X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=manipulating_trees_with_monads.mdwn;h=23abaa63ed33444ca1b3a6dd69fad974f4a8bb9d;hp=10c955647d4acee1c16ba9ff3e602382df66aae7;hb=d0c1b524d79cd88b2979d7825dc297bbbddc4f22;hpb=3d98e6d16929dbdeb122713b86171b86039dca4a
diff --git a/manipulating_trees_with_monads.mdwn b/manipulating_trees_with_monads.mdwn
index 10c95564..23abaa63 100644
--- a/manipulating_trees_with_monads.mdwn
+++ b/manipulating_trees_with_monads.mdwn
@@ -3,11 +3,13 @@
Manipulating trees with monads
------------------------------
-This topic develops an idea based on a detailed suggestion of Ken
-Shan's. We'll build a series of functions that operate on trees,
-doing various things, including replacing leaves, counting nodes, and
-converting a tree to a list of leaves. The end result will be an
-application for continuations.
+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, replacing leaves 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 transformer that
deals in monads. We can modify the behavior of the system by swapping
@@ -20,7 +22,7 @@ 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)
+ 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?]
@@ -30,7 +32,7 @@ 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)))
+ Leaf 11)))
.
___|___
| |
@@ -44,18 +46,18 @@ We'll be using trees where the nodes are integers, e.g.,
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 =
+ let rec tree_map (t : 'a tree) (leaf_modifier : 'a -> 'b): '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);;
+ | Node (l, r) -> Node (tree_map l leaf_modifier,
+ tree_map r leaf_modifier);;
-`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:
+`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;;
+ tree_map t1 double;;
- : int tree =
Node (Node (Leaf 4, Leaf 6), Node (Leaf 10, Node (Leaf 14, Leaf 22)))
@@ -71,14 +73,15 @@ structure of the tree unchanged. For instance:
14 22
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
-`tree_map`. For instance, we can easily square each leaf instead by
-supplying the appropriate `int -> int` operation in place of `double`:
+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 =ppp
+ tree_map t1 square;;
+ - : 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
@@ -92,8 +95,6 @@ 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 (. (. (f 2) (f 3)) (. (f 5) (. (f 7) (f 11))))
-
\f .
_____|____
| |
@@ -106,14 +107,25 @@ updated tree.
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 `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 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.
+tree`) into a reader monadic 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`.
+
+[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 `\f`) 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;; (* mnemonic: e for environment *)
let reader_unit (a : 'a) : 'a reader = fun _ -> a;;
@@ -131,11 +143,11 @@ 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 =
+ let rec tree_monadize (t : 'a tree) (f : 'a -> 'b reader) : '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' ->
+ | Node (l, r) -> reader_bind (tree_monadize l f) (fun l' ->
+ reader_bind (tree_monadize r f) (fun r' ->
reader_unit (Node (l', r'))));;
This function says: give me a function `f` that knows how to turn
@@ -173,17 +185,17 @@ Then we can expect that supplying it to our `int tree reader` will double all th
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 int_readerize t1 double;;
+ # tree_monadize t1 int_readerize 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
-int_readerize t1`) to a different `int -> int` function---say, the
+t1 int_readerize`) to a different `int -> int` function---say, the
squaring function, `fun i -> i * i`---we get an entirely different
result:
- # tree_monadize int_readerize t1 square;;
+ # tree_monadize t1 int_readerize square;;
- : int tree =
Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
@@ -201,16 +213,16 @@ 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 =
+ let rec tree_monadize (t : 'a tree) (f : 'a -> 'b state) : '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' ->
+ | Node (l, r) -> state_bind (tree_monadize l f) (fun l' ->
+ state_bind (tree_monadize r f) (fun r' ->
state_unit (Node (l', r'))));;
Then we can count the number of leaves in the tree:
- # tree_monadize (fun a -> fun s -> (a, s+1)) t1 0;;
+ # tree_monadize t1 (fun a -> fun s -> (a, s+1)) 0;;
- : int tree * int =
(Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11))), 5)
@@ -218,19 +230,57 @@ Then we can count the number of leaves in the tree:
___|___
| |
. .
- _|__ _|__
+ _|__ _|__ , 5
| | | |
2 3 5 .
_|__
| |
7 11
-Why does this work? Because the operation `fun a -> fun s -> (a, s+1)` takes an `int` and wraps it in an `int state` monadic box that increments the state. 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 state-incrementing for each of its leaves.
+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 `fun a -> fun s -> (a, s+1)`
+takes an `int` and wraps it in an `int state` monadic box that
+increments the state. 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 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 t1 (fun a -> fun s -> (s+1, s+1)) 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 (t : 'a tree) (f : 'a -> 'b state) : '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 r f) (fun r' -> (* R first *)
+ state_bind (tree_monadize l f) (fun l'-> (* Then L *)
+ state_unit (Node (l', r'))));;
+
+ # tree_monadize_rev t1 (fun a -> fun s -> (s+1, s+1)) 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
- # tree_monadize (fun i -> [ [i; square i] ]) t1;;
+ # tree_monadize t1 (fun i -> [ [i; square i] ]);;
- : int list tree list =
[Node
(Node (Leaf [2; 4], Leaf [3; 9]),
@@ -238,13 +288,13 @@ One more revealing example before getting down to business: replacing
Unlike the previous cases, instead of turning a tree into a function
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.
-
-
-
+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
+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
+box.]
Now for the main point. What if we wanted to convert a tree to a list
of leaves?
@@ -253,11 +303,11 @@ of leaves?
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 -> ('b, 'r) continuation) (t : 'a tree) : ('b tree, 'r) continuation =
+ let rec tree_monadize (t : 'a tree) (f : 'a -> ('b, 'r) continuation) : ('b tree, 'r) 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' ->
+ | Node (l, r) -> continuation_bind (tree_monadize l f) (fun l' ->
+ continuation_bind (tree_monadize r f) (fun r' ->
continuation_unit (Node (l', r'))));;
We use the Continuation monad described above, and insert the
@@ -265,10 +315,21 @@ We use the Continuation monad described above, and insert the
So for example, we compute:
- # tree_monadize (fun a -> fun k -> a :: k a) t1 (fun t -> []);;
+ # tree_monadize t1 (fun a k -> a :: k ()) (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 -> fun 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?
+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
@@ -276,7 +337,7 @@ 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);;
+ # tree_monadize t1 continuation_unit (fun t -> t);;
- : int tree =
Node (Node (Leaf 2, Leaf 3), Node (Leaf 5, Node (Leaf 7, Leaf 11)))
@@ -284,34 +345,129 @@ 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);;
+ # tree_monadize t1 (fun a -> fun k -> k (square a)) (fun t -> t);;
- : int tree =
Node (Node (Leaf 4, Leaf 9), Node (Leaf 25, Node (Leaf 49, Leaf 121)))
(* Simulating the int list tree list *)
- # tree_monadize (fun a -> fun k -> k [a; square a]) t1 (fun t -> t);;
+ # tree_monadize t1 (fun a -> fun k -> k [a; square a]) (fun t -> t);;
- : int list tree =
Node
(Node (Leaf [2; 4], Leaf [3; 9]),
Node (Leaf [5; 25], Node (Leaf [7; 49], Leaf [11; 121])))
(* Counting leaves *)
- # tree_monadize (fun a -> fun k -> 1 + k a) t1 (fun t -> 0);;
+ # tree_monadize t1 (fun a -> fun k -> 1 + k a) (fun t -> 0);;
- : int = 5
-We could simulate the tree state example too, but it would require
-generalizing the type of the Continuation monad to
+[To be fixed: exactly which kind of monad each of these computations simulates.]
+
+We could simulate the tree state example too by setting the relevant
+type to `('a, 'state -> 'result) continuation`.
+In fact, Andre Filinsky has suggested that the continuation monad is
+able to simulate any other monad (Google for "mother of all monads").
+
+We would eventually want to generalize the continuation type to
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 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 and in the same-fringe example that
+local properties of a tree (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;;
+
+let sentence1 = Node (Leaf "John",
+ Node (Node (Leaf "saw",
+ Leaf "everyone"),
+ Leaf "yesterday"));;
+
+
+Then we can crudely approximate quantification as follows:
+
+
+# tree_monadize sentence1 lex (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 sentence2 lex (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 sentence2 lex (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 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 Binary Tree monad
---------------------
Of course, by now you may have realized that we have discovered a new
-monad, the Binary Tree monad:
+monad, the Binary 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;;
@@ -330,8 +486,6 @@ 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 `f a`:
-\tree (. (f a1) (. (. (. (f a2) (f a3)) (f a4)) (f a5)))
-
. .
__|__ __|__
| | | |
@@ -361,9 +515,6 @@ As for the associative law,
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)))))
-
.
____|____
. . | |
@@ -394,10 +545,10 @@ called a
that is intended to represent non-deterministic computations as a tree.
-What's this have to do with tree\_mondadize?
+What's this have to do with tree\_monadize?
--------------------------------------------
-So we've defined a Tree monad
+So we've defined a Tree monad:
type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
let tree_unit (a: 'a) : 'a tree = Leaf a;;
@@ -408,121 +559,26 @@ So we've defined a Tree monad
What's this have to do with the `tree_monadize` functions we defined earlier?
- let rec tree_monadize (f : 'a -> 'b reader) (t : 'a tree) : 'b tree reader =
+ let rec tree_monadize (t : 'a tree) (f : 'a -> 'b reader) : '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' ->
+ | Node (l, r) -> reader_bind (tree_monadize l f) (fun l' ->
+ reader_bind (tree_monadize r f) (fun r' ->
reader_unit (Node (l', r'))));;
... 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 application, we had to figure out how to combine Reader, State, and Set monads in an ad-hoc way.) But 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)` 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 circuitious...
- * 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 that doesn't 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);;
+Well, notice that `tree\_monadizer` takes arguments whose types
+resemble that of a monadic `bind` function. Here's a schematic bind
+function compared with `tree\_monadizer`:
-Notice the key differences: where before we just returned `a`, now we return `M.unit a`. Where before we just supplied a 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 -> (fun b -> Leaf b) (f a)
- | Node (l, r) -> (fun l' r' -> Node (l', r')) (bind l f) (bind r f);;
-
- (* monadic operations for the TreeT monadic transformer *)
-
- 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'))));;
+ bind (u:'a Monad) (f: 'a -> 'b Monad): 'b Monad
+ tree\_monadizer (u:'a Tree) (f: 'a -> 'b Monad): 'b Tree Monad
+Comparing these types makes it clear that `tree\_monadizer` provides a
+way to distribute an arbitrary monad M across the leaves of any tree to
+form a new tree inside an M box.
+The more general 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]].