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On the probabilistic bisimulation spectrum with silent moves

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Abstract

In this paper we look at one of the seminal works of Rob van Glabbeek from a probabilistic angle. We develop the bisimulation spectrum with silent moves for probabilistic models, namely Markov decision processes. Especially the treatment of divergence makes this endeavour challenging. We provide operational as well as logical characterisations of a total of 32 bisimilarities.

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Notes

  1. When dealing with \(\xi \in \{\textsf {s},0,\lambda \}\) then \(u_1 \mathrel {\approx _{x}^{\xi }} u_2\) in \({\mathcal {N}}\) is possible, although \(u_1\) and \(u_2\) may not be \(\mathrel {\approx _{x}^{\xi }}\)-bisimilar in \({\mathcal {M}}\). This, however, is irrelevant for our purposes as we deal here only with the relation \(R_{{\mathcal {N}}}\) which enjoys the property that if \(u_1,u_2\in S\) then \((u_1,u_2) \in R\) iff \((u_1,u_2)\in R_{{\mathcal {N}}}\). Furthermore, with R also \(R_{{\mathcal {N}}}\) is \(\xi \)-respecting as all fresh states are terminal. Therefore, if \(\xi \in \{\textsf {s},0,\lambda \}\) then the \(\xi \)-predicate in \({\mathcal {N}}\) is \(\xi _{{\mathcal {N}}}=\{u_{{\mathcal {N}}}:u\in S\}\), while for \(\xi \in \{\Updelta ,\nabla \}\) the \(\xi \)-predicates in \({\mathcal {N}}\) and \({\mathcal {M}}\) agree in the sense that \(\Updelta _{{\mathcal {N}}}=\Updelta _{{\mathcal {M}}}\) and \(\nabla ^{R_{{\mathcal {N}}}}_{{\mathcal {N}}}=\nabla ^R_{{\mathcal {M}}}\).

  2. Note, however, that \({\text {supp}(\mu _n)}\) contains states \(u \in S\) that do not have R-equivalent states in \({\text {supp}(\mu )}\), unless \({\mathcal {T}}_n={\mathcal {T}}\). These are exactly the states that occur as labels of some inner node v of \({\mathcal {T}}\) with \(\textit{depth}(v)=n\). For these states u, we have \(\lim _{n \rightarrow \infty } \mu _n(u)=0\), but no monotonicity property can be guaranteed for the sequence \((\mu _n(u))_{n \geqslant 0}\).

  3. If there is no BSCC in \({\mathcal {C}}\) that consists of T-states then \(E=\varnothing \), in which case \(\mathrm {Pr}^{\sigma _0}_u\bigl ( \, (S \times \{\tau \})^{\omega } \, \bigr )=0\) for all states \(u\in S\).

  4. Recall that \(\mu \mathrel {\approx _{x}^{\xi }}\nu \) stands for \(\mu \equiv _{\mathrel {\approx _{x}^{\xi }}}\nu \).

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Authors and Affiliations

  1. Institute of Theoretical Computer Science, Technische Universität Dresden, Dresden, Germany

    Christel Baier

  2. FAMAF, Universidad Nacional de Córdoba, Buenos Aires, Argentina

    Pedro R. D’Argenio

  3. CONICET, Buenos Aires, Argentina

    Pedro R. D’Argenio

  4. Saarland University, Saarland Informatics Campus, Saarbrücken, Germany

    Pedro R. D’Argenio & Holger Hermanns

  5. Institute of Intelligent Software, Guangzhou, China

    Holger Hermanns

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  1. Christel Baier
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  2. Pedro R. D’Argenio
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  3. Holger Hermanns
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Correspondence to Christel Baier.

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This work has received financial support by DFG Grant 389792660 as part of TRR 248 (see perspicuous-computing.science), by ERC Advanced Grant 69561 (POWVER), by ANPCyT PICT-2017-3894 (RAFTSys), and by SeCyT-UNC 33620180100354CB (ARES), by the Cluster of Excellence EXC 2050/1 (CeTI, Project ID 390696704, as part of Germany’s Excellence Strategy), by the DFG-Projects BA-1679/11-1 and 1679/12-1 and by the CRC 912 HAEC.

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Baier, C., D’Argenio, P.R. & Hermanns, H. On the probabilistic bisimulation spectrum with silent moves. Acta Informatica 57, 465–512 (2020). https://doi.org/10.1007/s00236-020-00379-2

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