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Spatio-Temporal Domains: An Overview

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Theoretical Aspects of Computing – ICTAC 2018 (ICTAC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11187))

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Abstract

We consider the possibility of defining a general mathematical framework for the homogeneous modeling and analysis of heterogeneous spatio-temporal computations as they occur more and more in modern computerized systems of systems. It appears that certain fibrations of posets into posets, called here spatio-temporal domains, eventually provide a fully featured category that extends to space and time the category of cpos and continuous functions, aka Scott Domains, used in classical denotational semantics.

Work partially supported by Inria center Bordeaux-Sud-Ouest, from 09/2016 to 02/2017, long version at https://hal.archives-ouvertes.fr/hal-01634897.

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Notes

  1. 1.

    Strictly speaking, for \(\sim _\pi \) to be a symmetry, the timed poset P must be completed with sorts of “passing time” elements of the form (u, x) with \(x \le (u,x)\) and \(\pi (u,x) = u\), defined for all \(x \in P\) and \(u \in T\) such that there is no y above x with \(\pi (y) = u\).

  2. 2.

    Possibly gluing minimal elements when considering the subcategory of timed posets with a minimum elements.

  3. 3.

    One can easily verify that the monomorphisms in \( TPoset (T)\) are the injective synchronous functions. Then, as a consequence of the lemma, every injective synchronous function \(f : Q \rightarrow P\) is equivalent (as sub-object) with the inclusion synchronous function \( inc _{f(Q)} : f(Q) \rightarrow P\).

  4. 4.

    additionally proving that \( TPoset (T)\) also has all equalizers, which is easy since they are essentially defined as in \( Set \).

  5. 5.

    We call here a diagram functor a functor from the category freely generated by a graph G. As such a functor is fully determined by its value on graph vertices and edges it can simply be seen as a graph morphism from G into (the graph of) its codomain category.

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Acknowledgment

The author wishes to express his deep gratitude to Gordon Plotkin and Phil Scott for their early advice to look at the notion of presheaves, to Marek Zawadowski for his help in understanding Grothendieck topologies and sheaves, to referees for their numerous suggestions of improvement, and to Simon Archipoff, Michail Raskin and Bernard Serpette for many fruitful discussions on various aspects of this work.

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  1. LaBRI, Bordeaux INP, Université de Bordeaux, 351 cours de la Libération, 33405, Talence Cedex, France

    David Janin

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  1. Stellenbosch University, Stellenbosch, South Africa

    Bernd Fischer

  2. Reykjavík University, Reykjavik, Iceland

    Tarmo Uustalu

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Janin, D. (2018). Spatio-Temporal Domains: An Overview. In: Fischer, B., Uustalu, T. (eds) Theoretical Aspects of Computing – ICTAC 2018. ICTAC 2018. Lecture Notes in Computer Science(), vol 11187. Springer, Cham. https://doi.org/10.1007/978-3-030-02508-3_13

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  • DOI: https://doi.org/10.1007/978-3-030-02508-3_13

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