Asynchronous games 1: Uniformity by group invariance

Ten years ago, Abramsky, Jagadeesan, Malacaria (AJM) introduced a fully abstract game model of PCF, inspired by Girard’s Geometry of Interaction. The key ingredient of the model is a partial equivalence relation (per) on strategies, which captures the idea of a strategy “blind to the Opponent’s thread indexing”. We reveal the group-theoretic nature of this construction, and reformulate arena games accordingly. The justification pointers are replaced by thread indexing, modulo a left and right group action; a strategy is equivalent to itself (wrt. the per) when it verifies a bi-invariance condition, which replaces the familiar invariance condition of group theory.