Interactive Realizability for Classical Peano Arithmetic with Skolem Axioms

Abstract

Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is an approximation of some Skolem functions. The realizer interacts with the environment, which may provide a counter-proof, a counterexample invalidating the current construction of the realizer. But the realizer is always able to turn such a negative outcome into a positive information, which consists in some new piece of knowledge learned about the mentioned Skolem functions. The aim of this work is to extend Interactive realizability to a system which includes classical first-order Peano Arithmetic with Skolem axioms. For witness extraction, the learning capabilities of realizers will be exploited according to the paradigm of learning by levels. In particular, realizers of atomic formulas will be update procedures in the sense of Avigad and thus will be understood as stratified-learning algorithms