(*
* tree_monadize.ml
*
+ * 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 m = env -> 'a;;
+ let unit a : 'a m = fun e -> a;;
+ let bind (u : 'a m) (f : 'a -> 'b m) : 'b m =
+ fun e -> f (u e) e;;
+end
+
+module State_monad = struct
+ (* change this to suit your needs *)
+ type store = int;;
+
+ type 'a m = store -> 'a * store;;
+ let unit a : 'a m = fun s -> (a, s);;
+ let bind (u : 'a m) (f : 'a -> 'b m) : 'b m =
+ fun s -> (let (a, s') = u s in (f a) s');;
+end
+
+module List_monad = struct
+ type 'a m = 'a list;;
+ let unit a : 'a m = [a];;
+ let bind (u: 'a m) (f : 'a -> 'b m) : 'b m =
+ 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 M_monad
+ * module. Also we have to explicitly declare what commonality we're expecting
+ * from M_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:
* 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
+ * * module parameter that will choose
+ * * whether b = list or b = option or b = reader...
+ * * This module parameter may supply values as well as types *)
+ * M : 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 *)
+ * * What we're doing here is building a new module that makes
+ * * use of the module that was supplied as M *)
* struct
- * type ('a) t = 'a -> ('a) Parm.b
+ * type ('a) t = 'a -> ('a) M.b
* end
* And here's how you'd use it:
* module T_list = T_maker(struct type 'a b = 'a list 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 M_monad module.
*)
type 'a tree = Leaf of 'a | Node of ('a tree) * ('a tree);;
(Leaf 7, Leaf 11)));;
-module Tree_monadizer(Parm : sig
+module Tree_monadizer(M : 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 *)
+ * for unit and bind, as well as a monadic type constructor *)
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 Parm.m) (t: 'a tree) : 'b tree Parm.m =
+ let rec monadize (f: 'a -> 'b M.m) (t: 'a tree) : 'b tree M.m =
match t with
- | Leaf a -> Parm.bind (f a) (fun b -> Parm.unit (Leaf b))
+ | Leaf a -> M.bind (f a) (fun b -> M.unit (Leaf b))
| Node(l, r) ->
- Parm.bind (monadize f l) (fun l' ->
- Parm.bind (monadize f r) (fun r' ->
- Parm.unit (Node (l', r'))))
+ M.bind (monadize f l) (fun l' ->
+ M.bind (monadize f r) (fun r' ->
+ M.unit (Node (l', r'))))
end;;
-type env = int -> int;;
-
-type 'a reader = env -> 'a;;
-let reader_unit a : 'a reader = fun e -> a;;
-let reader_bind (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.
+(* Now we supply 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 = reader_unit
- let bind = reader_bind
-end);;
+module TreeReader = Tree_monadizer(Reader_monad);;
-type store = int;;
-
-type 'a state = store -> 'a * store;;
-let state_unit a : 'a state = fun s -> (a, s);;
-let state_bind (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 = state_unit
- let bind = state_bind
-end);;
+module TreeState = Tree_monadizer(State_monad);;
-let list_unit a = [a];;
-let list_bind (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 = list_unit
- let bind = list_bind
-end);;
+module TreeList = Tree_monadizer(List_monad);;
+
+(* The Continuation monad is a bit more complicated *)
+module Continuation_monad = struct
+ type ('r,'a) m = ('a -> 'r) -> 'r;;
+ let unit a : ('r,'a) m = fun k -> k a;;
+ let bind (u: ('r,'a) m) (f: 'a -> ('r,'b) m) : ('r,'b) m =
+ fun k -> u (fun a -> f a k);;
+end
-(* 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
+(* Since the Continuation monad is parameterized on two types---it's
+ * ('r,'a) 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,'x) m type constructor in its
+ * Tree_monadizer2 that takes a ('r,'a) monad 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
+module Tree_monadizer2(M : sig
+ type ('r,'a) m
+ val unit : 'a -> ('r,'a) m
+ val bind : ('r,'a) m -> ('a -> ('r,'b) m) -> ('r,'b) m
end) = struct
(* 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 =
+ let rec monadize (f: 'a -> ('r,'b) M.m) (t: 'a tree) : ('r,'b tree) M.m =
match t with
- | Leaf a -> Parm.bind (f a) (fun b -> Parm.unit (Leaf b))
+ | Leaf a -> M.bind (f a) (fun b -> M.unit (Leaf b))
| Node(l, r) ->
- Parm.bind (monadize f l) (fun l' ->
- Parm.bind (monadize f r) (fun r' ->
- Parm.unit (Node (l', r'))))
+ M.bind (monadize f l) (fun l' ->
+ M.bind (monadize f r) (fun r' ->
+ M.unit (Node (l', r'))))
end;;
-type ('a,'r) cont = ('a -> 'r) -> 'r;;
-let cont_unit a : ('a,'r) cont = fun k -> k a;;
-let cont_bind (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 = cont_unit
- let bind = cont_bind
-end);;
+module TreeCont = Tree_monadizer2(Continuation_monad);;
*)
-let int_readerize : int -> int reader =
+let asker : int -> int Reader_monad.m =
fun (a : int) -> fun (env : int -> int) -> env a;;
-(* int_readerize takes an int and returns a Reader monad that
+(* asker 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 *)
let env = fun i -> i + i in
-TreeReader.monadize int_readerize t1 env;;
+TreeReader.monadize asker t1 env;;
+
+(* You can also avoid declaring a separate toplevel TreeReader module
+ * (or even a separate Reader_monad module) by using one of these forms:
+ * ...
+ * let module T = Tree_monadizer(Reader_monad) in
+ * T.monadize asker t1 env;;
+ * or:
+ * ...
+ * let env = fun i -> i + i in
+ * let module Monad = struct
+ * type env = int -> int;;
+ * type 'a m = env -> 'a;;
+ * let unit a : 'a m = fun e -> a;;
+ * let bind (u : 'a m) (f : 'a -> 'b m) : 'b m =
+ * fun e -> f (u e) e;;
+ * end in
+ * let module T = Tree_monadizer(Monad) in
+ * T.monadize asker t1 env;;
+ * or:
+ * ...
+ * let module T = Tree_monadizer(struct
+ * type env = int -> int;;
+ * type 'a m = env -> 'a;;
+ * let unit a : 'a m = fun e -> a;;
+ * let bind (u : 'a m) (f : 'a -> 'b m) : 'b m =
+ * fun e -> f (u e) e;;
+ * end) in
+ * T.monadize asker t1 env;;
+ *)
+
(* square each leaf *)
let env = fun i -> i * i in
-TreeReader.monadize int_readerize t1 env;;
+TreeReader.monadize asker t1 env;;
-let incrementer : int -> int state =
- fun (a : int) -> fun s -> (a, s+1);;
+(* count leaves *)
+let incrementer : int -> int State_monad.m =
+ 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 *)
let initial_store = 0 in
TreeState.monadize incrementer t1 initial_store;;
+(* annotate leaves as they're visited *)
+
+let annotater : int -> (int * int) State_monad.m =
+ fun (a : int) -> fun s -> ((a,s+1), s+1);;
+
+let initial_store = 0 in
+TreeState.monadize annotater t1 initial_store;;
+
+
+(* copy tree with different choices for leaves *)
+let chooser i = if i = 2 then [20; 21] else [i];;
-(* replace leaves with list *)
-TreeList.monadize (fun i -> [ [i;i*i] ]) t1;;
+TreeList.monadize chooser t1;;
(* do nothing *)
let initial_continuation = fun t -> t in
-TreeCont.monadize cont_unit t1 initial_continuation;;
+TreeCont.monadize Continuation_monad.unit t1 initial_continuation;;
(* convert tree to list of leaves *)
let initial_continuation = fun t -> [] in
let initial_continuation = fun t -> t in
TreeCont.monadize (fun a k -> k (a*a)) t1 initial_continuation;;
-(* replace leaves with list, using continuation *)
-let initial_continuation = fun t -> t in
-TreeCont.monadize (fun a k -> k [a; a*a]) t1 initial_continuation;;
-
(* count leaves, using continuation *)
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, treeTC_reader =
- 'a tree reader;;
-
- let unit (a: 'a) : 'a tree reader =
- 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'));;