Games with Sequential Backtracking and Complete Game Semantics for Subclassical Logics

This paper introduces a game semantics for Arithmetic with various sub-classical logics that have implication as a primitive connective. This semantics clarifies the infinitary sequent calculus that the authors proposed for intuitionistic arithmetic with Excluded Middle for Sigma-0-1-formulas, a formal system motivated by proof mining and by the study of monotonic learning, for which no game semantics is known. This paper proposes games with Sequential Backtracking, and proves that they provide a sound and complete semantics for the logical system and other various subclassical logics. In order for that, this paper also defines a one-sided version of the logical system, whose proofs have a tree isomorphism with respect to the winning strategies of the game semantics.