Probabilistic game structures combine both nondeterminism and stochasticity, where players repeatedly take actions simultaneously to move to the next state of the concurrent game. Probabilistic alternating simulation is an important tool to compare the behaviour of different probabilistic game structures. In this paper, we present a sound and complete modal characterisation of this simulation relation by proposing a new logic based on probabilistic distributions. The logic enables a player to enforce a property in the next state or distribution. Its extension with fixpoints, which also characterises the simulation relation, can express a lot of interesting properties in practical applications.
Metric-based simulation on game structures have been studied by de @cite_15 regarding the probability of winning games whose goals are expressed in quantitative @math -calculus (qMu) @cite_1 . Two states are equivalent if the players can win the same games with the same probability from both states, and among states can thus be measured. Algorithmic verification complexities are further studied for MDP and turn-based games @cite_13 .
@cite_13 @cite_15 @cite_1
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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. The μ-calculus is a powerful tool for specifying and verifying transition systems, including those with both demonic (universal) and angelic (existential) choice; its quantitative generalization qMμ extends to include probabilistic choice.We make two major contributions to the theory of such systems. The first is to show that for a finite-state system, the logical interpretation of qMμ, via fixed points in a domain of real-valued functions into [0, 1], is equivalent to an operational interpretation given as a turn-based gambling game between two players.The second contribution is to show that each player in the gambling game has an optimal memoryless strategy---that is, a strategy which is independent of the game's history, and with which a player can achieve his optimal expected reward however his opponent chooses to play. Moreover, since qMμ is expressive enough to encode stochastic parity games, our result implies the existence of memoryless strategies in that framework, as well.As an additional feature, we include an extensive case study demonstrating the aforementioned duality between games and logic. Among other things, it shows that the use of algorithmic verification techniques is mathematically justified in the practical computation of probabilistic system properties. We consider two-player games played for an infinite number of rounds, with ω-regular winning conditions. The games may be concurrent, in that the players choose their moves simultaneously and independently, and probabilistic, in that the moves determine a probability distribution for the successor state. We introduce quantitative game µ-calculus, and we show that the maximal probability of winning such games can be expressed as the fixpoint formulas in this calculus. We develop the arguments both for deterministic and for probabilistic concurrent games; as a special case, we solve probabilistic turn-based games with ω-regular winning conditions, which was also open. We also characterize the optimality, and the memory requirements, of the winning strategies. In particular, we show that while memoryless strategies suffice for winning games with safety and reachability conditions, Buchi conditions require the use of strategies with infinite memory. The existence of optimal strategies, as opposed to e-optimal, is only guaranteed in games with safety winning conditions. A popular approach to solving large probabilistic systems relies on aggregating states based on a measure of similarity. Many approaches in the literature are heuristic. A number of recent methods rely instead on metrics based on the notion of bisimulation, or behavioral equivalence between states (, 2001, 2003; , 2004). An integral component of such metrics is the Kantorovich metric between probability distributions. However, while this metric enables many satisfying theoretical properties, it is costly to compute in practice. In this paper, we use techniques from network optimization and statistical sampling to overcome this problem. We obtain in this manner a variety of distance functions for MDP state aggregation, which differ in the tradeoff between time and space complexity, as well as the quality of the aggregation. We provide an empirical evaluation of these trade-offs.