A New Use of Friedman’s Translation: Interactive Realizability

Abstract

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".

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

“…The resulting notion of realizability is just Kreisel modified realizability (Kreisel 1959) extended with learning (Aschieri and Berardi 2012). The arithmetic HA + EM 1 + SK 1 is realized by adding an oracle for the halting problem to Gödel's system T, and by computationally interpreting EM 1 as a device that effectively learns oracle values during calculations.…”

Fluctuations, effective learnability and metastability in analysis

“…by Aschieri and Berardi (see e.g. [6,4,5]). By contrast to this, Σ 0 2 -DNE − does not allow such an interpretation and not even a (in this context) weaker monotone modified realizability interpretation (in the sense of the [24], see also [26]) as shown in [1], where this is used to prove that Σ Remark 1.1.…”

Strong Normalization for HA + EM1 by Non-Deterministic Choice

“…That is, we raise an exception only when we have falsified an hypothesis, while de Groote raises an exception when a proof of the contrary of the hypothesis is found (and given the presence of possibly many other hypotheses this does not imply that the hypothesis is actually false). Therefore, our reductions belong to the same family of Interactive realizability [1,2], while de Groote's belong to the same family of the Griffin, Krivine and Gödel double-negation approach to classical proofs. In this section we introduce the non-deterministic system HA + EM 1 , which is still a standard natural deduction system for Heyting Arithmetic with EM 1 .…”

“…Delimited exceptions were used by de Groote [16] in order to interpret the excluded middle in classical propositional logic with implication; by Herbelin [15], in order to pass witnesses to some existential formula when a falsification of its negation is encountered: in our setting they are used in a similar way, and our work may be seen as a modification and extension of some of de Groote's and Herbelin's techniques. Many of our ideas are inspired by Interactive realizability [1] for HA + EM 1 , which describes classical programs as programs that makes hypotheses, test them and learn by refuting the incorrect ones.…”