## Metric semantics for reactive probabilistic processes (1997)

Citations: | 6 - 1 self |

### BibTeX

@TECHREPORT{Norman97metricsemantics,

author = {Gethin Josiah Norman},

title = {Metric semantics for reactive probabilistic processes},

institution = {},

year = {1997}

}

### OpenURL

### Abstract

In this thesis we present three mathematical frameworks for the modelling of reac-tive probabilistic communicating processes. We first introduce generalised labelled transition systems as a model of such processes and introduce an equivalence, coarser than probabilistic bisimulation, over these systems. Two processes are identified with respect to this equivalence if, for all experiments, the probabilities of the respective processes passing a given experiment are equal. We next consider a probabilistic pro-cess calculus including external choice, internal choice, action-guarded probabilistic choice, synchronous parallel and recursion. We give operational semantics for this calculus be means of our generalised labelled transition systems and show that our equivalence is a congruence for this language. Following the methodology introduced by de Bakker & Zucker, we then give deno-tational semantics to the calculus by means of a complete metric space of probabilistic processes. The derived metric, although not an ultra-metric, satisfies the intuitive property that the distance between two processes tends to 0 if a measure of the dif-

### Citations

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Citation Context ...are two main process calculi that serve as the basis for probabilistic extensions, namely Milner’s Calculus of Communicating Systems (CCS) [Mil89] and Hoare’s Communicating Sequential Processes (CSP) =-=[BHR84]-=-. The difference between the theories of CCS and CSP arises from the types of equivalences and the approach used in modelling systems, and in particular the process constructors. CCS is modelled using... |

3463 |
Communication and Concurrency
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Citation Context ...d on and the kind of probabilistic behaviour modelled. There are two main process calculi that serve as the basis for probabilistic extensions, namely Milner’s Calculus of Communicating Systems (CCS) =-=[Mil89]-=- and Hoare’s Communicating Sequential Processes (CSP) [BHR84]. The difference between the theories of CCS and CSP arises from the types of equivalences and the approach used in modelling systems, and ... |

1385 | A structural approach to operational semantics
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(Show Context)
Citation Context ...theories of CCS and CSP arises from the types of equivalences and the approach used in modelling systems, and in particular the process constructors. CCS is modelled using labelled transition systems =-=[Plo81]-=-: a tuple (T, Act, −→), where T is a set of processes (or states), Act is a set of actions (or labels), and −→ ⊆ T × Act × T is a transition relation, where for any (P, a, Q) ∈ −→ we write P a −→ Q, d... |

1318 |
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Citation Context ...rties will hold at some time in the future) and precedence (one property must hold before another one becomes true). Temporal logics were first introduced to reason about concurrent systems by Pnueli =-=[Pnu77]-=-. Since then a variety of logics have been introduced. For example, Hennessy and Milner define a modal logic known as the Hennessy-Milner Logic (HML) [HM85] for expressing properties of labelled trans... |

1255 | Automatic verification of finite-state concurrent systems using temporal logic specifications
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(Show Context)
Citation Context ... Hennessy and Milner have then shown that two processes are bisimilar if and only if they satisfy the same HML formulae. Another example is Emerson, Clarke and Sistla’s Computational Tree Logic (CTL) =-=[CES83]-=- which includes the temporal operators: AXφ (φ is true in all immediately succeeding states), A[φUψ] (always φ holds until ψ holds), EXφ there exists an immediate successor state such that φ holds) an... |

899 | Dynamic Logic [M - Harel, Kozen, et al. - 2000 |

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(Show Context)
Citation Context ... where for any (P, a, Q) ∈ −→ we write P a −→ Q, denoting the process P performing the action a and then behaving as the process Q. Equivalences for these transition systems are based on bisimulation =-=[Par81]-=- and [Mil83]. A (strong) bisimulation is a relation ∼ such that for any P, Q ∈ T , P ∼ Q if and only if for all a ∈ Act: (i) if P a −→ P ′ then ∃ Q ′ ∈ T : Q a −→ Q ′ and P ′ ∼ Q ′ (ii) if Q a −→ Q ′ ... |

648 | A Compositional Approach to Performance Modelling
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Citation Context ...ucting a denotational model for stochastic process calculi; we have already derived a metric model for a subset of the stochastic process calculus called Performance Evaluation Process Algebra (PEPA) =-=[Hil96]-=- in [KN96a] (joint with M.Kwiatkowska), and to give a complete metric model for the full calculus we will need to add probabilistic behaviour, which may involve combining the work of this thesis and t... |

520 | Algebraic Laws for Nondeterminism and Concurrency
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Citation Context ... reason about concurrent systems by Pnueli [Pnu77]. Since then a variety of logics have been introduced. For example, Hennessy and Milner define a modal logic known as the Hennessy-Milner Logic (HML) =-=[HM85]-=- for expressing properties of labelled transition systems. HML extends classical propositional logic by the inclusion of the basic modal operator 〈a〉φ which holds true for a process if it can perform ... |

485 | Domain theory
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(Show Context)
Citation Context ...ursive programs, for example while statements, a solution of a recursive domain equation is used to give denotational semantics, and general techniques for such constructions, for example [Plo81] and =-=[AJ94]-=- for domains and [AR89] for metric spaces, are offered. However, in certain cases one approach can have advantages over the other. For example, when considering fixed points for metric spaces we have ... |

439 | Testing equivalences for processes
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- 1984
(Show Context)
Citation Context ... in Figure 2.2 and which we therefore view as too fine include: Segala and Lynch’s probabilistic simulation [SL94] and Wang Yi and Larsen’s testing equivalence [YL92], based on de Nicola and Hennessy =-=[NH84]-=- testing equivalences for non-deterministic processes and defined over Hansson and Jonsson’s [HJ90] alternating model. In Yi and Larsen’s model tests are represented by non-deterministic (and non-prob... |

389 | Process Algebra for Synchronous Communication
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Citation Context ...del. Although not based on CCS, Baeten, Bergsta and Smolka [BBS92] have considered probabilistic bisimulation over an extended version of Bergstra and Klop’s Algebra for Communicating Processes (ACP) =-=[BK84]-=- to allow generative probabilistic choice. Probabilistic bisimulation is shown to be a congruence over this calculus and also a sound and complete axiomatisation of this equivalence is given. As for d... |

380 | Propositional dynamic logic of regular programs - Fischer, Ladner - 1979 |

343 | Skip lists: a probabilistic alternative to balanced trees
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Citation Context ... algorithms cannot be proved to meet. The algorithms include: a symmetric distributed solution to the dining philosopher’s problem [LR81] and solutions to: consensus protocols [Sei92], load balancing =-=[Pug90]-=- and self-stabilisation [Her90]. Traditionally, the modelling of concurrent systems has abstracted away from quantitative aspects. As a result, there is no information about how frequently or with wha... |

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331 | Results on the propositional µ-calculus - Kozen - 1983 |

315 | The Linear Time – Branching Time Spectrum
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(Show Context)
Citation Context ...n as to when the processes make choices. For these reasons, the equivalences form a hierarchy, with bisimulation being the finest equivalence and the coarsest being trace equivalence (see [BKO88] and =-=[Gla90]-=-). To illustrate the differences between these equivalences consider the example given below: P� Q� R� ✚❩ ✚❩ a a✚ ❩❩❩ a a✚ a ❩❩❩ a ✚ ✚ ❄� �✚❂ � �✚❂ ❄� � ✚❩ b✚ ❩❩❩ c ✡❏ b c b b c c ✚ ✡ ❏❏❫ ✚❂ � � ❄�... |

295 | Probabilistic simulations for probabilistic processes
- Segala, Lynch
- 1994
(Show Context)
Citation Context ...n relation. Moreover, Jonsson and Larsen define a relation between specifications which leads to a probabilistic simulation relation over probabilistic processes. Based on this work, Segala and Lynch =-=[SL94]-=- extend both probabilistic bisimulation and simulation to a generalized reactive model of probabilistic processes. Processes of the model can be interpreted as allowing an internal choice between beha... |

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

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- 1995
(Show Context)
Citation Context ...nd the probability of acting as F is 0, but if the environment offers only actions that F can perform then E 1⊔⊓ 0 F will act as F . An alternative approach has been introduced by van Glabbeek et al. =-=[GSST90]-=- where three different models for probabilistic choice are presented, namely reactive, generative and stratified. In the reactive model, the model selected for consideration in this thesis, the enviro... |

154 |
Time and Probability in Formal Design of Distributed Systems
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- 1991
(Show Context)
Citation Context ... and action. Hence, his model can be regarded as a stratified model. He furthermore extends probabilistic bisimulation to this setting and shows it to be a congruence over his calculus. Also, Hansson =-=[Han94]-=- considers an extension of CCS with respect to the alternating model of [HJ90], an extension of the reactive model so named because states alternate between having an internal probabilistic choice or ... |

150 |
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Citation Context ...4, Han94, Sei92] and propositional dynamic logic (PDL) [Har79] with quantitative extensions [FH82, Koz83a, FL79]. 2.6 Other Related Research Other research into probabilistic behaviour includes Kozen =-=[Koz81]-=- where semantics for a probabilistic while-language is given in terms of linear continuous operators on partially ordered Banach spaces. Also, Jones and Plotkin [Jon90, JP89] construct a general frame... |

144 | Probabilistic Non-Determinism - Jones - 1990 |

133 | A probabilistic powerdomain of evaluations - Jones, Plotkin - 1989 |

125 | A simple approach to specifying concurrent systems
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- 1989
(Show Context)
Citation Context ...e processes. Also, Jonsson and Larsen [JL91] generalize probabilistic bisimulation by means of a specification formalism, which extends specifications for non-probabilistic processes (for example see =-=[Lam89]-=-). Specifications are represented by probabilistic transition systems where each transition is labelled with a set of probabilities. Using this they define a satisfaction relation between processes an... |

124 |
A Domain Equation For Bisimulation
- Abramsky
- 1990
(Show Context)
Citation Context ...ulation above. The classical denotational models for CCS are then based on Milner’s synchronization trees [Mil89] and can be divided into domain-theoretic and metric-space approaches, for example see =-=[Abr91a]-=- and [BZ82] respectively.s2 Related Work 6 In contrast, the models and equivalences for CSP are based on traces [Hoa85] and failures [BHR84]. The traces of a process are the possible sequences of acti... |

110 |
A Probabilistic PDL
- Kozen
- 1985
(Show Context)
Citation Context ...ence between the two approaches given above, we consider Larsen and Skou’s [LS91] qualitative extension to HML and Huth and Kwiatkowska’s [HK97] quantitative re-interpretation of the modal µ−calculus =-=[Koz83a]-=-. Recall that a process satisfies the formula 〈a〉φ of HML (and of the modal µ−calculus) if it can perform an a transition and the state reached satisfies the formula φ. Formally: ⎧ ⎨ true if ∃ Q such ... |

101 |
Specification and Refinement of Probabilistic Processes
- Jonsson, Larsen
- 1991
(Show Context)
Citation Context ... that if two processes are probabilistic bisimilar then there exists a testing algorithm that with probability 1 − ε, for ε arbitrarily small, will distinguish the processes. Also, Jonsson and Larsen =-=[JL91]-=- generalize probabilistic bisimulation by means of a specification formalism, which extends specifications for non-probabilistic processes (for example see [Lam89]). Specifications are represented by ... |

100 | Algebraic reasoning for probabilistic concurrent systems
- Giacalone, Jou, et al.
- 1990
(Show Context)
Citation Context ...e processes are restricted to non-probabilistic processes by letting all transitions occur with probability 1. Of approaches related to probabilistic bisimulation we mention Giacalone, Jou and Smolka =-=[GJS90]-=-, where a probabilistic version of Milner’s synchronous version of CCS (SCCS [Mil83]), called PCCS is considered. The difference from SCCS arises from the choice operator being replaced by a probabili... |

97 |
A calculus for communicating systems with time and probabilities
- Hansson, Jonsson
- 1990
(Show Context)
Citation Context ...more extends probabilistic bisimulation to this setting and shows it to be a congruence over his calculus. Also, Hansson [Han94] considers an extension of CCS with respect to the alternating model of =-=[HJ90]-=-, an extension of the reactive model so named because states alternate between having an internal probabilistic choice or an internal choice between actions. Probabilistic bisimulation is then extende... |

88 | Processes and the denotational semantics of concurrency - Bakker, Zucker - 1982 |

86 | Symbolic model checking for probabilistic processes - Baier, Clarke, et al. - 1997 |

81 |
On the Advantage of Free Choice: A Symmetric and Fully Distributed Solution to the Dining Philosophers Problem
- Lehmann, Rabin
- 1981
(Show Context)
Citation Context ...tructures, use less memory and can achieve results that deterministic algorithms cannot be proved to meet. The algorithms include: a symmetric distributed solution to the dining philosopher’s problem =-=[LR81]-=- and solutions to: consensus protocols [Sei92], load balancing [Pug90] and self-stabilisation [Her90]. Traditionally, the modelling of concurrent systems has abstracted away from quantitative aspects.... |

73 |
Solving reflexive domain equations in a category of complete metric spaces
- America, Rutten
- 1989
(Show Context)
Citation Context ...ample while statements, a solution of a recursive domain equation is used to give denotational semantics, and general techniques for such constructions, for example [Plo81] and [AJ94] for domains and =-=[AR89]-=- for metric spaces, are offered. However, in certain cases one approach can have advantages over the other. For example, when considering fixed points for metric spaces we have unique fixed points fro... |

73 |
Equivalences, congruences, and complete axiomatizations for probabilistic processes
- Jou, Smolka
- 1990
(Show Context)
Citation Context ...erational semantics for PCCS for each of the three models, where in each case probabilistic bisimulation is a congruence over all the usual operators of SCCS. For the generative model, Jou and Smolka =-=[JS90]-=- have given a complete axiomatisation of probabilistic bisimulation. Similarly, Larsen and Skou [LS92] have constructed a Calculus for Probabilistic Processes (CPP), also based on SCCS, which can be c... |

72 |
Automatic veri of concurrent systems using temporal logic
- Clarke, Emerson, et al.
- 1986
(Show Context)
Citation Context ... Hennessy and Milner have then shown that two processes are bisimilar if and only if they satisfy the same HML formulae. Another example is Emerson, Clarke and Sistla's Computational Tree Logic (CTL) =-=[-=-CES83] which includes the temporal operators: AX ( is true in all immediately succeeding states), A[U ] (always holds untilsholds), EX there exists an immediate successor state such that holds) and ... |

69 |
Observational equivalence as a testing equivalence
- Abramsky
- 1987
(Show Context)
Citation Context ...ra-metric semantic model which they show to be fully abstract with respect to probabilistic bisimulation. Both semantics are based on classical results for the non-probabilistic case, for example see =-=[Abr87]-=- and [AJ94] for the domain-theoretic and [BZ82] and [AR89] for the metric space construction and framework respectively. 2.3 Probabilistic Versions of CSP Relating to CSP, Lowe [Low93] considers a pro... |

68 |
Axiomatizing probabilistic processes: ACP with generative probabilities
- Baeten, Bergstra, et al.
- 1995
(Show Context)
Citation Context ...is shown to be a congruence and a complete axiomatisation of the equivalence is given. It should be noted that time is also present in his model. Although not based on CCS, Baeten, Bergsta and Smolka =-=[BBS92]-=- have considered probabilistic bisimulation over an extended version of Bergstra and Klop’s Algebra for Communicating Processes (ACP) [BK84] to allow generative probabilistic choice. Probabilistic bis... |

65 |
Post-graduate Lecture Notes in Advanced Domain Theory (incorporating the ‘Pisa notes
- Plotkin
- 1981
(Show Context)
Citation Context ...theories of CCS and CSP arises from the types of equivalences and the approach used in modelling systems, and in particular the process constructors. CCS is modelled using labelled transition systems =-=[Plo81]-=-: a tuple (T, Act, −→), where T is a set of processes (or states), Act is a set of actions (or labels), and −→ ⊆ T × Act × T is a transition relation, where for any (P, a, Q) ∈ −→ we write P a −→ Q, d... |

64 | Quantitative analysis and model checking
- Huth, Kwiatkowska
- 1997
(Show Context)
Citation Context ...h respect to our operational semantics. Chapter 7 investigates a logical approach based on the re-interpretation of the modal µ-calculus for probabilistic processes introduced by Huth and Kwiatkowska =-=[HK97]-=-, and we show a strong connection between this interpretation and our operational semantics. Finally, in Chapter 8 we conclude with an evaluation of the presented work and a discussion of possible fut... |

52 |
Acceptance trees
- Hennessy
- 1985
(Show Context)
Citation Context ...and only if max{R lub(P )((|r ′ |)), R lub(P )((|r ′′ |))} = 1. Proof. The proof follows by induction on σ ∈ A ∗ . ⊓⊔ Now comparing the third part of Proposition 4.6.2 with Hennessy’s acceptance sets =-=[Hen85]-=-, for any P ∈ S (using the notation from Proposition 4.6.2) the set: {(σ, X) | (σ, X) ∈ Act ∗ × Pf (Act) and max{R lub(P )((|r ′ |)) = 1, R lub(P )((|r ′′ |))} = 1} corresponds to the acceptance sets ... |

51 |
Testing preorders for probabilistic processes (extended abstract
- Cleaveland, Smolka, et al.
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(Show Context)
Citation Context ...w that these processes will not be distinguished, even considering probabilistic bisimulation.s2.5 Logics for Probabilistic Processes 18 In addition, Christoff [Chr90], Cleaveland, Smolka and Zwarico =-=[CSZ92]-=- and Yen et al. [YCDS94] have adapted de Nicola and Hennessy’s testing equivalences [NH84] to generative processes. However, as this work depends heavily on the generative nature of processes, and hen... |

48 | Compositional testing preorders for probabilistic processes - Jonsson, Yi - 1995 |

45 |
Testing Equivalences and Fully Abstract Models for Probabilistic Processes
- Christoff
- 1990
(Show Context)
Citation Context ...previously, it is straightforward to show that these processes will not be distinguished, even considering probabilistic bisimulation.s2.5 Logics for Probabilistic Processes 18 In addition, Christoff =-=[Chr90]-=-, Cleaveland, Smolka and Zwarico [CSZ92] and Yen et al. [YCDS94] have adapted de Nicola and Hennessy’s testing equivalences [NH84] to generative processes. However, as this work depends heavily on the... |

43 |
Concurrency and Automata on In Sequences
- Park
- 1980
(Show Context)
Citation Context ...n, where for any (P; a; Q) 2 ! we write P a ! Q, denoting the process P performing the action a and then behaving as the process Q. Equivalences for these transition systems are based on bisimulation =-=[Par-=-81] and [Mil83]. A (strong) bisimulation is a relation such that for any P; Q 2 T , P Q if and only if for all a 2 Act : (i) if P a ! P 0 then 9 Q 0 2 T : Q a ! Q 0 and P 0 Q 0 (ii) if Q a ! Q 0 th... |

41 |
Probability: an introduction
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- 1986
(Show Context)
Citation Context ... and v. 3.2 Probability Theory In this section we introduce the basic concepts and definitions of probability theory we shall need throughout this thesis. A more detailed introduction can be found in =-=[GW86]-=- or any good textbook on probability theory.s3.2 Probability Theory 22 3.2.1 Random Experiments and Events Games of chance such as tossing a coin or rolling a die are examples of “random experiments”.... |

40 | Compositional verification of probabilistic processes - Larsen, Skou |