## Compositional reasoning for probabilistic finite-state behaviors (2005)

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Venue: | In Processes, Terms and Cycles: Steps on the Road to Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday, LNCS 3838 |

Citations: | 17 - 4 self |

### BibTeX

@INPROCEEDINGS{Deng05compositionalreasoning,

author = {Yuxin Deng and Catuscia Palamidessi and Jun Pang},

title = {Compositional reasoning for probabilistic finite-state behaviors},

booktitle = {In Processes, Terms and Cycles: Steps on the Road to Infinity, Essays Dedicated to Jan Willem Klop, on the Occasion of His 60th Birthday, LNCS 3838},

year = {2005},

pages = {309--337},

publisher = {Springer}

}

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### Abstract

Abstract. We study a process algebra which combines both nondeterministic and probabilistic behavior in the style of Segala and Lynch’s simple probabilistic automata. We consider strong bisimulation and observational equivalence, and provide complete axiomatizations for a language that includes parallel composition and (guarded) recursion. The presence of the parallel composition introduces various technical difficulties and some restrictions are necessary in order to achieve complete axiomatizations. 1

### Citations

3639 | Communicating sequential processes
- Hoare
- 1978
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Citation Context ...d reasoning about the behaviors of distributed systems, algorithms and protocols in a compositional way. Some of the most prominent representants of these formalisms are CCS [27], ACP [8, 6], and CSP =-=[21]-=-. The axiomatic theories of process algebra provide an elegant way for proving properties of systems. Both a system and its desired external behavior can be expressed as process terms. The correctness... |

3427 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ...ssible to adapt those results to the framework of this paper. In [26] and [28] Milner gave complete axiomatizations for strong bisimilarity and observational equivalence, respectively, for a core CCS =-=[27]-=-. Our results in Section 5 and Section 6 extend [26] and [28] (for guarded expressions) respectively, to a strictly larger language with a probabilistic choice and a parallel composition operator. The... |

517 | Algebraic Laws for Nondeterminism and Concurrency
- Hennessy, Milner
- 1985
(Show Context)
Citation Context ...r) with both unguarded and guarded recursion. Bergstra and Klop [10] axiomatized observational equivalence in an alternative way by using an interesting graph rewriting technique. Hennessy and Milner =-=[20]-=- offered a complete equational axiomatization of strong bisimulation over the recursion free fragment of CCS. To deal with parallel composition, they used the so-called expansion law, which is an equa... |

428 | Bisimulation through Probabilistic Testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ...d analysis of probabilistic behaviors, as exhibited for instance in randomized, distributed and fault-tolerant systems. The notion of probabilistic bisimulation is introduced first by Larsen and Skou =-=[22]-=-. Later many variant behavioural equivalences have been defined for various probabilistic models. A representative model for analyzing probabilistic systems is provided by Segala and Lynch’s simple pr... |

384 | Process algebra for synchronous communication
- Bergstra, Klop
- 1984
(Show Context)
Citation Context ... representing and reasoning about the behaviors of distributed systems, algorithms and protocols in a compositional way. Some of the most prominent representants of these formalisms are CCS [27], ACP =-=[8, 6]-=-, and CSP [21]. The axiomatic theories of process algebra provide an elegant way for proving properties of systems. Both a system and its desired external behavior can be expressed as process terms. T... |

332 |
Calculi for synchrony and asynchrony
- Milner
- 1993
(Show Context)
Citation Context ...urse, impossible in our setting because of recursion. Giacalone, Jou and Smolka [18] axiomatized strong bisimulation for a fully probabilistic (i.e. without nondeterminism) extension of Milner’s SCCS =-=[25]-=-, where parallel composition is synchronous. In contrast, we consider an asynchronous parallel composition and we admit nondeterminism. Baeten, Bergstra and Smolka [4] proposed a probabilistic ACP by ... |

295 | Probabilistic simulations for probabilistic processes
- Segala, Lynch
- 1994
(Show Context)
Citation Context ...behavioural equivalences have been defined for various probabilistic models. A representative model for analyzing probabilistic systems is provided by Segala and Lynch’s simple probabilistic automata =-=[30]-=-, which take into account both probabilistic and nondeterministic behavior and which have been successfully adopted in the studies of distributed algorithms [24, 29] and practical communication protoc... |

128 |
A complete inference system for a class of regular behaviours
- Milner
- 1984
(Show Context)
Citation Context ...tion relation over a process algebra expressing finite-state processes has been a research focus for the process algebra community. This led to a wealth of classical results in the literature. Milner =-=[26, 28]-=- gave complete axiomatizations of both strong bisimilarity and observational equivalence for a core CCS (not containing the parallel composition operator) with both unguarded and guarded recursion. Be... |

98 | Algebraic reasoning for probabilistic concurrent systems
- Giacalone, Jou, et al.
- 1990
(Show Context)
Citation Context ...es of semantics. In both cases, their completeness proofs are done by structural induction on processes, which is, of course, impossible in our setting because of recursion. Giacalone, Jou and Smolka =-=[18]-=- axiomatized strong bisimulation for a fully probabilistic (i.e. without nondeterminism) extension of Milner’s SCCS [25], where parallel composition is synchronous. In contrast, we consider an asynchr... |

84 |
Decidability and Decomposition in Process Algebras
- Christensen
- 1993
(Show Context)
Citation Context ...put in parallel composition. In that language, strong bisimulation is decidable and they obtained a sound and complete sequent based equational theory, but observational equivalence is semi-decidable =-=[12]-=-. In this paper we follow [26, 28] and characterize recursion by laws concerning the explicit fixed point operator µ, while we capture by τ-laws the difference between observational equivalence and st... |

68 |
Axiomatizing probabilistic processes: ACP with generative probabilities
- Baeten, Bergstra, et al.
- 1995
(Show Context)
Citation Context ...ism) extension of Milner’s SCCS [25], where parallel composition is synchronous. In contrast, we consider an asynchronous parallel composition and we admit nondeterminism. Baeten, Bergstra and Smolka =-=[4]-=- proposed a probabilistic ACP by introducing a parameterized composition. They considered generative models, which are fully probabilistic, and axiomatized strong probabilistic bisimilarity for finite... |

60 |
A complete axiomatisation for observational congruence of finite-state behaviours
- Milner
- 1989
(Show Context)
Citation Context ...tion relation over a process algebra expressing finite-state processes has been a research focus for the process algebra community. This led to a wealth of classical results in the literature. Milner =-=[26, 28]-=- gave complete axiomatizations of both strong bisimilarity and observational equivalence for a core CCS (not containing the parallel composition operator) with both unguarded and guarded recursion. Be... |

45 | Verification of the Randomized Consensus Algorithm of Aspnes and Herlihy: a Case Study - Pogosyants, Segala, et al. |

42 |
Axiomatizations for probabilistic bisimulation
- Bandini, Segala
- 2001
(Show Context)
Citation Context ...ibuted algorithms [24, 29] and practical communication protocols [33]. An axiomatization for the finite sequential fragment of simple probabilistic automata has been provided by Bandini and Segala in =-=[7]-=-. Following this line of research, Deng and Palamidessi [16, 15] have given a sound and complete axiomatization for a larger language, which includes the recursion operator. In this paper, we improve ... |

40 |
Compositional verification of probabilistic processes
- Larsen, Skou
(Show Context)
Citation Context ...model, in which at each step the probabilistic choice ranges over the next state, while the action is fixed. In a sequel paper, Larsen and Skou also gave a complete axiomatization for the finite case =-=[23]-=-. 19sBandini and Segala [7] axiomatized two strong and two weak equivalences for a language similar to the fragment of our calculus without recursion and parallelism. They considered two types of sema... |

37 | On generative parallel composition - D’Argenio, Hermanns, et al. - 1999 |

36 | Proving time bounds for randomized distributed algorithms
- Lynch, Saias, et al.
- 1994
(Show Context)
Citation Context ... Lynch’s simple probabilistic automata [30], which take into account both probabilistic and nondeterministic behavior and which have been successfully adopted in the studies of distributed algorithms =-=[24, 29]-=- and practical communication protocols [33]. An axiomatization for the finite sequential fragment of simple probabilistic automata has been provided by Bandini and Segala in [7]. Following this line o... |

29 |
Veri of an alternating bit protocol by means of process algebra
- Bergstra, Klop
- 1985
(Show Context)
Citation Context ...en the protocol and its external behavior is established by equational reasoning based on axiomatization, hiding internal behavior, using fairness assumption, and the other feasible methods (see e.g. =-=[9, 17]-=-). Developing a both complete and sound axiomatization for a chosen bisimulation relation over a process algebra expressing finite-state processes has been a research focus for the process algebra com... |

29 | Linearization in µCRL
- Usenko
- 2002
(Show Context)
Citation Context ...cursive expression. To give a linearization algorithm for pCRL, Groote, Ponse and Usenko adopted a similar restriction for parallel composition [19]. Usenko extended this result to µCRL in his thesis =-=[34]-=-. In this paper our parallel composition operator allows communication and it can appear in the body of a recursive expression, though only in a restricted way. For example, the expression µX(a.X + a.... |

28 | A complete axiomatization for branching bisimulation congruence of finitestate behaviours
- Glabbeek
- 1993
(Show Context)
Citation Context ...st a single axiom. It remains open whether their results can be adapted to a probabilistic setting. Similarly, it might be interesting to extend van Glabbeek’s axiomatization for branching congruence =-=[35]-=- to a probabilistic setting. We believe that the general proof schema laid out in this paper could be reused for branching congruence, but the soundness proof of some axioms such as R2 ′ would be very... |

26 | Probabilistic automata: system types, parallel composition and comparison - Sokolova, Vink |

25 |
A complete inference system for regular processes with silent moves
- Bergstra, Klop
- 1986
(Show Context)
Citation Context ...xiomatizations of both strong bisimilarity and observational equivalence for a core CCS (not containing the parallel composition operator) with both unguarded and guarded recursion. Bergstra and Klop =-=[10]-=- axiomatized observational equivalence in an alternative way by using an interesting graph rewriting technique. Hennessy and Milner [20] offered a complete equational axiomatization of strong bisimula... |

25 | Axiomatizations for probabilistic finite-state behaviours
- Deng, Palamidessi
- 2005
(Show Context)
Citation Context ...ocols [33]. An axiomatization for the finite sequential fragment of simple probabilistic automata has been provided by Bandini and Segala in [7]. Following this line of research, Deng and Palamidessi =-=[16, 15]-=- have given a sound and complete axiomatization for a larger language, which includes the recursion operator. In this paper, we improve on [16, 15] by considering also the parallel composition. To our... |

25 | Root contention in IEEE 1394
- Stoelinga, Vaandrager
- 1999
(Show Context)
Citation Context ...ich take into account both probabilistic and nondeterministic behavior and which have been successfully adopted in the studies of distributed algorithms [24, 29] and practical communication protocols =-=[33]-=-. An axiomatization for the finite sequential fragment of simple probabilistic automata has been provided by Bandini and Segala in [7]. Following this line of research, Deng and Palamidessi [16, 15] h... |

22 | Linearization in parallel pCRL
- Groote, Ponse, et al.
(Show Context)
Citation Context ... that free merge should not appear in the body of any recursive expression. To give a linearization algorithm for pCRL, Groote, Ponse and Usenko adopted a similar restriction for parallel composition =-=[19]-=-. Usenko extended this result to µCRL in his thesis [34]. In this paper our parallel composition operator allows communication and it can appear in the body of a recursive expression, though only in a... |

19 |
A complete axiom system for finite-state probabilistic processes
- Stark, Smolka
- 2000
(Show Context)
Citation Context ...uation set in a certain format (standard format) that they both satisfy; (2) if two expressions satisfy the same standard equation set, then they can be proved equal by As. This schema is inspired by =-=[26, 32]-=-, but in our case the definition of standard format and the proof itself are more complicated due to the presence of both probabilistic and nondeterministic dimensions. 10sS1 E + 0 = E S2 E + E = E S3... |

18 | Equational Axioms for Probabilistic Bisimilarity
- Aceto, Esik, et al.
(Show Context)
Citation Context ...ude and discuss some related work not yet mentioned in the introduction. Detailed proofs of the main propositions in Section 4 are in the Appendix. 2 Preliminaries Let S be a set. A function η : S ↦→ =-=[0, 1]-=- is called a discrete probability distribution, or distribution for short, on S if the support of η, defined as spt(η) = {x ∈ S | η(x) > 0}, is finite or countably infinite and � x∈S η(x) = 1. We deno... |

13 | The decidable subsets of ccs
- Christensen, Moller
- 1994
(Show Context)
Citation Context ...sdeterministic features, we need to consider the branching structure of processes, which is ignored in observational congruence. Christensen, Hirshfeld and Moller studied a class of standard form CCS =-=[13]-=- where open expressions are allowed to be put in parallel composition. In that language, strong bisimulation is decidable and they obtained a sound and complete sequent based equational theory, but ob... |

13 |
de Pol. Verifying a sliding window protocol in CRL
- Fokkink, Groote, et al.
- 2004
(Show Context)
Citation Context ...en the protocol and its external behavior is established by equational reasoning based on axiomatization, hiding internal behavior, using fairness assumption, and the other feasible methods (see e.g. =-=[9, 17]-=-). Developing a both complete and sound axiomatization for a chosen bisimulation relation over a process algebra expressing finite-state processes has been a research focus for the process algebra com... |

11 | Deciding and axiomatizing weak ST bisimulation for a process algebra with recursion and action refinement
- Bravetti, Gorrieri
(Show Context)
Citation Context ...of this introduction) with CCSlike communication, while [5] considers a purely nondeterministic paradigm, but more general than our nondeterministic fragment. The same restriction already appeared in =-=[11]-=-, for a nondeterministic process algebra with CSP multiway synchronization. Recently there has been an increasing interest in the area of formal methods for the specification and analysis of probabili... |

10 |
Probabilistic Process Algebra
- Andova
- 2002
(Show Context)
Citation Context ...ucing a parameterized composition. They considered generative models, which are fully probabilistic, and axiomatized strong probabilistic bisimilarity for finite processes (without recursion). Andova =-=[3]-=- studied a different version of probabilistic ACP by allowing nondeterminism and a parallel composition which is not parameterized. She provided a sound and complete axiomatization for strong probabil... |

8 |
Compositional veri of probabilistic processes
- Larsen, Skou
- 1992
(Show Context)
Citation Context ...ive model, in which at each step the probabilistic choice ranges over the next state, while the action issxed. In a sequel paper, Larsen and Skou also gave a complete axiomatization for thesnite case =-=[23]-=-. 19 in ria -0 02 01 10 0,sv er sio ns1s- 2 3sDe cs20 07 Bandini and Segala [7] axiomatized two strong and two weak equivalences for a language similar to the fragment of our calculus without recursio... |

7 | A ground-complete axiomatization of finite state processes in process algebra
- Baeten, Bravetti
- 2005
(Show Context)
Citation Context ...expression. In this paper we relax this restriction by requiring, instead, that free variables do not appear in the scope of parallel composition. A similar restriction was adopted, independently, in =-=[5]-=-. In that paper, Baeten and Bravetti considered a generic process algebra of which CCS, CSP and ACP are subalgebras. Finitestateness is achieved by requiring that recursion variables do not occur in t... |

3 |
Axiomatisations and types for probabilistic and mobile processes
- Deng
- 2005
(Show Context)
Citation Context ...ocols [33]. An axiomatization for the finite sequential fragment of simple probabilistic automata has been provided by Bandini and Segala in [7]. Following this line of research, Deng and Palamidessi =-=[16, 15]-=- have given a sound and complete axiomatization for a larger language, which includes the recursion operator. In this paper, we improve on [16, 15] by considering also the parallel composition. To our... |

3 |
A complete axiom system for probabilistic processes
- Stark, Smolka
- 1999
(Show Context)
Citation Context ...uation set in a certain format (standard format) that they both satisfy; (2) if two expressions satisfy the same standard equation set, then they can be proved equal by As. This schema is inspired by =-=[26, 32]-=-, but in our case the denition of standard format and the proof itself are more complicated due to the presence of both probabilistic and nondeterministic dimensions. 10 in ria -0 02 01 10 0,sv er si... |

3 |
Linearization in CRL
- Usenko
- 2002
(Show Context)
Citation Context ...cursive expression. To give a linearization algorithm for pCRL, Groote, Ponse and Usenko adopted a similar restriction for parallel composition [19]. Usenko extended this result to CRL in his thesis =-=[34]-=-. In this paper our parallel composition operator allows communication and it can appear in the body of a recursive expression, though only in a restricted way. For example, the expression X(a:X + a:... |

1 | The quest for equational axiomatizations of parallel composition: Status and open problems
- Aceto, Fokkink
- 2005
(Show Context)
Citation Context ...rator (as the parallel composition in CCS) using the auxiliary left merge and communication merge operators. An interesting essay on equational axiomatizations of parallel composition can be found in =-=[2]-=-. Having both recursion and parallel composition in a process algebra complicates the matters to establish a complete axiomatization, mostly because this can give rise to infinite-state systems even w... |

1 |
A ground-complete axiomatization of state processes in process algebra
- Baeten, Bravetti
- 2005
(Show Context)
Citation Context ...expression. In this paper we relax this restriction by requiring, instead, that free variables do not appear in the scope of parallel composition. A similar restriction was adopted, independently, in =-=[5]-=-. In that paper, Baeten and Bravetti considered a generic process algebra of which CCS, CSP and ACP are subalgebras. Finitestateness is achieved by requiring that recursion variables do not occur in t... |

1 |
Axiomatizations for probabilistic behaviors
- Deng, Palamidessi
- 2005
(Show Context)
Citation Context ...otocols [33]. An axiomatization for thesnite sequential fragment of simple probabilistic automata has been provided by Bandini and Segala in [7]. Following this line of research, Deng and Palamidessi =-=[16, 15]-=- have given a sound and complete axiomatization for a larger language, which includes the recursion operator. In this paper, we improve on [16, 15] by considering also the parallel composition. To our... |