(*
* tree_monadize.ml
*
- * 'a and so on are type variables in OCaml; they stand for arbitrary types
+ * If you've got some block of code that uses `unit`s and `bind`s, and you
+ * want to interpret it alternately using this monad, that monad, or another
+ * monad, you can use OCaml's module system. You'd write your code like this:
+ *)
+
+module Reader_monad = struct
+ (* change this to suit your needs *)
+ type env = int -> int;;
+
+ type 'a monad = env -> 'a;;
+ let unit a : 'a monad = fun e -> a;;
+ let bind (u : 'a monad) (f : 'a -> 'b monad) : 'b monad =
+ fun e -> f (u e) e;;
+end
+
+module State_monad = struct
+ (* change this to suit your needs *)
+ type store = int;;
+
+ type 'a monad = store -> 'a * store;;
+ let unit a : 'a monad = fun s -> (a, s);;
+ let bind (u : 'a monad) (f : 'a -> 'b monad) : 'b monad =
+ fun s -> (let (a, s') = u s in (f a) s');;
+end
+
+module List_monad = struct
+ type 'a monad = 'a list;;
+ let unit a : 'a monad = [a];;
+ let bind (u: 'a monad) (f : 'a -> 'b monad) : 'b monad =
+ List.concat(List.map f u);;
+end
+
+(*
+ * Then you can replace code that looks like this:
+ * ... reader_bind ...
+ * with code that looks like this:
+ * ... Reader_monad.bind ...
+ * and the latter can be reformulated like this:
+ * let open Reader_monad in ... bind ...
+ * or equivalently, like this:
+ * Reader_monad.(... bind ...)
+ * Then you can use literally the same `... bind ...` code when writing instead:
+ * State_monad.(... bind ...)
+ *)
+
+(* That's great, however it still requires us to repeat the
+ * `... bind ...` code every time we want to change which monad we're working
+ * with. Shouldn't there be a way to _parameterize_ the `... bind ...` code
+ * on a monad, so that we only have to write the `... bind ...` code once,
+ * but can invoke it alternately with the Reader_monad supplied as an
+ * argument, or the State_monad, or another?
+ *
+ * There is a way to do this, but it requires putting the `... bind ...` code in
+ * its own module, and making that module parameterized on some X_monad
+ * module. Also we have to explicitly declare what commonality we're expecting
+ * from X_monad modules we're going to use as parameters. We'll explain how to
+ * do this in a moment.
+ *
+ * As preparation, a general observation:
+ * '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 *)
+ * X : 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 X *)
+ * struct
+ * type ('a) t = 'a -> ('a) X.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.
+ * I know, it seems unnecessarily complicated. Nonetheless, that's how it
+ * works. And that is also the technique we'll use to make our
+ * `... bind ...` code parametric on some X_monad module.
*)
type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
(Leaf 7, Leaf 11)));;
-module Tree_monadizer(S: sig
- type 'a m
- val unit : 'a -> 'a m
- val bind : 'a m -> ('a -> 'b m) -> 'b m
+module Tree_monadizer(X : 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 monad
+ val unit : 'a -> 'a monad
+ val bind : 'a monad -> ('a -> 'b monad) -> 'b monad
end) = struct
- let rec monadize (f: 'a -> 'b S.m) (t: 'a tree) : 'b tree S.m =
+ let rec monadize (f: 'a -> 'b X.monad) (t: 'a tree) : 'b tree X.monad =
match t with
- | Leaf a -> S.bind (f a) (fun b -> S.unit (Leaf b))
+ | Leaf a -> X.bind (f a) (fun b -> X.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'))))
+ X.bind (monadize f l) (fun l' ->
+ X.bind (monadize f r) (fun r' ->
+ X.unit (Node (l', r'))))
end;;
-type env = int -> int;;
-
-type 'a reader = env -> 'a;;
-let unit_reader a : 'a reader = fun e -> a;;
-let bind_reader (u : 'a reader) (f : 'a -> 'b reader) : 'b reader =
- fun e -> f (u e) e;;
-
-module TreeReader = Tree_monadizer(struct
- type 'a m = 'a reader
- let unit = unit_reader
- let bind = bind_reader
-end);;
+(* 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(Reader_monad);;
-type store = int;;
+(* Make a TreeState module containing monadize specialized to the State monad *)
+module TreeState = Tree_monadizer(State_monad);;
-type 'a state = store -> 'a * store;;
-let unit_state a : 'a state = fun s -> (a, s);;
-let bind_state (u : 'a state) (f : 'a -> 'b state) : 'b state =
- fun s -> (let (a, s') = u s in (f a) s');;
-module TreeState = Tree_monadizer(struct
- type 'a m = 'a state
- let unit = unit_state
- let bind = bind_state
-end);;
+(* Make a TreeList module containing monadize specialized to the List monad *)
+module TreeList = Tree_monadizer(List_monad);;
-let unit_list a = [a];;
-let bind_list (u: 'a list) (f : 'a -> 'b list) : 'b list =
- List.concat(List.map f u);;
+(* The Continuation monad is a bit more complicated *)
+module Continuation_monad = struct
+ type ('a,'r) monad = ('a -> 'r) -> 'r;;
+ let unit a : ('a,'r) monad = fun k -> k a;;
+ let bind (u: ('a,'r) monad) (f: 'a -> ('b,'r) monad) : ('b,'r) monad =
+ fun k -> u (fun a -> f a k);;
+end
-module TreeList = Tree_monadizer(struct
- type 'a m = 'a list
- let unit = unit_list
- let bind = bind_list
-end);;
-
-
-
-(* we need to a new module when the monad is parameterized on two types *)
-module Tree_monadizer2(S: sig
- type ('a,'x) m
- val unit : 'a -> ('a,'x) m
- val bind : ('a,'x) m -> ('a -> ('b,'x) m) -> ('b,'x) m
+(* Since the Continuation monad is parameterized on two types---it's
+ * ('a,'r) cont not ('a) cont---we can't match the type ('a) monad that
+ * Tree_monadizer expects in its parameter. So we have to make a different
+ * Tree_monadizer2 that takes a ('a,'z) monad type constructor in its
+ * parameter instead *)
+module Tree_monadizer2(X : sig
+ type ('a,'z) monad
+ val unit : 'a -> ('a,'z) monad
+ val bind : ('a,'z) monad -> ('a -> ('b,'z) monad) -> ('b,'z) monad
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) X.monad) (t: 'a tree) : ('b tree,'x) X.monad =
match t with
- | Leaf a -> S.bind (f a) (fun b -> S.unit (Leaf b))
+ | Leaf a -> X.bind (f a) (fun b -> X.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'))))
+ X.bind (monadize f l) (fun l' ->
+ X.bind (monadize f r) (fun r' ->
+ X.unit (Node (l', r'))))
end;;
-type ('a,'r) cont = ('a -> 'r) -> 'r;;
-let unit_cont a : ('a,'r) cont = fun k -> k a;;
-let bind_cont (u: ('a,'r) cont) (f: 'a -> ('b,'r) cont) : ('b,'r) cont =
- fun k -> u (fun a -> f a k);;
-
-module TreeCont = Tree_monadizer2(struct
- type ('a,'r) m = ('a,'r) cont
- let unit = unit_cont
- let bind = bind_cont
-end);;
+(* Make a TreeCont module containing monadize specialized to the Cont monad *)
+module TreeCont = Tree_monadizer2(Continuation_monad);;
* http://lambda.jimpryor.net/manipulating_trees_with_monads/
*)
-let int_readerize : int -> int reader =
- fun (a : int) (modifier : int -> int) -> modifier a;;
+
+let int_readerize : int -> int Reader_monad.monad =
+ 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;;
+
+(* You can also avoid declaring a separate toplevel TreeReader module
+ * (or even a separate Reader_monad module) by ysing one of these forms:
+ * ...
+ * let module T = Tree_monadizer(Reader_monad) in
+ * T.monadize int_readerize t1 env;;
+ * or:
+ * ...
+ * let env = fun i -> i + i in
+ * let module Monad = struct
+ * type env = int -> int;;
+ * type 'a monad = env -> 'a;;
+ * let unit a : 'a monad = fun e -> a;;
+ * let bind (u : 'a monad) (f : 'a -> 'b monad) : 'b monad =
+ * fun e -> f (u e) e;;
+ * end in
+ * let module T = Tree_monadizer(Monad) in
+ * T.monadize int_readerize t1 env;;
+ * or:
+ * ...
+ * let module T = Tree_monadizer(struct
+ * type env = int -> int;;
+ * type 'a monad = env -> 'a;;
+ * let unit a : 'a monad = fun e -> a;;
+ * let bind (u : 'a monad) (f : 'a -> 'b monad) : 'b monad =
+ * fun e -> f (u e) e;;
+ * end) in
+ * T.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_monad.monad =
+ 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 Continuation_monad.unit 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;;
+
+(*
+(* Tree monad *)
+(* type 'a tree defined above *)
+let tree_unit (a: 'a) : 'a tree = Leaf a;;
+let rec tree_bind (u : 'a tree) (f : 'a -> 'b tree) : 'b tree =
+ match u with
+ | Leaf a -> f a
+ | Node (l, r) -> Node (tree_bind l f, tree_bind r f);;
+type ('a) treeT_reader =
+ 'a tree reader;;
+
+let unit (a: 'a) : 'a tree reader =
+ reader_unit (Leaf a);;
+
+let rec bind (u : 'a tree_reader) (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'));;
+
+ *)