## Logical Characterizations of Bisimulations for Discrete Probabilistic Systems (2007)

Citations: | 11 - 0 self |

### BibTeX

@MISC{Parma07logicalcharacterizations,

author = {Augusto Parma and Roberto Segala},

title = {Logical Characterizations of Bisimulations for Discrete Probabilistic Systems },

year = {2007}

}

### OpenURL

### Abstract

We give logical characterizations of bisimulation relations for the probabilistic automata of Segala in terms of three Hennessy-Milner style logics. The three logics characterize strong, strong probabilistic and weak probabilistic bisimulation, and differ only for the kind of diamond operator used. Compared to the Larsen and Skou logic for reactive systems, these logics introduce a new operator that measures the probability of the set of states that satisfy a formula. Moreover, the satisfaction relation is defined on measures rather than single states. We rederive previous results of Desharnais et. al. by defining sublogics for Reactive and Alternating Models viewed as restrictions of probabilistic automata. Finally, we identify restrictions on probabilistic automata, weaker than those imposed by the Alternating Models, that preserve the logical characterization of Desharnais et. al. These restrictions require that each state either enables several ordinary transitions or enables a single probabilistic transition.

### Citations

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(Show Context)
Citation Context ...obability and nondeterminism interact together. The goal of this paper is to study logical characterizations of bisimulations for probabilistic automata in the context of Hennessy-Milner style logics =-=[8]-=-. Logical ⋆ Supported by INRIA project ProNoBiS and MIUR project AIDA.scharacterizations have been studied already by Larsen and Skou [11] for reactive systems [6] and by Desharnais et. al. [5] for la... |

414 | Bisimulation through probabilistic testing
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Citation Context ...listic automata in the context of Hennessy-Milner style logics [8]. Logical ⋆ Supported by INRIA project ProNoBiS and MIUR project AIDA.scharacterizations have been studied already by Larsen and Skou =-=[11]-=- for reactive systems [6] and by Desharnais et. al. [5] for labeled concurrent Markov chains (alternating model) [7, 13]. These logics are derived from the Hennessy-Milner logic by replacing the diamo... |

278 | Probabilistic Simulations for Probabilistic Processes - Segala, Lynch - 1995 |

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Citation Context ...Concluding Remarks We have studied logical characterizations, in terms of Hennessy-Milner style logics, of strong, strong probabilistic and weak probabilistic bisimulations for probabilistic automata =-=[14]-=-. Our logics are defined on measures over states rather than on single states, and add a new operator the classical Hennessy-Milner logic that measures the probability of the set of states that satisf... |

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Citation Context ... MIUR project AIDA.scharacterizations have been studied already by Larsen and Skou [11] for reactive systems [6] and by Desharnais et. al. [5] for labeled concurrent Markov chains (alternating model) =-=[7, 13]-=-. These logics are derived from the Hennessy-Milner logic by replacing the diamond operator with a probabilistic diamond operator that measures bounds on the probability of performing an externally vi... |

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Citation Context ... MIUR project AIDA.scharacterizations have been studied already by Larsen and Skou [11] for reactive systems [6] and by Desharnais et. al. [5] for labeled concurrent Markov chains (alternating model) =-=[7, 13]-=-. These logics are derived from the Hennessy-Milner logic by replacing the diamond operator with a probabilistic diamond operator that measures bounds on the probability of performing an externally vi... |

21 | R.: Decision algorithms for probabilistic bisimulation
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(Show Context)
Citation Context ...measure over (S, Σ) is a function µ: Σ → ℜ + such that, for every countable family {Ai}I ⊆ Σ of pairwise disjointsmeasurable sets, µ(∪IAi) = � I µ(Ai). A (sub-)probability measure is a measure µ: Σ → =-=[0, 1]-=- for which (µ(S) ≤ 1) µ(S) = 1. Probability measures are ranged over by µ, η, . . . and we propagate indices and primes where necessary. A set A ⊆ S is called a support for a measure µ on Σ if µ(S − A... |

16 | Weak bisimulation is sound and complete for pCTL
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- 2010
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Citation Context ...logics [8]. Logical ⋆ Supported by INRIA project ProNoBiS and MIUR project AIDA.scharacterizations have been studied already by Larsen and Skou [11] for reactive systems [6] and by Desharnais et. al. =-=[5]-=- for labeled concurrent Markov chains (alternating model) [7, 13]. These logics are derived from the Hennessy-Milner logic by replacing the diamond operator with a probabilistic diamond operator that ... |

13 | G.: Stochastic transition systems for continuous state spaces and non-determinism - Cattani, Segala, et al. - 2005 |

7 |
Comparative analysis of bisimulation relations on alternating and non-alternating probabilistic models
- Segala, Turrini
- 2005
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Citation Context ...e operator ♦· is: L N p ::= ⊤ | ¬ϕ | � ϕi | ♦· aϕ | [ϕ]p. I – µ |= ♦· aϕ iff for each s ∈ supp(µ) there exists a distribution η and a combined transition s a −→ η such that η |= ϕ. Since, as shown in =-=[16]-=-, ∼ p is weaker than ∼, adding the operator ♦· to L N does not change its expressivity. Hence, the operator ♦· is weaker than ♦. Soundness and completeness of L N p are stated by the following theorem... |

3 |
Axiomatisations and types for probabilistic and mobile processes
- Deng
- 2005
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Citation Context ...largest weak probabilistic bisimulation. There would be a fourth obvious relation that uses weak transitions induced by deterministic schedulers. However, this relation is not transitive, as shown in =-=[3]-=-, and thus it is not interesting. We recall an alternative way of defining bisimulation [8] as � i≥0 ∼n, where ∼0= S × S (all states are related) and for each pair of states s, r, s ∼n+1 r if for each... |

1 |
A logical chracterization of bisimulation for labelled markov processes
- Desharnais, Edalat, et al.
- 1998
(Show Context)
Citation Context ... ♦paϕ, where p is a rational number in [0, 1]. The only new operator is ♦p, whose semantics is: – s |= ♦paϕ iff there exists a transition s a −→ µ such that µ([ϕ]) ≥ p. Desharnais, Panangaden et. al. =-=[4]-=- have shown that negation is not necessary to characterize bisimulation for reactive systems. Moreover, they have shown that infinitary conjunction is not needed even if the state space is uncountable... |