## On the Foundations of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders (1998)

Venue: | PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE |

Citations: | 47 - 10 self |

### BibTeX

@INPROCEEDINGS{Rutten98onthe,

author = {Jan J. M. M. Rutten and Daniele Turi},

title = {On the Foundations of Final Semantics: Non-Standard Sets, Metric Spaces, Partial Orders},

booktitle = {PROCEEDINGS OF THE REX WORKSHOP ON SEMANTICS: FOUNDATIONS AND APPLICATIONS, VOLUME 666 OF LECTURE NOTES IN COMPUTER SCIENCE},

year = {1998},

pages = {477--530},

publisher = {Springer-Verlag}

}

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### Abstract

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 non-well-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.