Abstract. We consider the timed automata model of [3], which allows the analysis of real-time systems expressed in terms of quantitative timing constraints. Traditional approaches to real-time system description express the model purely in terms of nondeterminism; however, we may wish to express the likelihood of the system making certain transitions. In this paper, we present a model for real-time systems augmented with discrete probability distributions. Furthermore, using the algorithm of [5] with fairness, we develop a model checking method for such models against temporal logic properties which can refer both to timing properties and probabilities, such as, “with probability 0.6 or greater, the clock x remains below 5 until clock y exceeds 2”. 1

We survey various notions of probabilistic automata and probabilistic bisimulation, accumulating in an expressiveness hierarchy of probabilistic system types. The aim of this paper is twofold: On the one hand it provides an overview of existing types of probabilistic systems and, on the other hand, it explains the relationship between these models.

This paper presents various semantics in the branching-time spectrum of discrete-time and continuous-time Markov chains (DTMCs and CTMCs). Strong and weak bisimulation equivalence and simulation pre-orders are covered and are logically characterised in terms of the temporal logics PCTL (Probabilistic Computation Tree Logic) and CSL (Continuous Stochastic Logic). Apart from presenting various existing branching-time relations in a uniform manner, this paper presents the following new results: (i) strong simulation for CTMCs, (ii) weak simulation for CTMCs and DTMCs, (iii) logical characterizations thereof (including weak bisimulation for DTMCs), (iv) a relation between weak bisimulation and weak simulation equivalence, and (v) various connections between equivalences and pre-orders in the continuous- and discrete-time setting. The results are summarized in a branching-time spectrum for DTMCs and CTMCs elucidating their semantics as well as their relationship. Key Words: comparative semantics, Markov chain, (weak) simulation, (weak) bisimulation, temporal logic

We propose decision algorithms for bisimulation relations de ned on probabilistic automata, a model for concurrent nondeterministic systems with randomization. The algorithms decide both strong and weak bisimulation relations based on deterministic as well as randomized schedulers. These algorithms extend and complete other known algorithms for simpler relations and models. The algorithm we present for strong probabilistic bisimulation has polynomial time complexity, while the algorithm for weak probabilistic bisimulation is exponential; however we argue that the latter is feasible in practice.

This paper presents a simulation preorder for continuoustime Markov chains (CTMCs). The simulation preorder is a conservative extension of a weak variant of probabilistic simulation on fully probabilistic systems, i.e., discrete-time Markov chains. The main result of the paper is that the simulation preorder preserves safety and liveness properties expressed in continuous stochastic logic (CSL), a stochastic branching-time temporal logic interpreted over CTMCs.

Markov chains are one of the most popular models for the evaluation of performance and dependability of information processing systems. To obtain performance measures, typically long-run or transient state probabilities of Markov chains are determined. Sometimes the Markov chain at hand is equipped with rewards and computations involve determining long-run or instantaneous reward probabilities.

a family of behavioural pseudometrics for probabilistic transition systems. These pseudometrics are a quantitative analogue of probabilistic bisimilarity. Distance zero captures probabilistic bisimilarity. Each pseudometric has a discount factor, a real number in the interval (0, 1]. The smaller the discount factor, the more the future is discounted. If the discount factor is one, then the future is not discounted at all. Desharnais et al. showed that the behavioural distances can be calculated up to any desired degree of accuracy if the discount factor is smaller than one. In this paper, we show that the distances can also be approximated if the future is not discounted. A key ingredient of our algorithm is Tarski’s decision procedure for the first order theory over real closed fields. By exploiting the Kantorovich-Rubinstein duality theorem we can restrict to the existential fragment for which more efficient decision procedures exist. 1

In this paper, we consider action-labelled systems with non-deterministic and probabilistic choice. Using the concept of norm functions [GV98], we introduce two types of bisimulations that allow for delays when simulating a transition. The so obtained equivalences (called (strict) normed bisimulation equivalence) are strictly between strong and weak bisimulation equivalence `a la [LS89, SL94, SL95]. Using a suitable modification of the prominent splitter/partitioning technique [KS83, PT87], we present polynomial-time algorithms that constructs the quotient space of the (strict) normed bisimulation equivalence classes. Moreover, we briefly discuss other aspects such as the soundness for establishing linear time properties and compositiality.

The paper introduces symbolic bisimulations for a simple probabilistic π-calculus to overcome the infinite branching problem that still exists in checking ground bisimulations between probabilistic systems. Especially the definition of weak (symbolic) bisimulation does not rely on the random capability of adversaries and suggests a solution to the open problem on the axiomatization for weak bisimulation in the case of unguarded recursion. Furthermore, we present an efficient characterization of symbolic bisimulations for the calculus, which allows the ”on-the-fly ” instantiation of bound names and dynamic construction of equivalence relations for quantitative evaluation. This directly results in a local decision algorithm that can explore just a minimal portion of the state spaces of the probabilistic processes in question. 1

Abstract. We show how bisimulation equivalence and simulation preorder on probabilistic LTSs (PLTSs), namely the main behavioural relations on probabilistic nondeterministic processes, can be characterized by abstract interpretation. Both bisimulation and simulation can be obtained as completions of partitions and preorders, viewed as abstract domains, w.r.t. a pair of concrete functions that encode a PLTS. As a consequence, this approach provides a general framework for designing algorithms for computing bisimulation and simulation on PLTSs. Notably, (i) we show that the standard bisimulation algorithm by Baier et al. can be viewed as an instance of such a framework and (ii) we design a new efficient simulation algorithm that improves the state of the art. 1