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|
(*********************************************)
(* (Extended) Krivine Machine implementation *)
(* *)
(* Guillermo Ramos Gutiérrez (2015) *)
(*********************************************)
(*********)
(* Utils *)
(*********)
let print_with f x = print_endline (f x)
let concat_with sep f xs = String.concat sep (List.map f xs)
let rec find_idx_exn x = function
| [] -> raise Not_found
| (y::ys) -> if x = y then 0 else 1 + find_idx_exn x ys
let rec map_rev f xs =
let rec iter acc = function
| [] -> acc
| (x::xs) -> iter (f x :: acc) xs in
iter [] xs
type symbol = string * int
let show_symbol (s, _) = s
let symbol : string -> symbol =
let id = ref 0 in
let gensym c : symbol =
let newid = !id in
id := !id + 1;
(c, newid)
in
gensym
(**************************************)
(* (Extended) Untyped Lambda Calculus *)
(**************************************)
module Lambda = struct
type t = Var of symbol | App of t * t | Abs of symbol * t
| If of t * t * t | True | False
| Constr of t constr | Case of t * (symbol constr * t) list
| Fix of symbol * t
and 'a constr = symbol * 'a list
let rec show m =
let show_branch ((x, args), m) =
"(" ^ show_symbol x ^ "(" ^ concat_with "," show_symbol args ^ ") → "
^ show m ^ ")" in
match m with
| Var x -> show_symbol x
| App (m1, m2) -> "(" ^ show m1 ^ " " ^ show m2 ^ ")"
| Abs (x, m) -> "(λ" ^ show_symbol x ^ "." ^ show m ^ ")"
| If (m1, m2, m3) -> "if " ^ show m1
^ " then " ^ show m2
^ " else " ^ show m3
| True -> "True" | False -> "False"
| Constr (x, ms) -> show_symbol x ^ "(" ^ concat_with ", " show ms ^ ")"
| Case (m, bs) -> "(case " ^ show m ^ " of "
^ concat_with " " show_branch bs ^ ")"
| Fix (x, m) -> "fix(λ" ^ show_symbol x ^ "." ^ show m ^ ")"
let print = print_with show
(* Constants *)
let none = symbol "None"
let some = symbol "Some"
(* Peano arithmetic helpers *)
let z = symbol "z"
let s = symbol "s"
let rec peano_add n x =
if n == 0 then x else peano_add (n-1) (Constr (s, [x]))
let peano_of_int ?(base=Constr (z, [])) n = peano_add n base
(* Examples *)
(* (λx.((λy.y) x)) *)
let ex_m1 =
let x = symbol "x" in
let y = symbol "y" in
Abs (x, App (Abs (y, Var y), Var x))
(* (((λx.(λy.(y x))) (λz.z)) (λy.y)) *)
let ex_m2 =
let x = symbol "x" in
let y = symbol "y" in
let z = symbol "z" in
App (App (Abs (x, Abs (y, App (Var y, Var x))), Abs (z, Var z)), Abs (y, Var y))
(* (λc. case c of (Some(x) → x) (None → c)) Some(s(z))) *)
let ex_case_some =
let c = symbol "c" in
let x = symbol "x" in
App (Abs (c, Case (Var c, [((some, [x]), Var x); ((none, []), Var c)])),
Constr (some, [peano_of_int 1]))
(* (λc. case c of (Triple(x,y,z) → y)) Triple(1,2,3)) *)
let ex_case_tuple =
let c = symbol "c" in
let x = symbol "x" in
let y = symbol "y" in
let z = symbol "z" in
let triple = symbol "triple" in
App (Abs (c, Case (Var c, [((triple, [x;y;z]), Var y)])),
Constr (triple, List.map peano_of_int [1;2;3]))
(* fix(λf.λc. case c of (s(x) → s(s(f x))) (z → z)) s(s(s(z))) *)
let ex_fixpt_mul2 =
let f = symbol "f" in
let c = symbol "c" in
let x = symbol "x" in
App (Fix (f, Abs (c, Case (Var c,
[((s, [x]), peano_add 2 (App (Var f, Var x)));
((z, []), peano_of_int 0)]))),
peano_of_int 3)
(* fix(λf.λg.λc. case c of (s(x) → g (f g x)) (z → z)) (λy.s(s(s(y)))) s(s(s(z))) *)
let ex_fix_scale =
let f = symbol "f" in
let g = symbol "g" in
let c = symbol "c" in
let x = symbol "x" in
let y = symbol "y" in
App (App (Fix (f, Abs (g, Abs (c, Case (Var c,
[((s, [x]), App (Var g, (App (App (Var f, Var g), Var x))));
((z, []), peano_of_int 0)])))),
Abs (y, peano_add 3 (Var y))),
peano_of_int 3)
end
(**************************************************)
(* Untyped lambda calculus with De Bruijn indices *)
(**************************************************)
module DBILambda = struct
type dbi_symbol = int * symbol
type t = Var of dbi_symbol | App of t * t | Abs of symbol * t
| If of t * t * t | True | False
| Constr of t constr | Case of t * (symbol constr * t) list
| Fix of symbol * t
and 'a constr = symbol * 'a list
let dbi dbis x = (find_idx_exn x dbis, x)
let output_dbi = false
let show_dbi_symbol (n, x) =
if output_dbi then string_of_int n else show_symbol x
let show_dbi_param x =
if output_dbi then "" else show_symbol x
let rec show m =
match m with
| Var x -> show_dbi_symbol x
| App (m1, m2) -> "(" ^ show m1 ^ " " ^ show m2 ^ ")"
| Abs (x, m) -> "(λ" ^ show_dbi_param x ^ "." ^ show m ^ ")"
| If (m1, m2, m3) -> "if " ^ show m1
^ " then " ^ show m2
^ " else " ^ show m3
| True -> "True" | False -> "False"
| Constr (x, ms) -> show_symbol x ^ "(" ^ concat_with ", " show ms ^ ")"
| Case (m, bs) -> "(case " ^ show m ^ " of "
^ concat_with " " show_branch bs ^ ")"
| Fix (x, m) -> "fix(λ" ^ show_symbol x ^ "." ^ show m ^ ")"
and show_branch ((x, args), m) =
"(" ^ show_symbol x ^ "(" ^ concat_with "," show_symbol args ^ ") → "
^ show m ^ ")"
let print = print_with show
let of_lambda =
let rec of_lambda dbis = function
| Lambda.Var x -> let (n, x) = dbi dbis x in Var (n, x)
| Lambda.App (m1, m2) -> App (of_lambda dbis m1, of_lambda dbis m2)
| Lambda.Abs (x, m) -> Abs (x, of_lambda (x :: dbis) m)
| Lambda.If (m1, m2, m3) -> If (of_lambda dbis m1,
of_lambda dbis m2, of_lambda dbis m3)
| Lambda.True -> True | Lambda.False -> False
| Lambda.Constr (x, ms) -> Constr (x, List.map (of_lambda dbis) ms)
| Lambda.Case (m, bs) -> Case (of_lambda dbis m,
List.map (trans_br dbis) bs)
| Lambda.Fix (x, m) -> Fix (x, of_lambda (x :: dbis) m)
and trans_br dbis ((x, args), m) =
let dbis = List.rev args @ dbis in
((x, args), of_lambda dbis m) in
of_lambda []
end
(*******************)
(* Krivine Machine *)
(*******************)
module KM = struct
open DBILambda
type st_elm = Clos of DBILambda.t * stack
| IfCont of DBILambda.t * DBILambda.t
| CaseCont of (symbol DBILambda.constr * DBILambda.t) list * stack
| FixClos of symbol * DBILambda.t * stack
and stack = st_elm list
type state = DBILambda.t * stack * stack
let reduce m =
let rec reduce (st : state) : DBILambda.t =
match st with
(* Pure lambda calculus *)
| (Var (0, _), s, Clos (m, e') :: e) -> reduce (m, s, e')
| (Var (0, _), s, FixClos (f, m, e') :: e) ->
reduce (m, s, FixClos (f, m, e') :: e')
| (Var (n, x), s, _ :: e) -> reduce (Var (n-1, x), s, e)
| (App (m1, m2), s, e) -> reduce (m1, Clos (m2, e) :: s, e)
| (Abs (_, m), c :: s, e) -> reduce (m, s, c :: e)
(* Conditionals *)
| (If (m1, m2, m3), s, e) -> reduce (m1, IfCont (m2, m3) :: s, e)
| (True, IfCont (m2, m3) :: s, e) -> reduce (m2, s, e)
| (False, IfCont (m2, m3) :: s, e) -> reduce (m3, s, e)
(* Case expressions (+ constructors) *)
| (Case (m, bs), s, e) -> reduce (m, CaseCont (bs, e) :: s, e)
| (Constr (x, ms), CaseCont (((x', args), m) :: bs, e') :: s, e)
when x == x' && List.length ms == List.length args ->
reduce (List.fold_left (fun m x -> Abs (x, m)) m args,
map_rev (fun m -> Clos (m, e)) ms @ s, e')
| (Constr (x, ms), CaseCont (_ :: bs, e') :: s, e) ->
reduce (Constr (x, ms), CaseCont (bs, e') :: s, e)
| (Constr (x, ms), s, e) ->
Constr (x, List.map (fun m -> reduce (m, s, e)) ms)
(* Fixpoints *)
| (Fix (x, m), s, e) -> reduce (m, s, FixClos (x, m, e) :: e)
(* Termination checks *)
| (m, [], []) -> m
| (_, _, _) -> m in
reduce (m, [], [])
end
let dbi_and_red m =
let dbi_m = DBILambda.of_lambda m in
print_endline ("# Lambda term:\n " ^ DBILambda.show dbi_m);
let red_m = KM.reduce dbi_m in
print_endline ("# Reduced term:\n " ^ DBILambda.show red_m);
print_endline "----------------------------------------------------------\n"
let () =
let open Lambda in
List.iter dbi_and_red [ex_m1; ex_m2; ex_case_some;
ex_case_tuple; ex_fixpt_mul2; ex_fix_scale]
|
