## A linear process-algebraic format for probabilistic systems with data

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@MISC{Katoen_alinear,

author = {Joost-pieter Katoen and Jaco Van De Pol and Mariëlle Stoelinga and Mark Timmer},

title = {A linear process-algebraic format for probabilistic systems with data},

year = {}

}

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### Abstract

Abstract—This paper presents a novel linear processalgebraic format for probabilistic automata. The key ingredient is a symbolic transformation of probabilistic process algebra terms that incorporate data into this linear format while preserving strong probabilistic bisimulation. This generalises similar techniques for traditional process algebras with data, and — more importantly — treats data and data-dependent probabilistic choice in a fully symbolic manner, paving the way to the symbolic analysis of parameterised probabilistic systems. Keywords-probabilistic process algebra, linearisation, datadependent probabilistic choice, symbolic transformations I.

### Citations

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Citation Context ...′ can be obtained similarly. s7 B. Strong probabilistic bisimulation Strong probabilistic bisimulation [14] is a probabilistic extension of the traditional notion of bisimulation introduced by Milner =-=[29]-=-, equating any two processes that cannot be distinguished by an observer. Two states s, t of a PA A are strongly probabilistic bisimilar (denoted by s ≈ t) if there exists an equivalence relation R ⊆ ... |

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Citation Context ...estricted form of process equations akin to the Greibach normal form for string grammars. We prove that our transformation is correct, in the sense that it preserves strong probabilistic bisimulation =-=[14]-=-. Similar linearisations have been provided for plain µCRL [15] and a real-time variant thereof [16]. To motivate the expected advantage of a probabilistic linear format, we draw an analogy with the p... |

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Citation Context ...ependability and performance analysis. Major deficiencies of probabilistic model checking are the state explosion problem and the restricted treatment of data. As opposed to process calculi like µCRL =-=[1]-=- and ELOTOS, which support rich data types, the treatment of data in modelling formalisms for probabilistic systems is mostly neglected. Instead, the focus has been on understanding random phenomena a... |

48 | Reachability analysis of probabilistic systems by successive refinements
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Citation Context ...hoice is semantically well-defined [4], the incorporation of data yields a significant increase of, or even an infinite, state space. Aggressive abstraction techniques for probabilistic models (e.g., =-=[5]-=-, [6], [7], [8], [9]) obtain smaller models at the This research has been partially funded by NWO under grant 612.063.817 (SYRUP) and grant Dn 63-257 (ROCKS), and by the European Union under FP7-ICT-2... |

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Citation Context ...data-dependent probabilistic branching. To enable symbolic reductions, we provide a two-phase algorithm to transform prCRL terms into LPPEs: a probabilistic variant of linear process equations (LPEs) =-=[13]-=-, which is a restricted form of process equations akin to the Greibach normal form for string grammars. We prove that our transformation is correct, in the sense that it preserves strong probabilistic... |

39 | Focus points and convergent process operators: A proof strategy for protocol verification - Groote, Springintveld - 2001 |

32 | Game-based abstraction for Markov decision processes
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Citation Context ...y well-defined [4], the incorporation of data yields a significant increase of, or even an infinite, state space. Aggressive abstraction techniques for probabilistic models (e.g., [5], [6], [7], [8], =-=[9]-=-) obtain smaller models at the This research has been partially funded by NWO under grant 612.063.817 (SYRUP) and grant Dn 63-257 (ROCKS), and by the European Union under FP7-ICT-2007-1 grant 214755 (... |

29 |
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Citation Context ...hat our transformation is correct, in the sense that it preserves strong probabilistic bisimulation [14]. Similar linearisations have been provided for plain µCRL [15] and a real-time variant thereof =-=[16]-=-. To motivate the expected advantage of a probabilistic linear format, we draw an analogy with the purely functional case. There, LPEs have provided a uniform and simple format for a process algebra w... |

26 |
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Citation Context ...distributed model checking [22], symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a parameterised µ-calculus formula ([24], [25])), and confluence reduction =-=[26]-=- (a form of partial-order reduction). In all these cases, the LPE format enabled a smooth theoretical development with rigorous correctness proofs (often checked in PVS), and a unifying tool implement... |

24 | Three-valued abstraction for continuous-time markov chains
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Citation Context ...ically well-defined [4], the incorporation of data yields a significant increase of, or even an infinite, state space. Aggressive abstraction techniques for probabilistic models (e.g., [5], [6], [7], =-=[8]-=-, [9]) obtain smaller models at the This research has been partially funded by NWO under grant 612.063.817 (SYRUP) and grant Dn 63-257 (ROCKS), and by the European Union under FP7-ICT-2007-1 grant 214... |

22 | Verification of temporal properties of processes in a setting with data
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Citation Context ...eduction [20]), data abstraction [21], distributed model checking [22], symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a parameterised µ-calculus formula (=-=[24]-=-, [25])), and confluence reduction [26] (a form of partial-order reduction). In all these cases, the LPE format enabled a smooth theoretical development with rigorous correctness proofs (often checked... |

19 | Probabilistic CEGAR
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Citation Context ... the language level, minimising state spaces prior to their generation while preserving functional and quantitative properties. Other approaches that partially deal with data are probabilistic CEGAR (=-=[10]-=-, [11]) and the probabilistic GCL [12]. Our aim is to develop symbolic minimisation techniques — operating at the syntax level — for data-dependent probabilistic systems. The starting point for our wo... |

17 | Computer assisted manipulation of algebraic process specifications
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Citation Context ... cones and foci method for proof checking process equivalence ([17], [18]). It also enabled the application of model checking techniques to process algebra, such as optimisations from static analysis =-=[19]-=- (including dead variable reduction [20]), data abstraction [21], distributed model checking [22], symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a paramet... |

15 | Magnifying-lens abstraction for Markov decision processes
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Citation Context ... is semantically well-defined [4], the incorporation of data yields a significant increase of, or even an infinite, state space. Aggressive abstraction techniques for probabilistic models (e.g., [5], =-=[6]-=-, [7], [8], [9]) obtain smaller models at the This research has been partially funded by NWO under grant 612.063.817 (SYRUP) and grant Dn 63-257 (ROCKS), and by the European Union under FP7-ICT-2007-1... |

15 |
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Citation Context ...anguage level, minimising state spaces prior to their generation while preserving functional and quantitative properties. Other approaches that partially deal with data are probabilistic CEGAR ([10], =-=[11]-=-) and the probabilistic GCL [12]. Our aim is to develop symbolic minimisation techniques — operating at the syntax level — for data-dependent probabilistic systems. The starting point for our work is ... |

15 |
Translating a process algebra with symbolic data values to linear format
- Bosscher, Ponse
- 1995
(Show Context)
Citation Context ... form for string grammars. We prove that our transformation is correct, in the sense that it preserves strong probabilistic bisimulation [14]. Similar linearisations have been provided for plain µCRL =-=[15]-=- and a real-time variant thereof [16]. To motivate the expected advantage of a probabilistic linear format, we draw an analogy with the purely functional case. There, LPEs have provided a uniform and ... |

11 | MoDeST: A Compositional Modeling Formalism for Hard and Softly Timed Systems
- Bohnenkamp, D’Argenio, et al.
- 2006
(Show Context)
Citation Context ...., state-dependent) random choice. To model realistic systems, however, convenient means for data modelling are indispensable. Although parameterised probabilistic choice is semantically well-defined =-=[4]-=-, the incorporation of data yields a significant increase of, or even an infinite, state space. Aggressive abstraction techniques for probabilistic models (e.g., [5], [6], [7], [8], [9]) obtain smalle... |

10 | PROBMELA: a modeling language for communicating probabilistic processes, in
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Citation Context ...ss algebras typically allow a random choice over a fixed distribution, and input languages for model checkers such as the reactive module language of PRISM [2] or the probabilistic variant of Promela =-=[3]-=- only support basic data types, but neither support more advanced data structures or parameterised (i.e., state-dependent) random choice. To model realistic systems, however, convenient means for data... |

10 | V: Sliding Window Abstraction for Infinite Markov Chains
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(Show Context)
Citation Context ...emantically well-defined [4], the incorporation of data yields a significant increase of, or even an infinite, state space. Aggressive abstraction techniques for probabilistic models (e.g., [5], [6], =-=[7]-=-, [8], [9]) obtain smaller models at the This research has been partially funded by NWO under grant 612.063.817 (SYRUP) and grant Dn 63-257 (ROCKS), and by the European Union under FP7-ICT-2007-1 gran... |

9 |
A Database Approach to Distributed State Space Generation
- Blom, Lisser, et al.
- 2008
(Show Context)
Citation Context ...pplication of model checking techniques to process algebra, such as optimisations from static analysis [19] (including dead variable reduction [20]), data abstraction [21], distributed model checking =-=[22]-=-, symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a parameterised µ-calculus formula ([24], [25])), and confluence reduction [26] (a form of partial-order r... |

8 | Model-checking processes with data
- Groote, Willemse
(Show Context)
Citation Context ...on [20]), data abstraction [21], distributed model checking [22], symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a parameterised µ-calculus formula ([24], =-=[25]-=-)), and confluence reduction [26] (a form of partial-order reduction). In all these cases, the LPE format enabled a smooth theoretical development with rigorous correctness proofs (often checked in PV... |

5 |
de Pol, Cones and foci: A mechanical framework for protocol verification
- Fokkink, Pang, et al.
- 2006
(Show Context)
Citation Context ...y development and tool construction. It lead to elegant proof methods, like the use of invariants for process algebra [13], and the cones and foci method for proof checking process equivalence ([17], =-=[18]-=-). It also enabled the application of model checking techniques to process algebra, such as optimisations from static analysis [19] (including dead variable reduction [20]), data abstraction [21], dis... |

4 | Variations on Itai-Rodeh leader election for anonymous rings and their analysis in PRISM
- Fokkink, Pang
- 2006
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Citation Context ...xactly the LPPEs we expected. To illustrate the possible reductions for LPPEs, we model a protocol, inspired by the various leader election protocols that can be found in literature (e.g., Itai-Rodeh =-=[30]-=-), in prCRL. On this model we apply one reduction manually, and several more automatically. Future work will focus on defining and studying more reductions in detail. We consider a system consisting o... |

4 | de Pol (2002): State Space Reduction by Proving Confluence - Blom, van |

3 |
Probabilistic guarded commands mechanized
- Hurd, McIver, et al.
(Show Context)
Citation Context ...spaces prior to their generation while preserving functional and quantitative properties. Other approaches that partially deal with data are probabilistic CEGAR ([10], [11]) and the probabilistic GCL =-=[12]-=-. Our aim is to develop symbolic minimisation techniques — operating at the syntax level — for data-dependent probabilistic systems. The starting point for our work is laid down in this paper. We defi... |

3 |
de Pol, “Symbolic reachability for process algebras with recursive data types
- Blom, van
- 2008
(Show Context)
Citation Context ...ss algebra, such as optimisations from static analysis [19] (including dead variable reduction [20]), data abstraction [21], distributed model checking [22], symbolic model checking (either with BDDs =-=[23]-=- or by constructing the product of an LPE and a parameterised µ-calculus formula ([24], [25])), and confluence reduction [26] (a form of partial-order reduction). In all these cases, the LPE format en... |

2 | 2009): State Space Reduction of Linear Processes using Control Flow Reconstruction
- Pol, Timmer
(Show Context)
Citation Context ...g process equivalence ([17], [18]). It also enabled the application of model checking techniques to process algebra, such as optimisations from static analysis [19] (including dead variable reduction =-=[20]-=-), data abstraction [21], distributed model checking [22], symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a parameterised µ-calculus formula ([24], [25])),... |

2 |
de Pol, An abstract interpretation toolkit for µCRL
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- 2007
(Show Context)
Citation Context ...17], [18]). It also enabled the application of model checking techniques to process algebra, such as optimisations from static analysis [19] (including dead variable reduction [20]), data abstraction =-=[21]-=-, distributed model checking [22], symbolic model checking (either with BDDs [23] or by constructing the product of an LPE and a parameterised µ-calculus formula ([24], [25])), and confluence reductio... |

2 | Katoen (2006): MODEST: A Compositional Modeling Formalism for Hard and Softly Timed Systems - Bohnenkamp, D’Argenio, et al. |

1 | Größer (2004): PROBMELA: a modeling language for communicating probabilistic processes - Baier, Ciesinski, et al. |

1 | de Pol (2008): Symbolic Reachability for Process Algebras with Recursive Data Types - Blom, van |

1 | de Pol & Michael Weber (2009): A Database Approach to Distributed State-Space Generation - Blom, Lisser, et al. |

1 | K.G Larsen (2001): Reachability analysis of probabilistic systems by successive refinements - D’Argenio, Jeannet, et al. |

1 | de Pol (2007): An abstract interpretation toolkit for µCRL - Espada, van |

1 | de Pol (2006): Cones and foci: A mechanical framework for protocol verification - Fokkink, Pang, et al. |

1 | Springintveld (2001): Focus points and convergent process operators: a proof strategy for protocol verification - Groote, J |

1 | Willemse (2005): Model-checking processes with data - Groote, C |

1 | Zhang (2008): Probabilistic CEGAR - Hermanns, Wachter, et al. |