We give a constructive analysis of learning as it arises in various computational interpretations of classical Peano Arithmetic, such as Aschieri and Berardi learning based realizability, Avigad’s update procedures and epsilon substitution method. In particular, we show how to compute in Gödel’s system T upper bounds on the length of learning processes, which are themselves represented in T through learning based realizability. The result is achieved by the introduction of a new non standard model of Gödel’s T, whose new basic (...) objects are pairs of non standard natural numbers and moduli of convergence, where the latter are objects giving constructive information about the former. As a foundational corollary, we obtain that that learning based realizability is a constructive interpretation of Heyting Arithmetic plus excluded middle over Σ10 formulas and of all Peano Arithmetic when combined with Gödel’s double negation translation. As a byproduct of our approach, we also obtain a new proof of Avigad’s theorem for update procedures and thus of the termination of the epsilon substitution method for PA. (shrink)

We prove a soundness and completeness result for Aschieri and Berardiʼs learning based realizability for Heyting Arithmetic plus Excluded Middle over semi-decidable statements with respect to 1-Backtracking Coquand game semantics. First, we prove that learning based realizability is sound with respect to 1-Backtracking Coquand game semantics. In particular, any realizer of an implication-and-negation-free arithmetical formula embodies a winning recursive strategy for the 1-Backtracking version of Tarski games. We also give examples of realizers and winning strategy extraction for some classical proofs. (...) Secondly, we extend our notion of realizability to a total recursive learning based realizability and show that the notion is complete with respect to 1-Backtracking Coquand game semantics. (shrink)