(*
* tree_monadize.ml
*
- * 'a and so on are type variables in OCaml; they stand for arbitrary types
+ * 'a and so on are type variables in OCaml; they stand for arbitrary types.
* What if you want a variable for a type constructor? For example, you want to
* generalize this pattern:
- * type ('a) t1 = 'a -> 'a list
- * type ('a) t2 = 'a -> 'a option
- * type ('a) t3 = 'a -> 'a reader
+ * type ('a) t1 = 'a -> ('a) list
+ * type ('a) t2 = 'a -> ('a) option
+ * type ('a) t3 = 'a -> ('a) reader
* and so on? OCaml won't let you do this:
- * type ('a, 'b) t = 'a -> 'a 'b
- * to generalize on the 'b position, we instead have to use OCaml's modules,
+ * type ('a, 'b) t = 'a -> ('a) 'b
+ * To generalize on the 'b position, we instead have to use OCaml's modules,
* and in particular its ability to make modules parameterized on other modules
* (OCaml calls these parameterized modules Functors, but that name is also
* used in other ways in this literature, so I won't give in to it.)
*
* Here's how you'd have to define the t type from above:
- * module T_maker(S: sig
- * type 'a b
- * end) = struct
- * type 'a t = 'a -> 'a S.b
+ * module T_maker(
+ * (* A sig...end block specifies the type of a module
+ * * What we're doing here is specifying the type of the
+ * module parameter that will choose
+ * whether b = list or b = option or b = reader...
+ * This module parameter may supply values as well as types *)
+ * Parm: sig
+ * type ('a) b
+ * end
+ * ) =
+ * (* A struct...end block gives a module value
+ * What we're doing here is building a new module that makes
+ * use of the module that was supplied as Parm *)
+ * struct
+ * type ('a) t = 'a -> ('a) Parm.b
* end
* And here's how you'd use it:
* module T_list = T_maker(struct type 'a b = 'a list end);;
* type 'a t1 = 'a T_list.t;;
+ * module T_option = T_maker(struct type 'a b = 'a option end);;
+ * type 'a t2 = 'a T_option.t;;
+ * (* and so on *)
*
* I know, it seems unnecessarily complicated.
*)
(Leaf 7, Leaf 11)));;
-module Tree_monadizer(S: sig
+module Tree_monadizer(Parm : sig
+ (* the module we're using as a parameter has to supply function values
+ * for unit and bind, as well as a monadic type constructor m *)
type 'a m
val unit : 'a -> 'a m
val bind : 'a m -> ('a -> 'b m) -> 'b m
end) = struct
- let rec monadize (f: 'a -> 'b S.m) (t: 'a tree) : 'b tree S.m =
+ let rec monadize (f: 'a -> 'b Parm.m) (t: 'a tree) : 'b tree Parm.m =
match t with
- | Leaf a -> S.bind (f a) (fun b -> S.unit (Leaf b))
+ | Leaf a -> Parm.bind (f a) (fun b -> Parm.unit (Leaf b))
| Node(l, r) ->
- S.bind (monadize f l) (fun l' ->
- S.bind (monadize f r) (fun r' ->
- S.unit (Node (l', r'))))
+ Parm.bind (monadize f l) (fun l' ->
+ Parm.bind (monadize f r) (fun r' ->
+ Parm.unit (Node (l', r'))))
end;;
let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
fun e -> f (u e) e;;
+(* Now we supply the Reader monad as a parameter to Tree_monadizer.
+ * We'll get back a module TreeReader that contains a single value,
+ * the monadize function specialized to the Reader monad *)
module TreeReader = Tree_monadizer(struct
type 'a m = 'a reader
let unit = unit_reader
let bind_state (u : 'a state) (f : 'a -> 'b state) : 'b state =
fun s -> (let (a, s') = u s in (f a) s');;
+(* Make a TreeState module containing monadize specialized to the State monad *)
module TreeState = Tree_monadizer(struct
type 'a m = 'a state
let unit = unit_state
let bind_list (u: 'a list) (f : 'a -> 'b list) : 'b list =
List.concat(List.map f u);;
+(* Make a TreeList module containing monadize specialized to the List monad *)
module TreeList = Tree_monadizer(struct
type 'a m = 'a list
let unit = unit_list
end);;
-
-(* we need to a new module when the monad is parameterized on two types *)
-module Tree_monadizer2(S: sig
+(* since the continuation monad is parameterized on two types---it's
+ * ('a,'r) cont not ('a) cont---we can't match the type ('a) m that
+ * Tree_monadizer expects in its parameter. So we have to make a different
+ * Tree_monadizer2 that takes a ('a,'x) m type constructor in its
+ * parameter instead *)
+module Tree_monadizer2(Parm : sig
type ('a,'x) m
val unit : 'a -> ('a,'x) m
val bind : ('a,'x) m -> ('a -> ('b,'x) m) -> ('b,'x) m
end) = struct
- let rec monadize (f: 'a -> ('b,'x) S.m) (t: 'a tree) : ('b tree,'x) S.m =
- (* the definition is the same, the difference is only in the types *)
+ (* the body of the monadize function is the same; the only difference is in
+ * the types *)
+ let rec monadize (f: 'a -> ('b,'x) Parm.m) (t: 'a tree) : ('b tree,'x) Parm.m =
match t with
- | Leaf a -> S.bind (f a) (fun b -> S.unit (Leaf b))
+ | Leaf a -> Parm.bind (f a) (fun b -> Parm.unit (Leaf b))
| Node(l, r) ->
- S.bind (monadize f l) (fun l' ->
- S.bind (monadize f r) (fun r' ->
- S.unit (Node (l', r'))))
+ Parm.bind (monadize f l) (fun l' ->
+ Parm.bind (monadize f r) (fun r' ->
+ Parm.unit (Node (l', r'))))
end;;
type ('a,'r) cont = ('a -> 'r) -> 'r;;
let bind_cont (u: ('a,'r) cont) (f: 'a -> ('b,'r) cont) : ('b,'r) cont =
fun k -> u (fun a -> f a k);;
+(* Make a TreeCont module containing monadize specialized to the Cont monad *)
module TreeCont = Tree_monadizer2(struct
type ('a,'r) m = ('a,'r) cont
let unit = unit_cont
* http://lambda.jimpryor.net/manipulating_trees_with_monads/
*)
+
let int_readerize : int -> int reader =
- fun (a : int) (modifier : int -> int) -> modifier a;;
+ fun (a : int) -> fun (env : int -> int) -> env a;;
+
+(* int_readerize takes an int and returns a Reader monad that
+ * "looks up" that int in the environment (i.e. modifies it)
+ * this is structurally parallel to the function `lookup` we used
+ * before to "look up" variables in the environment *)
(* double each leaf *)
-TreeReader.monadize int_readerize t1 (fun i -> i + i);;
+let env = fun i -> i + i in
+TreeReader.monadize int_readerize t1 env;;
(* square each leaf *)
-TreeReader.monadize int_readerize t1 (fun i -> i * i);;
+let env = fun i -> i * i in
+TreeReader.monadize int_readerize t1 env;;
+
+
+
+let incrementer : int -> int state =
+ fun (a : int) -> fun s -> (a, s+1);;
+
+(* incrementer takes an 'a and returns it wrapped in a
+ * State monad that increments the store *)
(* count leaves *)
-TreeState.monadize (fun a s -> (a, s+1)) t1 0;;
+let initial_store = 0 in
+TreeState.monadize incrementer t1 initial_store;;
+
+
(* replace leaves with list *)
-TreeList.monadize (fun i -> [[i;i*i]]) t1;;
+TreeList.monadize (fun i -> [ [i;i*i] ]) t1;;
+
-(* convert tree to list of leaves *)
-TreeCont.monadize (fun a k -> a :: k a) t1 (fun t -> []);;
(* do nothing *)
-TreeCont.monadize unit_cont t1 (fun t-> t);;
+let initial_continuation = fun t -> t in
+TreeCont.monadize unit_cont t1 initial_continuation;;
+
+(* convert tree to list of leaves *)
+let initial_continuation = fun t -> [] in
+TreeCont.monadize (fun a k -> a :: k a) t1 initial_continuation;;
(* square each leaf using continuation *)
-TreeCont.monadize (fun a k -> k (a*a)) t1 (fun t -> t);;
+let initial_continuation = fun t -> t in
+TreeCont.monadize (fun a k -> k (a*a)) t1 initial_continuation;;
(* replace leaves with list, using continuation *)
-TreeCont.monadize (fun a k -> k [a; a*a]) t1 (fun t -> t);;
+let initial_continuation = fun t -> t in
+TreeCont.monadize (fun a k -> k [a; a*a]) t1 initial_continuation;;
(* count leaves, using continuation *)
-TreeCont.monadize (fun a k -> 1 + k a) t1 (fun t -> 0);;
+let initial_continuation = fun t -> 0 in
+TreeCont.monadize (fun a k -> 1 + k a) t1 initial_continuation;;