Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1

Abstract

We introduce a realizability semantics based on interactive learning for full second-order Heyting arithmetic with excluded middle and Skolem axioms over Σ10-formulas. Realizers are written in a classical version of Girard's System $\mathsf{F}$ and can be viewed as programs that learn by interacting with the environment. We show that the realizers of any Π20-formula represent terminating learning processes whose outcomes are numerical witnesses for the existential quantifier of the formula.