The metric closure powerspace construction

  • Robert E. Kent
Part II Structure Theory Of Continuous Posets And Related Objects
Part of the Lecture Notes in Computer Science book series (LNCS, volume 298)


In this paper we develop a natural powerobject construction in the context of enriched categories, a context which generalizes the traditional order-theoretic and metric space contexts. This powerobject construction is a subobject transformer involving the dialectical flow of closed subobjects of enriched categories. It is defined via factorization of a comprehension schema over metrical predicates, followed by the fibrational inverse image of metrical predicates along character, the left adjoint in the comprehension schema. A fundamental continuity property of this metrical powerobject construction vis-a-vis greatest fixpoints is established by showing that it preserves the limit of any Cauchy ωop-diagram. Using this powerobject construction we unify two well-known fixpoint semantics for concurrent interacting processes.


Inverse Image Normed Category Comprehension Schema Enrich Category Adjoint Pair 
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  1. 1.
    J. Adamek and V. Koubek, Least Fixed Point of a Functor, J. Comput. System Sci. 19 (1979) 163.CrossRefGoogle Scholar
  2. 2.
    M.A. Arbib and E.G. Manes, Parameterized Data Types Do Not Need Highly Constrained Parameters, Info. and Contr. 52 (1982) 139–158.CrossRefGoogle Scholar
  3. 3.
    S. Bernow and P. Raskin, Ecology of Scientific Consciousness, Telos 28, Summer (1976).Google Scholar
  4. 4.
    J.W. DeBakker and J.I. Zucker, Processes and the Denotational Semantics of Concurrency, Info. and Contr. 54 (1982) 70–120.CrossRefGoogle Scholar
  5. 5.
    W. Golson and W. Rounds, Connections between Two Theories of Concurrency: Metric Spaces and Synchronization Trees, Tech. Rep. TR-3-83, Computing Research Laboratory, University of Michigan, 1983.Google Scholar
  6. 6.
    J. Gray, Fibred and Cofibred Categories, Conference on Categorical Algebra (1965).Google Scholar
  7. 7.
    R.E. Kent, Observational Equivalence of Concurrent Processes is the Kernel of a Multiplicity Morphism of Abstract Tree Data Types, International Computer Symposium 1984, Taipei, Taiwan (1984).Google Scholar
  8. 8.
    R.E. Kent, The Metric Powerspace Construction, Tech. Rep. UIC-EECS-84-14, EECS Dept., University of Illinois at Chicago, 1984.Google Scholar
  9. 9.
    R.E. Kent, Synchronization Trees and Milner's Strong Congruence: a Fixpoint Approach, Tech. Rep. UIC-EECS-85-5, EECS Dept., University of Illinois at Chicago, 1985.Google Scholar
  10. 10.
    R.E. Kent, Dialectical Systems: Interactions and Combinations, manuscript (1986).Google Scholar
  11. 11.
    R.E. Kent, Dialectical Development in First Order Logic: I. Relational Database Semantics (1987), submitted for publication.Google Scholar
  12. 12.
    K. Kuratowski, Topology, Vol.1, (Academic Press, New York, N.Y., 1966).Google Scholar
  13. 13.
    F.W. Lawvere, Metric Spaces, Generalized Logic, and Closed Categories, Seminario Matematico E. Fisico. Rendiconti. Milan. 43 (1973) 135–166.Google Scholar
  14. 14.
    D.J. Lehmann and M.B. Smyth, Algebraic Specification of Data Types: A Synthetic Approach, Math. Systems Theory 14 (1981) 97–139.CrossRefGoogle Scholar
  15. 15.
    M.G. Main, Free Constructions of Powerdomains, in: Proceedings of the Conference on Mathematical Foundations of Programming Language Semantics, Lec. Notes in Comp. Sci. 239, (Springer-Verlag, New York).Google Scholar
  16. 16.
    M.G. Main, Semiring Module Powerdomains, Tech. Rep. CU-CS-286-84, Department of Computer Science, University of Colorado at Boulder, 1984.Google Scholar
  17. 17.
    H. Marcuse, Zum Problem der Dialektik, Die Gesellschaft, Volume VII (1930–31); On the Problem of the Dialectic, Telos 27, Spring (1976).Google Scholar
  18. 18.
    R. Milner, A Calculus of Communicating Systems, Lec. Notes in Comp. Sci. 92, (Springer-Verlag, New York, 1980).Google Scholar
  19. 19.
    R. Milner, Calculi for Synchrony and Asynchrony, Theo. Comp. Sci. 25 (1983) 267–310.CrossRefGoogle Scholar
  20. 20.
    M. Nivat, Infinite Words, Infinite Trees, Infinite Computations, Mathematical Centre Tracts 109 (1979) 1–52.Google Scholar
  21. 21.
    G.D. Plotkin, A Powerdomain Construction, SIAM J. Comput. 5 (1976) 452–487.CrossRefGoogle Scholar
  22. 22.
    K. Popper, What is Dialectic?, Mind 49 (1940) 403–426.Google Scholar
  23. 23.
    M.B. Smyth, Power Domains, J. Comput. System Sci. 16 (1978) 23–36.CrossRefGoogle Scholar
  24. 24.
    M.B. Smyth, Power Domains and Predicate Transformers: a Topological View, 10th ICALP, Barcelona, Spain, Lec. Notes in Comp. Sci. 154, (Springer-Verlag, New York, 1983).Google Scholar
  25. 25.
    M. Steenstrup, M.A. Arbib and E. Manes, Port Automata and the Algebra of Concurrent Processes, COINS Tech. Rep. 81-25, University of Massuchusetts, 1981.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • Robert E. Kent
    • 1
  1. 1.Department of Electrical Engineering and Computer ScienceUniversity of Illinois at ChicagoChicago

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