Abstract
In a seminal article, Kahn has introduced the notion of process network and given a semantics for those using Scott domains whose elements are (possibly infinite) sequences of values. This model has since then become a standard tool for studying distributed asynchronous computations. From the beginning, process networks have been drawn as particular graphs, but this syntax is never formalized. We take the opportunity to clarify it by giving a precise definition of these graphs, that we call nets. The resulting category is shown to be a fixpoint category, i.e. a cartesian category which is traced wrt the monoidal structure given by the product, and interestingly this structure characterizes the category: we show that it is the free fixpoint category containing a given set of morphisms, thus providing a complete axiomatics that models of process networks should satisfy. We then use these tools to build a model of networks in which data vary over a continuous time, in order to elaborate on the idea that process networks should also be able to encompass computational models such as hybrid systems or electric circuits. We relate this model to Kahn's semantics by introducing a third model of networks based on nonstandard analysis, whose elements form an internal complete partial order for which many properties of standard domains can be reformulated. The use of hyperreals in this model allows it to formally consider the notion of infinitesimal, and thus to make a bridge between discrete and continuous time: time is "discrete", but the duration between two instants is infinitesimal. Finally, we give some examples of uses of the model by describing some networks implementing common constructions in analysis.
BibTeX  Entry
@InProceedings{beauxis_et_al:LIPIcs:2011:3221,
author = {Romain Beauxis and Samuel Mimram},
title = {{A NonStandard Semantics for Kahn Networks in Continuous Time}},
booktitle = {Computer Science Logic (CSL'11)  25th International Workshop/20th Annual Conference of the EACSL},
pages = {3550},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897323},
ISSN = {18688969},
year = {2011},
volume = {12},
editor = {Marc Bezem},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2011/3221},
URN = {urn:nbn:de:0030drops32212},
doi = {10.4230/LIPIcs.CSL.2011.35},
annote = {Keywords: Kahn network, nonstandard analysis, fixpoint category, internal cpo}
}
Keywords: 

Kahn network, nonstandard analysis, fixpoint category, internal cpo 
Collection: 

Computer Science Logic (CSL'11)  25th International Workshop/20th Annual Conference of the EACSL 
Issue Date: 

2011 
Date of publication: 

31.08.2011 