Relating Apartness and Branching Bisimulation Games

Geuvers and Jacobs (LMCS 2021) formulated the notion of apartness relation on state-based systems modelled as coalgebras. In this context apartness is formally dual to bisimilarity, and gives an explicit proof system for showing that certain states are not bisimilar. In the current paper, we relate apartness to another classical element of the theory of behavioural equivalences: that of turn-based two-player games. Studying both strong and branching bisimilarity, we show that winning configurations for the Spoiler player correspond to apartness proofs, for transition systems that are image-finite (in the case of strong bisimilarity) and finite (in the case of branching bisimilarity).