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The Role of Auxiliary Variables in the Formal Development of Concurrent Programs

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Reflections on the Work of C.A.R. Hoare

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Abstract

So called “auxiliary variables” are often used in reasoning about concurrent programs. They can be useful – but they can also be undesirable in that they can undermine the hard won property of “compositionality”. This paper explores the issue of auxiliary variables and tries to set concerns about overuse in a wider context; it concludes with an attempt to recommend constraints on their use.

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Notes

  1. 1.

    My view of the importance of Hoare’s paper led me to take [22] as the “fulcrum” for [39]; that discussion links the prior work of Floyd, Naur and van Wijngaarden (and remarks on the lack of what could have been an interesting link back to Turing’s work).

  2. 2.

    Both points were true not only in the early book on program development in VDM [34] but also the earlier IBM reports [32, 31].

  3. 3.

    In the rule,Pis a predicate of one state;Wa predicate of two that is well founded (thus establishing termination without the need for a “variant function”);W ∗is the reflexive and transitive closure ofW. See [37] for the honest form of this rule which has an additional hypothesis on definedness – but this paper is not about partial functions.

  4. 4.

    Rely/guarantee conditions are quite capable of recording “no change” but Section 8.2.3 discusses how the read/write frames of VDM simplify such descriptions.

  5. 5.

    Colin Stirling was interested in meta results more than usability in applications.

  6. 6.

    There are those who argue that the root of the problem is, in fact, shared variable concurrency. Another of Tony Hoare’s major contributions is, of course, the development of CSP [26, 27]. Although the concept of communicating processes has yielded considerable insight into the nature of concurrency, it is by no means immune from interference. The interference just comes from communication. This is manifest in any process algebra in which shared variables can be simulated by a process that holds their current value.

  7. 7.

    Far too much of the Vienna Lab’s work was only published as technical reports.

  8. 8.

    The community was denied a journal version of this paper because it was rejected by JACM.

  9. 9.

    In the case that there is no such value, the program can either return an indicator or add a sentinel that does have the property.

  10. 10.

    VDM notation [37] is used; the only item that might be unfamiliar is the use of s for the initial (and undecorated s for the final) state in relational postconditions. Furthermore, the predicate { is} − { prime} should be obvious and the function mults delivers the set of multiples (by 2 and above) of its argument.

  11. 11.

    Some papers from the 2004 workshop appear in Journal of Universal Computing Science, Vol. 11, No.5; similarly (and in the same journal), Vol. 13, No. 8 for the 2006 workshop.

  12. 12.

    My valued friends working on Separation Logic [55, 28] for concurrency [56, 49, 10, 51, 52] should remember that this was back in the early 1990s.

  13. 13.

    This is the source of the difficulty in [50] in proving that two parallel instances of < x ← x + 1 > achieve the obvious result.

  14. 14.

    Amir was of course co-author of the first paper to apply rely/guarantee thinking to temporal logic [9].

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Acknowledgements

I have had the pleasure of knowing Tony Hoare since the 1960s and my DPhilresearch was done under his supervision in 1979–1981. The process of editing“Essays” [21] enhanced our collaboration after I left Oxford. As was said(repeatedly) at the Cambridge meeting in April, 2009 Tony has inspired andsupported many of us over decades.

This paper was not actually presented at the Cambridge meeting to markTony’s birthday because Bill Roscoe and I had held ours as “makeweights” in caseany speakers could not get there. The material was actually presented at the PSYworkshop at CAV Grenoble (June 2009).

I am grateful for comments on drafts of this paper from Joey Coleman, LinasLaibinis, Thai Son Hoang and Bill Roscoe; and to my ever-patient proof readerMs. Allison. This is also a nice opportunity to give belated thanks to SchloßDagstuhl for (among other pleasurable visits) the two on “Atomicty”. The staff inDagstuhl, the environment and the stimulating participants always make tripsthere rewarding and refreshing.

My research is currently funded by the EU “Deploy” project, the (UK) EPSRC “TrAmS” platform grant and the ARC project (that brings together Ian Hayes, Keith Clark, Alan Burns and myself) “Time Bands for Teleo-Reactive Programs”.

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  1. School of Computing Science, Newcastle University, Newcastle, UK

    C. B. Jones

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  1. Oxford University Computing Laboratory, Parks Road, Oxford, OX13QD, United Kingdom

    A.W. Roscoe

  2. Dept. Computing Science, University of Newcastle upon Tyne, Newcastle upon Tyne, NE17RU, United Kingdom

    Cliff B. Jones

  3. Microsoft Research Ltd., J. J. Thomson Avenue 7, Cambridge, CB30FB, United Kingdom

    Kenneth R. Wood

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Jones, C.B. (2010). The Role of Auxiliary Variables in the Formal Development of Concurrent Programs. In: Roscoe, A., Jones, C., Wood, K. (eds) Reflections on the Work of C.A.R. Hoare. Springer, London. https://doi.org/10.1007/978-1-84882-912-1_8

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