## Symbolic model checking for probabilistic processes (1997)

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Venue: | IN PROCEEDINGS OF ICALP '97 |

Citations: | 85 - 30 self |

### BibTeX

@INPROCEEDINGS{Baier97symbolicmodel,

author = {Christel Baier and Edmund M. Clarke and Vasiliki Hartonas-garmhausen and Marta Kwiatkowska and Mark Ryan},

title = {Symbolic model checking for probabilistic processes},

booktitle = {IN PROCEEDINGS OF ICALP '97},

year = {1997},

pages = {430--440},

publisher = {SpringerVerlag}

}

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### Abstract

We introduce a symbolic model checking procedure for Probabilistic Computation Tree Logic PCTL over labelled Markov chains as models. Model checking for probabilistic logics typically involves solving linear equation systems in order to ascertain the probability of a given formula holding in a state. Our algorithm is based on the idea of representing the matrices used in the linear equation systems by Multi-Terminal Binary Decision Diagrams (MTBDDs) introduced in Clarke et al [14]. Our procedure, based on the algorithm used by Hansson and Jonsson [24], uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space construction. A PCTL model checker is being implemented in Verus [9].

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Citation Context ...ctical use when verifying realistic systems. As a result, efficiency of probabilistic analysis lags behind efficient model checking techniques for conventional logics, such as symbolic model checking =-=[11, 12, 10, 8, 15, 28]-=-, for which tools capable of tackling industrial scale applications are available (cf. smv). This is undesirable as probabilistic approaches allow one to establish that certain properties hold (in som... |

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Citation Context ...ctical use when verifying realistic systems. As a result, efficiency of probabilistic analysis lags behind efficient model checking techniques for conventional logics, such as symbolic model checking =-=[11, 12, 10, 8, 15, 28]-=-, for which tools capable of tackling industrial scale applications are available (cf. smv). This is undesirable as probabilistic approaches allow one to establish that certain properties hold (in som... |

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Citation Context ..., and the Nuffield Foundation.Much has been published concerning the verification methods for probabilistic logics. Probabilistic extensions of dynamic logic [26] and temporal and modal logics, e.g. =-=[2, 6, 17, 24, 21, 27, 30, 31, 34]-=-, and automatic procedures for checking satisfaction for such logics have been proposed. The latter are based on reducing the calculation of the probability of formulas being satisfied to a linear alg... |

234 |
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Citation Context ... such as faulttolerant systems, randomized distributed systems and communication protocols. Models for such systems are variants of probabilistic automata (such as labelled Markov chains used in e.g. =-=[24, 34, 35, 17]-=-), in which the usual (boolean) transition relation is replaced with its probabilistic version given in the form of a Markov probability transition matrix. The probabilistic logics are typically obtai... |

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204 |
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Yannakakis: The complexity of probabilistic verification
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Citation Context ...s the size of the state space. Optimal methods are known (for sequential Markov chains, the lower bound is single exponential in the size of the formula and polynomial in the size of the Markov chain =-=[18]-=-), but these algorithms are not of much practical use when verifying realistic systems. As a result, efficiency of probabilistic analysis lags behind efficient model checking techniques for convention... |

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Multi-terminal binary decision diagrams: An efficient data structure for matrix representation. Formal Methods in System Design
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Citation Context ...ent for the construction of the state space. This paper proposes an alternative, which is based on expressing the probability calculations in terms of Multi-Terminal Binary Decision Diagrams (MTBDDs) =-=[16]-=-. MTBDDs are a generalization of (ordered) BDDs in the sense that they allow arbitrary real numbers in the terminal nodes instead of just 0 and 1, and so can provide a compact representation for matri... |

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Citation Context ...umberpin the[0;1]-interval the formula[]pin whichpacts as a threshold for truth in the sense that for the formula[]pto be satisfied (in the states) the probability that holds insmust be at leastp(see =-=[26, 32, 25]-=- for a different approach). With such logics one can express quantitative properties such as “the probability of the message being delivered withinttime steps is at least0:75” (see e.g. the timing or ... |

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Citation Context ...the matrices used in the linear equation systems by Multi-Terminal Binary Decision Diagrams (MTBDDs) introduced in Clarke et al [14]. Our procedure, based on the algorithm used by Hansson and Jonsson =-=[24]-=-, uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space construction. A PCTL model checker is being implemented in Verus [9]. 1... |

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Citation Context ..., 23, 5, 6, 2]) or (the more prevalent) qualitative properties, for which is required to be satisfied by almost all executions (which amounts to showing that is satisfied with probability 1, see e.g. =-=[1, 17, 23, 24, 21, 22, 29, 30, 34]-=-). ?This research was sponsored in part by the National Science Foundation under grant no. CCR8722633, by the Semiconductor Research Corporation under contract 92-DJ-294, and by the Wright Laboratory,... |

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Citation Context ..., 23, 5, 6, 2]) or (the more prevalent) qualitative properties, for which is required to be satisfied by almost all executions (which amounts to showing that is satisfied with probability 1, see e.g. =-=[1, 17, 23, 24, 21, 22, 29, 30, 34]-=-). ?This research was sponsored in part by the National Science Foundation under grant no. CCR8722633, by the Semiconductor Research Corporation under contract 92-DJ-294, and by the Wright Laboratory,... |

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Citation Context ..., 23, 5, 6, 2]) or (the more prevalent) qualitative properties, for which is required to be satisfied by almost all executions (which amounts to showing that is satisfied with probability 1, see e.g. =-=[1, 17, 23, 24, 21, 22, 29, 30, 34]-=-). ?This research was sponsored in part by the National Science Foundation under grant no. CCR8722633, by the Semiconductor Research Corporation under contract 92-DJ-294, and by the Wright Laboratory,... |

36 | D.L.: Verifying Automata Specifications of Probabilistic Real-time Systems
- Alur, Courcoubetis, et al.
- 1991
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Citation Context ...probability transition matrix. The probabilistic logics are typically obtained by “lifting” a nonprobabilistic logic to the probabilistic case by constructing for each formula and a real numberpin the=-=[0;1]-=--interval the formula[]pin whichpacts as a threshold for truth in the sense that for the formula[]pto be satisfied (in the states) the probability that holds insmust be at leastp(see [26, 32, 25] for ... |

36 |
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- 1988
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Citation Context ... such as faulttolerant systems, randomized distributed systems and communication protocols. Models for such systems are variants of probabilistic automata (such as labelled Markov chains used in e.g. =-=[24, 34, 35, 17]-=-), in which the usual (boolean) transition relation is replaced with its probabilistic version given in the form of a Markov probability transition matrix. The probabilistic logics are typically obtai... |

35 |
Generating BDDs for Symbolic Model Checking in CCS
- Enders, Filkorn, et al.
- 1991
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Citation Context ...s useful in timing analysis of systems. We expect that MTBDDs can be used to derive PCTL model checking by applying the methods of [18]. Likewise, testing of probabilistic bisimulation and simulation =-=[3, 19]-=- can be implemented using MTBDDs. An extension to the case of infinite state systems, perhaps by appropriate combination with induction, as well as a generalization to allow non-determinism, would be ... |

33 |
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- 1984
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29 | Verus: A tool for quantitative analysis of finite-state real-time systems
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- 1996
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Citation Context ...n [24], uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space construction. A PCTL model checker is being implemented in Verus =-=[9]-=-. 1 Introduction Probabilistic techniques, and in particular probabilistic logics, have proved successful in the specification and verification of systems that exhibit uncertainty, such as faulttolera... |

26 |
Multi-Terminal Binary Decision Diagrams and Hybrid Decision Diagrams
- Clarke, Fujita, et al.
- 1996
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Citation Context ... holding in a state. Our algorithm is based on the idea of representing the matrices used in the linear equation systems by Multi-Terminal Binary Decision Diagrams (MTBDDs) introduced in Clarke et al =-=[14]-=-. Our procedure, based on the algorithm used by Hansson and Jonsson [24], uses BDDs to represent formulas and MTBDDs to represent Markov chains, and is efficient because it avoids explicit state space... |

20 | Polynomial time algorithms for testing probabilistic bisimulation and simulation
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- 1996
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Citation Context ... the path operatorsAandEin CTL have been replaced by the operator []wp. The usual derived constants and operators are: :2),1!2=:1_2. Operators for modelling “eventually” or “always” can be derived by:=-=[3]-=-p=[ttU]p,[2]p=:[3:]>1?p, and similarly for[]>p. LetM=(S;P;L)be a labelled Markov chain. The satisfaction relationj= sj=[1U2]wpiffProbf2Path!(s):j=1U2gwp sj=[X]wpiffProbf2Path!(s):j=Xgwp alls2S sj=:iff... |

20 |
Time and Probability
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Citation Context ...tive properties such as “the probability of the message being delivered withinttime steps is at least0:75” (see e.g. the timing or average-case analysis of real-time or randomized distributed systems =-=[24, 23, 5, 6, 2]-=-) or (the more prevalent) qualitative properties, for which is required to be satisfied by almost all executions (which amounts to showing that is satisfied with probability 1, see e.g. [1, 17, 23, 24... |

6 |
Formal Verification of Performance and Reliability of Real-Time Systems
- Alfaro
- 1996
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Citation Context .... In the case of[X]wpthe probability is calculated by multiplying the transition matrix by the boolean vector set to 1 iff the state satisfies , whereas B[:]=:B[] B[tt]=TRUE for[1U2]wpwe B[1^2]=B[1]^B=-=[2]-=- B[ai]=x[xi] derive an operator calledUNTIL(), based on [24], which we express in terms of MTBDDs. LetM=(S;P;L)be a labelled Markov chain which is represented by a MTBDD Pover2nvariables as described ... |

6 |
Verfication Tools for Finite-State Concurrent Programs
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- 1993
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2 |
Probabilistic model checking using multi terminal binary decision diagrams
- Baier, Campos, et al.
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Citation Context ...nd MTBDDs for probabilistic formulas. Currently, we are implementing the PCTL symbolic model checking in Verus [9]. For reasons of space we omit much detail from this paper, which will be reported in =-=[4]-=-. We assume some familiarity with BDDs, automata on infinite sequences, probability and measure theory [8, 33, 20].2 Labelled Markov chains We use discrete time Markov chains as models (we do not con... |