The paper starts by reviewing the notion of cobisimilarity, which is defined by dualising coalgebraic bisimulations. The authors show that for certain functors, such as the subdistribution functor, the cobisimilarity proof system requires reasoning about an infinite set of couplings, making it less practical.
To address this issue, the authors introduce behavioural apartness, which is defined by dualising behavioural equivalence rather than bisimulations. They provide a proof system for behavioural apartness, prove it to be sound and complete, and show how it can be optimised to only require reasoning about states that are reachable in one step.
The paper demonstrates the benefits of the behavioural apartness proof system through several examples, including labelled Markov processes and stream systems. In the subdistribution functor example, the behavioural apartness proof system allows for finite proofs of distinguishability, in contrast to the cobisimilarity approach.
The authors also discuss potential future work, such as investigating the connection between proofs of behavioural apartness and distinguishing formulas in a corresponding modal logic, as well as exploring the notion of codensity apartness and its applications to quantitative settings.
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