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## Symbolic model checking for probabilistic processes (1997)

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Venue: | IN PROCEEDINGS OF ICALP '97 |

Citations: | 97 - 29 self |

### Citations

3471 | Graph-Based Algorithms for Boolean Function Manipulation
- Bryant
- 1986
(Show Context)
Citation Context ...andPsatisfies[3a]pthenP terminates with probability at leastp. j=Xiff(1)j= existsk0with(i)j=1,i=0;1;:::;k?1and(k)j=2. 4 Multi-terminal binary decision diagrams Ordered Binary Decision Diagrams (BDDs) =-=[7, 8, 15, 28]-=- are a compact representation of boolean functionsf:f0;1gn!f0;1g. They are based on the canonical representation of the binary tree of the function as a directed graph obtained through folding 4For si... |

1373 | Automatic verification of finite-state concurrent systems using temporal logic specifications
- Clarke, Emerson, et al.
- 1986
(Show Context)
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... |

692 |
An Automata-Theoretic Approach to Automatic Program Verification (Preliminary Report
- Vardi, Wolper
- 1986
(Show Context)
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... |

599 |
Symbolic Model Checking: An Approach to the State Explosion Problem
- McMillan
- 1992
(Show Context)
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... |

562 |
Measure Theory
- HALMOS
- 1974
(Show Context)
Citation Context ...erus [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 consider nondeterminism). LetAPdenote a finite set of atomic propositions. A labelled Markov chain over a set of atom... |

557 |
Automata on infinite objects
- Thomas
- 1990
(Show Context)
Citation Context ...erus [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 consider nondeterminism). LetAPdenote a finite set of atomic propositions. A labelled Markov chain over a set of atom... |

521 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
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... |

359 | N.: Probabilistic simulations for probabilistic processes
- Segala, Lynch
- 1995
(Show Context)
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... |

310 |
Automatic verification of probabilistic concurrent finite-state programs
- Vardi
- 1985
(Show Context)
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... |

284 | Model checking of probabilistic and nondeterministic systems
- Bianco, Alfaro
- 1995
(Show Context)
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... |

226 |
Emerson: Synthesis of synchronization skeletons for branching time temporal logic
- Clarke, E
(Show Context)
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... |

216 |
The complexity of probabilistic verification
- Courcoubetis, Yannakakis
- 1995
(Show Context)
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... |

184 |
Multiterminal binary decision diagrams: an efficient data structure for matrix representation
- Fujita, McGeer, et al.
- 1997
(Show Context)
Citation Context ... 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 matrices. As a matter of fact, in =-=[13]-=- MTBDDs have been shown to perform no worse than sparse matrices. Thus, converting to MTBDDs ensures smooth integration with a symbolic model checker such as smv and has the potential to outperform sp... |

142 | Spectral transforms for large Boolean functions with application to technology mapping
- Clarke, McMillan, et al.
- 1993
(Show Context)
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... |

135 | Probabilistic predicate transformers
- Morgan, McIver, et al.
- 1996
(Show Context)
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 ... |

131 | Model checking for a probabilistic branching time logic with fairness
- Baier, Kwiatkowska
- 1998
(Show Context)
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... |

124 |
A probabilistic PDL
- Kozen
- 1985
(Show Context)
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 ... |

78 |
A logic for reasoning about time and probability
- Hansen, Jonsson
- 1994
(Show Context)
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... |

73 | Termination of probabilistic concurrent programs
- Hart, Sharir, et al.
- 1983
(Show Context)
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,... |

69 | Quantitative analysis and model checking - Huth, Kwiatkowska - 1997 |

54 |
Verification of Multiprocess Probabilistic Protocols
- Pnueli, Zuck
- 1986
(Show Context)
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,... |

47 |
Probabilistic Verification
- Pnueli, Zuck
- 1993
(Show Context)
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,... |

41 | Verifying Automata Specifications of Probabilistic Real-time Systems
- Alur, Courcoubetis, et al.
- 1992
(Show Context)
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 ... |

39 |
Verifying temporal properties of finite-state probabilistic programs
- Courcoubetis, Yannakakis
- 1988
(Show Context)
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... |

36 |
Generating BDDs for symbolic model checking in CCS
- Enders, Filkorn, et al.
- 1993
(Show Context)
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 ... |

34 | Probabilistic temporal logics for finite and bounded models
- Hart, Sharir
- 1984
(Show Context)
Citation Context |

31 | Verus: A Tool for Quantitative Analysis of Finite-State Real-Time Systems
- Campos, Clarke, et al.
- 1995
(Show Context)
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
(Show Context)
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... |

22 | Polynomial Time Algorithms for Testing Probabilistic Bisimulation and Simulation
- Baier
- 1996
(Show Context)
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... |

21 |
Time and Probability
- Hansson
- 1994
(Show Context)
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... |

9 |
Formal verification of performance and reliability of real-time systems
- Alfaro
- 1996
(Show Context)
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
- Clarke, Grumberg, et al.
- 1993
(Show Context)
Citation Context |

2 |
Probabilistic model checking using multi terminal binary decision diagrams
- Baier, Campos, et al.
(Show Context)
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... |