Link to this article: http://journals.cambridge.org/abstract_S0960129513000455 How to cite this article: FEDERICO ASCHIERI (2014). Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1.We introduce a realizability semantics based on interactive learning for full second-order Heyting arithmetic with excluded middle and Skolem axioms over Σ 0 1 -formulas. Realizers are written in a classical version of Girard's System F and can be viewed as programs that learn by interacting with the environment. We show that the realizers of any Π 0 2 -formula represent terminating learning processes whose outcomes are numerical witnesses for the existential quantifier of the formula.

Abstract-Propositional Gödel logic G extends intuitionistic logic with the non-constructive principle of linearity (A → B) ∨ (B → A). We introduce a Curry-Howard correspondence for G and show that a simple natural deduction calculus can be used as a typing system. The resulting functional language extends the simply typed λ-calculus via a synchronous communication mechanism between parallel processes, which increases its expressive power. The normalization proof employs original termination arguments and proof transformations implementing forms of code mobility. Our results provide a computational interpretation of G, thus proving A. Avron's 1991 thesis.

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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".

Friedman's translation is a well-known transformation of formulas. The Friedman translation has two properties: i) it validates intuitionistic theorems -if a formula is intuitionistically provable, then so it is its Friedman translation; ii) it is suitable for program extraction from classical proofs -the intuitionistic provability of the Friedman translation of the negative translation of a for-all-exist-formula implies the intuitionistic provability of the formula itself. However, the Friedman translation does not validate classical principles, like the Excluded Middle.Here, we define a restricted Friedman translation which both validates the Excluded Middle and Skolem axiom schemata restricted to Σ 0 1 -formulas and it is also suitable for program extraction from classical proofs using such principles: the intuitionistic provability of the restricted Friedman translation of a for-all-exist-formula implies the intuitionistic provability of the formula itself. Then we introduce a learning-based Realizability Semantics for Heyting Arithmetic with all finite types, extended with the two previous axiom schemata. We call this semantics "Interactive Realizability", and we characterize it as the composition of our restricted Friedman translation with Kreisel modified realizability. As a corollary, we show that Interactive Realizability is, in a sense, "axiom-driven", while the other Realizability Semantics for Classical Arithmetic, like the semantics of Krivine, are "goal-driven".