Semantics for Intuitionistic Arithmetic Based on Tarski Games with Retractable Moves

We define an effective, sound and complete game semantics for HAinf, Intuitionistic Arithmetic with ω-rule. Our semantics is equivalent to the original semantics proposed by Lorentzen [6], but it is based on the more recent notions of ”backtracking” ([5], [2]) and of isomorphism between proofs and strategies ([8]). We prove that winning strategies in our game semantics are tree-isomorphic to the set of proofs of some variant of HAinf, and that they are a sound and complete interpretation of HAinf. 1 Why game semantics of Intuitionistic Arithmetic? In [7], S.Hayashi proposed the use of an effective game semantics in his Proof Animation project. The goal of the project is ”animating” (turning into algorithms) proofs of program specifications, in order to find bugs in the way a specification is formalized. Proofs are formalized in classical Arithmetic, and the method chosen for “animating” proofs is a simplified version of Coquand’s game interpretation ([4], [5]) of PAinf, classical arithmetic with ω-rule. The interest of the game interpretation is that it interprets rules of classical arithmetic by very simple operations, like arithmetical operation, reference to a pointer, adding and removing elements to a stack. Coquand, however, defined implication A → B as classical implication, as “A is false or B is true”. In real proofs, instead, we often use the constructive definition of implication A ⇒ B, which is: ”assume A in order to prove B”. A ⇒ B is classically equivalent to “A is false or B is true”, but this means that in order to interpret a proof in Coquand’s semantics we have first to modify it. If we want some control and understanding of the algorithm we extract from a proof, instead, it is crucial to animate