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## Probabilistic Reachability for Parametric Markov Models

Citations: | 29 - 4 self |

### Citations

4818 |
Introduction to automata theory, languages, and computation
- Hopcroft, Ullman
- 1979
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Citation Context ... solve this problem. In this approach, the transition probabilities are considered as letters of an alphabet. Thus, the model can be viewed as a finite automaton. Then, based on the state elimination =-=[15]-=- method, the regular expression describing the language of such an automaton is calculated. In a postprocessing step, this regular expression is recursively evaluated resulting in a rational function ... |

837 | Crowds: Anonymity for web transactions
- Reiter, Rubin
- 1998
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Citation Context ...responding Prism models. All experiments were run on a Linux machine with an AMD Athlon(tm) XP 2600+ processor at 2 GHz equipped with 2GB of RAM. Crowds Protocol. The intention of the Crowds protocol =-=[21]-=- is to protect the anonymity of Internet users. The protocol hides each user’s communications via random routing. Assume that we have N honest Crowd members, and M dishonest members. Moreover, assume ... |

752 | Introduction to the numerical solution of Markov chains - Stewart - 1994 |

370 | A Logic for Reasoning about Time and Reliability
- Hansson, Jonsson
- 1994
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Citation Context ...all maximal well-defined evaluation u, s1 ∼D s2 implies that s1 ∼Du s2, and s1 ≈D s2 implies that s1 ≈Du s2. Both strong and weak bisimulation preserve the probabilistic reachability for ordinary MCs =-=[13,3]-=-. By the above lemma, for PMCs, both strong and weak bisimulation preserve probabilistic reachability for all maximal well-defined evaluations. Similar result holds for PMRMs: if two states s1,s2 of M... |

290 | Model Checking of Probabilistic and Nondeterministic Systems
- Bianco, Alfaro
- 1995
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Citation Context ...mpute the maximum maxA Pr Mu,A (s0,B) for each strictly well-defined valuation u, with A ranging over all schedulers. For the ordinary MDP case (e.g. Mu where u is strictly well-defined), as shown in =-=[5]-=-, the class of stationary schedulers is sufficient to achieve this maximum probabilistic reachability. For PMDPs, different stationary schedulers are needed for different evaluations: Example 1. Consi... |

282 | PRISM: A tool for automatic verification of probabilistic systems
- Hinton, Kwiatkowska, et al.
- 2006
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Citation Context ...the state-elimination algorithm as well as the bisimulation minimisation algorithm. Param allows guarded-commands based input language supporting MC, MRM and MDPs. The language is extended from Prism =-=[14]-=- with unknown parameters. Properties are specified by PCTL formulae without nesting.0.8 0.4 152 148 144 0 0 0.2 0.4 P 0.6 0.8 1 B 0.20.40.60.8 1 0 140 0 0.2 p 0.4 0.6 0 0.2 0.4 0.6 q Fig. 2. Left: Cr... |

272 |
Algorithms for Computer Algebra
- Geddes, Czapor, et al.
- 1992
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Citation Context ...s3,a = 0. Complexity. Since our algorithm is dealing with rational functions, we first discuss briefly the complexity of arithmetic for polynomials and rational functions. For more detail we refer to =-=[10]-=-. For a polynomial f, let mon(f) denote the number of monomials. Addition and subtraction of two polynomials f and g are performed by adding or subtracting coefficients of like monomials, which take t... |

122 |
Specification and refinement of probabilistic processes
- Jonsson, Larsen
- 1991
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Citation Context ...bility measure in the PMC (MA)u.2.1 Bisimulation Relations A bisimulation is an equivalence relation on states which subsumes states satisfying the same properties. Now we extend the standard strong =-=[17]-=- and weak bisimulation [4] relations for Markov models to our parametric setting in an obvious way. Definition 8. Let D = (S,s0,P,V ) be a PMC and R be an equivalence relation on S. R is a strong bisi... |

107 | Stochastic model checking. In:
- Kwiatkowska, Norman, et al.
- 2007
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Citation Context ...t holds that Pr Du (s0,B) = f[V/u]. Parametric MRMs. Let R = (D,r) be a PMRM with D = (S,s0,P,V ). Let B ⊆ S be a set of target states. We are interested in the parametric expected accumulated reward =-=[18]-=- until B, which is denoted by R(s0,B). Formally, R(s0,B) is the expectation of the random variable X : σ ∈ Path D (s0) → R≥0 which is defined by: X(σ) equals 0 if first(σ) ∈ B, ∞ if σ[i] /∈ B for all ... |

74 | Weak bisimulation for fully probabilistic processes.
- Baier, Hermanns
- 1997
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Citation Context ...MC. This is then submitted to the dedicated algorithm for parametric MCs. In all settings, we reduce the state space prior to state elimination, by extending standard strong [9] and weak bisimulation =-=[3]-=- quotioning techniques to parametric MCs and MRMs. A very important observation is that for parametric MDPs we can apply the quotioning on the encoded parametric MC, since this preserves the maximal p... |

64 | Comparative Branching-Time Semantics for Markov Chains.
- Baier, Katoen, et al.
- 2005
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Citation Context ...MA)u.2.1 Bisimulation Relations A bisimulation is an equivalence relation on states which subsumes states satisfying the same properties. Now we extend the standard strong [17] and weak bisimulation =-=[4]-=- relations for Markov models to our parametric setting in an obvious way. Definition 8. Let D = (S,s0,P,V ) be a PMC and R be an equivalence relation on S. R is a strong bisimulation on D with respect... |

61 |
Stochastic Petri net models of polling systems
- Ibe, Trivedi
- 1990
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Citation Context ...isimulation optimisation was not of any use, as the quotient equals the original model. For n = 140, the analysis took 64 seconds and 50 MB of memory. Cyclic Polling Server. The cyclic polling server =-=[16]-=- consists of a number of N stations which are handled by the polling server. Process i is allowed to send a job to the server if he owns the token, circulating around the stations in a round robin man... |

55 |
Verification of multiprocess probabilistic protocols.
- Pnueli, Zuck
- 1986
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Citation Context ...12 8 4 0 0 0.2 0.4 p1 0.6 0.8 1 0 0.2 0.4 0.6 p2 0.8 1 Fig. 3. Left: Cyclic Polling Server. Right: Randomised Mutual Exclusion Randomised Mutual Exclusion. In the randomised mutual exclusion protocol =-=[20]-=- several processes try to enter a critical section. We consider the protocol with two processes i = 1,2. Process i tries to enter the critical section with pi, and with probability 1 − pi, it waits un... |

36 |
Optimal StateSpace Lumping
- Derisavi, Hermanns, et al.
- 2003
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Citation Context ...the non-determinism occurring in the final rational function. We also discuss briefly the complexity of the bisimulation minimisation algorithms. For ordinary MCs, strong bisimulation can be computed =-=[9]-=- in O(mlog n) where n,m denote the number of states and transitions respectively. The complexity of deciding weak bisimulation [3] is O(mn). These algorithms can be extended to PMCs directly, with the... |

32 | Symbolic and parametric model checking of discrete-time markov chains,”
- Daws
- 2005
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Citation Context ...stungsgarantien für Rechnersysteme”, and has received funding from the European Community’s Seventh Framework Programme under grant agreement n o 214755.For (discrete-time) Markov chains (MCs), Daws =-=[8]-=- has devised a languagetheoretic approach to solve this problem. In this approach, the transition probabilities are considered as letters of an alphabet. Thus, the model can be viewed as a finite auto... |

28 | Optimal state-space lumping in Markov chains. - Derisavi, Hermanns, et al. - 2003 |

26 | Model-checking Markov chains in the presence of uncertainties - Sen, Viswanathan, et al. - 2006 |

19 |
Signal flow graph techniques for sequential circuit state diagrams.
- Brzozowski, Jr
- 1963
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Citation Context ...MRMs) which takes O(n 2 ) polynomial operations in worst case. Thus, all together O(n 3 ) many operations are needed to get the final function, which is the same as in the state elimination algorithm =-=[6]-=-. The complexity of arithmetic for polynomials depends on the degrees. For PMDPs, we first encode the non-deterministic choices via new binary variables. Then, the encoding PMC is submitted to the ded... |

16 | Approximate parameter synthesis for probabilistic time-bounded reachability.
- Han, Katoen, et al.
- 2008
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Citation Context ...evaluation is undecidable. Moreover, the problem to minimise or maximise unbounded probabilistic reachability is also shown to be undecidable. For parametric continuous-time Markov chains, Han et al. =-=[12]-=- have provided approximation algorithms to find valid valuations with respect to time bounded reachability properties. Recently, Damman et al. [7] have extended the approach of [8] to generate counter... |

13 | Optimal lower bounds on regular expression size using communication complexity.
- Gruber, Johannsen
- 2008
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Citation Context ...n automaton is calculated. In a postprocessing step, this regular expression is recursively evaluated resulting in a rational function over the parameters of the model. Recently, Gruber and Johannsen =-=[11]-=- have shown, however, that the size of the regular expression of a finite automaton explodes: it is n Θ(log n) where n is the number of states. This excessive growth is not only a theoretical insight,... |

13 |
Parametric probabilistic transition systems for system design and analysis.
- Lanotte, Maggiolo-Schettini, et al.
- 2007
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Citation Context ...uotioning. We illustrate the feasibility of the entire approach on a number of non-trivial parametric MCs, MRMs and MDPs. Our work has connections to several other recent scientific contributions. In =-=[19]-=-, Lanotte et al. considered parametric MCs, showing that the problem whether there exists a well-defined evaluation is undecidable. Moreover, the problem to minimise or maximise unbounded probabilisti... |

10 |
ProbMela and verification of Markov decision processes.
- Baier, Ciesinski, et al.
- 2005
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Citation Context ...ossy channel, where whenever a package is sent, it is received with probability x but lost with probability 1 − x. Such a network can be specified in a probabilistic variation of the PROMELA language =-=[2]-=-. In this context, we might aim, for instance, at determining parametric reachability probabilities, i.e., the probability to reach a given set of target states. This probability is a rational functio... |

7 |
Regular expressions for PCTL counterexamples
- Damman, Han, et al.
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Citation Context ...parametric continuous-time Markov chains, Han et al. [12] have provided approximation algorithms to find valid valuations with respect to time bounded reachability properties. Recently, Damman et al. =-=[7]-=- have extended the approach of [8] to generate counterexamples for MC. The regular expressions generated can be seen as a more compact and structured representation of counterexamples than providing a... |

5 | Symbolic partition refinement with dynamic balancing of time and space.
- Wimmer, Derisavi, et al.
- 2008
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Citation Context ...plan to look into continuous time models – with clocks. Other possible directions include the use of symbolic model representations, such as MTBDD-based techniques, symbolic bisimulation minimisation =-=[22]-=-, and also a symbolic variant of the state elimination algorithm. All relevant material (tool inputs and outputs) of the case studies is available at: http://d.cs.uni-sb.de/~zhang/parametricReference... |

3 |
The design of cocoalib
- Abbott
- 2006
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Citation Context ...e the transition matrix Pu : S × S → FV is defined by Pu(s,s ′ ) = P(s,s ′ )[Dom(u)/u]. We introduce the notion of well-defined evaluations. A total evaluation u is well-defined for D if Pu(s,s ′ ) ∈ =-=[0,1]-=- for all s,s ′ ∈ S, and Pu(s,S) ∈ [0,1] for all s ∈ S where Pu(s,S) denotes the sum ∑ s ′ ∈S Pu(s,s ′). Intuitively, u is well-defined if and only if the resulting PMC Du is then an ordinary MC withou... |