+++ /dev/null
-type primFunction = Succ | Pred | IsZero | Leq | Leq_partially_applied of int
-
-type constant = Num of int | Bool of bool | Funct of primFunction
-
-type identifier = string
-
-type lambdaTerm = Constant of constant | Var of identifier | Abstract of identifier * lambdaTerm | App of lambdaTerm * lambdaTerm | IfThenElse of lambdaTerm * lambdaTerm * lambdaTerm | Let of identifier * lambdaTerm * lambdaTerm
-
-let rec free_in (ident:identifier) (term:lambdaTerm) : bool =
- match term with
- | Constant _ -> false
- | Var(var_ident) -> var_ident = ident
- | Abstract(bound_ident, body) -> bound_ident <> ident && free_in ident body
- | App(head, arg) -> free_in ident head || free_in ident arg
- | IfThenElse(test, yes, no) -> free_in ident test || free_in ident yes || free_in ident no
- | Let(bound_ident, arg, body) -> free_in ident arg || (bound_ident <> ident && free_in ident body)
-
-let fresh_var (base : identifier) (term:lambdaTerm) =
- let rec all_vars term vs = match term with
- | Constant _ -> vs
- | Var(var_ident) -> var_ident :: vs
- | Abstract(bound_ident, body) -> all_vars body (bound_ident :: vs)
- | App(head, arg) -> let vs' = all_vars head vs
- in all_vars arg vs'
- | IfThenElse(test, yes, no) -> let vs' = all_vars test vs
- in let vs'' = all_vars yes vs'
- in all_vars no vs''
- | Let(bound_ident, arg, body) -> let vs' = all_vars arg vs
- in all_vars body (bound_ident :: vs')
- in let current = all_vars term []
- in let rec check ident = if List.mem ident current then check (ident ^ "'") else ident
- in check (base ^ "'") (* keep adding primes until we find a variable unused (either free or bound) in term *)
-
-let rec substitute (term:lambdaTerm) (ident:identifier) (replacement:lambdaTerm) : lambdaTerm =
- match term with
- | Constant _ -> term
- | Var(var_ident) when var_ident = ident -> replacement
- | Var _ -> term
- | App(head, arg) -> let head' = substitute head ident replacement
- in let arg' = substitute arg ident replacement
- in App(head', arg')
- | IfThenElse(test, yes, no) -> let test' = substitute test ident replacement
- in let yes' = substitute yes ident replacement
- in let no' = substitute no ident replacement
- in IfThenElse(test', yes', no')
- | Abstract(bound_ident, body) when bound_ident = ident || not (free_in ident body) ->
- (* vacuous substitution *)
- term
- | Abstract(bound_ident, body) when not (free_in bound_ident replacement) ->
- (* can substitute without renaming bound_ident *)
- let body' = substitute body ident replacement
- in Abstract(bound_ident, body')
- | Abstract(bound_ident, body) ->
- (* find a fresh variable unused in either body or replacement (which we hack by specifying their App) *)
- let bound_ident' = fresh_var bound_ident (App(body,replacement))
- in let body' = substitute body bound_ident (Var bound_ident')
- in let body'' = substitute body' ident replacement
- in Abstract(bound_ident', body'')
- | Let(bound_ident, arg, body) when bound_ident = ident || not (free_in ident body) ->
- let arg' = substitute arg ident replacement
- in Let(bound_ident, arg', body)
- | Let(bound_ident, arg, body) when not (free_in bound_ident replacement) ->
- (* can substitute without renaming bound_ident *)
- let body' = substitute body ident replacement
- in let arg' = substitute arg ident replacement
- in Let(bound_ident, arg', body')
- | Let(bound_ident, arg, body) ->
- (* find a fresh variable unused in either body or replacement (which we hack by specifying their App) *)
- let bound_ident' = fresh_var bound_ident (App(body,replacement))
- in let body' = substitute body bound_ident (Var bound_ident')
- in let body'' = substitute body' ident replacement
- in let arg' = substitute arg ident replacement
- in Let(bound_ident', arg', body'')
-
-type reduceOutcome = AlreadyResult | ReducedTo of lambdaTerm | StuckAt of lambdaTerm
-
-exception Stuck of lambdaTerm
-
-let rec reduce1 (term:lambdaTerm) : reduceOutcome =
- match term with
- (* notice we never evaluate a yes/np branch until it is chosen *)
- | IfThenElse(Constant(Bool true), yes, _) -> ReducedTo yes
- | IfThenElse(Constant(Bool false), _, no) -> ReducedTo no
- | IfThenElse(test, yes, no) -> (match reduce1 test with
- | AlreadyResult -> StuckAt term (* if test was not reducible, neither is IfThenElse *)
- | ReducedTo test' -> ReducedTo (IfThenElse (test', yes, no))
- | StuckAt _ as outcome -> outcome)
- (* notice we never evaluate the body except after substituting, and that happens only after arg is reduced to a result *)
- | Let(bound_var, arg, body) -> (match reduce1 arg with
- | AlreadyResult -> (* if arg was not reducible, we can substitute *)
- ReducedTo (substitute body bound_var arg)
- | ReducedTo arg' -> ReducedTo (Let(bound_var, arg', body))
- | StuckAt _ as outcome -> outcome)
- (* notice we only substitute after arg is reduced to a result *)
- | App(Abstract(bound_var, body) as head, arg) -> (match reduce1 arg with
- | AlreadyResult -> (* if arg was not reducible, we can substitute *)
- ReducedTo (substitute body bound_var arg)
- | ReducedTo arg' -> ReducedTo (App(head, arg'))
- | StuckAt _ as outcome -> outcome)
- (* applications of primFunctions are reduced only when their arguments have been reduced to THE RIGHT TYPES of result *)
- | App(Constant(Funct Succ), Constant(Num n)) -> ReducedTo (Constant(Num (n+1)))
- | App(Constant(Funct Pred), Constant(Num n)) -> ReducedTo (Constant(Num (if n = 0 then 0 else n-1)))
- | App(Constant(Funct IsZero), Constant(Num n)) -> ReducedTo (Constant(Bool (n=0)))
- (* binary primFunctions are curried, have to be reduced in two steps *)
- | App(Constant(Funct Leq), Constant(Num n)) -> ReducedTo (Constant(Funct (Leq_partially_applied n)))
- | App(Constant(Funct (Leq_partially_applied m)), Constant(Num n)) -> ReducedTo (Constant(Bool (m<=n)))
- (* first the head should be reduced, next the arg *)
- | App(head, arg) -> (match reduce1 head with
- | ReducedTo head' -> ReducedTo (App(head', arg))
- | StuckAt _ as outcome -> outcome
- | AlreadyResult -> (* head was not reducible, was arg? *)
- (match reduce1 arg with
- | ReducedTo arg' -> ReducedTo (App(head, arg'))
- (* else the reducible cases of App(result, result) were caught above; this must be stuck *)
- | AlreadyResult -> StuckAt term
- | StuckAt _ as outcome -> outcome))
- | Var _ -> StuckAt term (* free variables are stuck *)
- | Constant _ -> AlreadyResult
- | Abstract(_, _) -> AlreadyResult
-
-let rec check_numbers (term:lambdaTerm) : unit =
- match term with
- | Constant(Num n) when n < 0 -> failwith ("Bad Number: " ^ string_of_int n)
- | Constant _ -> ()
- | Var _ -> ()
- | Abstract(_, body) -> check_numbers body
- | App(head, arg) -> let () = check_numbers head
- in check_numbers arg
- | Let(_, arg, body) -> let () = check_numbers arg
- in check_numbers body
- | IfThenElse(test, yes, no) -> let () = check_numbers test
- in let () = check_numbers yes
- in check_numbers no
-
-let reduce (term:lambdaTerm) : lambdaTerm =
- (* scan to verify that term doesn't have any Const(Num (negative)) *)
- let () = check_numbers term
- in let rec aux term = match reduce1 term with
- | AlreadyResult -> term
- | ReducedTo term' -> aux term' (* keep trying *)
- | StuckAt stuck_term -> raise (Stuck stuck_term) (* fail by raising exception *)
- in aux term
-