In this paper we present a novel abstraction technique for Markov decision processes (MDPs), which are widely used for modelling systems that exhibit both probabilistic and nondeterministic behaviour. In the field of model checking, abstraction has proved an extremely successful tool to combat the state-space explosion problem. In the probabilistic setting, however, little practical progress has been made in this area. We propose an abstraction method for MDPs based on stochastic two-player games. The key idea behind this approach is to maintain a separation between nondeterminism present in the original MDP and nondeterminism introduced through abstraction, each type being represented by a different player in the game. Crucially, this allows us to obtain distinct lower and upper bounds for both the best and worst-case performance (minimum or maximum probabilities) of the MDP. We have implemented our techniques and illustrate their practical utility by applying them to a quantitative analysis of the Zeroconf dynamic network configuration protocol. 1

Abstract. This paper proposes a novel abstraction technique for continuous-time Markov chains (CTMCs). Our technique fits within the realm of three-valued abstraction methods that have been used successfully for traditional model checking. The key idea is to apply abstraction on uniform CTMCs that are readily obtained from general CTMCs, and to abstract transition probabilities by intervals. It is shown that this provides a conservative abstraction for both true and false for a threevalued semantics of the branching-time logic CSL (Continuous Stochastic Logic). Experiments on an infinite-state CTMC indicate the feasibility of our abstraction technique. 1

Abstract. We present a novel abstraction technique which allows the analysis of reachability and safety properties of Markov decision processes with very large state spaces. The technique, called magnifyinglens abstraction, copes with the state-explosion problem by partitioning the state-space into regions, and by computing upper and lower bounds for reachability and safety properties on the regions, rather than on the states. To compute these bounds, magnifying-lens abstraction iterates over the regions, considering the concrete states of each region in turn, as if one were sliding across the abstraction a magnifying lens which allowed viewing the concrete states. The algorithm adaptively refines the regions, using smaller regions where more detail is needed, until the difference between upper and lower bounds is smaller than a specified accuracy. We provide experimental results illustrating that magnifying-lens abstractions can provide accurate answers, with drastic savings in memory requirements, in many cases where previous abstraction techniques yield no benefit. 1

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Abstract. This paper studies the effect of bisimulation minimisation in model checking of monolithic discrete-time and continuous-time Markov chains as well as variants thereof with rewards. Our results show that—as for traditional model checking—enormous state space reductions (up to logarithmic savings) may be obtained. In contrast to traditional model checking, in many cases, the verification time of the original Markov chain exceeds the quotienting time plus the verification time of the quotient. We consider probabilistic bisimulation as well as versions thereof that are tailored to the property to be checked. 1

models whose verification results transfer to the abstracted models for a logic with unrestricted use of negation and quantification. This framework is novel in that its models have quantitative or probabilistic observables and state transitions. Properties of a quantitative temporal logic have measurable denotations in these models. For probabilistic models such denotations approximate the probabilistic semantics of full LTL. We show how predicate-based abstractions specify abstract quantitative and probabilistic models with finite state space. 1

Abstract. Modeling and analysis techniques are presented for real-time, safety-critical software. Software analysis is the task of verifying whether the computer code will execute safely, free of run-time errors. The critical properties that prove safe execution include bounded-ness of variables and termination of the program in a finite number of steps. In this paper, dynamical system representations of computer programs along with specific models that are pertinent to analysis via an optimization-based search for system invariants are developed. It is shown that the automatic search for system invariants that establish the desired properties of computer code, can be formulated as a convex optimization problem, such as linear programming, semidefinite programming, and/or sum of squares programming. 1

Abstract. This paper investigates relative precision and optimality of analyses for concurrent probabilistic systems. Aiming at the problem at the heart of probabilistic model checking – computing the probability of reaching a particular set of states – we leverage the theory of abstract interpretation. With a focus on predicate abstraction, we develop the first abstract-interpretation framework for Markov decision processes which admits to compute both lower and upper bounds on reachability probabilities. Further, we describe how to compute and approximate such abstractions using abstraction refinement and give experimental results. 1

Probabilistic model checking enjoys a rapid increase of interest from different communities. Software tools such as PRISM [13] (with about 4,000 downloads), MRMC [12], and LiQuor [2] support the verification of Markov chains or variants thereof that exhibit nondeterminism. They have been applied to case studies

In this paper, we combine abstraction-refinement and symbolic techniques to fight the state-space explosion problem when model checking Markov Decision Processes (MDPs). The abstract-refinement technique, called magnifying-lens abstraction (MLA), partitions the statespace into regions and computes upper and lower bounds for reachability and safety properties on the regions, rather than states. To compute such bounds, MLA iterates over the regions, analysing the concrete states of each region in turn- as if one was sliding a magnifying lens across the system to view the states. The algorithm adaptively refines the regions, using smaller regions where more detail is required, until the difference between the bounds is below a specified accuracy. The symbolic technique is based on Multi-Terminal Binary Decision Diagrams (MTBDDs) which have been used extensively to provide compact encodings of probabilistic models. We introduce a symbolic version of the MLA algorithm, called symbolic MLA, which combines the power of both practical techniques when verifying MDPs. An implementation of symbolic MLA in the probabilistic model checker PRISM and experimental results to illustrate the advantages of our approach are presented. 1