Document Open Access Logo

Aczel-Mendler Bisimulations in a Regular Category

Author Jérémy Dubut

PDF
Thumbnail PDF

File

LIPIcs.CALCO.2023.19.pdf
  • Filesize: 0.77 MB
  • 18 pages

Document Identifiers

Author Details

Jérémy Dubut
  • National Institute of Advanced Science and Technology, Tokyo, Japan

Funding

The author is supported by JST, CREST Grant Number JPMJCR22M1, Japan.

Cite As Get BibTex

Jérémy Dubut. Aczel-Mendler Bisimulations in a Regular Category. In 10th Conference on Algebra and Coalgebra in Computer Science (CALCO 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 270, pp. 19:1-19:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.CALCO.2023.19

Abstract

Aczel-Mendler bisimulations are a coalgebraic extension of a variety of computational relations between systems. It is usual to assume that the underlying category satisfies some form of axiom of choice, so that the theory enjoys desirable properties, such as closure under composition. In this paper, we accommodate the definition in a general regular category - which does not necessarily satisfy any form of axiom of choice. We show that this general definition 1) is closed under composition without using the axiom of choice, 2) coincides with other types of coalgebraic formulations under milder conditions, 3) coincides with the usual definition when the category has the regular axiom of choice. We then develop the particular case of toposes, where the formulation becomes nicer thanks to the power-object monad, and extend the formalism to simulations. Finally, we describe several examples in Stone spaces, toposes for name-passing, and modules over a ring.

Subject Classification

ACM Subject Classification
  • Theory of computation → Models of computation
  • Theory of computation → Semantics and reasoning
Keywords
  • Regular Categories
  • Toposes
  • Bisimulations
  • Coalgebra

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Samson Abramsky. Coalgebras, Chu spaces, and representations of physical systems. Journal of Philosophical Logic, 42:551-574, 2013. Google Scholar
  2. Peter Aczel and Nax Mendler. A final coalgebra theorem. In Proceedings of Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science. Springer, 1999. Google Scholar
  3. Emilio Jesús Gallego Arias and James B. Lipton. Logic Programming in Tabular Alegories. In Technical Communications of the 28th International Conference on Logic Programming (ICLP'12), Leibniz International Proceedings in Informatics, pages 334-347. Schloss Dagstuhl, 2012. Google Scholar
  4. Roland Backhouse and Paul Hoogendijk. Final dialgebras: from categories to allegories. Theoretical Informatics and Application, 33:401-426, 1999. Google Scholar
  5. Nick Bezhanishvili, Gaelle Fontaine, and Yde Yenema. Vietoris bisimulations. Journal of Logic and Computation, 20(5):1017-1040, 2010. Google Scholar
  6. Filippo Bonchi, Marcello Bonsangue, Michele Boreale, Jan Rutten, and Alexandra Silva. A coalgebraic perspective on linear weighted automata. Information and Computation, 211:77-105, 2012. Google Scholar
  7. Vincent Danos, Josée Desharnais, Francois Laviolette, and Prakash Panangaden. Bisimulation and cocongruence for probabilistic systems. Information and Computation, 204(4):503-523, 2006. URL: https://doi.org/10.1016/j.ic.2005.02.004.
  8. Josée Desharnais, Abbas Edalat, and Prakash Panangaden. Bisimulation for Labelled Markov Processes. Information and Computation, 179(2):163-193, 2003. URL: https://doi.org/10.1006/inco.2001.2962.
  9. Peter Freyd and Andre Scedrov. Categories, Allegories, volume 39 of Mathematical Library. North-Holland, 1990. Google Scholar
  10. Chris Heunen, Ohad Kammar, Sam Staton, and Hongseok Yang. A convenient category for higher-order probability theory. In Proceedings of the 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS'17), 2017. Google Scholar
  11. Jesse Hughes and Bart Jacobs. Simulations in coalgebra. Theoretical Computer Science, 327(1-2):71-108, 2004. Google Scholar
  12. J. Martin E. Hyland. The effective topos. In The L.E.J. Brouwer Centenary Symposium, volume 110 of Studies in Logic and the Foundations of Mathematics, pages 165-216, 1982. Google Scholar
  13. Bart Jacobs. Introduction to Coalgebra: Towards Mathematics of States and Observation, volume 29 of Cambridge Tracts in Theoretical Computer Science. Cambridge University Press, 2017. Google Scholar
  14. Peter T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium, volume 1. Oxford University Press, 2002. Google Scholar
  15. Peter T. Johnstone. Sketches of an Elephant: A Topos Theory Compendium, volume 2. Oxford University Press, 2002. Google Scholar
  16. André Joyal, Mogens Nielsen, and Glynn Winskel. Bisimulation from Open maps. Information and Computation, 127:164-185, 1996. Google Scholar
  17. F. William Lawvere. Equality in hyperdoctrines and comprehension schema as an adjoint functor. In Proceedings of the AMS Symposium on Pure Mathematics XVII, pages 1-14, 1970. Google Scholar
  18. F. William Lawvere. Qualitative Distinctions between some Toposes of Generalized Graphs. Contemporary Mathematics, 92, 1989. Google Scholar
  19. Saunders MacLane and Ieke Moerdijk. Sheaves in Geometry and Logic: A First Introduction to Topos Theory. Springer New-York, 1992. Google Scholar
  20. Davide Sangiorgi. A theory of bisimulations for the π-calculus. Acta Informatica, 33:69-97, 1996. Google Scholar
  21. Alex Simpson. Measure, randomness and sublocales. Annals of Pure and Applied Logic, 163(11):1642-1659, 2012. Google Scholar
  22. Sam Staton. Name-passing process calculi: operational models and structural operational semantics. PhD thesis, University of Cambridge, 2008. Google Scholar
  23. Sam Staton. Relating Coalgebraic Notions of Bisimulation. Logical Methods in Computer Science, 7:1-21, 2011. Google Scholar
  24. Paulo Tabuada. Open Maps, Alternating Simulations and Control Synthesis. In Proceedings of the 15th International Conference in Concurrency Theory (CONCUR'04), volume 3170 of Lecture Notes in Computer Science, pages 466-480. Springer, 2004. Google Scholar
  25. Michael Winter. Goguen Categories: A Categorical Approach to L-fuzzy Relations. Number 25 in Trends in Logic. Springer, 2007. Google Scholar
  26. Thorsten Wißman, Jérémy Dubut, Shin-ya Katsumata, and Ichiro Hasuo. Path Category for Free: Open Morphisms from Coalgebras with Non-deterministic Branching. In Proceedings of the 22nd International Conference on Fondations of Software Science and Computation Structures (FoSSaCS'19), volume 11425 of Lecture Notes in Computer Science. Springer, 2019. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact