147 lines
4.5 KiB
OCaml
147 lines
4.5 KiB
OCaml
(* $Id$ *)
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open StdLabels
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open MoreLabels
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(* Use maps for substitutions and sets for free variables *)
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module Subst = Map.Make(struct type t = string let compare = compare end)
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module Names = Set.Make(struct type t = string let compare = compare end)
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(* Variables are common to lambda and expr *)
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type var = [`Var of string]
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let subst_var ~subst : var -> _ =
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function `Var s as x ->
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try Subst.find s subst
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with Not_found -> x
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let free_var : var -> _ = function `Var s -> Names.singleton s
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(* The lambda language: free variables, substitutions, and evaluation *)
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type 'a lambda = [`Var of string | `Abs of string * 'a | `App of 'a * 'a]
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let free_lambda ~free_rec : _ lambda -> _ = function
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#var as x -> free_var x
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| `Abs (s, t) -> Names.remove s (free_rec t)
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| `App (t1, t2) -> Names.union (free_rec t1) (free_rec t2)
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let map_lambda ~map_rec : _ lambda -> _ = function
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#var as x -> x
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| `Abs (s, t) as l ->
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let t' = map_rec t in
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if t == t' then l else `Abs(s, t')
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| `App (t1, t2) as l ->
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let t'1 = map_rec t1 and t'2 = map_rec t2 in
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if t'1 == t1 && t'2 == t2 then l else `App (t'1, t'2)
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let next_id =
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let current = ref 3 in
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fun () -> incr current; !current
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let subst_lambda ~subst_rec ~free ~subst : _ lambda -> _ = function
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#var as x -> subst_var ~subst x
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| `Abs(s, t) as l ->
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let used = free t in
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let used_expr =
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Subst.fold subst ~init:[]
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~f:(fun ~key ~data acc ->
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if Names.mem s used then data::acc else acc) in
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if List.exists used_expr ~f:(fun t -> Names.mem s (free t)) then
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let name = s ^ string_of_int (next_id ()) in
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`Abs(name,
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subst_rec ~subst:(Subst.add ~key:s ~data:(`Var name) subst) t)
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else
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map_lambda ~map_rec:(subst_rec ~subst:(Subst.remove s subst)) l
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| `App _ as l ->
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map_lambda ~map_rec:(subst_rec ~subst) l
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let eval_lambda ~eval_rec ~subst l =
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match map_lambda ~map_rec:eval_rec l with
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`App(`Abs(s,t1), t2) ->
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eval_rec (subst ~subst:(Subst.add ~key:s ~data:t2 Subst.empty) t1)
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| t -> t
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(* Specialized versions to use on lambda *)
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let rec free1 x = free_lambda ~free_rec:free1 x
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let rec subst1 ~subst = subst_lambda ~subst_rec:subst1 ~free:free1 ~subst
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let rec eval1 x = eval_lambda ~eval_rec:eval1 ~subst:subst1 x
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(* The expr language of arithmetic expressions *)
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type 'a expr =
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[`Var of string | `Num of int | `Add of 'a * 'a
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| `Neg of 'a | `Mult of 'a * 'a]
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let free_expr ~free_rec : _ expr -> _ = function
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#var as x -> free_var x
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| `Num _ -> Names.empty
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| `Add(x, y) -> Names.union (free_rec x) (free_rec y)
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| `Neg x -> free_rec x
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| `Mult(x, y) -> Names.union (free_rec x) (free_rec y)
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(* Here map_expr helps a lot *)
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let map_expr ~map_rec : _ expr -> _ = function
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#var as x -> x
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| `Num _ as x -> x
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| `Add(x, y) as e ->
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let x' = map_rec x and y' = map_rec y in
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if x == x' && y == y' then e
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else `Add(x', y')
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| `Neg x as e ->
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let x' = map_rec x in
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if x == x' then e else `Neg x'
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| `Mult(x, y) as e ->
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let x' = map_rec x and y' = map_rec y in
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if x == x' && y == y' then e
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else `Mult(x', y')
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let subst_expr ~subst_rec ~subst : _ expr -> _ = function
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#var as x -> subst_var ~subst x
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| #expr as e -> map_expr ~map_rec:(subst_rec ~subst) e
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let eval_expr ~eval_rec e =
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match map_expr ~map_rec:eval_rec e with
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`Add(`Num m, `Num n) -> `Num (m+n)
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| `Neg(`Num n) -> `Num (-n)
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| `Mult(`Num m, `Num n) -> `Num (m*n)
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| #expr as e -> e
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(* Specialized versions *)
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let rec free2 x = free_expr ~free_rec:free2 x
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let rec subst2 ~subst = subst_expr ~subst_rec:subst2 ~subst
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let rec eval2 x = eval_expr ~eval_rec:eval2 x
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(* The lexpr language, reunion of lambda and expr *)
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type lexpr =
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[ `Var of string | `Abs of string * lexpr | `App of lexpr * lexpr
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| `Num of int | `Add of lexpr * lexpr | `Neg of lexpr
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| `Mult of lexpr * lexpr ]
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let rec free : lexpr -> _ = function
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#lambda as x -> free_lambda ~free_rec:free x
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| #expr as x -> free_expr ~free_rec:free x
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let rec subst ~subst:s : lexpr -> _ = function
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#lambda as x -> subst_lambda ~subst_rec:subst ~subst:s ~free x
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| #expr as x -> subst_expr ~subst_rec:subst ~subst:s x
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let rec eval : lexpr -> _ = function
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#lambda as x -> eval_lambda ~eval_rec:eval ~subst x
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| #expr as x -> eval_expr ~eval_rec:eval x
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(* A few examples:
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eval1 (`App(`Abs("x",`Var"x"), `Var"y"));;
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eval2 (`Add(`Mult(`Num 3,`Neg(`Num 2)), `Var"x"));;
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eval (`Add(`App(`Abs("x",`Mult(`Var"x",`Var"x")),`Num 2), `Num 5));;
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*)
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