## An Introduction to Probabilistic Automata (2002)

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Venue: | Bulletin of the European Association for Theoretical Computer Science |

Citations: | 11 - 0 self |

### BibTeX

@ARTICLE{Stoelinga02anintroduction,

author = {Mariëlle Stoelinga},

title = {An Introduction to Probabilistic Automata},

journal = {Bulletin of the European Association for Theoretical Computer Science},

year = {2002},

volume = {78},

pages = {176--198}

}

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### Abstract

This paper provides an elementary introduction to the probabilistic automaton (PA) model, which has been developed by Segala. We describe how distributed systems with discrete probabilities can be modeled and analyzed by means of PAs. We explain how the basic concepts for the analysis of nonprobabilistic automata can be extended to probabilistic systems. In particular, we treat the parallel composition operator on PAs, the semantics of a PA as a set of trace distributions, an extension of the PA model with time and simulation relations for PAs. Finally, we give an overview of various other state based models that are used for the analysis of probabilistic systems.

### Citations

1963 | A theory of timed automata
- Alur, Dill
- 1994
(Show Context)
Citation Context ...ributions over time. Moreover, there is a second model that extends PAs with nondeterministic timing. The automata introduced in [KNSS01] --- also called PTAs --- augment the classical timed automata =-=[AD94]-=- with discrete probabilistic choice. They allow timing constraints to be specified via real--valued clocks. 2.4 Parallel Composition The parallel composition operator k allows one to construct a PA fr... |

1064 |
Computer Networks
- Tanenbaum
- 1996
(Show Context)
Citation Context ...s (such as the randomized dining philosophers [LR81]), probabilistic communication protocols (such as the IEEE 1394 Root Contention protocol [IEE96]) and the Binary Exponential Back Off protocol, see =-=[Tan81]-=-); and fault tolerant systems (such as unreliable communication channels). PAs are based on state transition systems and make a clear distinction between probabilistic and nondeterministicschoice. We ... |

580 |
Markov Decision Processes
- Puterman
- 1994
(Show Context)
Citation Context ... output or internal action is purely probabilistic. Input actions, on the other hand, are always enabled and can be taken before the sojourn time has expired. Semi--Markov decision processes Puterman =-=[Put94]-=- discusses Semi--Markov decision processes (SMDPs), which are basically Semi--Markov chains with external nondeterministic choice. 6.3 Probabilistic Models with Full Nondeterminism Probabilistic autom... |

471 | An introduction to Input/Output automata
- Lynch, Tuttle
- 1989
(Show Context)
Citation Context ...on exist. Some definitions require, for instance, a unique start state rather than a set of start states or allow only a single internal action, rather than a set of these. In the I/O automaton model =-=[LT89]-=-, external actions are divided into input and output actions. Input actions, not 2 being under the control of the NA, are required to be enabled in any state. The basic concepts are similar for the va... |

402 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...s. These are labeled by probabilities and specify a probability distribution over the nondeterministic states. The alternating model and the PA model are isomorphic upto so-called strong bisimulation =-=[LS91]-=-. This means that all notions defined on PAs that respect strong bisimulation can be translated into the alternating model and vice versa. In order to translate an alternating model into a PA, one rem... |

265 | Probabilistic simulations for probabilistic processes - Segala, Lynch - 1995 |

245 | A logic for reasoning about time and reliability - Hansson, Jonsson - 1994 |

230 |
Modeling and verification of randomized distributed real-time systems
- Segala
- 1995
(Show Context)
Citation Context ...the PTA is well--defined at each point in time and that, conversely, two subsequent time passage actions can be combined into a single one. The PTA model presented here is a simplification of the one =-=[Seg95b-=-], which is based on a generalization of Wang's axiom. Definition 2.10 A PTA A is a PA enriched with a partition of Act nfg into a set of discrete actions Act D and the set R >0 of positive real numbe... |

205 | An old-fashioned recipe for real time
- Abadi, Lamport
- 1994
(Show Context)
Citation Context ... the area of probability theory, the reader is referred to [Coh89]. 2.3 Timing Timing can be incorporated in the PA model in a similar way as in the NA model (c.f. the "old fashioned recipe for t=-=ime" [AL92]-=-). A probabilistic timed automaton (PTA) is a PA with time passage actions. These are actions d 2 R >0 that indicate the passage of d time units. While time elapses, no other actions take place and, i... |

147 |
Time and probability in formal design of distributed systems
- Hansson
- 1994
(Show Context)
Citation Context ...reof combine nondeterministic and discrete probabilistic choice. Several process algebras, such as ACP [And99b, And99a], the probabilistic --calculus and the probabilistic process algebra defined in [=-=Han94]-=-, allow one to describe such models algebraically. Interactive Markov chains Interactive Markov Chains (IMCs) [Her99] combine exponential distributions with full nondeterminism. The definition of a pa... |

115 |
Formal Verification of Probabilistic Systems
- Alfaro
- 1998
(Show Context)
Citation Context ...vior. If we base the notion of behavior based on deterministic adversaries, then it is not possible to implement nondeterministic with randomized algorithms, unlike a notion of randomized adversaries =-=[Alf97]-=-. However, it has been proven [Alf99] that, if one is only interested in the minimal and maximal probability of a certain event, then it suffices to consider only deterministic adversaries. 3.2.3 The ... |

78 |
Interactive Markov Chains
- Hermanns
- 2002
(Show Context)
Citation Context ...a], the probabilistic --calculus and the probabilistic process algebra defined in [Han94], allow one to describe such models algebraically. Interactive Markov chains Interactive Markov Chains (IMCs) [=-=Her99]-=- combine exponential distributions with full nondeterminism. The definition of a parallel composition operator poses the same problems as when one combines exponential distributions with external nond... |

78 |
On the advantages of free choice: A symmetric and fully distributed solution to the dining philosophers problem
- Rabin, Lehmann
- 1981
(Show Context)
Citation Context ...hronous, concurrent systems with discrete probabilistic choice in a formal and precise way. Examples of such systems are randomized, distributed algorithms (such as the randomized dining philosophers =-=[LR81]-=-), probabilistic communication protocols (such as the IEEE 1394 Root Contention protocol [IEE96]) and the Binary Exponential Back Off protocol, see [Tan81]); and fault tolerant systems (such as unreli... |

75 |
Measure Theory
- Cohn
- 1997
(Show Context)
Citation Context ...measure on FE such that PE [C ] = Q E () for all 2 Path (s; A). The fact that ( E ; FE ; PE ) is a probability space follows from standard measure theory arguments, see for instance [Hal50] or [Coh=-=80-=-]. Note that E and FE do not depend on E but only on A, and that PE is fully determined by the function Q E . Note that the cone C is contained in FE for every finite path in E, but that the cone its... |

72 | Automatic verification of real-time systems with discrete probability distributions. Theoretical Computer Science 282
- Kwiatkowska, Norman, et al.
- 2002
(Show Context)
Citation Context ...obability, see Section 6, including models dealing with continuous distributions over time. Moreover, there is a second model that extends PAs with nondeterministic timing. The automata introduced in =-=[KNSS01]-=- --- also called PTAs --- augment the classical timed automata [AD94] with discrete probabilistic choice. They allow timing constraints to be specified via real--valued clocks. 2.4 Parallel Compositio... |

61 |
Real-time behaviour of asynchronous agents
- Yi
- 1990
(Show Context)
Citation Context ...ular, no (internally) nondeterministic or probabilistic choices can be specified within time passage actions, see requirements 1 and 2 in the definition below. The third condition below, Wang's Axiom =-=[Yi90]-=-, requires that, while time advances, the state of the PTA is well--defined at each point in time and that, conversely, two subsequent time passage actions can be combined into a single one. The PTA m... |

52 | Composition and behaviors of probabilistic I/O automata
- Wu, Smolka, et al.
- 1997
(Show Context)
Citation Context ...1 2 1 2 1 2 1 2 e e e e u @ @ @R @ @ @R a b 1 2 1 2 Figure 9: A PA model and two alternating models equivalent to it occur only on transitions labeled by a single action. This approach is followed in =-=[WSS97]-=- and in [HP00]. Surprisingly, the latter reverses the role of input and output actions. In our experience, many practical systems can be modeled conveniently with PAs (see Remark 2.7; deadlocks can be... |

36 | Compositional trace–based semantics for probabilistic automata - Segala - 1995 |

33 | Process algebra with probabilistic choice - Andova |

32 | Computing minimum and maximum reachability times in probabilistic systems
- Alfaro
- 1999
(Show Context)
Citation Context ...or based on deterministic adversaries, then it is not possible to implement nondeterministic with randomized algorithms, unlike a notion of randomized adversaries [Alf97]. However, it has been proven =-=[Alf99]-=- that, if one is only interested in the minimal and maximal probability of a certain event, then it suffices to consider only deterministic adversaries. 3.2.3 The probability space associated to an ad... |

25 | Compositional methods for probabilistic systems
- Alfaro, Henzinger, et al.
- 2001
(Show Context)
Citation Context ...vPTD with the principal context C. Theorem 4.3 ([Seg95b]) Let C be the PA shown in Figure 12, where we suppose that the actions pleft and pright are fresh. Then A vPTD B iff A k C vTD B k C. The work =-=[AHJ01]-=- develops notion of behavior for a variable-based, synchronous probabilistic model. The key idea is to retain the variable dependencies when composing two systems. That is, an adversary of 17 e e e @ ... |

19 |
An Introduction to the Philosophy of Induction and Probability
- Cohen
- 1989
(Show Context)
Citation Context ...e appropriate for describing and predicting certain phenomena in system behavior. For more information on the philosophical issues arising in the area of probability theory, the reader is referred to =-=[Coh89]. 2.3 Timi-=-ng Timing can be incorporated in the PA model in a similar way as in the NA model (c.f. the "old fashioned recipe for time" [AL92]). A probabilistic timed automaton (PTA) is a PA with time p... |

16 |
and backward simulations, ii: Timing-based systems
- Lynch, Vaandrager, et al.
- 1996
(Show Context)
Citation Context ...A s 0 and s d !A s 00 then s 0 = s 00 . 3. (Wang's Axiom) s d !A s 0 iff 9s 00 : s d 0 !A s 00 and s 00 d d 0 !A s 0 . 1 For simplicity the conditions here are slightly more restrictive than those in [LV96]. 6 s0 ; 0 s2 ; 0 s1 ; 0 s2 ; 2 s1 ; 1 s4 ; 2 s3 ; 1 sH H Hj * - - - - 1 2 a a 1 2 1 2 Figure 6: A part of a PTA As PTAs are a special kind of PAs, we can use the not... |

5 |
Probabilistic asynchronous -calculus
- Herescu, Palamidessi
- 2000
(Show Context)
Citation Context ... e e e e u @ @ @R @ @ @R a b 1 2 1 2 Figure 9: A PA model and two alternating models equivalent to it occur only on transitions labeled by a single action. This approach is followed in [WSS97] and in =-=[HP00]-=-. Surprisingly, the latter reverses the role of input and output actions. In our experience, many practical systems can be modeled conveniently with PAs (see Remark 2.7; deadlocks can be modeled by mo... |

4 |
Measure Theory, Van Nostrand Reinhold
- Halmos
- 1969
(Show Context)
Citation Context .... A fundamental theorem from measure theory now states that, under the conditions met here, we can give a probability measure on all sets in FE by specifying it on the sets C only, see for instance [=-=Hal50-=-] and [Seg95b]. Definition 3.12 The probability space associated to a partial adversary E starting in s 0 is the probability space given by 1. E = Path max (E), 2. FE is the smallest --field that cont... |

3 | Process algebra with interleaving probabilistic parellel composition - Andova - 1999 |

2 |
jacta est: verification of probabilistic, real-time and parametric systems
- Alea
- 2002
(Show Context)
Citation Context ...tions for PAs. In Section 6, we give several other models dealing with probabilistic choice and finally, Section 7 presents a summary of the paper. This paper is a revised version of a Chapter 2 in [=-=Sto-=-02], which was written while the author working at the Computing Science Institute, University of Nijmegen, the Netherlands. 1 0 " 1 n n - snd(0) snd(1) - rec(0) rec(1) Figure 1: A channe... |

1 |
Probabilistic asynchronous�-calculus
- Herescu, Palamidessi
- 2000
(Show Context)
Citation Context ...� Figure 8: Parallel composition � � Figure 9: A PA model and two alternating models equivalent to it occur only on transitions labeled by a single action. This approach is followed in [WSS97] and in =-=[HP00]-=-. Surprisingly, the latter reverses the role of input and output actions. In our experience, many practical systems can be modeled conveniently with PAs (see Remark 2.7; deadlocks can be modeled by mo... |