Learning based realizability for HA + EM1 and 1-Backtracking games: Soundness and completeness

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.