1 type primFunction = Succ | Pred | IsZero | Leq | Leq_partially_applied of int
3 type constant = Num of int | Bool of bool | Funct of primFunction
5 type identifier = string
7 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
9 let rec free_in (ident:identifier) (term:lambdaTerm) : bool =
12 | Var(var_ident) -> var_ident = ident
13 (* | Abstract(bound_ident, body) -> COMPLETE THIS LINE *)
14 (* | App(head, arg) -> COMPLETE THIS LINE *)
15 | IfThenElse(test, yes, no) -> free_in ident test || free_in ident yes || free_in ident no
16 | Let(bound_ident, arg, body) -> free_in ident arg || (bound_ident <> ident && free_in ident body)
18 let fresh_var (base : identifier) (term:lambdaTerm) =
19 let rec all_vars term vs = match term with
21 | Var(var_ident) -> var_ident :: vs
22 | Abstract(bound_ident, body) -> all_vars body (bound_ident :: vs)
23 | App(head, arg) -> let vs' = all_vars head vs
25 | IfThenElse(test, yes, no) -> let vs' = all_vars test vs
26 in let vs'' = all_vars yes vs'
28 | Let(bound_ident, arg, body) -> let vs' = all_vars arg vs
29 in all_vars body (bound_ident :: vs')
30 in let current = all_vars term []
31 in let rec check ident = if List.mem ident current then check (ident ^ "'") else ident
32 in check (base ^ "'") (* keep adding primes until we find a variable unused (either free or bound) in term *)
34 let rec substitute (term:lambdaTerm) (ident:identifier) (replacement:lambdaTerm) : lambdaTerm =
37 | Var(var_ident) when var_ident = ident -> replacement
39 | App(head, arg) -> let head' = substitute head ident replacement
40 in let arg' = substitute arg ident replacement
42 | IfThenElse(test, yes, no) -> let test' = substitute test ident replacement
43 in let yes' = substitute yes ident replacement
44 in let no' = substitute no ident replacement
45 in IfThenElse(test', yes', no')
46 | Abstract(bound_ident, body) when bound_ident = ident || not (free_in ident body) ->
47 (* vacuous substitution *)
49 | Abstract(bound_ident, body) when not (free_in bound_ident replacement) ->
50 (* can substitute without renaming bound_ident *)
51 let body' = substitute body ident replacement
52 in (* COMPLETE THIS LINE *)
53 | Abstract(bound_ident, body) ->
54 (* find a fresh variable unused in either body or replacement (which we hack by specifying their App) *)
55 let bound_ident' = fresh_var bound_ident (App(body,replacement))
56 in let body' = substitute body bound_ident (Var bound_ident')
57 in let body'' = substitute body' ident replacement
58 in Abstract(bound_ident', body'')
59 | Let(bound_ident, arg, body) when bound_ident = ident || not (free_in ident body) ->
60 let arg' = substitute arg ident replacement
61 in Let(bound_ident, arg', body)
62 | Let(bound_ident, arg, body) when not (free_in bound_ident replacement) ->
63 (* can substitute without renaming bound_ident *)
64 let body' = substitute body ident replacement
65 in let arg' = substitute arg ident replacement
66 in Let(bound_ident, arg', body')
67 | Let(bound_ident, arg, body) ->
68 (* find a fresh variable unused in either body or replacement (which we hack by specifying their App) *)
69 let bound_ident' = fresh_var bound_ident (App(body,replacement))
70 in let body' = substitute body bound_ident (Var bound_ident')
71 in let body'' = substitute body' ident replacement
72 in let arg' = substitute arg ident replacement
73 in Let(bound_ident', arg', body'')
75 type reduceOutcome = AlreadyResult | ReducedTo of lambdaTerm | StuckAt of lambdaTerm
77 exception Stuck of lambdaTerm
79 let rec reduce1 (term:lambdaTerm) : reduceOutcome =
81 (* notice we never evaluate a yes/np branch until it is chosen *)
82 | IfThenElse(Constant(Bool true), yes, _) -> ReducedTo yes
83 | IfThenElse(Constant(Bool false), _, no) -> ReducedTo no
84 | IfThenElse(test, yes, no) -> (match reduce1 test with
85 | AlreadyResult -> StuckAt term (* if test was not reducible, neither is IfThenElse *)
86 | ReducedTo test' -> ReducedTo (IfThenElse (test', yes, no))
87 | StuckAt _ as outcome -> outcome)
88 (* notice we never evaluate the body except after substituting, and that happens only after arg is reduced to a result *)
89 | Let(bound_var, arg, body) -> (match reduce1 arg with
90 | AlreadyResult -> (* if arg was not reducible, we can substitute *)
91 ReducedTo (substitute body bound_var arg)
92 | ReducedTo arg' -> ReducedTo (Let(bound_var, arg', body))
93 | StuckAt _ as outcome -> outcome)
94 (* notice we only substitute after arg is reduced to a result *)
95 | App(Abstract(bound_var, body) as head, arg) -> (match reduce1 arg with
96 | AlreadyResult -> (* if arg was not reducible, we can substitute *)
97 ReducedTo (substitute body bound_var arg)
98 | ReducedTo arg' -> ReducedTo (App(head, arg'))
99 | StuckAt _ as outcome -> outcome)
100 (* applications of primFunctions are reduced only when their arguments have been reduced to THE RIGHT TYPES of result *)
101 | App(Constant(Funct Succ), Constant(Num n)) -> ReducedTo (Constant(Num (n+1)))
102 | App(Constant(Funct Pred), Constant(Num n)) -> ReducedTo (Constant(Num (if n = 0 then 0 else n-1)))
103 | App(Constant(Funct IsZero), Constant(Num n)) -> ReducedTo (Constant(Bool (n=0)))
104 (* binary primFunctions are curried, have to be reduced in two steps *)
105 | App(Constant(Funct Leq), Constant(Num n)) -> ReducedTo (Constant(Funct (Leq_partially_applied n)))
106 | App(Constant(Funct (Leq_partially_applied m)), Constant(Num n)) -> ReducedTo (Constant(Bool (m<=n)))
107 (* first the head should be reduced, next the arg *)
108 | App(head, arg) -> (match reduce1 head with
109 | ReducedTo head' -> ReducedTo (App(head', arg))
110 | StuckAt _ as outcome -> outcome
111 | AlreadyResult -> (* head was not reducible, was arg? *)
112 (match reduce1 arg with
113 | ReducedTo arg' -> ReducedTo (App(head, arg'))
114 (* else the reducible cases of App(result, result) were caught above; this must be stuck *)
115 | AlreadyResult -> StuckAt term
116 | StuckAt _ as outcome -> outcome))
117 | Var _ -> StuckAt term (* free variables are stuck *)
118 | Constant _ -> AlreadyResult
119 | Abstract(_, _) -> AlreadyResult
121 let rec check_numbers (term:lambdaTerm) : unit =
123 | Constant(Num n) when n < 0 -> failwith ("Bad Number: " ^ string_of_int n)
126 | Abstract(_, body) -> check_numbers body
127 | App(head, arg) -> let () = check_numbers head
129 | Let(_, arg, body) -> let () = check_numbers arg
130 in check_numbers body
131 | IfThenElse(test, yes, no) -> let () = check_numbers test
132 in let () = check_numbers yes
135 let reduce (term:lambdaTerm) : lambdaTerm =
136 (* scan to verify that term doesn't have any Const(Num (negative)) *)
137 let () = check_numbers term
138 in let rec aux term = match reduce1 term with
139 | AlreadyResult -> term
140 | ReducedTo term' -> aux term' (* keep trying *)
141 | StuckAt stuck_term -> raise (Stuck stuck_term) (* fail by raising exception *)