[[!toc]]
Manipulating trees with monads
------------------------------
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, copying the tree 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 machine 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 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);;
[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 tree_map (leaf_modifier : 'a -> 'b) (t : 'a tree) : 'b tree =
match t with
| Leaf i -> Leaf (leaf_modifier i)
| Node (l, r) -> Node (tree_map leaf_modifier l,
tree_map leaf_modifier r);;
`tree_map` takes a tree and 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;;
tree_map 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 `tree_map`
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 =
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.
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)
tree that is ready to accept any `int -> int` function and produce the
updated tree.
fun e -> .
_____|____
| |
. .
__|___ __|___
| | | |
e 2 e 3 e 5 .
__|___
| |
e 7 e 11
That is, we want to transform the ordinary tree `t1` (of type `int
tree`) into a reader monadic object of type `(int -> int) -> int
tree`: something that, when you apply it to an `int -> int` function
`e` returns an `int tree` in which each leaf `i` has been replaced
with `e 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 `e`) 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;;
let reader_unit (a : 'a) : 'a reader = fun _ -> a;;
let reader_bind (u: 'a reader) (f : 'a -> 'b reader) : 'b reader =
fun e -> f (u e) e;;
It would be a simple matter to turn an *integer* into an `int reader`:
let asker : int -> int reader =
fun (a : int) ->
fun (modifier : int -> int) -> modifier a;;
asker 2 (fun i -> i + i);;
- : int = 4
`asker a` is a monadic box that waits for an an environment (here, the argument `modifier`) and returns what that environment maps `a` to.
How do we do the analagous transformation when our `int`s are scattered over the leaves of a tree? How do we turn an `int tree` into a reader?
A tree is not the kind of thing that we can apply a
function of type `int -> int` to.
But we can do this:
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'))));;
This function says: give me a function `f` that knows how to turn
something of type `'a` into an `'b reader`---this is a function of the same type that you could bind an `'a reader` to, such as `asker` or `reader_unit`---and I'll show you how to
turn an `'a tree` into an `'b tree reader`. That is, if you show me how to do this:
------------
1 ---> | 1 |
------------
then I'll give you back the ability to do this:
____________
. | . |
__|___ ---> | __|___ |
| | | | | |
1 2 | 1 2 |
------------
And how will that boxed tree behave? Whatever actions you perform on it will be transmitted down to corresponding operations on its leaves. For instance, our `int reader` expects an `int -> int` environment. If supplying environment `e` to our `int reader` doubles the contained `int`:
------------
1 ---> | 1 | applied to e ~~> 2
------------
Then we can expect that supplying it to our `int tree reader` will double all the leaves:
____________
. | . | .
__|___ ---> | __|___ | applied to e ~~> __|___
| | | | | | | |
1 2 | 1 2 | 2 4
------------
In more fanciful terms, the `tree_monadize` function builds plumbing that connects all of the leaves of a tree into one connected monadic network; it threads the
`'b reader` monad through the original tree's leaves.
# tree_monadize asker t1 double;;
- : 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, `tree_monadize
asker t1`) to a different `int -> int` function---say, the
squaring function, `fun i -> i * i`---we get an entirely different
result:
# tree_monadize asker t1 square;;
- : int tree =
Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
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
the tree.
type 'a state = int -> 'a * int;;
let state_unit a = fun s -> (a, s);;
let state_bind u f = fun s -> let (a, s') = u s in f a s';;
Gratifyingly, we can use the `tree_monadize` function without any
modification whatsoever, except for replacing the (parametric) type
`'b reader` with `'b state`, and substituting in the appropriate unit and bind:
let rec tree_monadize (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 l) (fun l' ->
state_bind (tree_monadize f r) (fun r' ->
state_unit (Node (l', r'))));;
Then we can count the number of leaves in the tree:
# let incrementer = fun a ->
fun s -> (a, s+1);;
# tree_monadize incrementer t1 0;;
- : int tree * int =
(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5)
.
___|___
| |
. .
( _|__ _|__ , 5 )
| | | |
2 3 5 .
_|__
| |
7 11
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 `incrementer`
takes an argument `a` and wraps it in an State monadic box that
increments the store and leaves behind a wrapped `a`. 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 store-incrementing for each of its leaves.
We can use the state monad to annotate 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 -> ((a,s+1), s+1)) t1 0;;
- : int tree * int =
(Node
(Node (Leaf (2, 1), Leaf (3, 2)),
Node
(Leaf (5, 3),
Node (Leaf (7, 4), Leaf (11, 5)))),
5)
The key thing to notice is that instead of just wrapping `a` in the
monadic box, we wrap a pair of `a` and the current store.
Reversing the annotation 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' -> (* R first *)
state_bind (tree_monadize f l) (fun l'-> (* Then L *)
state_unit (Node (l', r'))));;
# tree_monadize_rev (fun a -> fun s -> ((a,s+1), s+1)) t1 0;;
- : int tree * int =
(Node
(Node (Leaf (2, 5), Leaf (3, 4)),
Node
(Leaf (5, 3),
Node (Leaf (7, 2), Leaf (11, 1)))),
5)
Later, we will talk more about controlling the order in which nodes are visited.
One more revealing example before getting down to business: replacing
`state` everywhere in `tree_monadize` with `list` lets us do:
# let decider i = if i = 2 then [20; 21] else [i];;
# tree_monadize decider t1;;
- : int tree List_monad.m =
[
Node (Node (Leaf 20, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)));
Node (Node (Leaf 21, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
]
Unlike the previous cases, instead of turning a tree into a function
from some input to a result, this monadized tree gives us back a list of trees,
one for each choice of `int`s for its leaves.
Now for the main point. What if we wanted to convert a tree to a list
of leaves?
type ('r,'a) continuation = ('a -> 'r) -> 'r;;
let continuation_unit a = fun k -> k a;;
let continuation_bind u f = fun k -> u (fun a -> f a k);;
let rec tree_monadize (f : 'a -> ('r,'b) continuation) (t : 'a tree) : ('r,'b tree) continuation =
match t with
| Leaf a -> continuation_bind (f a) (fun b -> continuation_unit (Leaf b))
| Node (l, r) -> continuation_bind (tree_monadize f l) (fun l' ->
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.
So for example, we compute:
# tree_monadize (fun a k -> a :: k ()) t1 (fun _ -> []);;
- : int list = [2; 3; 5; 7; 11]
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 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?
Soon we'll return to the same-fringe problem. Since the
simple but inefficient way to solve it is to map each tree to a list
of its leaves, this transformation is on the path to a more efficient
solution. We'll just have to figure out how to postpone computing the
tail of the list until it's needed...
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
apply the result to the identity function:
# tree_monadize continuation_unit t1 (fun t -> t);;
- : 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 `tree_monadize`:
(* Simulating the tree reader: distributing a operation over the leaves *)
# tree_monadize (fun a -> fun k -> k (square a)) t1 (fun t -> t);;
- : int tree =
Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
(* Counting leaves *)
# tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 0);;
- : int = 5
It's not immediately obvious to us how to simulate the List monadization of the tree using this technique.
We could simulate the tree annotating example by setting the relevant
type to `(store -> 'result, 'a) continuation`.
Andre Filinsky has proposed that the continuation monad is
able to simulate any other monad (Google for "mother of all monads").
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 idea of using continuations to characterize natural language meaning
------------------------------------------------------------------------
We might a philosopher or a linguist be interested in continuations,
especially if efficiency of computation is usually not an issue?
Well, the application of continuations to the same-fringe problem
shows that continuations can manage order of evaluation in a
well-controlled manner. In a series of papers, one of us (Barker) and
Ken Shan have argued that a number of phenomena in natural langauge
semantics are sensitive to the order of evaluation. We can't
reproduce all of the intricate arguments here, but we can give a sense
of how the analyses use continuations to achieve an analysis of
natural language meaning.
**Quantification and default quantifier scope construal**.
We saw in the copy-string example ("abSd") and in the same-fringe example that
local properties of a structure (whether a character is `'S'` or not, which
integer occurs at some leaf position) can control global properties of
the computation (whether the preceeding string is copied or not,
whether the computation halts or proceeds). Local control of
surrounding context is a reasonable description of in-situ
quantification.
(1) John saw everyone yesterday.
This sentence means (roughly)
forall x . yesterday(saw x) john
That is, the quantifier *everyone* contributes a variable in the
direct object position, and a universal quantifier that takes scope
over the whole sentence. If we have a lexical meaning function like
the following:
let lex (s:string) k = match s with
| "everyone" -> Node (Leaf "forall x", k "x")
| "someone" -> Node (Leaf "exists y", k "y")
| _ -> k s;;
Then we can crudely approximate quantification as follows:
# let sentence1 = Node (Leaf "John",
Node (Node (Leaf "saw",
Leaf "everyone"),
Leaf "yesterday"));;
# tree_monadize lex sentence1 (fun x -> x);;
- : string tree =
Node
(Leaf "forall x",
Node (Leaf "John", Node (Node (Leaf "saw", Leaf "x"), Leaf "yesterday")))
In order to see the effects of evaluation order,
observe what happens when we combine two quantifiers in the same
sentence:
# let sentence2 = Node (Leaf "everyone", Node (Leaf "saw", Leaf "someone"));;
# tree_monadize lex sentence2 (fun x -> x);;
- : string tree =
Node
(Leaf "forall x",
Node (Leaf "exists y", Node (Leaf "x", Node (Leaf "saw", Leaf "y"))))
The universal takes scope over the existential. If, however, we
replace the usual `tree_monadizer` with `tree_monadizer_rev`, we get
inverse scope:
# tree_monadize_rev lex sentence2 (fun x -> x);;
- : string tree =
Node
(Leaf "exists y",
Node (Leaf "forall x", Node (Leaf "x", Node (Leaf "saw", Leaf "y"))))
There are many crucially important details about quantification that
are being simplified here, and the continuation treatment used here is not
scalable for a number of reasons. Nevertheless, it will serve to give
an idea of how continuations can provide insight into the behavior of
quantifiers.
The Tree monad
==============
Of course, by now you may have realized that we are working with a new
monad, the binary, leaf-labeled 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) : '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);;
For once, let's check the Monad laws. The left identity law is easy:
Left identity: bind (unit a) f = bind (Leaf a) f = f a
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 the tree returned by `f a`:
. .
__|__ __|__
| | /\ |
a1 . f a1 .
_|__ __|__
| | | /\
. a5 . f a5
bind _|__ f = __|__
| | | /\
. a4 . f a4
__|__ __|___
| | /\ /\
a2 a3 f a2 f a3
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 (f a) g)
we'll give an example that will show how an inductive proof would
proceed. Let `f a = Node (Leaf a, Leaf a)`. Then
.
____|____
. . | |
bind __|__ f = __|_ = . .
| | | | __|__ __|__
a1 a2 f a1 f a2 | | | |
a1 a1 a1 a1
Now when we bind this tree to `g`, we get
.
_____|______
| |
. .
__|__ __|__
| | | |
g a1 g a1 g a1 g a1
At this point, it should be easy to convince yourself that
using the recipe on the right hand side of the associative law will
build 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.
What's this have to do with tree\_monadize?
--------------------------------------------
Our different implementations of `tree_monadize` above were different *layerings* of the Tree monad with other monads (Reader, State, List, and Continuation). We'll explore that further here: [[Monad Transformers]].