## Algebraic Reasoning for Probabilistic Concurrent Systems (1990)

Venue: | Proc. IFIP TC2 Working Conference on Programming Concepts and Methods |

Citations: | 94 - 5 self |

### BibTeX

@INPROCEEDINGS{Giacalone90algebraicreasoning,

author = {Alessandro Giacalone and Chi-chang Jou and Scott A. Smolka},

title = {Algebraic Reasoning for Probabilistic Concurrent Systems},

booktitle = {Proc. IFIP TC2 Working Conference on Programming Concepts and Methods},

year = {1990},

pages = {443--458},

publisher = {North-Holland}

}

### Years of Citing Articles

### OpenURL

### Abstract

We extend Milner's SCCS to obtain a calculus, PCCS, for reasoning about communicating probabilistic processes. In particular, the nondeterministic process summation operator of SCCS is replaced with a probabilistic one, in which the probability of behaving like a particular summand is given explicitly. The operational semantics for PCCS is based on the notion of probabilistic derivation, and is given structurally as a set of inference rules. We then present an equational theory for PCCS based on probabilistic bisimulation, an extension of Milner's bisimulation proposed by Larsen and Skou. We provide the first axiomatization of probabilistic bisimulation, a subset of which is relatively complete for finite-state probabilistic processes. In the probabilistic case, a notion of processes with almost identical behavior (i.e., with probability 1 \Gamma ffl, for ffl sufficiently small) appears to be more useful in practice than a notion of equivalence, since the latter is often too restricti...

### Citations

3218 |
Communication and Concurrency
- Milner
- 1989
(Show Context)
Citation Context ..., and to give a probabilistic semantics to a language like CCS. Our paper investigates both of these issues. In [PS87], Purushothaman and Subrahmanyam explore modifying CCS (the asynchronous calculus =-=[Mi89]) by rep-=-lacing the hidden action �� with the family of actions f�� p j p 2 [0; 1]g. They do not provide an operational semantics for their resulting calculus nor do they consider the notion of a metri... |

1289 | A Structured Approach to Operational Semantics
- Plotkin
- 1981
(Show Context)
Citation Context ...highlight the technical results contained in the paper. The operational semantics for PCCS based on probabilistic derivation is given in Section 2 as a set of inference rules, in the style of Plotkin =-=[Pl81]-=- and Milner [Mi83, Mi89]. Of particular interest is the rule for synchronous composition (product), which interprets interprocess synchronization/communication as simultaneous occurrence of independen... |

823 | Dynamic Logic
- Harel, Kozen, et al.
- 2000
(Show Context)
Citation Context ...family of PCCS processes: P i def j [1 \Gamma f(i)]1 : P i+1 + [f(i)]a : Q with f(i) = 1 \Delta e \Gamma i Quantitative approaches have also been used for probabilistic sequential programs, including =-=[FH82] and [Ko83]. In the -=-latter, Kozen proposes a probabilistic version of Propositional Dynamic Logic (PDL). Propositions are generalized to measurable functions, "and" and "or" to linear combination, sta... |

405 | Bisimulation through probabilistic testing
- Larsen, Skou
- 1991
(Show Context)
Citation Context ... such as: ffl Specification and Verification: PCCS is a succinct yet expressive specification language for probabilistic concurrent systems. Its equational theory, based on probabilistic bisimulations=-=[LS89], provides-=- a formal framework for reasoning algebraically about probabilistic systems. ffl Reliability Analysis: For "deterministic" PCCS processes, our framework allows us to address questions such a... |

313 |
Calculi for synchrony and asynchrony
- Milner
- 1983
(Show Context)
Citation Context ...probabilitysp. Note the recursion in the definition. In [JS89], we show that probabilistic bisimulation is a congruence in PCCS. We write M �� for Milner's (nonprobabilistic) bisimulation equivale=-=nce [Mi83]. As in -=-the case of M ��, we have that P �� = S f R j R is a probabilistic bisimulationg, and can be found by fixed point iteration. As noted in the Section 1, probabilistic bisimulation is a consider... |

215 | Automatic verification of probabilistic concurrent finite state programs - Vardi - 1985 |

172 |
Probabilistic automata
- Rabin
- 1963
(Show Context)
Citation Context ...odel. We have chosen the "generative model" for PCCS in that, for a stochastic PCCS process, the probabilities of all (singlestep) derivations sum to 1. This is in contrast to the "reac=-=tive model" of [Ra63] and [LS89-=-], where for each action symbol ff 2 Act, a process either possesses no ff-derivations or else the probabilities of these derivations sum to 1. In terms of "button-pushing experiments", a bu... |

153 | Reactive, generative, and stratified models of probabilistic processes
- Glabbeek, Smolka, et al.
- 1995
(Show Context)
Citation Context ...process, as the observer may attempt to depress more than one button at a time. The process itself will decide which button goes down! The reactive and generative models are studied more carefully in =-=[GSST90]-=-, where it is shown that for any single-button experiment, a generative process behaves exactly the same as its reactive counterpart. We close with an interesting open problem. Our metric space is def... |

125 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ...neural networks, which are concurrent systems known to exhibit probabilistic behavior. Section 7 concludes the paper and poses an interesting open problem concerning the metric space. Related Work In =-=[JP89]-=-, Jones and Plotkin investigate a probabilistic powerdomain of evaluations which they use to give the semantics of a language with a probabilistic parallel construct. They note that it would be intere... |

102 | Priorities in process algebras
- Cleaveland, Hennessy
- 1990
(Show Context)
Citation Context ...rograms to real-valued functions. He thus obtains a logic in which logical connectives have been replaced with arithmetic ones. In other related work, Baeten et al. [BBK86] and Cleveland and Hennessy =-=[CH88] have inve-=-stigated the notion of priority in process algebra. In the latter, a dialect of CCS is presented where actions in a "+" context may be assigned priority, preventing actions without priority ... |

96 |
A Probabilistic PDL
- Kozen
- 1985
(Show Context)
Citation Context ...CCS processes: P i def j [1 \Gamma f(i)]1 : P i+1 + [f(i)]a : Q with f(i) = 1 \Delta e \Gamma i Quantitative approaches have also been used for probabilistic sequential programs, including [FH82] and =-=[Ko83]. In the latter, Koz-=-en proposes a probabilistic version of Propositional Dynamic Logic (PDL). Propositions are generalized to measurable functions, "and" and "or" to linear combination, states to meas... |

74 |
Syntax and defining equations for an interrupt mechanism in process algebra
- Baeten, Bergstra, et al.
(Show Context)
Citation Context ...unction transformers, and finally programs to real-valued functions. He thus obtains a logic in which logical connectives have been replaced with arithmetic ones. In other related work, Baeten et al. =-=[BBK86] and Cleve-=-land and Hennessy [CH88] have investigated the notion of priority in process algebra. In the latter, a dialect of CCS is presented where actions in a "+" context may be assigned priority, pr... |

51 |
Algèbre de processus et synchronisations
- AUSTRY, BOUDOL
- 1984
(Show Context)
Citation Context ... PCCS, which lend themselves as targets of future exploration. One is our choice in using SCCS as the basis for PCCS, rather than a calculus with asynchronous composition, such as CCS [Mi89] or Meije =-=[AB84]-=-. In fact we initially proceeded with CCS but soon encountered a specific need to quantify the probability of a process not making a transition in the asynchronous calculus. The situation is easily re... |

51 |
Verification of Multiprocess Proba-bilistic Protocols
- Pnueli, Zuck
- 1986
(Show Context)
Citation Context ...developed qualitative approaches to verification of probabilistic concurrent programs based on temporal logic, including [HS84, HSP83, LS83, Va85, PZ86]. For example, in the method of Pnueli and Zuck =-=[PZ86] for provi-=-ng liveness properties, the quantitative notion of probabilistic computation is represented by the qualitative abstraction of "extreme fairness". A well-known quantitative approach to analyz... |

40 |
On the consistency of Koomen’s fair abstraction rule
- Baeten, Bergstra, et al.
- 1987
(Show Context)
Citation Context ...are relatively complete for finite state processes. Rule (2) resembles Koomen's fair abstraction rule (KFAR), which permits the replacement of a cycle of internal actions by an unobservable �� act=-=ion [BBK87]. Rule (-=-2), on the other hand, rules out the possibility of infinite syntactic substitution in an expression containing unguarded recursion. Zero Process (1) P \Theta 0 P �� 0 (2) 0 . . . . . A P �� 0... |

34 | Reasoning with time and chance - Lehman, Shelah |

30 | Probabilistic temporal logics for finite and bounded models, in: R. De Millo (Ed - Hart, Sharir - 1984 |

14 |
Bounds on the size and transmission rate of communication protocols. Computers and Mathematics with Applications 8(3), 205–214. This is a later version of [Aho, Ullman, and Yannakakis
- Aho, Ullman, et al.
- 1982
(Show Context)
Citation Context ...erized by p. We then use standard Markov chain analysis techniques to prove liveness of the protocol, and to compare the protocol's average transmission time with the theoretical lower bound given in =-=[AUWY82]-=-. We have also used PCCS to model several variations of a physical system of randomly colliding particles [GJS88]. Future applications of PCCS will include neural networks, which are concurrent system... |

14 |
A remark on bisimulation between probabilistic processes
- Bloom, Meyer
- 1989
(Show Context)
Citation Context ...tic processes, and exhibit a testing algorithm that, with probability 1 \Gamma ffl, where ffl is arbitrarily small, can distinguish processes that are not probabilistically bisimilar. Bloom and Meyer =-=[BM89]-=- further show that if nondeterministic bounded-branching processes P and Q are bisimilar, then there is an assignment of probabilities to the edges of P and Q, yielding processes P 0 and Q 0 such that... |

8 |
Modeling communication protocols by automata
- Aho, Ullman, et al.
- 1979
(Show Context)
Citation Context ...w that most syntactic contexts of PCCS are safe in this sense (Theorem 1). An example application of PCCS is treated in Section 5, where we consider the AUY communication protocol for deletion errors =-=[AUY79]-=-. The protocol is modeled as four PCCS processes, one each for the transmitter, receiver, and two communication channels. The channel definitions are parameterized by p, the probability of deletion er... |

6 |
Complete Axiomatizations for Probabilistic Processes
- Jou, Smolka, et al.
- 1990
(Show Context)
Citation Context ...simulation [Mi83] proposed by Larsen and Skou [LS89]. We provide the first equational axiomatization of probabilistic bisimulation, a subset of which is relatively complete for finite-state processes =-=[JS89]-=-. Probabilistic bisimulation is a considerably stronger notion than Milner's bisimulation, yet most of the equational laws of SCCS find probabilistic analogues in PCCS. In the context of probabilistic... |

5 |
Reasoning about probabilistic behavior in concurrent systems
- Purushothaman, Subrahmanyam
- 1987
(Show Context)
Citation Context ...They note that it would be interesting to compare this semantics with an operational one, and to give a probabilistic semantics to a language like CCS. Our paper investigates both of these issues. In =-=[PS87], Purush-=-othaman and Subrahmanyam explore modifying CCS (the asynchronous calculus [Mi89]) by replacing the hidden action �� with the family of actions f�� p j p 2 [0; 1]g. They do not provide an opera... |

3 | Termination of concurrent probabilistic programs - HART, SHARIR, et al. - 1983 |

3 |
Performance Analysis Using Stochastic Petri Nets
- Molly
- 1982
(Show Context)
Citation Context ...robabilistic computation is represented by the qualitative abstraction of "extreme fairness". A well-known quantitative approach to analyzing probabilistic concurrent systems is Stochastic P=-=etri Nets [Mo82]-=-. Let a be an SPN transition from place P to Q, the firing of which is governed by a geometric distribution parameterized by p. We can exactly model this transition in PCCS by the process P def j [1 \... |

1 |
Probabilities in processes: an algebraic /operational framework
- Giacalone, Jou, et al.
- 1988
(Show Context)
Citation Context ...ior of processes, and thus a concrete model for simulating and visualizing physical systems. For example, we have used PCCS to describe the behavior of a closed system of randomly colliding particles =-=[GJS88]-=-. We now highlight the technical results contained in the paper. The operational semantics for PCCS based on probabilistic derivation is given in Section 2 as a set of inference rules, in the style of... |