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.
We give logical characterizations of bisimulation relations for the probabilistic automata of Segala in terms of three Hennessy-Milner style logics. The three logics characterize strong, strong probabilistic and weak probabilistic bisimulation, and differ only in the kind of diamond operator used. Compared to the Larsen and Skou logic for reactive systems, these logics introduce a new operator that measures the probability of the set of states that satisfy a formula. Moreover, the satisfaction relation is defined on measures rather than single states. We rederive previous results of by defining sublogics for Reactive and Alternating Models viewed as restrictions of probabilistic automata. Finally, we identify restrictions on probabilistic automata, weaker than those imposed by the Alternating Models, that preserve the logical characterization of These restrictions require that each state either enables several ordinary transitions or enables a single probabilistic transition. Probabilistic automata (PAs) constitute a general framework for modeling and analyzing discrete event systems that exhibit both nondeterministic and probabilistic behavior, such as distributed algorithms and network protocols. The behavior of PAs is commonly defined using schedulers (also called adversaries or strategies), which resolve all nondeterministic choices based on past history. From the resulting purely probabilistic structures, trace distributions can be extracted, whose intent is to capture the observable behavior of a PA. However, when PAs are composed via an (asynchronous) parallel composition operator, a global scheduler may establish strong correlations between the behavior of system components and, for example, resolve nondeterministic choices in one PA based on the outcome of probabilistic choices in the other. It is well known that, as a result of this, the (linear-time) trace distribution precongruence is not compositional for PAs. In his 1995 Ph.D. thesis, Segala has shown that the (branching-time) probabilistic simulation preorder is compositional for PAs. In this paper, we establish that the simulation preorder is, in fact, the coarsest refinement of the trace distribution preorder that is compositional. We prove our characterization result by providing (1) a context of a given PA @math , called the tester, which may announce the state of @math to the outside world, and (2) a specific global scheduler, called the observer, which ensures that the state information that is announced is actually correct. Now when another PA @math is composed with the tester, it may generate the same external behavior as the observer only when it is able to simulate @math in the sense that whenever @math goes to some state @math , @math can go to a corresponding state @math , from which it may generate the same external behavior. Our result shows that probabilistic contexts together with global schedulers are able to exhibit the branching structure of PAs.  In 1992 Wang & Larsen extended the may- and must preorders of De Nicola and Hennessy to processes featuring probabilistic as well as nondeterministic choice. They concluded with two problems that have remained open throughout the years, namely to find complete axiomatisations and alternative characterisations for these preorders. This paper solves both problems for finite processes with silent moves. It characterises the may preorder in terms of simulation, and the must preorder in terms of failure simulation. It also gives a characterisation of both preorders using a modal logic. Finally it axiomatises both preorders over a probabilistic version of CSP. We extend the trace semantics for labeled transition systems to a randomized model of concurrent computation. The main objective is to obtain a compositional semantics. The role of a trace in the randomized model is played by a probability distribution over traces, called a trace distribution. We show that the preorder based on trace distribution inclusion is not a precongruence, and we build an elementary context, called the principal context, that is sufficiently powerful to characterize the coarsest precongruence that is contained in the trace distribution preorder. Finally, we introduce a notion of a probabilistic forward simulation and we prove that it is sound for the trace distribution precongruence. An important characteristic of probabilistic forward simulations is that they relate states to probability distributions over states. Several probabilistic simulation relations for probabili stic systems are defined and evaluated according to two criteria: compositionality and preservation of "interesting" properties. Here, the interesting properties of a system are identified with those that are expressible in an untimed version of the Timed Probabilistic concurrent Computation Tree Logic (TPCTL) of Hansson. The definitions are made, and the evaluations carried out, in terms of a general labeled transition system model for concurrent probabilistic computation. The results cover weak simulations, which abstract from internal computation, as well as strong simulations, which do not 