Question

For a lambda expression (x (λx y. x y) z h), I did not understand how free variables x (outer x), z and h can be converted in C code?

Regards,
darkie

Answer 1

Free variables are a special case in the Lambda Calculus — they cannot be “converted” to anything, so usually they’re taken as just symbols. In your example, and given some fictitious constructors for lambda expressions, you’d translate it to the following (which includes dealing with the implicit currying) C-like expression:

mk_app(mk_app(mk_app(mk_sym("x"),
                     mk_lam("x",
                            mk_lam("y",
                                   mk_app(mk_var("x"),
                                          mk_var("y"))))),
              mk_sym("z")),
       mk_sym("h"));

Obviously, you can use mk_var() for those symbols too, but that would be misleading since they’re not really variables, since they’re not bound. In other words, if you do any alpha conversion on the expression, they will have to stay the same.

(BTW, the relevant part here is Barendregt’s free variable assumption.)