Abstract
This paper studies presheaf models for concurrent computation. An aim is to harness the general machinery around presheaves for the purposes of process calculi. Traditional models like synchronisation trees and event structures have been shown to embed fully and faithfully in particular presheaf models in such a way that bisimulation, expressed through the presence of a span of open maps, is conserved. As is shown in the work of Joyal and Moerdijk, presheaves are rich in constructions which preserve open maps, and so bisimulation, by arguments of a very general nature. This paper contributes similar results but biased towards questions of bisimulation in process calculi. It is concerned with modelling process constructions on presheaves, showing these preserve open maps, and with transferring such results to traditional models for processes. One new result here is that a wide range of left Kan extensions, between categories of presheaves, preserve open maps. As a corollary, this also implies that any colimit-preserving functor between presheaf categories preserves open maps. A particular left Kan extension is shown to coincide with a refinement operation on event structures. A broad class of presheaf models is proposed for a general process calculus. General arguments are given for why the operations of a presheaf model preserve open maps and why for specific presheaf models the operations coincide with those of traditional models.
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References
Adámek, J., and Rosicky, J., Locally Presentable and Accessible Categories. London Mathematical Society Lecture Note Series, 189. Cambridge University Press, 1994.
Bednarczyk, M., Hereditary history preserving bisimulation or what is the power of the future perfect in program logics. Technical report, Polish Academy of Sciences, Gdansk, 1991.
Borceux, F., Handbook of Categorical Algebra, vol. 1, 2. Encyclopedia of Mathematics and its Applications, vol. 50, 51. Cambridge University Press, 1994.
Cheng, A., and Nielsen, M., Open Maps, Behavioural Equivalences, and Congruences. Report Series RS-96-2, BRICS, University of Aarhus. Denmark, January 1996.
Fiore, M.P., Axiomatic Domain Theory in Categories of Partial Maps. Ph.D Thesis. Distinguished Dissertations in Computer Science, Cambridge University Press, 1996.
Fiore, M.P., Moggi, E., and Sangiorgi, D., A Fully-Abstract Model for the π-calculus In Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science, pages. 43–54. IEEE Computer Society Press, 1996.
Joyal, A., and Moerdijk, I., A completeness theorem for open maps. In Annals of Pure and Applied Logic 70, 51–86, 1994.
Joyal, A., Nielsen, M., and Winskel, G., Bisimulation from open maps. In LICS'93 special issue of Information and Computation, vol. 127, pp. 164–185, 1996.
van Glabbeek, R.J., and Goltz, U., Equivalence notions for concurrent systems and refinement of actions. In Proceedings of Mathematical foundations of computer science 1989, pp. 237–248, Lecture Notes in Computer Science vol. 379, 1989.
Mac Lane, S., Categories for the Working Mathematician. Springer-Verlag, 1971.
Mac Lane, S., Moerdijk, I., Sheaves in Geometry and Logic. Springer-Verlag, 1992.
Nielsen, M., and Winskel, G., Petri nets and bisimulations. Theoretical Computer Science 153, pp. 211–244, 1996.
Plotkin, G., Algebraic Completeness and Compactness in an Enriched Setting. Transparencies from the Invited Lecture given at the Workshop on Logic, Domains and Programming Languages. Darmstadt, 1995.
Rabinovitch, A., and Traktenbrot, B., Behaviour structures and nets. Fundamenta Informatica, 11(4), pp. 357–404, 1988.
Stark, I., A Fully Abstract Domain Model for the π-Calculus. In Proceedings of the Eleventh Annual IEEE Symposium on Logic in Computer Science, pages. 36–42. IEEE Computer Society Press, 1996.
Winskel, G., A Presheaf Semantics of Value-Passing Processes. In Proceedings of the Seventh International Conference on Concurrency Theory: CONCUR'96, pp. 98–114, Lecture Notes in Computer Science vol. 1119, 1996.
Winskel, G., and Nielsen, M., Models for Concurrency. In the Handbook of Logic in Computer Science, vol. IV pp. 1–148, ed. Abramsky, Gabbay and Maibaum, Oxford University Press, 1995.
Wyler, O., Lecture Notes on Topoi and Quasitopoi. World Scientific Publishing Co., 1991.
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Cattani, G.L., Winskel, G. (1997). Presheaf models for concurrency. In: van Dalen, D., Bezem, M. (eds) Computer Science Logic. CSL 1996. Lecture Notes in Computer Science, vol 1258. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63172-0_32
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DOI: https://doi.org/10.1007/3-540-63172-0_32
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