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CCS: It’s not fair!

Fair schedulers cannot be implemented in CCS-like languages even under progress and certain fairness assumptions

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Abstract

In the process algebra community it is sometimes suggested that, on some level of abstraction, any distributed system can be modelled in standard process-algebraic specification formalisms like CCS. This sentiment is strengthened by results testifying that CCS, like many similar formalisms, is Turing powerful and provides a mechanism for interaction. This paper counters that sentiment by presenting a simple fair scheduler—one that in suitable variations occurs in many distributed systems—of which no implementation can be expressed in CCS, unless CCS is enriched with a fairness assumption. Since Dekker’s and Peterson’s mutual exclusion protocols implement fair schedulers, it follows that these protocols cannot be rendered correctly in CCS without imposing a fairness assumption. Peterson expressed this algorithm correctly in pseudocode without resorting to a fairness assumption, so it furthermore follows that CCS lacks the expressive power to accurately capture such pseudocode.

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Notes

  1. In [4] a form of reasoning using a particularly strong global fairness assumption was integrated in the axiomatic framework of ACP, and shown to be compatible with the notion of weak bisimulation commonly taken as the semantic basis for CCS.

  2. The process \(E'\) with \(E' \mathop {=}\limits ^{{ def}} a.a.E' + b.0\) on the other hand really needs strong fairness.

  3. Also known as First Come First Served (FCFS).

  4. http://en.wikipedia.org/wiki/Scheduling_(computing).

  5. http://en.wikipedia.org/wiki/Completely_Fair_Scheduler.

  6. This relaxed requirement only serves to increase the range of acceptable schedulers, thereby strengthening our impossibility result. It by no means rules out a scheduler that schedules task \(t_1\) exactly once for each request \(r_1\) received.

  7. Our specification places no restrictions on the presence or absence of any additional occurrences of \(e\). This again increases the range of acceptable implementations.

  8. When assuming that this formula holds, \(F_{0}\) trivially satisfies the other properties required in Sect. 7: the system will always return to a state where it remains ready to accept the next request \(r\) until it arrives; in any partial run there are no more occurrences of \(t\) than of \(r\), and between each two occurrences of \(t\) the action \(e\) is scheduled.

  9. The output and internal actions of CCS\(^!\) are similar to the output and internal action of I/O automata [39]. However, the remaining actions of I/O automata are input actions that are totally under the control of the environment of the modelled system. In CCS, on the other hand, the default type of action is a synchronisation that can happen only in cooperation between a system and its environment.

  10. When assuming that these formulas hold, \(F_1|F_2\) trivially satisfies the other properties required of it: the system will always return to a state where it remains ready to accept the next request \(r_i\) until it arrives—hence for any numbers \(n_1\) and \(n_2\in \mathbb {N}\cup \{\infty \}\) there is at least one run of the system in which exactly that many requests of type \(r_1\) and \(r_2\) are received—and in any partial run there are no more occurrences of \(t_i\) than of \(r_i\).

  11. By definition \(Y\) does not contain non-blocking action.

  12. The CCS process \(a.0 | b.0\) has two transitions labelled \(a\), namely \(a.0 | b.0 \mathrel {\xrightarrow {\,\, a \,\,}} 0|b.0\) and \(a.0 | 0 \mathrel {\xrightarrow {\,\, a \,\,}} 0|0\). The only difference between these two transitions is that one occurs before the action \(b\) is performed by the parallel component and the other afterwards. In [57] we formalise a notion of an abstract transition that identifies these two concrete transitions.

  13. We only give the liveness property for process A; the one for process B is similar.

  14. Whether weak fairness suffices depends on the interpretation of enabledness (cf. Sect. 2).

  15. It differs in a crucial way, however, namely by treating each action as output. As a consequence, under fairness of actions the process \(F_1|F_2\) of Sect. 10 is guaranteed to perform each of the actions \(r_1\) and \(r_2\) infinitely often. To model a protocol where the action \(r_i\) is not forced to occur, a \(\tau \)-loop is inserted at each location where \(r_{i}\) is enabled.

  16. Fairness of components is a form of weak fairness, requiring that if a component from some point onwards is enabled in each state, an action from that component will eventually be scheduled. Here a component is enabled if an action from that component is enabled, possibly in synchronisation with an action from outside that component. Under this notion of fairness, the system \(E\) from Sect. 2, defined by \(E \mathop {=}\limits ^{{ def}} a.E + b.0\), is not ensured to do a \(b\) eventually. However, the composition \((E|\bar{b}.c.0)\backslash b\) is ensured to do a \(c\) eventually, because the component \(\bar{b}.c.0\) is enabled in every state.

  17. Here, we slightly deviate from standard notation [52], where \({}^\bullet u\) and \({u}^\bullet \) are usually plain sets, obtained from our multisets by abstracting from the multiplicities of their elements. We prefer to retain this information, so as to shorten various formulas.

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Acknowledgments

We gratefully thank the anonymous referees. Their reports showed deep insights in the material, and helped a lot to improve the quality of the paper. In particular, the link between our fair scheduler and Peterson’s mutual exclusion protocol was made by one of the referees.

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Authors and Affiliations

  1. NICTA, Sydney, Australia

    Rob J. van Glabbeek & Peter Höfner

  2. UNSW, Sydney, Australia

    Rob J. van Glabbeek & Peter Höfner

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  1. Rob J. van Glabbeek
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Correspondence to Rob J. van Glabbeek or Peter Höfner.

Additional information

It is our great pleasure to dedicate this paper to Walter Vogler on the occasion of his 60th birthday. We have combined two of Walter’s main interests: Petri nets and process algebra. In fact, we proved a result about Petri nets that had been proven before by Walter, but in a restricted form, as we discovered only after finishing our proof. We also transfer this result to the process algebra CCS. Beyond foundational research in the theory of concurrent systems, Walter achieved excellent results in related subjects such as temporal logic and efficiency. In addition to being a dedicated researcher, he is also meticulous in all of his endeavours, including his writing. As a consequence his scientific papers tend to contain no flaws, which is just one of the reasons that makes reading them so enjoyable. It’s fair to say: “CCS Walter!”—Congratulations and Continuous Success!

NICTA is funded by the Australian Government through the Department of Communications and the Australian Research Council through the ICT Centre of Excellence Program.

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van Glabbeek, R., Höfner, P. CCS: It’s not fair!. Acta Informatica 52, 175–205 (2015). https://doi.org/10.1007/s00236-015-0221-6

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