A metric characterization of fair computations in CCS

  • Gerardo Costa
Colloquium On Trees In Algebra And Programming Concurrency
Part of the Lecture Notes in Computer Science book series (LNCS, volume 185)


We address the problem of characterizing fair (infinite) behaviours of concurrent systems as limits of finite approximations. The framework chosen is Milner's Calculus of Communicating Systems. The results can be summarized as follows. On the set FD of all finite derivations in the calculus we define three distances: da, dw, ds. Then the metric completion of (FD,da) yields the space of all derivations, while the completion of (FD,dw), resp. (FD,ds), yields the space of all finite derivations together with all — and only — the weakly, resp. strongly, fair computations (i.e. non-extendable derivations). The results concerning da and dw are a reformulation of previously known ones, while that concerning ds is — we believe — new.


Cauchy Sequence Common Prefix Fairness Constraint Finite Approximation Null String 


  1. AN.
    A. ARNOLD, M. NIVAT. Metric interpretations of infinite trees and semantics of non-deterministic recursive procedures. T.C.S. 11 (1980) 181–205.CrossRefGoogle Scholar
  2. BZ.
    J.W. DE BAKKER, J.I. ZUCKER. Processes and a fair semantics for the ADA rendez-vous. ICALP'83, LNCS 154 (1983) 52–66.Google Scholar
  3. C.
    G. COSTA. A metric characterization of fair computations in CCS. Technical Rep. CSR-169-84, Dept. Comput. Sci. Univ. Edinburgh (1984).Google Scholar
  4. CS1.
    G. COSTA, C. STIRLING. A fair calculus of communicating systems. To appear in Acta Informatica; shortened version: FCT'83, LNCS 158 (1983) 94–105.Google Scholar
  5. CS2.
    G. COSTA, C. STIRLING. Weak and strong fairness in CCS. Technical Rep. CSR-167-84, Dept. Comput. Sci. Univ. Edinburgh (1984); shortened version: MFCS'84, LNCS 176 (1984) 245–254.Google Scholar
  6. DM.
    P. DEGANO, U. MONTANARI. Liveness properties as convergence in metric spaces. Proc. 16th ACM STOC (1984).Google Scholar
  7. H.
    M. HENNESSY. Private communication.Google Scholar
  8. M1.
    R. MILNER. A calculus of communicationg systems. LNCS 92 (1980).Google Scholar
  9. M2.
    R. MILNER. A finite delay operator in synchronous CCS. Technical Rep. CSR-116-82 Dept. Comput. Sci. Univ. Edinburgh (1982).Google Scholar
  10. Pal.
    D. PARK. On the semantics of fair parallelism. LNCS 86 (1980) 504–526.Google Scholar
  11. Pa2.
    D. PARK. A predicate transformer for weak fair iteration. Proc. 6th IBM Symp. on Mathematical Foundat. of Comput. Sci., Hakone, Japan (1981).Google Scholar
  12. Pa3.
    D. PARK. The "fairness" problem and nondeterministic computing networks. Foundat. of Comput. Sci. IV, De Bakker — Van Leuven edit. Amsterdam (1982).Google Scholar
  13. P1.
    G. PLOTKIN. A powerdomain for countable nondeterminism. ICALP'82, LNCS 140 (1982) 418–482.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1985

Authors and Affiliations

  • Gerardo Costa
    • 1
  1. 1.Istituto di Matematica — Università di GenovaGenovaItaly

Personalised recommendations