Bisimilarity as an equivalence notion of systems has been central to process theory. Due to the recent rise of interest in quantitative systems (probabilistic, weighted, hybrid, etc.), bisimilarity has been extended in various ways: notably, bisimulation metric between probabilistic systems. An important feature of bisimilarity is its game-theoretic characterization, where Spoiler and Duplicator play against each other; extension of bisimilarity games to quantitative settings has been actively pursued too. In this paper, we present a general framework that uniformly describes game characterizations of bisimilarity-like notions. Our framework is formalized categorically using fibrations and coalgebras. In particular, our characterization of bisimilarity in terms of fibrational predicate transformers allows us to derive codensity bisimilarity games: a general categorical game characterization of bisimilarity. Our framework covers known bisimilarity-like notions (such as bisimulation metric) as well as new ones (including what we call bisimulation topology).
@cite_24: 
@cite_29: We tackle the problem of non robustness of simulation and bisimulation when dealing with probabilistic processes. It is important to ignore tiny deviations in probabilities because these often come from experiments or estimations. A few approaches have been proposed to treat this issue, for example metrics to quantify the non bisimilarity (or closeness) of processes. Relaxing the definition of simulation and bisimulation is another avenue which we follow. We define a new semantics to a known simple logic for probabilistic processes and show that it characterises a notion of epsi-simulation. We also define two-players games that correspond to these notions: the existence of a winning strategy for one of the players determines epsi-(bi)simulation. Of course, for all the notions defined, letting epsi = 0 gives back the usual notions of logical equivalence, simulation and bisimulation. However, in contrast to what happens in fully probabilistic systems when epsi = 0, two-way e-simulation for epsi > 0 is not equal to epsi-bisimulation. Next we give a polynomial time algorithm to compute a naturally derived metric: distance between states s and t is defined as the smallest epsi such that s and t are epsi-equivalent. This is the first polynomial algorithm for a non-discounted metric. Finally we show that most of these notions can be extended to deal with probabilistic systems that allow non-determinism as well.
@cite_13: Labelled Markov processes are probabilistic versions of labelled transition systems. In general, the state space of a labelled Markov process may be a continuum. Logical characterizations of probabilistic bisimulation and simulation were given by These results hold for systems defined on analytic state spaces and assume that there are countably many labels in the case of bisimulation and finitely many labels in the case of simulation. In this paper, we first revisit these results by giving simpler and more streamlined proofs. In particular, our proof for simulation has the same structure as the one for bisimulation, relying on a new result of a topological nature. This departs from the known proof for this result, which uses domain theory techniques and falls out of a theory of approximation of Labelled Markov processes. Both our proofs assume the presence of countably many labels. We investigate the necessity of this assumption, and show that the logical characterization of bisimulation may fail when there are uncountably many labels. However, with a stronger assumption on the transition functions (continuity instead of just measurability), we can regain the logical characterization result, for arbitrarily many labels. These new results arose from a new game-theoretic way of understanding probabilistic simulation and bisimulation.
The categorical generalization @cite_24 is based on the game notion in @cite_29 , while ours is based on that in @cite_13 . Therefore, for some bisimulation notions (including the bisimulation metric), we obtain a game notion with a smaller arena. Compare Table (an instance of ours) and Table (an instance of @cite_24 ).