Formal Aspects of Computing

, Volume 6, Issue 6, pp 696–715 | Cite as

Infinite Concurrent Systems — I. The Relationship between Metric and Order Convergence

  • Mike Stannett
Formal Aspects of Computing
  • 31 Downloads

Abstract

In recent years, Mazurkiewicz trace theory has become increasingly popular as a model of semantics for non-interleaving concurrency, and the theory is here further extended to allow descriptions of infinite behaviours based on alphabets of arbitrary cardinality. We describe an unconventional but nonetheless natural, ordering on trace space, and show that under this ordering, every nonempty set of traces has a greatest lower bound. Consequently, the ordering is consistently complete, i.e. if a set of traces has an upper bound, it has a least such bound. This parallels the result that traces over finite alphabets form a domain.

Kwiatkowska has demonstrated an ultrametric, definable on (standard) trace spaces, under which they become compact, and hence complete, topological spaces. For finite alphabets, thus ultrametric essentially agrees with that of Comyn and Dauchet, but for infinite alphabets they differ in behaviour. We investigate the structure of trace space under various topologies, demonstrate the relationship between order convergence and the various metric convergence regimes, and thereby explain apparent behaviourial anomalies.

Concurrency Concurrent semantics Mazurkiewicz trace theory 

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References

  1. [BMP90]
    Bonizzoni, P., Mauri, G. and Pighizzini, G.: About Infinite Traces. In: Proceedings of the Workshop of the EBRA Working Group No. 3166, V. Diekert (ed.), volume TUM-I9002 of Technical Reports, Technical University Munich, 1990.Google Scholar
  2. [CoD85]
    Comyn, G. and Dauchet, M.: Metric Approximations in Ordered Domains. In: Algebraic Methods in Semantics, Nivat and Reynolds (eds), Cambridge University Press, 1985.Google Scholar
  3. [Die91]
    Diekert, V.: On the Concatenation of Infinite Traces. In: Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Springer-Verlag, 1991.Google Scholar
  4. [Kwi89]
    Kwiatkowska, M. S.: Fairness for non-interleaving concurrency. PhD Thesis, Technical report CSD-22, Dept of Computer Studies, Leicester University, 1989.Google Scholar
  5. [Kwi90]
    Kwiatkowska, M. Z.: A Metric for Traces. Information Processing Letters, 35(3), 129–135 1990.CrossRefMATHMathSciNetGoogle Scholar
  6. [KwS91]
    Kwiatkowska, M. Z. and Stannett, M.: On Transfinite Traces. Research report CS-91-06, Dept of Computer Science, Sheffield University, 1991. (Also available as technical report CSD-45, Leicester University.)Google Scholar
  7. [Maz77]
    Mazurkiewicz, A.: Concurrent Program Schemes and Their Interpretations. Technical report, Aarhus University, 1977.Google Scholar
  8. [Sta92]
    Stannett, M.: Trace Convergence over Infinite Alphabets I — Metric vs Order Convergence. In: Proc. ASMICS Workshop Infinite Traces, V. Diekert and E. Ebinger (eds), Bericht 4/92, Universität Stuttgart Fakultät Informatik, pp. 46–71, 1992.Google Scholar

Copyright information

© BCS 1994

Authors and Affiliations

  • Mike Stannett
    • 1
  1. 1.13 Ibbotson RoadSheffieldUK

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