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)
+ 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?]
let square i = i * i;;
tree_map square t1;;
- - : int tree =ppp
+ - : 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
`tree_monadize` function, it then wraps an `int tree` in a box, one
that does the same state-incrementing for each of its leaves.
+We can use the state monad to replace 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 -> (s+1, s+1)) t1 0;;
+ - : int tree * int =
+ (Node (Node (Leaf 1, Leaf 2), Node (Leaf 3, Node (Leaf 4, Leaf 5))), 5)
+
+The key thing to notice is that instead of copying `a` into the
+monadic box, we throw away the `a` and put a copy of the state in
+instead.
+
+Reversing the 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' ->
+ state_bind (tree_monadize f l) (fun l' ->
+ state_unit (Node (l', r'))));;
+
+ # tree_monadize_rev (fun a -> fun s -> (s+1, s+1)) t1 0;;
+ - : int tree * int =
+ (Node (Node (Leaf 5, Leaf 4), Node (Leaf 3, Node (Leaf 2, Leaf 1))), 5)
+
+We will need below to depend on controlling the order in which nodes
+are visited when we use the continuation monad to solve the
+same-fringe problem.
+
One more revealing example before getting down to business: replacing
`state` everywhere in `tree_monadize` with `list` gives us
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.
-[Why is the argument to tree_monadize `int -> int list list` instead
+[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
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).
+Using continuations to solve the same fringe problem
+----------------------------------------------------
+
+We've seen two solutions to the same fringe problem so far.
+The simplest is to map each tree to a list of its leaves, then compare
+the lists. But if the fringes differ in an early position, we've
+wasted our time visiting the rest of the tree.
+
+The second solution was to use tree zippers and mutable state to
+simulate coroutines. We would unzip the first tree until we found the
+next leaf, then store the zipper structure in the mutable variable
+while we turned our attention to the other tree. Because we stop as
+soon as we find the first mismatched leaf, this solution does not have
+the flaw just mentioned of the solution that maps both trees to a list
+of leaves before beginning comparison.
+
+Since zippers are just continuations reified, we expect that the
+solution in terms of zippers can be reworked using continuations, and
+this is indeed the case. To make this work in the most convenient
+way, we need to use the fully general type for continuations just mentioned.
+
+tree_monadize (fun a k -> a, k a) t1 (fun t -> 0);;
+
+
The Binary Tree monad
---------------------
... 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'))));;
-
+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. We discuss that further here: [[Monad Transformers]].