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 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));;
+ | Node (l, r) -> Node (treemap newleaf l,
+ treemap newleaf r);;
`treemap` takes a function that transforms old leaves into new leaves,
and maps that function over all the leaves in the tree, leaving 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
+more flexible. So the next step---combining the tree transformer 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
+tree that is ready to accept any `int -> int` function and produce the
updated tree.
-\tree (. (. (f2) (f3))(. (f5) (.(f7)(f11))))
- \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 `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`.
+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`:
- let int2int_reader (x:'a): 'b reader = fun (op:'a -> 'b) -> op x;;
+ let int2int_reader : 'a -> 'b reader = fun (a : 'a) -> fun (op : 'a -> 'b) -> op a;;
int2int_reader 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.
+function of type `int -> int` to.
- let rec treemonadizer (f:'a -> 'b reader) (t:'a tree):('b tree) reader =
+ 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 ->
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.
+`'b reader` monad through the leaves.
# treemonadizer int2int_reader t1 (fun i -> i + i);;
- : int tree =
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
+int2int_reader t1`) to a different `int -> int` function---say, the
squaring function, `fun i -> i * i`---we get an entirely different
result:
- : 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
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 i -> (a, i);;
+ let state_bind u f = fun i -> let (a, i') = u i in f a (i' + 1);;
Gratifyingly, we can use the `treemonadizer` 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 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 ->
Notice that we've counted each internal node twice---it's a good
exercise to adjust the code to count each node once.
+<!--
+A tree with n leaves has 2n - 1 nodes.
+This function will currently return n*1 + (n-1)*2 = 3n - 2.
+To convert b = 3n - 2 into 2n - 1, we can use: let n = (b + 2)/3 in 2*n -1
+
+But I assume Chris means here, adjust the code so that no corrections of this sort have to be applied.
+-->
+
+
One more revealing example before getting down to business: replacing
`state` everywhere in `treemonadizer` with `list` gives us
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 =
+ 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 ->
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 (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));;