From: Chris Date: Sat, 14 Mar 2015 13:54:47 +0000 (-0400) Subject: edits X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=5043cc69749b1e0ba308691cf3b2320fc953485a edits --- diff --git a/code/reduction_with_substitution.ml b/code/reduction_with_substitution.ml new file mode 100644 index 00000000..184081e9 --- /dev/null +++ b/code/reduction_with_substitution.ml @@ -0,0 +1,142 @@ +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) -> COMPLETE THIS LINE *) +(* | App(head, arg) -> COMPLETE THIS LINE *) + | 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 (* COMPLETE THIS LINE *) + | 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