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).
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 ).
@cite_24 @cite_29 @cite_13
2038193812 2102794165 2890208395 2108886305
We consider two-player games played over finite state spaces for an infinite number of rounds. At each state, the players simultaneously choose moves; the moves determine a successor state. It is often advantageous for players to choose probability distributions over moves, rather than single moves. Given a goal, for example, reach a target state, the question of winning is thus a probabilistic one: what is the maximal probability of winning from a given state? On these game structures, two fundamental notions are those of equivalences and metrics. Given a set of winning conditions, two states are equivalent if the players can win the same games with the same probability from both states. Metrics provide a bound on the difference in the probabilities of winning across states, capturing a quantitative notion of state similarity. We introduce equivalences and metrics for two-player game structures, and we show that they characterize the difference in probability of winning games whose goals are expressed in the quantitative mu-calculus. The quantitative mu-calculus can express a large set of goals, including reachability, safety, and omega-regular properties. Thus, we claim that our relations and metrics provide the canonical extensions to games, of the classical notion of bisimulation for transition systems. We develop our results both for equivalences and metrics, which generalize bisimulation, and for asymmetrical versions, which generalize simulation. Part I of this paper has described a new theory for the analysis of games with incomplete information. It has been shown that, if the various players' subjective probability distributions satisfy a certain mutual-consistency requirement, then any given game with incomplete information will be equivalent to a certain game with complete information, called the “Bayes-equivalent” of the original game, or briefly a “Bayesian game.” Part II of the paper will now show that any Nash equilibrium point of this Bayesian game yields a “Bayesian equilibrium point” for the original game and conversely. This result will then be illustrated by numerical examples, representing two-person zero-sum games with incomplete information. We shall also show how our theory enables us to analyze the problem of exploiting the opponent's erroneous beliefs. However, apart from its indubitable usefulness in locating Bayesian equilibrium points, we shall show it on a numerical example the Bayes-equivalent of a two-person cooperative game that the normal form of a Bayesian game is in many cases a highly unsatisfactory representation of the game situation and has to be replaced by other representations e.g., by the semi-normal form. We shall argue that this rather unexpected result is due to the fact that Bayesian games must be interpreted as games with “delayed commitment” whereas the normal-form representation always envisages a game with “immediate commitment.” Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we present an abstract categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicates—following the seminal work by Hermida and Jacobs—but our focus is on the structural aspect of fibrational frameworks. Specifically we use morphisms of fibrations as well as canonical liftings of functors via Kan extensions. We apply this categorical framework by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory. 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.