On the foundations of final semantics: Non-standard sets, metric spaces, partial orders

  • Jan J. M. M. Rutten
  • Daniele Turi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 666)


Canonical solutions of domain equations are shown to be final coalgebras, not only in a category of non-standard sets (as already known), but also in categories of metric spaces and partial orders. Coalgebras are simple categorical structures generalizing the notion of post-fixed point. They are also used here for giving a new comprehensive presentation of the (still) non-standard theory of nonwell-founded sets (as non-standard sets are usually called). This paper is meant to provide a basis to a more general project aiming at a full exploitation of the finality of the domains in the semantics of programming languages — concurrent ones among them. Such a final semantics enjoys uniformity and generality. For instance, semantic observational equivalences like bisimulation can be derived as instances of a single ‘coalgebraic’ definition (introduced elsewhere), which is parametric of the functor appearing in the domain equation. Some properties of this general form of equivalence are also studied in this paper.


final semantics category functor coalgebra domain equation fixed point non-well-founded sets non-standard set theory metric spaces partial orders concurrency (F-)bisimulation ordered F-bisimulation 


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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Jan J. M. M. Rutten
    • 1
  • Daniele Turi
    • 1
  1. 1.CWIAmsterdam

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