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)))
.
___|___
| |
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
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;;
+ 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
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 =
- : 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 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 =
match t with
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