Interactive Learning-Based Realizability for Heyting Arithmetic with EM1

Abstract

We apply to the semantics of Arithmetic the idea of ``finite approximation''used to provide computational interpretations of Herbrand's Theorem, and weinterpret classical proofs as constructive proofs (with constructive rules for$\vee, \exists$) over a suitable structure \StructureN for the language ofnatural numbers and maps of G\"odel's system \SystemT. We introduce a newRealizability semantics we call ``Interactive learning-based Realizability'',for Heyting Arithmetic plus \EM_1 (Excluded middle axiom restricted to$\Sigma^0_1$ formulas). Individuals of \StructureN evolve with time, andrealizers may ``interact'' with them, by influencing their evolution. We buildour semantics over Avigad's fixed point result, but the same semantics may bedefined over different constructive interpretations of classical arithmetic(Berardi and de' Liguoro use continuations). Our notion of realizabilityextends intuitionistic realizability and differs from it only in the atomiccase: we interpret atomic realizers as ``learning agents''