## It Usually Works: The Temporal Logic of Stochastic Systems (1995)

Citations: | 88 - 0 self |

### BibTeX

@INPROCEEDINGS{Aziz95itusually,

author = {Adnan Aziz and Vigyan Singhal and Felice Balarin and Robert K. Brayton and Alberto L. Sangiovanni-vincentelli},

title = {It Usually Works: The Temporal Logic of Stochastic Systems},

booktitle = {},

year = {1995},

pages = {155--165},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

. In this paper the branching time logic pCTL is defined. pCTL expresses quantitative bounds on the probabilities of correct behavior; it can be interpreted over discrete Markov processes. A bisimulation relation is defined on finite Markov processes, and shown to be sound and complete with respect to pCTL . We extend the universe of models to generalized Markov processes in order to support notions of refinement, abstraction, and parametrization. Model checking pCTL over generalized Markov processes is shown to be elementary by a reduction to RCF. We conclude by describing practical and theoretical avenues for further work. 1 Introduction The study of formal methods to specify and prove properties of finite state systems has been the subject of intense research. Various methodologies have been proposed; some of the most fruitful, in both theory and practise, have been based on temporal logic [10]. Properties are expressed using formulae which are built out of operators ...

### Citations

1153 | Temporal and modal logic
- Emerson
- 1990
(Show Context)
Citation Context ...s of finite state systems has been the subject of intense research. Various methodologies have been proposed; some of the most fruitful, in both theory and practise, have been based on temporal logic =-=[10]-=-. Properties are expressed using formulae which are built out of operators on the computation paths. The goal of this paper is to take verification techniques developed for temporal logic, and apply t... |

353 |
Real Analysis
- Royden
- 1988
(Show Context)
Citation Context ... s0 , i.e. the class of subsets of U s0 to which probabilities can be assigned. The transition probability matrix T yields the probability measure s0 : C s0 ! [0; 1]; by the measure extension theorem =-=[19]-=-,ss0 : C s0 ! [0; 1] is well defined. Given a set fi of sequences over the alphabet 2 AP , we will abuse notation and refer to the probability of fi when we mean the probability of the set of all stat... |

268 | A logic for reasoning about time and reliability
- Hansson, Jonsson
- 1994
(Show Context)
Citation Context ...n a fair timed automaton; the proof of correctness is non trivial due to the density of the underlying model of time. Formulations of quantitative approaches to probabilistic verification also exist. =-=[12]-=- describes an extension of CTL, referred to as PCTL, capable of expressing numerical bounds on the probability of specified properties; it is essentially identical to the logic pCTL defined in x 3. [8... |

164 |
Introduction to probabilistic automata
- Paz
- 1971
(Show Context)
Citation Context ...g systems where the transition probabilities are not completely specified; these systems allow notions of abstraction and refinement. Using characterizations of discrete Markov processes developed in =-=[15]-=- and results on the decidability of real closed fields [5] we derive an elementary decision procedure for model checking pCTL over generalized discrete Markov processes. There is a body of past work o... |

119 |
The complexity of elementary algebra and geometry
- Ben-Or, Kozen, et al.
- 1986
(Show Context)
Citation Context ...tely specified; these systems allow notions of abstraction and refinement. Using characterizations of discrete Markov processes developed in [15] and results on the decidability of real closed fields =-=[5]-=- we derive an elementary decision procedure for model checking pCTL over generalized discrete Markov processes. There is a body of past work on applying formal methods to stochastic systems. An early ... |

82 |
Markov chains
- Revuz
- 1984
(Show Context)
Citation Context ...ge of the linear specification, and composing it with the Markov process -- the measure of the set of accepting states of this composed structure can then be obtained by using accumulation techniques =-=[18]-=-. It should be noted that pCTL can be model checked in polynomial time. 4 Bisimulation for finite Markov processes Developing a notion of bisimulation is integral to the analysis of state based dynami... |

48 | Sparse Elimination and Applications in Kinematics - Emiris - 1994 |

45 |
Probabilistic Verification
- Pnueli, Zuck
(Show Context)
Citation Context ...entional temporal logic analysis on a model with fairness constraints that eliminate events of measure zero; the actual probabilities of events is immaterial. Examples of such approaches are given in =-=[13, 16, 20, 1]-=-. [16] introduces the notion of ff-fairness; this construct embodies the fact that if a given state is visited infinitely often, then with probability one each state which the given state can make a t... |

33 |
Probabilistic Temporal Logic for Finite and Bounded Models
- Hart, Sharir
- 1984
(Show Context)
Citation Context ...entional temporal logic analysis on a model with fairness constraints that eliminate events of measure zero; the actual probabilities of events is immaterial. Examples of such approaches are given in =-=[13, 16, 20, 1]-=-. [16] introduces the notion of ff-fairness; this construct embodies the fact that if a given state is visited infinitely often, then with probability one each state which the given state can make a t... |

19 |
Characterizing Kripke structures in temporal logic
- Browne, Clarke, et al.
- 1987
(Show Context)
Citation Context ...teness. States which agree in their truth values on all formulae are probabilistically bisimilar i.e. (8OE 2 pCTL ) s j= OE $ t j= OE ) E prob (s; t) The proof of this lemma proceeds in the spirit of =-=[6]-=-: inductively build up pCTL formula eventually characterizing states upto E prob equivalence; details are given in [2]. It should be noted that the states can in fact be characterized in pCTL; this is... |

19 |
Verification of probabilistic programs
- Sharir, Pnueli, et al.
- 1984
(Show Context)
Citation Context ...n elementary decision procedure for model checking pCTL over generalized discrete Markov processes. There is a body of past work on applying formal methods to stochastic systems. An early approach is =-=[14]-=- which computes the expected time for a probabilistic program to terminate. Programs are modeled as Markov chains; the techniques are standard in Markov chain theory. [9] describes an approach to the ... |

11 |
Verifying Temporal Properties of Finite State Probabilistic Programs
- Courcoubetis, Yannakakis
- 1988
(Show Context)
Citation Context ...2] describes an extension of CTL, referred to as PCTL, capable of expressing numerical bounds on the probability of specified properties; it is essentially identical to the logic pCTL defined in x 3. =-=[8]-=- describes an algorithm for determining the measure of a set of paths in a discrete Markov process satisfying a PLTL [10] formula; this immediately provides a procedure for model checking the logic pC... |

10 | Equivalences for fair kripke structures
- Aziz, Singhal, et al.
- 1994
(Show Context)
Citation Context ...uivalence; details are given in [2]. It should be noted that the states can in fact be characterized in pCTL; this is surprising given that bisimulation under fairness can not be characterized by CTL =-=[3], and that-=- probabilistic transitions are in some loose sense "fair" transitions. However, there is no contradiction -- quite simply, arbitrary fairness can not be injected into a system by making cert... |

8 |
An Automata-Theoretic Approach to Program Veri - cation
- Vardi, Wolper
- 1986
(Show Context)
Citation Context ...entional temporal logic analysis on a model with fairness constraints that eliminate events of measure zero; the actual probabilities of events is immaterial. Examples of such approaches are given in =-=[13, 16, 20, 1]-=-. [16] introduces the notion of ff-fairness; this construct embodies the fact that if a given state is visited infinitely often, then with probability one each state which the given state can make a t... |

4 |
Model Checking for Probabilistic Real Time Systems
- Alur, Courcoubetis, et al.
- 1991
(Show Context)
Citation Context |

2 |
Testing Preorders for Probabilistic Processes
- Cleavland, Smolka, et al.
- 1992
(Show Context)
Citation Context ...g a PLTL [10] formula; this immediately provides a procedure for model checking the logic pCTL , defined in x 3. Definitions of probabilistic simulation/bisimulation exist in the CCS/CSP community eg =-=[7]-=-, which are essentially the same as ours. However, the relationship between probabilistic bisimulation and the expressiveness of probabilistic temporal logic has not been explored in the past. 2 Defin... |

2 |
Automatic Verification of Finite State Programs
- Courcoubetis, Yannakakis
- 1990
(Show Context)
Citation Context ...ystems. An early approach is [14] which computes the expected time for a probabilistic program to terminate. Programs are modeled as Markov chains; the techniques are standard in Markov chain theory. =-=[9]-=- describes an approach to the control of finite state Markov processes subject to !-regular specifications. Criterion for optimality are developed; finding an optimal control strategy reduces to solvi... |

1 | Efficient Decision Procedures for Fragments of R.C.F (in preparation - Rege - 1995 |