## A Survey of Markovian Behavioral Equivalences

Citations: | 5 - 0 self |

### BibTeX

@MISC{Bernardo_asurvey,

author = {Marco Bernardo},

title = {A Survey of Markovian Behavioral Equivalences},

year = {}

}

### OpenURL

### Abstract

Abstract. Markovian behavioral equivalences are a means to relate and manipulate the formal descriptions of systems with an underlying CTMC semantics. There are three fundamental approaches to their definition: bisimilarity, testing, and trace. In this paper we survey the major results appeared in the literature about Markovian bisimilarity, Markovian testing equivalence, and Markovian trace equivalence. The objective is to compare these equivalences with respect to a number of criteria such as their discriminating power, the exactness of the CTMC-level aggregations they induce, the achievement of the congruence property, the existence of sound and complete axiomatizations, the existence of logical characterizations, and the existence of efficient verification algorithms. 1

### Citations

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Citation Context ...rties. Among the various proposals appeared in the literature [25], there are three fundamental approaches to the definition of behavioral equivalences: bisimilarity [43, 41], testing [21], and trace =-=[33]-=-. In the first approach, two models are considered to be equivalent if they are able to mimic each other’s behavior stepwise. In the second approach, two models are considered to be equivalent if an e... |

3360 |
Communication and Concurrency
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(Show Context)
Citation Context ...r functional and performance properties. Among the various proposals appeared in the literature [25], there are three fundamental approaches to the definition of behavioral equivalences: bisimilarity =-=[43, 41]-=-, testing [21], and trace [33]. In the first approach, two models are considered to be equivalent if they are able to mimic each other’s behavior stepwise. In the second approach, two models are consi... |

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Citation Context ...r functional and performance properties. Among the various proposals appeared in the literature [25], there are three fundamental approaches to the definition of behavioral equivalences: bisimilarity =-=[43, 41]-=-, testing [21], and trace [33]. In the first approach, two models are considered to be equivalent if they are able to mimic each other’s behavior stepwise. In the second approach, two models are consi... |

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Citation Context ...ative designs on the basis of their expected QoS guarantees. These notations range from more theoretical ones – like queueing networks [38], stochastic Petri nets [1], and stochastic process algebras =-=[32]-=- – to more applicative ones – like formal modeling languages (Modest [12]), architectural description languages (Æmilia [5]), coordination languages (StoKlaim [22]), and object-oriented modeling langu... |

583 | Queueing Systems
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(Show Context)
Citation Context ...ertain performance requirements as well as to choose among alternative designs on the basis of their expected QoS guarantees. These notations range from more theoretical ones – like queueing networks =-=[38]-=-, stochastic Petri nets [1], and stochastic process algebras [32] – to more applicative ones – like formal modeling languages (Modest [12]), architectural description languages (Æmilia [5]), coordinat... |

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Citation Context ... : P 3.5 Logical Characterization Table 1. Axiomatization of ∼MB over PC Markovian bisimilarity has a modal logic characterization over PC,pc based on a Markovian variant of the Hennessy-Milner logic =-=[29]-=-, in which the diamond operator is decorated with a rate lower bound. Definition 14. The set of the formulas of HMLMB is generated by the following syntax: φ ::= true | ¬φ | φ ∧ φ | ∇a | 〈a〉λφ where a... |

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Citation Context ...ition 11. Markovian bisimilarity, denoted by ∼MB, is the union of all the Markovian bisimulations. Obviously, ∼MB is strictly finer than classical bisimilarity [43, 41] and probabilistic bisimilarity =-=[40]-=-. We conclude with an easy-to-check necessary condition. Proposition 1. Let P1, P2 ∈ PC. Whenever P1 ∼MB P2, then for all a ∈ Name and l ∈ {0, −1}: rate(P1, a, l, PC) = rate(P2, a, l, PC)s3.2 Exactnes... |

377 |
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Citation Context ...g run in some state that satisfies a certain formula. 3.6 Verification Complexity Markovian bisimilarity can be decided in polynomial time by means of a partition refinement algorithm in the style of =-=[42]-=-. This algorithm exploits the fact that ∼MB can be characterized as a fixed point of successively finer relations. In fact we have: ∼MB = � i∈ NI ∼MB,i where ∼MB,0 = PC × PC and ∼MB,i for i ≥ 1 is def... |

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Citation Context ...nts as well as to choose among alternative designs on the basis of their expected QoS guarantees. These notations range from more theoretical ones – like queueing networks [38], stochastic Petri nets =-=[1]-=-, and stochastic process algebras [32] – to more applicative ones – like formal modeling languages (Modest [12]), architectural description languages (Æmilia [5]), coordination languages (StoKlaim [22... |

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Citation Context ...late models that are syntactically different, but also to manipulate models in a way that preserves their functional and performance properties. Among the various proposals appeared in the literature =-=[25]-=-, there are three fundamental approaches to the definition of behavioral equivalences: bisimilarity [43, 41], testing [21], and trace [33]. In the first approach, two models are considered to be equiv... |

210 |
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Citation Context ..., 9, 49] their properties. Although behavioral equivalences abstract from specific kinds of models, most of their properties can be better investigated and understood in a process algebraic framework =-=[41, 33, 2, 6]-=-. For this reason, the three Markovian behavioral equivalences mentioned above will be defined in this paper over Markovian process calculi. Many such calculi have been proposed in the literature, lik... |

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Citation Context ... 5. Let P1, P2 ∈ PC,pc. Then: P1 ∼MB P2 ⇐⇒ (∀φ ∈ HMLMB. P1 |=MB φ ⇐⇒ P2 |=MB φ) We also mention that Markovian bisimilarity has a temporal logic characterization based on the branching-time logic CSL =-=[3]-=-. Besides the usual propositional connectives, this logic comprises: – a probability operator, which replaces the universal and existential computation quantifiers and allows to refer to the probabili... |

131 |
Dynamic Probabilistic Systems
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Citation Context ... distinguish between ordinarily lumpable states by ascribing them a different meaning. The usual approach to the specification of performance measures for CTMCbased models relies on reward structures =-=[34]-=-. This requires associating real numbers with model behaviors and activities, which are then transferred to the proper states (as yield rewards) and transitions (as bonus rewards) of the underlying CT... |

118 | Multiprocessor and distributed system design : the integration of functional specification and performance analysis using stochastic process algebras, volume 729
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(Show Context)
Citation Context ...his reason, the three Markovian behavioral equivalences mentioned above will be defined in this paper over Markovian process calculi. Many such calculi have been proposed in the literature, like TIPP =-=[27]-=-, PEPA [32], MPA [16], EMPAgr [10], Sπ [44], IMC [30], and PIOA [46]. They differ for the action representation – durational actions (TIPP, PEPA, MPA, EMPAgr, Sπ, PIOA) vs. instantaneous actions separ... |

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(Show Context)
Citation Context ...equivalences mentioned above will be defined in this paper over Markovian process calculi. Many such calculi have been proposed in the literature, like TIPP [27], PEPA [32], MPA [16], EMPAgr [10], Sπ =-=[44]-=-, IMC [30], and PIOA [46]. They differ for the action representation – durational actions (TIPP, PEPA, MPA, EMPAgr, Sπ, PIOA) vs. instantaneous actions separated from time passing (IMC) – as well as f... |

73 | Exact and ordinary lumpability in finite Markov chains
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(Show Context)
Citation Context ...arbitrarily closely. This has resulted in the development of the Markovian versions of bisimilarity, testing equivalence, and trace equivalence, which will be surveyed in this paper by recalling from =-=[32, 15, 31, 14, 18, 4, 23, 10, 30, 8, 7, 9, 49]-=- their properties. Although behavioral equivalences abstract from specific kinds of models, most of their properties can be better investigated and understood in a process algebraic framework [41, 33,... |

66 | Testing Equivalence as a Bisimulation Equivalence
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(Show Context)
Citation Context ...reas this is not the case for the verification algorithms for ∼MT and ∼MTr. As far as testing is concerned, it should be investigated whether the algorithm for verifying classical testing equivalence =-=[20]-=- can be adapted to the Markovian framework. The reason is that this algorithm reduces the verification of classical testing equivalence over standard state-transition models to the verification of (a ... |

52 | An efficient algorithm for aggregating PEPA models
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Citation Context ...perties, this can be exploited in practice. For instance, such Markovian behavioral equivalences can be used to aggregate the state space of a model by taking advantage of symmetries within the model =-=[24]-=-, or to reduce the state space of a model before applying analysis techniques like model checking [45], without altering the performance properties to be assessed. We conclude by mentioning some open ... |

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Citation Context ...ication of classical testing equivalence over standard state-transition models to the verification of (a generalization of) classical bisimilarity over transformed models inspired by acceptance trees =-=[28]-=-, hence it can be exploited for minimization purposes as well. – We would like to assess whether ∼MT can be used also for quantitative analysis. So far, it supports a merely qualitative analysis, in t... |

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Citation Context ... Markovian behavioral equivalences mentioned above will be defined in this paper over Markovian process calculi. Many such calculi have been proposed in the literature, like TIPP [27], PEPA [32], MPA =-=[16]-=-, EMPAgr [10], Sπ [44], IMC [30], and PIOA [46]. They differ for the action representation – durational actions (TIPP, PEPA, MPA, EMPAgr, Sπ, PIOA) vs. instantaneous actions separated from time passin... |

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Citation Context ...ences θ ∈ ( RI >0) ∗ of average amounts of time: prob(SC≤θ(P1, T )) = prob(SC≤θ(P2, T ))sObviously, ∼MT is strictly finer than classical testing equivalence [21] and probabilistic testing equivalence =-=[17, 19]-=-. On the other hand, it is strictly coarser than ∼MB as it is less sensitive to branching points. A consequence of this fact is that the derivatives of two Markovian testing equivalent terms are not n... |

25 |
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Citation Context ...arbitrarily closely. This has resulted in the development of the Markovian versions of bisimilarity, testing equivalence, and trace equivalence, which will be surveyed in this paper by recalling from =-=[32, 15, 31, 14, 18, 4, 23, 10, 30, 8, 7, 9, 49]-=- their properties. Although behavioral equivalences abstract from specific kinds of models, most of their properties can be better investigated and understood in a process algebraic framework [41, 33,... |

25 |
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Citation Context ...joint union of [[P1]] and [[P2]] such that their two sets of (original) action names labeling their outgoing transitions coincide. 3. For each equivalence class R induced by R, apply the algorithm of =-=[47]-=- to check the embedded discrete-time versions of [[P1]] and [[P2]] for probabilistic language equivalence by considering R as the set of accepting states. The time complexity is O(n 5 ), where n is th... |

21 | Specifying Performance Measures for PEPA
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Citation Context ...arbitrarily closely. This has resulted in the development of the Markovian versions of bisimilarity, testing equivalence, and trace equivalence, which will be surveyed in this paper by recalling from =-=[32, 15, 31, 14, 18, 4, 23, 10, 30, 8, 7, 9, 49]-=- their properties. Although behavioral equivalences abstract from specific kinds of models, most of their properties can be better investigated and understood in a process algebraic framework [41, 33,... |

19 | A testing equivalence for reactive probabilistic processes
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Citation Context ...MLMB, where the syntax is decorated with rate lower bounds and the satisfaction relation is qualitative, here there is no extra information in the syntax and a quantitative interpretation inspired by =-=[39]-=- is adopted. This establishes the probability with which a process term satisfies a formula quickly enough on average, i.e. within a given sequence of average amounts of time. Definition 45. The set o... |

17 |
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Citation Context ...[1], and stochastic process algebras [32] – to more applicative ones – like formal modeling languages (Modest [12]), architectural description languages (Æmilia [5]), coordination languages (StoKlaim =-=[22]-=-), and object-oriented modeling languages (UML SPT/MARTE [48]). An important feature shared by most of the performance-oriented notations mentioned above is that of providing behavioral models of the ... |

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Citation Context ...erformance properties. Among the various proposals appeared in the literature [25], there are three fundamental approaches to the definition of behavioral equivalences: bisimilarity [43, 41], testing =-=[21]-=-, and trace [33]. In the first approach, two models are considered to be equivalent if they are able to mimic each other’s behavior stepwise. In the second approach, two models are considered to be eq... |

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Citation Context ...ons range from more theoretical ones – like queueing networks [38], stochastic Petri nets [1], and stochastic process algebras [32] – to more applicative ones – like formal modeling languages (Modest =-=[12]-=-), architectural description languages (Æmilia [5]), coordination languages (StoKlaim [22]), and object-oriented modeling languages (UML SPT/MARTE [48]). An important feature shared by most of the per... |

11 | A process-algebraic language for probabilistic I/O automata
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Citation Context ...ove will be defined in this paper over Markovian process calculi. Many such calculi have been proposed in the literature, like TIPP [27], PEPA [32], MPA [16], EMPAgr [10], Sπ [44], IMC [30], and PIOA =-=[46]-=-. They differ for the action representation – durational actions (TIPP, PEPA, MPA, EMPAgr, Sπ, PIOA) vs. instantaneous actions separated from time passing (IMC) – as well as for the action synchroniza... |

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Citation Context ...g networks [38], stochastic Petri nets [1], and stochastic process algebras [32] – to more applicative ones – like formal modeling languages (Modest [12]), architectural description languages (Æmilia =-=[5]-=-), coordination languages (StoKlaim [22]), and object-oriented modeling languages (UML SPT/MARTE [48]). An important feature shared by most of the performance-oriented notations mentioned above is tha... |

6 | A Note on the Congruence Proof for Recursion in Markovian Bisimulation Equivalence
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Citation Context ...ve ones – like formal modeling languages (Modest [12]), architectural description languages (Æmilia [5]), coordination languages (StoKlaim [22]), and object-oriented modeling languages (UML SPT/MARTE =-=[48]-=-). An important feature shared by most of the performance-oriented notations mentioned above is that of providing behavioral models of the systems under construction. Given two such models, establishi... |

1 |
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Citation Context ... finer-than/coarser-than relation, thus providing information about the linear-time/branching-time spectrum in the Markovian case. As we shall see, similarly to what happens in the probabilistic case =-=[36, 35]-=-, the Markovian spectrum is more condensed than the nondeterministic one [25].s2. Exactness. Each of the three Markovian behavioral equivalences induces an aggregation at the CTMC level. In general, a... |

1 |
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Citation Context ...e used to aggregate the state space of a model by taking advantage of symmetries within the model [24], or to reduce the state space of a model before applying analysis techniques like model checking =-=[45]-=-, without altering the performance properties to be assessed. We conclude by mentioning some open problems in the field of Markovian behavioral equivalences: – In our framework based on an asymmetric ... |