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21
Probabilistic CEGAR
 University of Saarland
, 2007
"... Abstract. Counterexampleguided abstraction refinement (CEGAR) has been en vogue for the automatic verification of very large systems in the past years. When trying to apply CEGAR to the verification of probabilistic systems, various foundational questions arise. This paper explores them in the cont ..."
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Cited by 36 (2 self)
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Abstract. Counterexampleguided abstraction refinement (CEGAR) has been en vogue for the automatic verification of very large systems in the past years. When trying to apply CEGAR to the verification of probabilistic systems, various foundational questions arise. This paper explores them in the context of predicate abstraction. 1
B.: Counterexample generation in probabilistic model checking
 IEEE Trans. on Software Engineering
"... Abstract—Providing evidence for the refutation of a property is an essential, if not the most important, feature of model checking. This paper considers algorithms for counterexample generation for probabilistic CTL formulas in discretetime Markov chains. Finding the strongest evidence (i.e., the m ..."
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Cited by 33 (9 self)
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Abstract—Providing evidence for the refutation of a property is an essential, if not the most important, feature of model checking. This paper considers algorithms for counterexample generation for probabilistic CTL formulas in discretetime Markov chains. Finding the strongest evidence (i.e., the most probable path) violating a (bounded) untilformula is shown to be reducible to a singlesource (hopconstrained) shortest path problem. Counterexamples of smallest size that deviate most from the required probability bound can be obtained by applying (small amendments to) kshortest (hopconstrained) paths algorithms. These results can be extended to Markov chains with rewards, to LTL model checking, and are useful for Markov decision processes. Experimental results show that, typically, the size of a counterexample is excessive. To obtain much more compact representations, we present a simple algorithm to generate (minimal) regular expressions that can act as counterexamples. The feasibility of our approach is illustrated by means of two communication protocols: leader election in an anonymous ring network and the Crowds protocol. Index Terms—Diagnostic feedback, Markov chain, model checking, regular expression, shortest path. Ç 1
J.P.: Counterexamples in probabilistic model checking
 In: Tools and Algorithms for the Construction and Analysis of Systems (TACAS), 13th International Conference. (2007
"... ..."
Significant diagnostic counterexamples in probabilistic model checking
 Proc. of HVC’08, volume 5394 of LNCS
, 2009
"... Abstract. This paper presents a novel technique for counterexample generation in probabilistic model checking of Markov Chains and Markov Decision Processes. (Finite) paths in counterexamples are grouped together in witnesses that are likely to provide similar debugging information to the user. We l ..."
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Cited by 25 (3 self)
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Abstract. This paper presents a novel technique for counterexample generation in probabilistic model checking of Markov Chains and Markov Decision Processes. (Finite) paths in counterexamples are grouped together in witnesses that are likely to provide similar debugging information to the user. We list five properties that witnesses should satisfy in order to be useful as debugging aid: similarity, accuracy, originality, significance, and finiteness. Our witnesses contain paths that behave similar outside strongly connected components. This papers shows how to compute these witnesses by reducing the problem of generating counterexamples for general properties over Markov Decision Processes, in several steps, to the easy problem of generating counterexamples for reachability properties over acyclic Markov Chains. 1
Counterexamples for Model Checking of Markov Decision Processes
"... Abstract. The debugging of stochastic system models relies on the availability of diagnostic information. Classic probabilistic model checkers, which are based on iterated numerical probability matrix operations, do not provide such diagnostic information. In precursory work, we have devised counter ..."
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Cited by 17 (5 self)
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Abstract. The debugging of stochastic system models relies on the availability of diagnostic information. Classic probabilistic model checkers, which are based on iterated numerical probability matrix operations, do not provide such diagnostic information. In precursory work, we have devised counterexample generation methods for continuous and discretetime Markov Chains based on heuristics guided explicit state space search. In this paper we address the problem of generating diagnostic information, or counterexamples, for Markov Decision Processes (MDPs), which are a convenient formalism for modelling concurrent stochastic systems. We define the notion of counterexamples for MDPs in relation to an upwardsbounded PCTL formula. Next we present our approach to counterexample generation. We first use an adoption of Eppstein’s algorithm for kshortest paths in order to collect the most probable MDP execution traces contributing to a violation of the PCTL formula. We then use the data structure of AND/OR trees in order to adequately extract from the collected execution sequences the most informative counterexample and to compute its probability. In our experimental evaluation we show that our approach scales to models of realistic size, and that the collected diagnostic information is helpful in system debugging. 1
S.: Debugging of Dependability Models Using Interactive Visualization of Counterexamples
 In: QEST, IEEE Computer Society
, 2008
"... We present an approach to support the debugging of stochastic system models using interactive visualization. The goal of this work is to facilitate the identification of causal factors in the potentially very large sets of execution paths that form counterexamples in stochastic model checking. The v ..."
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Cited by 15 (8 self)
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We present an approach to support the debugging of stochastic system models using interactive visualization. The goal of this work is to facilitate the identification of causal factors in the potentially very large sets of execution paths that form counterexamples in stochastic model checking. The visualization is interactive and allows the user to focus on the most meaningful aspects of a counterexample. We present the application of the visualization method as implemented in our prototype tool DIPRO to two case studies. 1
Extended directed search for probabilistic timed reachability
 In FORMATS’06, volume 4202 of LNCS
, 2006
"... Abstract. Current numerical model checkers for stochastic systems can efficiently analyse stochastic models. However, the fact that they are unable to provide debugging information constrains their practical use. In precursory work we proposed a method to select diagnostic traces, in the parlance of ..."
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Cited by 14 (7 self)
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Abstract. Current numerical model checkers for stochastic systems can efficiently analyse stochastic models. However, the fact that they are unable to provide debugging information constrains their practical use. In precursory work we proposed a method to select diagnostic traces, in the parlance of functional model checking commonly referred to as failure traces or counterexamples, for probabilistic timed reachability properties on discretetime and continuoustime Markov chains. We applied directed explicitstate search algorithms, like Z ∗ , to determine a diagnostic trace which carries large amount of probability. In this paper we extend this approach to determining sets of traces that carry large probability mass, since properties of stochastic systems are typically not violated by single traces, but by collections of those. To this end we extend existing heuristics guided search algorithms so that they select sets of traces. The result is provided in the form of a Markov chain. Such diagnostic Markov chains are not just essential tools for diagnostics and debugging but, they also allow the solution of timed reachability probability to be approximated from below. In particular cases, they also provide real counterexamples which can be used to show the violation of the given property. Our algorithms have been implemented in the stochastic model checker PRISM. We illustrate the applicability of our approach using a number of case studies. 1
Survey on Directed Model Checking
, 2009
"... Abstract. This article surveys and gives historical accounts to the algorithmic essentials of directed model checking, a promising bughunting technique to mitigate the state explosion problem. In the enumeration process, successor selection is prioritized. We discuss existing guidance and methods t ..."
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Cited by 9 (1 self)
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Abstract. This article surveys and gives historical accounts to the algorithmic essentials of directed model checking, a promising bughunting technique to mitigate the state explosion problem. In the enumeration process, successor selection is prioritized. We discuss existing guidance and methods to automatically generate them by exploiting system abstractions. We extend the algorithms to feature partialorder reduction and show how liveness problems can be adapted by lifting the search space. For deterministic, finite domains we instantiate the algorithms to directed symbolic, external and distributed search. For realtime domains we discuss the adaption of the algorithms to timed automata and for probabilistic domains we show the application to counterexample generation. Last but not least, we explain how directed model checking helps to accelerate finding solutions to scheduling problems. 1
Counterexamples in Probabilistic LTL Model Checking for Markov Chains
 In Proceedings of the 20th International Conference on Concurrency Theory
, 2009
"... Abstract. We propose a way of presenting and computing a counterexample in probabilistic LTL model checking for discretetime Markov chains. In qualitative probabilistic model checking, we present a counterexample as a pair (α,γ), where α,γ are finite words such that all paths that extend α and hav ..."
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Cited by 7 (0 self)
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Abstract. We propose a way of presenting and computing a counterexample in probabilistic LTL model checking for discretetime Markov chains. In qualitative probabilistic model checking, we present a counterexample as a pair (α,γ), where α,γ are finite words such that all paths that extend α and have infinitely many occurrences of γ violate the specification. In quantitative probabilistic model checking, we present a counterexample as a pair (W,R), where W is a set of such finite words α and R is a set of such finite words γ. Moreover, we suggest how the counterexample presented helps the user identify the underlying error in the system by means of an interactive game with the model checker. 1
K : A heuristic search algorithm for finding the k shortest paths
 Artificial Intelligence
"... We present a directed search algorithm, called K ∗ , for finding the k shortest paths between a designated pair of vertices in a given directed weighted graph. K ∗ has two advantages compared to current kshortestpaths algorithms. First, K ∗ operates onthefly, which means that it does not require ..."
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Cited by 7 (1 self)
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We present a directed search algorithm, called K ∗ , for finding the k shortest paths between a designated pair of vertices in a given directed weighted graph. K ∗ has two advantages compared to current kshortestpaths algorithms. First, K ∗ operates onthefly, which means that it does not require the graph to be explicitly available and stored in main memory. Portions of the graph will be generated as needed. Second, K ∗ can be guided using heuristic functions. We prove the correctness of K ∗ and determine its asymptotic worstcase complexity when using a consistent heuristic to be the same as the state of the art, O(m + n log n + k), with respect to both runtime and space, where n is the number of vertices and m is the number of edges of the graph. We present an experimental evaluation of K ∗ by applying it to route planning problems as well as counterexample generation for stochastic model checking. The experimental results illustrate that due to the use of heuristic, onthefly search K ∗ can use less time and memory compared to the most efficient kshortestpaths algorithms known so far. Key words: kshortestpaths problem; K ∗ ; heuristic search; onthefly search 1.