Interactive Learning-Based Realizability for Heyting Arithmetic with EM1

Abstract

Abstract. We apply to the semantics of Arithmetic the idea of "finite approximation" used to provide computational interpretations of Herbrand's Theorem, and we interpret classical proofs as constructive proofs (with constructive rules for ∨, ∃) over a suitable structure N for the language of natural numbers and maps of Gödel's system T . We introduce a new Realizability semantics we call "Interactive learning-based Realizability", for Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to Σ 0 1 formulas). Individuals of N evolve with time, and realizers may "interact" with them, by influencing their evolution. We build our semantics over Avigad's fixed point result, but the same semantics may be defined over different constructive interpretations of classical arithmetic (Berardi and de' Liguoro use continuations). Our notion of realizability extends intuitionistic realizability and differs from it only in the atomic case: we interpret atomic realizers as "learning agents".

A New Use of Friedman’s Translation: Interactive Realizability

“…In section §4, we extend Interactive Realizability, as described in [1], to HA ω + EM 1 + SK 1 , an arithmetical system with functional variables.…”

“…The content of this section is based on Aschieri and Berardi [1], with a few simplifications, namely in the notion of state. We shall review the typed lambda calculi T and T Class in which learning-based realizers are written in [1].…”

“…We shall review the typed lambda calculi T and T Class in which learning-based realizers are written in [1]. T is a completely standard extension of Gödel's system T (see Girard [16]) with some syntactic sugar.…”

“…We define first the relation t s C by induction and by cases according to the form of C: Remark. The ideas behind the definition of s in the case of HA + EM 1 + SK 1 are those we already explained in [3], [1]. The system HA ω + EM 1 + SK 1 has, w.r.t.…”

“…Among the direct interpretations, there are different versions of Classical Realizability (Krivine's [22] and Avigad's [6]), there is Coquand game semantics [11], cut-elimination and normalization of classical proofs [14] (under Curry-Howard correspondence of not), and the epsilon substitution method [23] (the Kreisel nocounterexample interpretation [20] is an easy corollary of the other ones). More recently, another Classical Realizability interpretation for Heyting Arithmetic with Excluded Middle and Skolem axioms over Σ 0 1 -formulas has been introduced by Aschieri, Berardi and de' Liguoro [8], [1], [9]: it is based on the notion of learning and it is called the "Interactive Realizability". The goal of this paper is to compare Interactive Realizability with the other notions of Classical Realizability, using Friedman's translation and Kreisel modified Realizability as tools.…”