Dependent choice, 'quote' and the clock

When using the Curry-Howard correspondence in order to obtain executable programs from mathematical proofs, we are faced with a difficult problem : to interpret each axiom of our axiom system for mathematics (which may be, for example, second order classical logic, or classical set theory) as an instruction of our programming language. This problem has been solved for the axiom of excluded middle by T. Griffin[5], who found that it can be interpreted by means of control instructions like call-with-current-continuation in SCHEME, catch and throw in LISP or try ... with ... in ML. The solution for the comprehension axiom scheme for second order logic was essentially given by G. Takeuti[14], who gave a formulation of this scheme by means of an elimination rule of the second order universal quantifier, and J.Y. Girard who showed in [4] that this rule can be interpreted by the identity instruction λx x. In [11], this problem is solved for the axioms of classical Zermelo-Fraenkel set theory, with the axiom of foundation, but without the axiom of choice. Indeed the problem remains for the axiom of choice, which is essential in many proofs. In most cases, we know how to avoid it, but at the price of much longer proofs. The weaker axiom of countable choice is fundamental in calculus, measure theory, . . . (this very important part of mathematics which can be formalized in second order classical logic). It is certainly unavoidable. In this paper, we give an interpretation of the axiom of countable choice (and even the slightly stronger axiom of dependent choice) in classical second order logic, as a programming instruction. Using the results of [11], the same method applies indeed in ZF set theory. Our solution is to introduce a new instruction in λ-calculus, which uses an enumeration of λ-terms. We give two different computational interpretations and in fact implementations, for this instruction : the first is similar to the quote instruction of LISP and the second is given in terms of a clock. Such an instruction is rather difficult to handle, because it is incompatible with βreduction. Therefore, we must set very precisely our framework in order to be able to