Article

# Branching bisimulation for probabilistic systems: Characteristics and decidability

Department of Mathematics and Computer Science, Technische Universiteit Eindhoven, Eindhoven, North Brabant, Netherlands
(Impact Factor: 0.66). 05/2006; 356(3):325-355. DOI: 10.1016/j.tcs.2006.02.010
Source: DBLP

ABSTRACT

We address the concept of abstraction in the setting of probabilistic reactive systems, and study its formal underpinnings for the strictly alternating model of Hansson. In particular, we define the notion of branching bisimilarity and study its properties by studying two other equivalence relations, viz. coloured trace equivalence and branching bisimilarity using maximal probabilities. We show that both alternatives coincide with branching bisimilarity. The alternative characterisations have their own merits and focus on different aspects of branching bisimilarity. Coloured trace equivalence can be understood without knowledge of probability theory and is independent of the notion of a scheduler. Branching bisimilarity, rephrased in terms of maximal probabilities gives rise to an algorithm of polynomial complexity for deciding the equivalence. Together they give a better understanding of branching bisimilarity. Furthermore, we show that the notions of branching bisimilarity in the alternating model of Hansson and in the nonalternating model of Segala differ: branching bisimilarity in the latter setting turns out to discriminate between systems that are intuitively branching bisimilar.

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• "Definition 4.6 [Branching bisimulation according to [2]] An equivalence relation R ⊆ S × S is a branching bisimulation iff, for every (s, t) ∈ R the following holds: (i) if s a − → s for some a ∈ A and s ∈ S, then there exists a scheduler σ, such that P rob(Paths m (σ) /t a ⇒[s ] R ) = 1; and (ii) if s ∈ S p , then there is a scheduler σ such that µ R (s, D) = P rob(Paths m (σ) /t⇒D ) for all D ∈ S /R , D = [s] R . "
##### Article: Branching bisimulation congruence for probabilistic systems
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ABSTRACT: A notion of branching bisimilarity for the alternating model of probabilistic systems, compatible with parallel composition, is defined. For a congruence result, an internal transition immediately followed by a non-trivial probability distribution is not considered inert. A weaker definition of branching bisimilarity for the same model has been given earlier. Here we show that our branching bisimulation is the coarsest congruence for parallel composition that is included in the weaker version. To support the use of the present equivalence as a reduction technique, we also show that probabilistic CTL formulae are preserved by our equivalence, and we provide a polynomial-time algorithm deciding branching bisimilarity.
Preview · Article · Jan 2012 · Theoretical Computer Science
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• "The same authors also provided a metric analogue of weak bisimulation [13]. Recently, Andova and Willemse studied branching bisimulation for the alternating model [4] [5], and together with Baeten [3] provided a complete axiomatization of this process equivalence in a process algebra setting. However, the alternating probabilistic automata are not coalgebras (see [40]) and therefore do not qualify for our definition. "
##### Article: Coalgebraic Weak Bisimulation for Action-Type Systems
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ABSTRACT: We propose a coalgebraic deflnition of weak bisimulation for classes of coalgebras obtained from bifunctors in the category Set. Weak bisimilarity for a system is obtained as strong bisimilarity of a transformed system. The particular transformation consists of two steps: First, the behavior on actions is lifted to behavior on flnite words. Second, the behavior on flnite words is taken modulo the hiding of internal or invisible actions, yielding behavior on equivalence classes of words closed under silent steps. The coalgebraic deflnition is validated by two correspondence results: one for the classical notion of weak bisimulation of Milner, another for the notion of weak bisimulation for generative probabilistic transition systems as advocated by Baier and Hermanns.
Full-text · Article · Jan 2009
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• "Definition 4.6 [Branching bisimulation according to [2]] An equivalence relation R ⊆ S × S is a branching bisimulation iff, for every (s, t) ∈ R the following holds: (i) if s a − → s for some a ∈ A and s ∈ S, then there exists a scheduler σ, such that P rob(Paths m (σ) /t a ⇒[s ] R ) = 1; and (ii) if s ∈ S p , then there is a scheduler σ such that µ R (s, D) = P rob(Paths m (σ) /t⇒D ) for all D ∈ S /R , D = [s] R . "
##### Article: Branching Bisimulation Congruence for Probabilistic Systems
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ABSTRACT: The notion of branching bisimulation for the alternating model of probabilistic systems is not a congruence with respect to parallel composition. In this paper we first define another branching bisimulation in the more general model allowing consecutive probabilistic transitions, and we prove that it is compatible with parallel composition. We then show that our bisimulation is actually the coarsest congruence relation included in the existing branching bisimulation when restricted to the alternating model.
Full-text · Article · Dec 2008 · Electronic Notes in Theoretical Computer Science