Classical proofs as programs

Abstract

We present an extension of the correspondence between intuitionistic proofs and functional programs to classical proofs, and more precisely to second order classical proofs. The advantage of classical logic in this context is that it allows to model imperative features of programming languages too (cf [5]). But there is an intrinsic difficulty with classical logic which lies in certain non-determinism of its computational interpretations. The use of a natural deduction system removes a part of this non determinism by fixing the inputs to the left of the sequents (cf [10] and [11]). However a conflict remains between the confluence of the computation mechanism and the uniqueness of the representation of data (for instance the uniqueness of the representation of the natural number 1). In this paper we develop the solution to this problem proposed in [11]: we show how to extract the intuitionistic representation of a data from a classical one using an “output” operator, while keeping a confluent computation mechanism. This result allows to extend in a sound way the proofs-as-programs paradigm to classical proofs in a framework where all the usual theoretical properties of intuitionistic proofs still hold.