path: root/proofs/monads.ml

module type Type = sig
  type t
end

module type S = sig
  type +'a t

  val return : 'a -> 'a t
  val bind : 'a t -> ('a -> 'b t) -> 'b t
  val seq : unit t -> 'a t -> 'a t
  val ignore : 'a t -> unit t
(* spiwack: these will suffice for now, if we would use monads more
   globally, then I would add map/join/List.map and such functions,
   plus default implementations *)
end

module type State = sig
  type s (* type of the state *)
  include S

  val set : s -> unit t
  val get : s t
end

module type Writer = sig
  type m (* type of the messages *)
  include S

  val put : m -> unit t
end

module type IO = sig
  include S

  type 'a ref

  val ref : 'a -> 'a ref t
  val (:=) : 'a ref -> 'a -> unit t
  val (!) : 'a ref -> 'a t

  val raise : exn -> 'a t
  val catch : 'a t -> (exn -> 'a t) -> 'a t

  (** In the basic IO monad, [timeout n x] acts as [x] unless it runs for more than [n]
      seconds in which case it raises [IO.Timeout]. *)
  val timeout : int -> 'a t -> 'a t
end

module IO : sig
  include IO

  (** To help distinguish between exceptions raised by the [IO] monad
      from the one used natively by Coq, the former are wrapped in
      [Exception]. *)
  exception Exception of exn
  (** This exception is used to signal abortion in [timeout] functions. *)
  exception Timeout
  (** runs the suspension for its effects *)
  val run : 'a t -> 'a
end = struct
  type 'a t = unit -> 'a

  let run x = x ()

  let return x () = x
  let bind x k () = k (x ()) ()
  let seq x k () = x (); k ()
  let ignore x () = ignore (x ())

  type 'a ref = 'a Pervasives.ref

  let ref x () = Pervasives.ref x
  let (:=) r x () = Pervasives.(:=) r x
  let (!) r () = Pervasives.(!) r

  exception Exception of exn
  let raise e () = raise (Exception e)
  let catch s h () =
    try s ()
    with Exception e -> h e ()

  exception Timeout
  let timeout n t () =
    let timeout_handler _ = Pervasives.raise (Exception Timeout) in
    let psh = Sys.signal Sys.sigalrm (Sys.Signal_handle timeout_handler) in
    Pervasives.ignore (Unix.alarm n);
    let restore_timeout () =
      Pervasives.ignore (Unix.alarm 0);
      Sys.set_signal Sys.sigalrm psh
    in
    try
      let res = t () in
      restore_timeout ();
      res
    with
    | e -> restore_timeout (); Pervasives.raise e

  let _ = Errors.register_handler begin function
    | Timeout -> Errors.errorlabstrm "Monads.IO.timeout" (Pp.str"Timeout!")
    | Exception e -> Errors.print e
    | _ -> Pervasives.raise Errors.Unhandled
  end
end

(* State monad transformer, adapted from Haskell's StateT *)
module State (X:Type) (T:S) : sig
  (* spiwack: it is not essential that both ['a result] and ['a t] be
     private (or either, for that matter). I just hope it will help
     catch more programming errors. *)
  type +'a result = private { result : 'a ; state : X.t }
  include State with type s = X.t and type +'a t = private X.t -> 'a result T.t
  (* a function version of the coercion from the private type ['a t].*)
  val run : 'a t -> s -> 'a result T.t
  val lift : 'a T.t -> 'a t
end = struct
  type s = X.t
  type 'a result = { result : 'a ; state : s }
  type 'a t = s -> 'a result T.t

  let run x = x
  (*spiwack: convenience notation, waiting for ocaml 3.12 *)
  let (>>=) = T.bind
  let return x s = T.return { result=x ; state=s }
  let bind x k s =
    x s >>= fun { result=a ; state=s } ->
    k a s
  let ignore x s =
    x s >>= fun x' ->
    T.return { x' with result=() }
  let seq x t s =
    (x s) >>= fun x' ->
    t x'.state
  let lift x s =
    x >>= fun a ->
    T.return { result=a ; state=s }

  let set s _ =
    T.return { result=() ; state=s }
  let get s =
    T.return { result=s ; state=s }
end

module type Monoid = sig
  type t

  val zero : t
  val ( * ) : t -> t -> t
end

module Writer (M:Monoid) (T:S) : sig
  include Writer with type +'a t = private ('a*M.t) T.t and type m = M.t

  val lift : 'a T.t -> 'a t
  (* The coercion from private ['a t] in function form. *)
  val run : 'a t -> ('a*m) T.t
end = struct

  type 'a t = ('a*M.t) T.t
  type m = M.t

  let run x = x

  (*spiwack: convenience notation, waiting for ocaml 3.12 *)
  let (>>=) = T.bind

  let bind x k = 
    x >>= fun (a,m) ->
    k a >>= fun (r,m') ->
    T.return (r,M.( * ) m m')

  let seq x k =
    x >>= fun ((),m) ->
    k >>= fun (r,m') ->
    T.return (r,M.( * ) m m')

  let return x =
    T.return (x,M.zero)

  let ignore x =
    x >>= fun (_,m) ->
    T.return ((),m)

  let lift x =
    x >>= fun r ->
    T.return (r,M.zero)

  let put m =
    T.return ((),m)
end

(* Double-continuation backtracking monads are reasonable folklore for
   "search" implementations (including Tac interactive prover's
   tactics). Yet it's quite hard to wrap your head around these.  I
   recommand reading a few times the "Backtracking, Interleaving, and
   Terminating Monad Transformers" paper by O. Kiselyov, C. Chen,
   D. Fridman.  The peculiar shape of the monadic type is reminiscent
   of that of the continuation monad transformer.

   The paper also contains the rational for the [split] abstraction.

   An explanation of how to derive such a monad from mathematical
   principles can be found in "Kan Extensions for Program
   Optimisation" by Ralf Hinze.

   One way to think of the [Logic] functor is to imagine ['a
   Logic(X).t] to represent list of ['a] with an exception at the
   bottom (leaving the monad-transforming issues aside, as they don't
   really work well with lists). Each of the element is a valid
   result, sequentialising with a [f:'a -> 'b t] is done by applying
   [f] to each element and then flatten the list, [plus] is
   concatenation, and [split] is pattern-matching. *)
(* spiwack: I added the primitives from [IO] directly in the [Logic]
   signature for convenience. It doesn't really make sense, strictly
   speaking, as they are somewhat deconnected from the semantics of
   [Logic], but without an appropriate language for composition (some
   kind of mixins?) I would be going nowhere with a gazillion
   functors. *)
module type Logic = sig
  include IO

  (* [zero] is usually argument free, but Coq likes to explain errors,
     hence error messages should be carried around. *)
  val zero : exn -> 'a t
  val plus : 'a t -> (exn -> 'a t) -> 'a t
(** Writing (+) for plus and (>>=) for bind, we shall require that

    x+(y+z) = (x+y)+z

    zero+x = x

    x+zero = x

    (x+y)>>=k = (x>>=k)+(y>>=k) *)

  (** [split x] runs [x] until it either fails with [zero e] or finds
      a result [a].  In the former case it returns [Inr e], otherwise
      it returns [Inl (a,y)] where [y] can be run to get more results
      from [x]. It is a variant of prolog's cut. *)
  val split : 'a t -> ('a * (exn -> 'a t) , exn ) Util.union t
end
(* The [T] argument represents the "global" effect: it is not
   backtracked when backtracking occurs at a [plus]. *)
(* spiwack: hence, [T] will not be instanciated with a state monad
   representing the proofs, we will use a surrounding state transformer
   to that effect. In fact at the time I'm writing this comment, I have
   no immediate use for [T]. We might, however, consider instantiating it
   with a "writer" monad around [Pp] to implement [idtac "msg"] for
   instance. *)
module Logic (T:IO) : sig
  include Logic

  (** [run x] raises [e] if [x] is [zero e]. *)
  val run : 'a t -> 'a T.t

  val lift : 'a T.t -> 'a t
end = struct
(* spiwack: the implementation is an adaptation of the aforementionned
   "Backtracking, Interleaving, and Terminating Monad Transformers"
   paper *)
  (* spiwack: [fk] stands for failure continuation, and [sk] for success
     continuation. *)
  type +'r fk = exn -> 'r
  type (-'a,'r) sk = 'a -> 'r fk -> 'r
  type 'a t = { go : 'r. ('a,'r T.t) sk -> 'r T.t fk -> 'r T.t }

  let return x = { go = fun sk fk ->
    sk x fk
                 }
  let bind x k = { go = fun sk fk ->
    x.go (fun a fk -> (k a).go sk fk) fk
                 }
  let ignore x = { go = fun sk fk ->
    x.go (fun _ fk -> sk () fk) fk
                 }
  let seq x t = { go = fun sk fk ->
    x.go (fun () fk -> t.go sk fk) fk
                }

  let zero e = { go = fun _ fk -> fk e }

  let plus x y = { go = fun sk fk ->
    x.go sk (fun e -> (y e).go sk fk)
                 }

  let lift x = { go = fun sk fk ->
    T.bind x (fun a -> sk a fk)
               }

  let run x =
    let sk a _ = T.return a in
    let fk e = raise e in
    x.go sk fk

  (* For [reflect] and [split] see the "Backtracking, Interleaving,
     and Terminating Monad Transformers" paper *)
  let reflect : ('a*(exn -> 'a t) , exn) Util.union -> 'a t = function
    | Util.Inr e -> zero e
    | Util.Inl (a,x) -> plus (return a) x

  let lower x =
    let fk e = T.return (Util.Inr e) in
    let sk a fk = T.return (Util.Inl (a,fun e -> bind (lift (fk e)) reflect)) in
    x.go sk fk  

  let split x =
    lift (lower x)

  (*** IO ***)

  type 'a ref = 'a T.ref
  let ref x = lift (T.ref x)
  let (:=) r x = lift (T.(:=) r x)
  let (!) r = lift (T.(!) r)

  let raise e = lift (T.raise e)
  let catch t h =
    let h' e = lower (h e) in
    bind (lift (T.catch (lower t) h')) reflect

  (** [timeout n x] can have several success. It succeeds as long as,
      individually, each of the past successes run in less than [n]
      seconds.
      In case of timeout if fails with [zero Timeout]. *)
  let rec timeout n x =
    (* spiwack: adds a [T.return] in front of [x] in order to force
       [lower] to go into [T] before running [x]. The problem is that
       in a continuation-passing monad transformer, the monadic
       operations don't make use of the underlying ones. Hence, when
       going back to a lower monad, much computation can be done
       before returning (and running the lower monad). It is
       undesirable for timeout, obviously. *)
    let x = seq (lift (T.return ())) x in
    let x' =
      (* spiwack: this could be a [T.map] if provided *)
      T.bind (lower x) begin function
        | Util.Inr _ as e -> T.return e
        | Util.Inl (a,h) -> T.return (Util.Inl (a, fun e -> timeout n (h e)))
      end
    in
    (* spiwack: we report timeouts as resumable failures. *)
    bind (catch (lift (T.timeout n x')) begin function
      | IO.Timeout -> zero IO.Timeout
      | e -> raise e
    end) reflect
end


(* [State(X)(T:Logic)] can be lifted to [Logic] by backtracking on state on [plus].*)
module StateLogic(X:Type)(T:Logic) : sig
  (* spiwack: some duplication from interfaces as we need ocaml 3.12
     to substitute inside signatures. *)
  type s = X.t
  type +'a result = private { result : 'a ; state : s }
  include Logic with type +'a t = private X.t -> 'a result T.t

  val set : s -> unit t
  val get : s t

  val lift : 'a T.t -> 'a t
  val run : 'a t -> s -> 'a result T.t
end = struct

  module S = State(X)(T)
  include S

  let zero e = lift (T.zero e)
  let plus x y =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    get >>= fun initial ->
    lift begin
      (T.plus (run x initial) (fun e -> run (y e) initial)) 
    end >>= fun { result = a ; state = final } ->
    set final >>
    return a
  (* spiwack: the definition of [plus] is essentially [plus x y = fun s
     -> T.plus (run x s) (run y s)]. But the [private] annotation
     prevents us from writing that. Maybe I should remove [private]
     around [+'a t] (it would be unnecessary to remove that of ['a
     result]) after all. I'll leave it like that for now. *)

  let split x =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    get >>= fun initial ->
    lift (T.split (run x initial)) >>= function
      | Util.Inr _ as e -> return e
      | Util.Inl (a,y) ->
          let y' e =
            lift (y e) >>= fun b ->
            set b.state >>
            return b.result
          in
          set a.state >>
          return (Util.Inl(a.result,y'))


  (*** IO ***)

  type 'a ref = 'a T.ref
  let ref x = lift (T.ref x)
  let (:=) r x = lift (T.(:=) r x)
  let (!) r = lift (T.(!) r)

  let raise e = lift (T.raise e)
  let catch t h =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    get >>= fun initial ->
    let h' e = run (h e) initial in
    lift (T.catch (run t initial) h') >>= fun a ->
    set a.state >>
    return a.result

  exception Timeout
  let timeout n t =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    get >>= fun initial ->
    lift (T.timeout n (run t initial)) >>= fun a ->
    set a.state >>
    return a.result
end

(* [Writer(M)(T:Logic)] can be lifted to [Logic] by backtracking on state on [plus].*)
module WriterLogic(M:Monoid)(T:Logic) : sig
  (* spiwack: some duplication from interfaces as we need ocaml 3.12
     to substitute inside signatures. *)
  type m = M.t
  include Logic with type +'a t = private ('a*m) T.t

  val put : m -> unit t

  val lift : 'a T.t -> 'a t
  val run : 'a t -> ('a*m) T.t
end = struct
  module W = Writer(M)(T)
  include W

  let zero e = lift (T.zero e)
  let plus x y =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    lift begin
      (T.plus (run x) (fun e -> run (y e))) 
    end >>= fun (r,m) ->
    put m >>
    return r

  let split x =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    lift (T.split (run x)) >>= function
      | Util.Inr _ as e -> return e
      | Util.Inl ((a,m),y) ->
          let y' e =
            lift (y e) >>= fun (b,m) ->
            put m >>
            return b
          in
          put m >>
          return (Util.Inl(a,y'))


  (*** IO ***)

  type 'a ref = 'a T.ref
  let ref x = lift (T.ref x)
  let (:=) r x = lift (T.(:=) r x)
  let (!) r = lift (T.(!) r)

  let raise e = lift (T.raise e)
  let catch t h =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    let h' e = run (h e) in
    lift (T.catch (run t) h') >>= fun (a,m) ->
    put m >>
    return a

  exception Timeout
  let timeout n t =
    (* spiwack: convenience notation, waiting for ocaml 3.12 *)
    let (>>=) = bind in
    let (>>) = seq in
    lift (T.timeout n (run t)) >>= fun (a,m) ->
    put m >>
    return a
end