+
+ let unit (a: 'a) : ('a, M) tree =
+ M.unit (Leaf a);;
+
+ let rec bind (u : ('a, M) tree) (f : 'a -> ('b, M) tree) : ('b, M) tree =
+ match u with
+ | Leaf a -> M.bind (f a) (fun b -> M.unit (Leaf b))
+ | Node (l, r) -> M.bind (bind l f) (fun l' ->
+ M.bind (bind r f) (fun r' ->
+ M.unit (Node (l', r'));;
+
+Compare this definition of `bind` for the TreeT monadic transformer to our earlier definition of `tree_monadize`, specialized for the Reader monad:
+
+ 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'))));;
+
+