The main challenge in using abstractions effectively, is to construct a suitable abstraction for the system being verified. One approach that tries to address this problem is that of counterexample guided abstraction-refinement (CEGAR) , wherein one starts with a coarse abstraction of the system, and progressively refines it, based on invalid counterexamples seen in prior model checking runs, until either an abstraction proves the correctness of the system or a valid counterexample is generated. While CEGAR has been successfully used in verifying non-probabilistic systems automatically, CEGAR has not been applied in the context of probabilistic systems. The main issues that need to be tackled in order to extend the approach to probabilistic systems is a suitable notion of counterexample'', algorithms to generate counterexamples, check their validity, and then automatically refine an abstraction based on an invalid counterexample. In this paper, we address these issues, and present a CEGAR framework for Markov Decision Processes.
@cite_30: 
@cite_18: In this paper the abstraction-refinement paradigm based on 3-valued logics is extended to the setting of probabilistic systems. We define a notion of abstraction for Markov chains. To be able to relate the behavior of abstract and concrete systems, we equip the notion of abstraction with the concept of simulation. Furthermore, we present model checking for abstract probabilistic systems (abstract Markov chains) with respect to specifications in probabilistic temporal logics, interpreted over a 3-valued domain. More specifically, we introduce a 3-valued version of probabilistic computation-tree logic (PCTL) and give a model checking algorithm w.r.t. abstract Markov chains.
@cite_14: 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.
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@cite_8: Probabilistic models are widely used to analyze embedded, networked, and more recently biological systems. Existing numerical analysis techniques are limited to finite- state models and suffer from the state explosion problem. As a consequence, the user often has to manually abstract the intended model to get a tractable one. To this end, we propose the predicate abstraction model checker PASS which automates this process. We leverage recent advances in automatic theorem proving to compute tractable finite-state models. Experiments show the feasibility of our approach. To the best of our knowledge, this is the first time that properties of probabilistic infinite-state models have been verified at this level of automation.
@cite_28: 
@cite_21: 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 three-valued semantics of the branching-time logic CSL (Continuous Stochastic Logic). Experiments on an infinite-state CTMC indicate the feasibility of our abstraction technique.
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@cite_3: 
@cite_23: 
@cite_31: We propose a formal language for the specification of trace properties of probabilistic, nondeterministic transition systems, encompassing the properties expressible in Linear Time Logic. Those formulas are in general undecidable on infinite deterministic transition systems and thus on infinite Markov decision processes. This language has both a semantics in terms of sets of traces, as well as another semantics in terms of measurable functions; we give and prove theorems linking the two semantics. We then apply abstract interpretation-based techniques to give upper bounds on the worst-case probability of the studied property. We propose an enhancement of this technique when the state space is partitioned -- for instance along the program points -- allowing the use of faster iteration methods.
@cite_10: We generalize the familiar semantics for probabilistic computation tree logic from finite-state to infinite-state labelled Markov chains such that formulas are interpreted as measurable sets. Then we show how to synthesize finite-state abstractions which are sound for full probabilistic computation tree logic and in which measures are approximated by monotone set functions. This synthesis of sound finite-state approximants also applies to finite-state systems and is a probabilistic analogue of predicate abstraction. Sufficient and always realizable conditions are identified for obtaining optimal such abstractions for probabilistic propositional modal logic.
@cite_12: 
Abstraction Schemes: Abstractions have been extensively studied in the context of probabilistic systems. General issues in defining good abstractions as well as specific proposals for families of abstract models are presented in @cite_23 @cite_6 @cite_7 @cite_10 @cite_12 @cite_30 @cite_18 @cite_21 @cite_31 @cite_14 @cite_3 . Recently, theorem-prover based algorithms for constructing abstractions of probabilistic systems based on predicates have been presented @cite_8 . Another notion that has been recently proposed is the notion of a magnifying-lens abstraction'' @cite_28 , which can be used to assist in the model checking process, by approximating the measure of the satisfaction of path formulas for sets of concrete states; the method is not an abstraction in the traditional sense in that neither is an abstract model explicitly constructed, nor is the model used for reasoning, one that simulates the concrete model.