Skip to main content
SpringerLink
Log in

Spatio-Temporal Domains: An Overview

  • Conference paper
  • First Online:
Theoretical Aspects of Computing – ICTAC 2018 (ICTAC 2018)

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

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 69.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 89.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

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.

References

  1. Abrial, J.R.: Modeling in Event-B: System and Software Design. Cambridge University Press, Cambridge (2010). https://doi.org/10.1017/cbo9781139195881

  2. Archipoff, S., Janin, D.: Structured reactive programming with polymorphic temporal tiles. In: Proceedings of 4th ACM SIGPLAN International Workshop on Functional Art, Music, Modeling and Design FARM 2016, pp. 29–40. ACM Press, New York (2016). https://doi.org/10.1145/2975980.2975984

  3. Archipoff, S., Janin, D.: Unified media programming: an algebraic approach. In: Proceedings of 5th ACM SIGPLAN International Workshop on Functional Art, Music, Modeling and Design, FARM 2017, pp. 36–47. ACM Press, New York (2017). https://doi.org/10.1145/3122938.3122943

  4. Barr, M., Wells, C.: Category Theory for Computing Science, 3rd edn. Centre de Recherche Mathématique (CRM), Montréal (1999)

    Google Scholar 

  5. Benveniste, A., Caspi, P., Edwards, S.A., Halbwachs, N., Le Guernic, P., De Simone, R.: The synchronous languages twelve years later. Proc. IEEE 91(1), 64–83 (2003)

    Article  Google Scholar 

  6. Berry, G.: Stable models of typed \(\lambda \)-calculi. In: Ausiello, G., Böhm, C. (eds.) ICALP 1978. LNCS, vol. 62, pp. 72–89. Springer, Heidelberg (1978). https://doi.org/10.1007/3-540-08860-1_7

    Chapter  Google Scholar 

  7. Berry, G., Gonthier, G.: The Esterel synchronous programming language: design, semantics, implementation. Sci. Comput. Program. 19(2), 87–152 (1992). https://doi.org/10.1016/0167-6423(92)90005-v

    Article  MATH  Google Scholar 

  8. Cattani, G.L., Stark, I., Winskel, G.: Presheaf models for the \(\pi \)-calculus. In: Moggi, E., Rosolini, G. (eds.) CTCS 1997. LNCS, vol. 1290, pp. 106–126. Springer, Heidelberg (1997). https://doi.org/10.1007/BFb0026984

    Chapter  Google Scholar 

  9. Cattani, G.L., Winskel, G.: Presheaf models for CCS-like languages. Theor. Comput. Sci. 300(1–3), 47–89 (2003). https://doi.org/10.1016/s0304-3975(01)00209-2

    Article  MathSciNet  MATH  Google Scholar 

  10. Chapiro, D.M.: Globally-asynchronous locally-synchronous systems. Ph.D. thesis, Department of Computer Science, Stanford University (1985)

    Google Scholar 

  11. Colaço, J.L., Girault, A., Hamon, G., Pouzet, M.: Towards a higher-order synchronous data-flow language. In: Proceedings of 4th ACM International Conference on Embedded Software, EMSOFT 2004, Pisa, Septemebr 2004, pp. 230–239. ACM Press, New York (2004). https://doi.org/10.1145/1017753.1017792

  12. Colaço, J.-L., Pouzet, M.: Clocks as first class abstract types. In: Alur, R., Lee, I. (eds.) EMSOFT 2003. LNCS, vol. 2855, pp. 134–155. Springer, Heidelberg (2003). https://doi.org/10.1007/978-3-540-45212-6_10

    Chapter  Google Scholar 

  13. Cousot, P., Cousot, R., Mauborgne, L.: Logical abstract domains and interpretations. In: Nanz, S. (ed.) The Future of Software Engineering (Meyer Festschrift), pp. 48–71. Springer, Heidelberg (2010). https://doi.org/10.1007/BFb0026984

    Chapter  Google Scholar 

  14. Elliott, C., Hudak, P.: Functional reactive animation. In: Proceedings of 2nd ACM International Conference on Functional Programming, ICFP 1997, Amsterdam, June 1997, pp. 263–273. ACM Press, New York (1997). https://doi.org/10.1145/258948.258973

  15. Elliott, C.M.: Push-pull functional reactive programming. In: Proceedings of 2nd ACM SIGPLAN Symposium on Haskell, Haskell 2009, Edinburgh, September 2009, pp. 25–36. ACM Press, New York (2009) https://doi.org/10.1145/1596638.1596643

  16. Girard, J.Y.: Linear logic. Theor. Comput. Sci. 50, 1–102 (1987). https://doi.org/10.1016/0304-3975(87)90045-4

    Article  MathSciNet  MATH  Google Scholar 

  17. Hudak, P.: A sound and complete axiomatization of polymorphic temporal media. Technical report, RR-1259, Department of Computer Science, Yale University (2008)

    Google Scholar 

  18. Hudak, P.: The Haskell School of Music: From Signals to Symphonies. Department of Computer Science, Yale University (2013)

    Google Scholar 

  19. Hughes, J.: Programming with arrows. In: Vene, V., Uustalu, T. (eds.) AFP 2004. LNCS, vol. 3622, pp. 73–129. Springer, Heidelberg (2005). https://doi.org/10.1007/11546382_2

    Chapter  MATH  Google Scholar 

  20. Jacobs, B.: Categorical Logic and Type Theory. Studies in Logic and the Foundations of Mathematics, vol. 141. North Holland, Amsterdam (1999). https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141/

  21. Jeffrey, A.: Functional reactive types. In: Proceedings of EACSL Annual Conference and 29th Ann ACM/IEEE Symposium on Logic in Computer Science, CSL-LICS 2014, Vienna, July 2014, Article 54. ACM Press, New York (2014). https://doi.org/10.1145/2603088.2603106

  22. Jeltsch, W.: An abstract categorical semantics for functional reactive programming with processes. In: Proceedings of 2014 ACM SIGPLAN Workshop on Programming Languages Meets Program Verification, PLPV 2014, San Diego, CA, January 2014, pp. 47–58. ACM Press, New York (2014). https://doi.org/10.1145/2541568.2541573

  23. Kaynar, D.K., Lynch, N., Segala, R., Vaandrager, F.: The Theory of Timed I/O Automata. Synthesis Lectures on Computer Science. Morgan & Claypool Publishers (2006). https://doi.org/10.2200/s00006ed1v01y200508csl001

    Article  Google Scholar 

  24. Krishnaswami, N.R.: Higher-order functional reactive programming without spacetime leaks. In: Proceedings of 18th ACM SIGPLAN International Conference on Functional Programming, ICFP 2013, Boston, MA, September 2013, pp. 221–232. ACM Press, New York (2013). https://doi.org/10.1145/2500365.2500588

  25. Krishnaswami, N.R., Benton, N.: Ultrametric semantics of reactive programs. In: Proceedings of 26th Annual IEEE Symposium on Logic in Computer Science, LICS 2011, Toronto, ON, June 2011, pp. 257–266. IEEE CS Press, Washington, DC (2011). https://doi.org/10.1109/lics.2011.38

  26. Liu, X., Lee, E.A.: CPO semantics of timed interactive actor networks. Theor. Comput. Sci. 409(1), 110–125 (2008). https://doi.org/10.1016/j.tcs.2008.08.044

    Article  MathSciNet  MATH  Google Scholar 

  27. Mac Lane, S., Moerdijk, I.: Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Universitext. U. Springer, New York (1992). https://doi.org/10.1007/978-1-4612-0927-0

    Book  MATH  Google Scholar 

  28. Matsikoudis, E., Lee, E.A.: The fixed-point theory of strictly causal functions. Theor. Comput. Sci. 574, 39–77 (2015)

    Article  MathSciNet  Google Scholar 

  29. Streicher, T.: Fibred categories à la Jean Bénabou. Revised notes of a course on fibred categories given at a spring school in Munich 1999 (2014)

    Google Scholar 

  30. Teehan, P., Greenstreet, M.R., Lemieux, G.G.: A survey and taxonomy of GALS design styles. IEEE Des. Test. Comput. 24(5), 418–428 (2007). https://doi.org/10.1109/mdt.2007.151

    Article  Google Scholar 

  31. Winskel, G.: Event structures. In: Brauer, W., Reisig, W., Rozenberg, G. (eds.) ACPN 1986. LNCS, vol. 255, pp. 325–392. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-17906-2_31

    Chapter  Google Scholar 

  32. Winskel, G.: Events, causality and symmetry. In: Proceedings of BCS International Academic Conference on Visions of Computer Science, London, September 2008, pp. 111–127. Electronic Workshops in Computing. British Computer Society (2008). https://ewic.bcs.org/content/ConWebDoc/22872

Download references

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.

Author information

Authors and Affiliations

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

    David Janin

Authors
  1. David Janin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Janin .

Editor information

Editors and Affiliations

  1. Stellenbosch University, Stellenbosch, South Africa

    Bernd Fischer

  2. Reykjavík University, Reykjavik, Iceland

    Tarmo Uustalu

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

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

Download citation

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

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-02507-6

  • Online ISBN: 978-3-030-02508-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation