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))))))
+ let t1 = Node (Node (Leaf 2, Leaf 3),
+ Node (Leaf 5, Node (Leaf 7,
+ Leaf 11)))
.
___|___
| |
. .
- _|__ _|__
+ _|_ _|__
| | | |
2 3 5 .
_|__
Our first task will be to replace each leaf with its double:
- let rec treemap (newleaf:'a -> 'b) (t:'a tree):('b tree) =
+ let rec tree_map (leaf_modifier : '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));;
+ | Leaf i -> Leaf (leaf_modifier i)
+ | Node (l, r) -> Node (tree_map leaf_modifier l,
+ tree_map leaf_modifier r);;
-`treemap` takes a function that transforms old leaves into new leaves,
+`tree_map` 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;;
+ tree_map double t1;;
- : int tree =
Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
| |
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
+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
-`treemap`. For instance, we can easily square each leaf instead by
+`tree_map`. 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;;
+ let square i = i * i;;
+ tree_map 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
+Note that what `tree_map` 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
+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 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
+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.
-\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
+\tree (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11))))
- \f .
- ____|____
- | |
- . .
- __|__ __|__
- | | | |
- f2 f3 f5 .
- __|___
- | |
- f7 f11
+ \f .
+ _____|____
+ | |
+ . .
+ __|___ __|___
+ | | | |
+ f 2 f 3 f 5 .
+ __|___
+ | |
+ 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 returns an `int
-tree` in which each leaf `x` has been replaced with `(f x)`.
+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 this situation, it will be
-enough for now to expect that our reader will expect a function of
-type `int->int`.
+Jacobson-inspired link monad), etc. In the present case, it will be
+enough to expect that our "environment" will be some 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;;
+ type 'a reader = (int -> int) -> 'a;; (* mnemonic: e for environment *)
+ 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's easy to figure out how to turn an `int` into an `int reader`:
+It would be a simple matter to turn an *integer* into an `int reader`:
- let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
- int2int_reader 2 (fun i -> i + i);;
+ let int_readerize : int -> int reader = fun (a : int) -> fun (modifier : int -> int) -> modifier a;;
+ int_readerize 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.
+But 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 treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
+ let rec tree_monadize (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 ->
+ | Leaf i -> reader_bind (f i) (fun i' -> reader_unit (Leaf i'))
+ | Node (l, r) -> reader_bind (tree_monadize f l) (fun x ->
+ reader_bind (tree_monadize 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
+turn an `'a tree` into an `'b tree reader`. 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
-monad through the leaves.
+`'b reader` monad through the original tree's leaves.
- # treemonadizer int2int_reader t1 (fun i -> i + i);;
+ # tree_monadize int_readerize 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, `treemonadizer
-int2int_reader t1`) to a different `int->int` function---say, the
+we apply the very same `int tree reader` (namely, `tree_monadize
+int_readerize 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);;
+ # tree_monadize int_readerize t1 square;;
- : 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
+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 nodes in
+
+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 x i = (x, i+.5);;
- let state_bind u f i = let (a, i') = u i in f a (i'+.5);;
+ 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 `treemonadizer` function without any
+Gratifyingly, we can use the `tree_monadize` function without any
modification whatsoever, except for replacing the (parametric) type
-`reader` with `state`:
+`'b reader` with `'b state`, and substituting in the appropriate unit and bind:
- let rec treemonadizer (f:'a -> 'b state) (t:'a tree):('b tree) state =
+ let rec tree_monadize (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 ->
+ | Leaf i -> state_bind (f i) (fun i' -> state_unit (Leaf i'))
+ | Node (l, r) -> state_bind (tree_monadize f l) (fun x ->
+ state_bind (tree_monadize f r) (fun y ->
state_unit (Node (x, y))));;
-Then we can count the number of nodes in the tree:
+Then we can count the number of leaves in the tree:
- # treemonadizer state_unit t1 0;;
+ # tree_monadize (fun a s -> (a, s+1)) t1 0;;
- : int tree * int =
- (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 13)
+ (Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 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
+`state` everywhere in `tree_monadize` with `list` gives us
- # treemonadizer (fun x -> [ [x; square x] ]) t1;;
+ # tree_monadize (fun i -> [ [i; square i] ]) t1;;
- : int list tree list =
[Node
(Node (Leaf [2; 4], Leaf [3; 9]),
from some input to a result, this transformer replaces each `int` with
a list of `int`'s.
+<!--
+FIXME: We don't make it clear why the fun has to be int -> int list list, instead of int -> int list
+-->
+
+
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 continuation_unit a = fun k -> k a;;
+ let continuation_bind u f = fun k -> u (fun a -> f a k);;
- let rec treemonadizer (f:'a -> ('b, 'r) continuation) (t:'a tree):(('b tree), 'r) continuation =
+ let rec tree_monadize (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 ->
+ | Leaf i -> continuation_bind (f i) (fun i' -> continuation_unit (Leaf i'))
+ | Node (l, r) -> continuation_bind (tree_monadize f l) (fun x ->
+ continuation_bind (tree_monadize 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.
+`continuation` type in the appropriate place in the `tree_monadize` code.
We then compute:
- # treemonadizer (fun a c -> a :: (c a)) t1 (fun t -> []);;
+ # tree_monadize (fun a k -> a :: (k 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
+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:
- # treemonadizer continuation_unit t1 (fun x -> x);;
+ # tree_monadize continuation_unit t1 (fun i -> i);;
- : 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`:
+interesting functions for the first argument of `tree_monadize`:
(* Simulating the tree reader: distributing a operation over the leaves *)
- # treemonadizer (fun a c -> c (square a)) t1 (fun x -> x);;
+ # tree_monadize (fun a k -> k (square a)) t1 (fun i -> i);;
- : 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);;
+ # tree_monadize (fun a k -> k [a; square a]) t1 (fun i -> i);;
- : 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);;
+ # tree_monadize (fun a k -> 1 + k a) t1 (fun i -> 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;;
+ 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
---------------------
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 =
+ let tree_unit (a: 'a) = Leaf a;;
+ let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
match u with
- | Leaf x -> f x
+ | 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 = fa
+ 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 `fa`:
+except that each leaf `a` has been replaced with `f a`:
-\tree (. (fa1) (. (. (. (fa2)(fa3)) (fa4)) (fa5)))
+\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5)))
. .
__|__ __|__
| | | |
- a1 . fa1 .
+ a1 . f a1 .
_|__ __|__
| | | |
- . a5 . fa5
+ . a5 . f a5
bind _|__ f = __|__
| | | |
- . a4 . fa4
+ . a4 . f a4
__|__ __|___
| | | |
- a2 a3 fa2 fa3
+ a2 a3 f a2 f a3
Given this equivalence, the right identity law
As for the associative law,
- Associativity: bind (bind u f) g = bind u (\a. bind (fa) g)
+ 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
-\tree (. (. (. (. (a1)(a2)))))
-\tree (. (. (. (. (a1) (a1)) (. (a1) (a1))) ))
+\tree (. (. (. (. (a1) (a2)))))
+\tree (. (. (. (. (a1) (a1)) (. (a1) (a1)))))
.
____|____
. . | |
bind __|__ f = __|_ = . .
| | | | __|__ __|__
- a1 a2 fa1 fa2 | | | |
+ a1 a2 f a1 f a2 | | | |
a1 a1 a1 a1
Now when we bind this tree to `g`, we get
- .
- ____|____
- | |
- . .
- __|__ __|__
- | | | |
- ga1 ga1 ga1 ga1
+ .
+ _____|______
+ | |
+ . .
+ __|__ __|__
+ | | | |
+ 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