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## On probabilistic model checking (1996)

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Citations: | 107 - 25 self |

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5192 |
Probability and Measure
- Billingsley
- 1995
(Show Context)
Citation Context ...nd case studies that use stochastic model checking. Section 6 concludes the tutorial. 2 Preliminaries In the following, we assume some familiarity with probability and measure theory, see for example =-=[16]-=-. Definition 1. Let Ω be an arbitrary non-empty set and F a family of subsets of Ω. We say that F is a field on Ω if: 1. the empty set ∅ is in F; 2. whenever A is an element of F, then the complement ... |

3943 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...probabilistic bisimulation was also proposed and studied in a testing scenario. The definition of probabilistic bisimulation was derived from an alternative characterization of strong bisimulation in =-=[Mil89]-=- as an equivalence class on states subject to certain conditions. While this concept is natural and has been proven an important operational tool for probabilistic setting, see e.g. [SL94], it lacks a... |

2648 | A theory of timed automata - Alur, Dill - 1994 |

1489 |
Symbolic Model Checking
- McMillan
- 1993
(Show Context)
Citation Context ...domain of truth values [0; 1] (`fuzzy' logic). We chose the modal mu-calculus as a specification language since a variety of temporal logics, like CTL, have straightforward translations into ��M (=-=see [McM93]-=- for details). Using the definition of the \Sigma-algebra D P and its least and greatest fixed point semantics it is now easy to write a probabilistic model checking algorithm based on the semantics [... |

1387 | Automatic verification of finite-state concurrent systems using temporal logic specifications
- Clarke, Emerson, et al.
- 1986
(Show Context)
Citation Context ...ccur. As a specification language for DTMCs we use the temporal logic called Probabilistic Computation Tree Logic (PTCL) [29], which is based on well-known branching-time Computation Tree Logic (CTL) =-=[20]-=-. In the case of CTMCs, we employ the temporal logic Continuous Stochastic Logic (CSL) developed originally by Aziz et al. [4, 5] and since extended by Baier et al. [10], also based on CTL. Algorithms... |

1020 | Design and synthesis of synchronization skeletons using branching-time temporal logic - Clarke, Emerson - 1981 |

843 | Counterexample-Guided Abstraction Refinement - Clarke, Grumberg, et al. - 2000 |

777 | Concurrency and automata on infinite sequences, in: - Park - 1981 |

757 | A Compositional Approach to Performance Modelling
- Hillston
- 1994
(Show Context)
Citation Context ...oach focuses on the actions, which is in line with the development of process algebras and their logics, it can therefore be seen as a natural development for stochastic process algebras such as PEPA =-=[Hil96]-=-, allowing finer analysis than the state-based methodology. It has been shown in [Hil96] that PEPA gives rise to a continuous time Markov chain, but little is known about the verification of such mode... |

594 | A randomized protocol for signing contracts
- Even, Goldreich, et al.
- 1985
(Show Context)
Citation Context ...oes.s5.2 Case Study 1: Probabilistic Contract Signing Stochastic Model Checking 39 This case study, taken from [51], concerns the probabilistic contract signing protocol of Even, Goldreich and Lempel =-=[25]-=-. The protocol is designed to allow two parties, A and B, to exchange commitments to a contract. In an asynchronous setting, it is difficult to perform this task in a way that is fair to both parties,... |

562 | Automata on Infinite Objects - Thomas - 1990 |

540 | Domain theory, in
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...urational probabilistic transition systems, the point being that these concepts are independent structural enrichments of labeled transition 13 Our approach is similar to taking the lower powerdomain =-=[AJ94] of -=-T . systems. 14 The first design question in giving OE 2 ��M meaning over such structures is that of the type of its denotation. Here we think of time as a transformer of probability transformers:... |

527 | Bisimulation through Probabilistic Testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...s [Koz83], in its positive form, and the probabilistic aspect only comes into play by giving this logic a non-standard semantics over probabilistic transition systems, basically the models studied in =-=[LS91]-=-. The standard interpretation of a formula OE with respect to a labeled transition system (Q; \Delta) is a subset of states, namely those q 2 Q which are deemed to satisfy OE. Thinking of the powerset... |

446 | A Kripke-Kleene semantics for logic programs
- Fitting
- 1985
(Show Context)
Citation Context ...has been given in [Sei96]; there, meanings are functions of type Q ! T with T being an appropriate domain modeling discrete time. In logic programming the range 2 has also been modified to bilattices =-=[Fit91]-=- or even the unit interval [vE86]. One consequence of our choice of the meaning function is that the process of evaluating a formula in a given model must involve numerical calculations and approximat... |

420 | Principles of Model Checking - Baier, Katoen - 2007 |

370 | A Logic for Reasoning about Time and Reliability
- Hansson, Jonsson
- 1994
(Show Context)
Citation Context ...riod during execution, rather than whether it is possible for such a fault to occur. As a specification language for DTMCs we use the temporal logic called Probabilistic Computation Tree Logic (PTCL) =-=[29]-=-, which is based on well-known branching-time Computation Tree Logic (CTL) [20]. In the case of CTMCs, we employ the temporal logic Continuous Stochastic Logic (CSL) developed originally by Aziz et al... |

329 |
Results on the propositional mu-calculus
- Kozen
- 1983
(Show Context)
Citation Context ...duction The title of this paper might suggest that we are telling a story about some kind of probabilistic logic --- in which case the reader has been misled. This work is about the modal mu-calculus =-=[Koz83]-=-, in its positive form, and the probabilistic aspect only comes into play by giving this logic a non-standard semantics over probabilistic transition systems, basically the models studied in [LS91]. T... |

323 | Reactive modules
- Alur, Henzinger
- 1999
(Show Context)
Citation Context ...ition. Proposition 1. For any non-empty set Ω and A a family of subsets of Ω, there exists a unique smallest σ-algebra containing A. Definition 2. Let (Ω, F) be a measurable space. A function µ : F → =-=[0, 1]-=- is a probability measure on (Ω, F) and (Ω, F, µ) a probability space, if µ satisfies the following properties: 1. µ(Ω) = 1 2. µ(∪iAi) = � i µ(Ai) for any countable disjoint sequence A1, A2, . . . of ... |

319 | F.: Algebraic decision diagrams and their applications
- Bahar, Frohm, et al.
(Show Context)
Citation Context ...dels because they are described in a structured, high-level modelling language. More specifically, since we need to store numerical values, PRISM uses multi-terminal binary decision diagrams (MTBDDs) =-=[21, 7]-=- and a number of variants [46, 52, 48] developed to improve the efficiency of probabilistic analysis, which involve combinations of symbolic data structures such as MTBDDs and conventional explicit st... |

308 |
Automatic verification of probabilistic concurrent finite-state Programs
- Vardi
- 1985
(Show Context)
Citation Context ... this yields ffi at p, 0 at q and 1 at r. 5. It may seem that the probability of a single infinite sequence is always 0. As the example of the system below (which is not bounded in the terminology of =-=[Var85]-=-) demonstrates, this is not necessarily the case. Let (r i ) i0 be a sequence of positive reals where P i0 r i is convergent, i.e. 0 ! P i0 r i ! 1, and consider the system below where all transitions... |

288 | Model Checking of Probabilistic and Nondeterministic Systems
- Bianco, Alfaro
- 1995
(Show Context)
Citation Context ...astic model checking not covered in this tutorial and below we attempt to give brief pointers to related and further work. More expressive logics than PCTL have been proposed, including LTL and PCTL* =-=[6, 15]-=-. For the corresponding model checking algorithms see [62, 22, 6, 15, 13]. We also mention the alternative reward extension of PCTL given in [2]. With regards to CTMCs, a number of extensions of CSL h... |

235 | Model-checking algorithms for continuous-time markov chains.
- Baier, Haverkort, et al.
- 2003
(Show Context)
Citation Context ...e Computation Tree Logic (CTL) [20]. In the case of CTMCs, we employ the temporal logic Continuous Stochastic Logic (CSL) developed originally by Aziz et al. [4, 5] and since extended by Baier et al. =-=[10]-=-, also based on CTL. Algorithms for stochastic model checking were originally introduced in [62, 23, 29, 5, 10], derive from conventional model checking, numerical linear algebra and standard techniqu... |

215 |
M.: The complexity of probabilistic verification
- Courcoubetis, Yannakakis
- 1995
(Show Context)
Citation Context ...us Stochastic Logic (CSL) developed originally by Aziz et al. [4, 5] and since extended by Baier et al. [10], also based on CTL. Algorithms for stochastic model checking were originally introduced in =-=[62, 23, 29, 5, 10]-=-, derive from conventional model checking, numerical linear algebra and standard techniques for Markov chains. We describe algorithms for PCTL and CSL and for extensions of these logics to specify rew... |

183 |
Multi-terminal binary decision diagrams: an efficient data structure for matrix representation
- Clarke, Fujita, et al.
- 1993
(Show Context)
Citation Context ...dels because they are described in a structured, high-level modelling language. More specifically, since we need to store numerical values, PRISM uses multi-terminal binary decision diagrams (MTBDDs) =-=[21, 7]-=- and a number of variants [46, 52, 48] developed to improve the efficiency of probabilistic analysis, which involve combinations of symbolic data structures such as MTBDDs and conventional explicit st... |

165 | Semantic Domains, - Gunter, Scott - 1990 |

163 |
Time and Probability in Formal Design of Distributed Systems
- Hansson
- 1994
(Show Context)
Citation Context ...s of the exact durations but we will ask about the likelihood of some formula being realized within a given time limit 13 . A temporal logic expressing similar properties is the logic pCTL of Hansson =-=[Han94]-=-. This semantic design decision motivates to change \Gamma1 to the more familiar ? which merely plays a technical role in our semantics. We may re-construct not being realizable at all by letting it b... |

156 | Approximate Symbolic Model Checking of ContinuousTime Markov Chains
- Baier, Katoen, et al.
- 1999
(Show Context)
Citation Context ...⇔ � s ′ |= Φ πC s (s ′ ) ∼ p Prob C (s, φ) def = Pr s{ω ∈ Path C (s) | ω |= φ} ω |= X Φ ⇔ ω(1) is defined and ω(1) |= Φ ω |= Φ U I Ψ ⇔ ∃t ∈ I. ( ω@t |= Ψ ∧ ∀x ∈ [0, t). (ω@x |= Φ) ) . As discussed in =-=[12]-=-, for any path formula Φ, the set {ω ∈ Path C (s) | ω |= φ} is a measurable set of (Path C (s), Σ Path C (s)), and hence Pr s is well defined over this set. In addition the steady-state probabilities ... |

134 | Policy optimization for dynamic power management,
- Benini, Paleologo, et al.
- 1999
(Show Context)
Citation Context ...ults demonstrate, as expected, that increasing N decreases the power consumption, while increasing both the queue size and the number of lost requests. For further details about DPM see, for example, =-=[14, 55]-=- and for probabilistic model checking of DPM [50]. 5.4 Case Study 3: Fibroblast Growth Factors The final case study concerns a biological pathway for Fibroblast Growth Factors taken from [32]. Fibrobl... |

133 | Fast randomized consensus using shared memory.
- Aspnes, Herlihy
- 1990
(Show Context)
Citation Context ...In distributed algorithms electronic coin tossing is used as a symmetry breaker and as a means to derive efficient algorithms, for example in randomised leader election [38, 26], randomised consensus =-=[3, 18]-=- and root contention in IEEE 1394 FireWire [37, 47]. Traditionally, probability has also been used as a tool to analyse system performance, where typically queueing theory is applied to obtain steady-... |

129 | Model Checking for a Probabilistic Branching Time Logic with Fairness
- Baier, Kwiatkowska
- 1996
(Show Context)
Citation Context ...e attempt to give brief pointers to related and further work. More expressive logics than PCTL have been proposed, including LTL and PCTL* [6, 15]. For the corresponding model checking algorithms see =-=[62, 22, 6, 15, 13]-=-. We also mention the alternative reward extension of PCTL given in [2]. With regards to CTMCs, a number of extensions of CSL have been proposed in the literature, along with associated model checking... |

125 | Verifying continuous time Markov chains.
- Aziz, Sanwal, et al.
- 1996
(Show Context)
Citation Context ...ng finer analysis than the state-based methodology. It has been shown in [Hil96] that PEPA gives rise to a continuous time Markov chain, but little is known about the verification of such models, see =-=[ASSB96]-=-. We expect that our framework can be further developed to derive action-based logics and verification methods for continuous time models. Finally, we study the semantics for time and probability, and... |

123 | A Probabilistic PDL - Kozen - 1985 |

116 | Exact and ordinary lumpability in finite Markov chains.
- Buchholz
- 1994
(Show Context)
Citation Context ..., respectively. This tutorial concentrated on stochastic model checking. Related topic include: probabilistic generalisations of bisimulation and simulation relations for DTMCs [49, 57] and for CTMCs =-=[17, 11]-=-; and approximate methods for stochastic model checking based on discrete event simulation [33, 63]. Stochastic model checkers SMART [19], E TMC 2 [34] and MRMC [39] have similarities with the PRISM m... |

105 | Model-checking for probabilistic real-time systems. - Alur, Courcoubetis, et al. - 1991 |

105 | Quantitative deduction and its fixpoint theory
- Emden
- 1986
(Show Context)
Citation Context ...meanings are functions of type Q ! T with T being an appropriate domain modeling discrete time. In logic programming the range 2 has also been modified to bilattices [Fit91] or even the unit interval =-=[vE86]-=-. One consequence of our choice of the meaning function is that the process of evaluating a formula in a given model must involve numerical calculations and approximation, a fact interesting in itself... |

102 | It Usually Works: The Temporal Logic of Stochastic Systems
- Aziz, Singhal, et al.
- 1995
(Show Context)
Citation Context ...ecifications for DTMC models can be written in PCTL (Probabilistic Computation Tree Logic) [29], a probabilistic extension of the temporal logic CTL. PCTL is essentially the same as the logic pCTL of =-=[6]-=-. Definition 6. The syntax of PCTL is as follows: Φ ::= true � � a � � ¬Φ � � Φ ∧ Φ � � P∼p[φ] φ ::= X Φ � � Φ U �k Φ where a is an atomic proposition, ∼ ∈{<, �, �, >}, p ∈ [0, 1] and k ∈ N ∪ {∞}. PCT... |

94 | Probabilistic model checking of complex biological pathways
- Heath, Kwiatkowska, et al.
(Show Context)
Citation Context ...ple, [14, 55] and for probabilistic model checking of DPM [50]. 5.4 Case Study 3: Fibroblast Growth Factors The final case study concerns a biological pathway for Fibroblast Growth Factors taken from =-=[32]-=-. Fibroblast Growth Factors (FGF) are a family of proteins which play a key role in the process of cell signalling in a variety of contexts, for example wound healing. The model is a CTMC and it incor... |

94 | M.: Symbolic model checking for probabilistic processes - Baier, Clarke, et al. - 1997 |

91 | Model checking continuous-time Markov chains by transient analysis.
- Baier, Haverkort, et al.
- 2000
(Show Context)
Citation Context ...CTL bounded until operator. Originally, [12] proposed to do this via approximate solution of Volterra integral equation systems. Experiments in [34] showed that this method was generally slow and, in =-=[8]-=-, a simpler alternative was presented which reduces the problem to transient analysis. This approach is outlined below. Definition 15. For any CTMC C = (S, ¯s, R, L) and CSL formula Φ, let CTMC C[Φ] =... |

83 | Model checking continuous time Markov chains
- Aziz, Sanwal, et al.
- 2000
(Show Context)
Citation Context ...which is based on well-known branching-time Computation Tree Logic (CTL) [20]. In the case of CTMCs, we employ the temporal logic Continuous Stochastic Logic (CSL) developed originally by Aziz et al. =-=[4, 5]-=- and since extended by Baier et al. [10], also based on CTL. Algorithms for stochastic model checking were originally introduced in [62, 23, 29, 5, 10], derive from conventional model checking, numeri... |

83 |
Computing Poisson Probabilities.
- Fox, Glynn
- 1988
(Show Context)
Citation Context ...In particular, Q contains both positive and negative values which can cause severe round-off errors. Secondly, the infinite sum is now easier to truncate. For example, the techniques of Fox and Glynn =-=[27]-=-, which allow efficient computation of the Poissons22 Marta Kwiatkowska, Gethin Norman, and David Parker probabilities γi,q·t, also produce an upper and lower bound (Lε, Rε), for some desired precisio... |

75 | The Ins and Outs of the Probabilistic Model Checker MRMC. - Kaoten, Zapreev, et al. - 2009 |

74 | B.: A logic for reasoning about time and probability - Hansson, Jonsson - 1994 |

73 | Reachability analysis of probabilistic systems by successive refinements. - D’Argenio, Jeannet, et al. - 2001 |

71 | Approximate Probabilistic Model Checking. In - Herault, Lassaigne, et al. - 2004 |

70 | Performance Analysis of Communication Systems: Modeling with Non-Markovian Stochastic Petri Nets. - German - 2000 |

70 | Termination of probabilistic concurrent programs - Hart, Sharir, et al. - 1983 |

69 |
Performance of Computer Communication Systems: A Model-Based Approach
- Haverkort
- 1998
(Show Context)
Citation Context ...ool to analyse system performance, where typically queueing theory is applied to obtain steady-state probabilities in order to arrive at estimates of measures such as throughput and mean waiting time =-=[30, 61]-=-. Probability is also used to model unreliable or unpredictable behaviour, as in e.g. fault-tolerant systems and multi-media protocols, where properties such as frame loss of 1 in every 100 can be des... |

62 | Domain theory and integration.
- Edalat
- 1995
(Show Context)
Citation Context ...t the process of evaluating a formula in a given model must involve numerical calculations and approximation, a fact interesting in itself in the light of recent work on computation and approximation =-=[Eda95b]-=-. Probabilistic logics of programs, mainly temporal, have been considered by a number of authors, see e.g. [HSP83, Var85, Koz85, CY88, PZ93, ACD91, LS91, HJ94, Han94, BK96]. The prevailing method is b... |

58 | A Markov Chain Model Checker. - Hermanns, Katoen, et al. - 2000 |

57 | Liquor: A tool for qualitative and quantitative linear time analysis of reactive systems - Ciesinski, Baier - 2006 |

52 | On the Logical Characterisation of Performability Properties”,
- Baier, Haverkort, et al.
- 2000
(Show Context)
Citation Context ... With regards to CTMCs, a number of extensions of CSL have been proposed in the literature, along with associated model checking algorithms. For example, [35] proposes an action based version of CSL; =-=[31, 9]-=- introduce the logics CRL and CSRL which added support for reward-based properties [42]; and [44] augment CSL with random time-bounded until and random expected-time operators, respectively. This tuto... |

45 | Probabilistic Verification - Pnueli, Zuck |

40 | Automated verification techniques for probabilistic systems. - Forejt, Kwiatkowska, et al. - 2011 |

39 | Bisimulation Minimisation Mostly Speeds Up Probabilistic Model Checking - Katoen, Kemna, et al. - 2007 |

36 | Model checking performability properties. In:
- Haverkort, Cloth, et al.
- 2003
(Show Context)
Citation Context ... With regards to CTMCs, a number of extensions of CSL have been proposed in the literature, along with associated model checking algorithms. For example, [35] proposes an action based version of CSL; =-=[31, 9]-=- introduce the logics CRL and CSRL which added support for reward-based properties [42]; and [44] augment CSL with random time-bounded until and random expected-time operators, respectively. This tuto... |

36 | Probabilistic CEGAR. - Hermanns, Wachter, et al. - 2008 |

34 | A simple population protocol for fast robust approximate majority. Distributed Computing - Angluin, Aspnes, et al. - 2008 |

34 | Symbolic model checking of probabilistic processes using mtbdds and the Kronecker representation - Alfaro, Kwiatkowska, et al. - 2000 |

32 | B.: Counterexample generation in probabilistic model checking - Han, Katoen, et al. - 2009 |

30 | F.: Partial order reduction for probabilistic systems - Baier, Grosser, et al. - 2004 |

29 | Probabilistic reachability for parametric markov models. Model Checking Software, - Hahn, Hermanns, et al. - 2009 |

28 | P.: Partial order reduction on concurrent probabilistic programs - D’Argenio, Niebert - 2004 |

27 | Automatic verification of the IEEE 1394 root contention protocol with
- Daws, Kwiatkowska, et al.
- 2004
(Show Context)
Citation Context ...Cs and MDPs, and CSL for CTMCs. Probabilistic timed automata have a logic PTCTL, an extension of TCTL, a subset of which is supported via a connection to the timed automata model checking tool Kronos =-=[24]-=-. Tool Overview. PRISM first parses the model description and constructs an internal representation of the probabilistic model, computing the reachable state space of the model and discarding any unre... |

27 | Quantitative synthesis for concurrent programs - Cerný, Chatterjee, et al. - 2011 |

26 |
Verifying Temporal Properties of Systems
- Bradfield
- 1992
(Show Context)
Citation Context ...l model are complete lattices D which are also monotone \Sigma-algebras. 1 For such algebras the semantics is completely determined and well-defined as a standard least/greatest fixed point semantics =-=[Bra91]-=-. As a motivating example let us take Kozen's standard interpretation [Koz83]. Given a labeled transition system L = (Q; \Delta) where Q is a set and \Delta ` Q \Theta Act \Theta Q Kozen defined a sem... |

25 | Significant diagnostic counterexamples in probabilistic model checking - Andrés, D’Argenio, et al. - 2009 |

25 | Magnifying-lens abstraction for Markov decision processes. - Alfaro, Roy - 2007 |

24 | Directed explicit state-space search in the generation of counterexamples for stochastic model checking - Aljazzar, Leue - 2009 |

23 | Abstraction refinement for probabilistic software,” in - Kattenbelt, Kwiatkowska, et al. - 2009 |

22 | Measuring and synthesizing systems in probabilistic environments - Chatterjee, Henzinger, et al. - 2010 |

21 | H.: A modest approach to checking probabilistic timed automata - Hartmanns, Hermanns - 2009 |

20 | Model checking probabilistic real time systems”, - Jensen - 1996 |

19 |
Discrete-time rewards model-checked.
- Andova, Hermanns, et al.
- 2003
(Show Context)
Citation Context ...s ′ )) equals: Pr s(C(s, I, . . . , In−1, sn)) · P emb(C) (sn, s ′ ) · � −E(sn)·inf I′ −E(sn)·sup e − e I′� . Example 13. Consider the CTMC C1 from Fig. 4 and the sequence of states and intervals s0, =-=[0, 2]-=-, s1 (i.e. taking I0 = [0, 2] in the notation of the previous paragraph). Using the probability measure Pr s0 over (PathC1 (s0), Σ Path C (s0)), for the cylinder set C(s0, [0, 2], s1), we have: Pr s0(... |

18 |
Logic and stochastic modeling with smart, Perform
- CIARDO, JONES, et al.
(Show Context)
Citation Context ... and simulation relations for DTMCs [49, 57] and for CTMCs [17, 11]; and approximate methods for stochastic model checking based on discrete event simulation [33, 63]. Stochastic model checkers SMART =-=[19]-=-, E TMC 2 [34] and MRMC [39] have similarities with the PRISM model checker described here. Finally, we mention a challenging direction of research is into the verification of models which allow more ... |

17 | Hybrid numerical solution of the chemical master equation. - Henzinger, Mikeev, et al. - 2010 |

16 |
Verifying Temporal Properties of Finite State Probabilistic Programs,
- Courcoubetis, Yannakakis
- 1988
(Show Context)
Citation Context ...ations can be seen in practice in Section 5. c1 c2 1s3.3 Model checking PCTL. Stochastic Model Checking 9 We now summarise a model checking algorithm for PCTL over DTMCs, which was first presented in =-=[22, 29, 23]-=-. The inputs to the algorithm are a labelled DTMC D = (S, ¯s, P, L) and a PCTL formula Φ. The output is the set of states Sat(Φ) = {s ∈ S | s |= Φ}, i.e. the set containing all the states of the model... |

16 | Approximate parameter synthesis for probabilistic time-bounded reachability. - Han, Katoen, et al. - 2008 |

12 | Quantitative analysis under fairness constraints - Baier, Größer, et al. - 2009 |

12 | ProbDiVinE-MC: Multi-core LTL model checker for probabilistic systems - Barnat, Brim, et al. - 2008 |

12 | Symmetry reduction for probabilistic model checking using generic representatives - Donaldson, Miller - 2006 |

11 | J.P.: Probably on time and within budget: On reachability in priced probabilistic timed automata - Berendsen, Jansen, et al. - 2006 |

10 | Domain theory in stochastic processes - Edalat - 1995 |

10 | Randomized wait-free consensus using an atomicity assumption
- Cheung
- 2005
(Show Context)
Citation Context ...In distributed algorithms electronic coin tossing is used as a symmetry breaker and as a means to derive efficient algorithms, for example in randomised leader election [38, 26], randomised consensus =-=[3, 18]-=- and root contention in IEEE 1394 FireWire [37, 47]. Traditionally, probability has also been used as a tool to analyse system performance, where typically queueing theory is applied to obtain steady-... |

9 | Integration in real PCF.
- Edalat, Escardo
- 1996
(Show Context)
Citation Context ... DP ) q 0 : Setting L �� (A) = fa 0 a 1 : : : a n 2 Act j n 2 !; [[ha 0 iha 1 i : : : ha n ic]] DP q 0 ? ��g 12 Error propagation can possibly be avoided by means of techniques developed for R=-=eal PCF [Eda95], but it rem-=-ains to be seen how this approach performs in empirical studies. for all �� 2 [0; 1] we therefore see that L(A) = L 0 (A). Can we characterize the formal languages L �� (A), �� 2 (0; 1], o... |

9 | F.: Fortuna: Model checking priced probabilistic timed automata - Berendsen, Jansen, et al. - 2010 |

9 | Pass: Abstraction refinement for infinite probabilistic models - Hahn, Hermanns, et al. - 2010 |

9 | D.: Game-based probabilistic predicate abstraction - Kattenbelt, Kwiatkowska, et al. |

8 | Yannakakis: Verifying Temporal Properties of Finite-State - Courcoubetis, M - 1988 |

8 | Simulation for continuoustime Markov chains.
- Baier, Katoen, et al.
- 2002
(Show Context)
Citation Context ..., respectively. This tutorial concentrated on stochastic model checking. Related topic include: probabilistic generalisations of bisimulation and simulation relations for DTMCs [49, 57] and for CTMCs =-=[17, 11]-=-; and approximate methods for stochastic model checking based on discrete event simulation [33, 63]. Stochastic model checkers SMART [19], E TMC 2 [34] and MRMC [39] have similarities with the PRISM m... |

7 | Variations on Itai-Rodeh leader election for anonymous rings and their analysis in PRISM
- Fokkink, Pang
- 2006
(Show Context)
Citation Context ...software and hardware systems. In distributed algorithms electronic coin tossing is used as a symmetry breaker and as a means to derive efficient algorithms, for example in randomised leader election =-=[38, 26]-=-, randomised consensus [3, 18] and root contention in IEEE 1394 FireWire [37, 47]. Traditionally, probability has also been used as a tool to analyse system performance, where typically queueing theor... |

6 | Pnueli A. Termination of Probabilistic Concurrent Programs - Hart, Sharir - 1983 |

5 | Probabilistic abstractions with arbitrary domains - Esparza, Gaiser - 2011 |

5 | Game-based abstraction and controller synthesis for probabilistic hybrid systems. - Hahn, Norman, et al. - 2011 |

4 | On The Approximation of Denotational Mu-semantics
- Huth
- 1998
(Show Context)
Citation Context ... = (Q; \Delta) where Q is a set and \Delta ` Q \Theta Act \Theta Q Kozen defined a semantics [[ ]] DL on the power set, P(Q), of Q. There is a natural 1 It suffices that D is merely a bicomplete dcpo =-=[Hut96]. is-=-omorphism P(Q) �� = Q ! 2 where the latter is the complete lattice of all functions f : Q ! 2 into the two point lattice 2 = f0 ! 1g with the order fsf 0 if and only if f(q)sf 0 (q) for all q 2 Q.... |

4 |
Probabilistic Simulations for Probabilistic
- Segala, Lynch
- 1994
(Show Context)
Citation Context ...ulation in [Mil89] as an equivalence class on states subject to certain conditions. While this concept is natural and has been proven an important operational tool for probabilistic setting, see e.g. =-=[SL94]-=-, it lacks a co-inductive formulation, which raises a number of questions, for example, how can we generate such equivalence relations and what about the existence of a greatest such relation as the c... |

4 | M.: A counterexample guided abstraction-refinement framework for markov decision processes - Chadha, Viswanathan - 2010 |

3 |
A Modal {Calculus for Durational Transition Systems
- Seidl
- 1995
(Show Context)
Citation Context ...difying the range in Q ! 2 is not new, but the application to probability seems to be. For example, a durational semantics to the modal mucalculus over durational transition systems has been given in =-=[Sei96]-=-; there, meanings are functions of type Q ! T with T being an appropriate domain modeling discrete time. In logic programming the range 2 has also been modified to bilattices [Fit91] or even the unit ... |

1 | Model Checking for a Probabilistic Temporal Logic with Fairness - Baier, Kwiatkowska - 1996 |

1 | Games and modal mu-calculus. volume 1055 - Stirling - 1996 |