Abstract
We study equivalences of concurrent processes represented by objects of algebraic topology. We use methods of category theory and consider precubical sets (analogs of semisimplicial sets) and precubical spaces (analogs of cell complexes). In particular, we consider categories of these objects and construct subcategories of path-objects. We define open morphisms with respect to these subcategories and formulate criteria for a morphism to be open. We prove that the equivalence of precubical sets (spaces) based on open morphisms coincides with a behavioral equivalence of concurrent processes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Berger and B. Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces (Springer-Verlag, New York, 1988) [Géométrie Différentielle: Varietés, Courbes et Surfaces (Presses Universitaires de France, Paris, 1987)].
L. Fajstrup, “Dicovering spaces,” Homology Homotopy Appl. 5(2), 1–17 (2003).
L. Fajstrup, “Dipaths and dihomotopies in a cubical complex,” Adv. Appl. Math. 35(2), 188–206 (2005).
U. Fahrenberg, “Directed homology,” Electronic Notes in Theoret. Comput. Sci. 100, 111–125, (2004).
Fahrenberg U. “A category of higher-dimensional automata,” in Foundations of Software Science and Computation Structures, Lecture Notes in Comput. Sci. 3441 (Springer, Berlin, 2005), pp. 187–201.
R. J. van Glabbeek, “On the expressiveness of higher dimensional automata,” Theoret. Comput. Sci. 356(3), 265–290, (2006).
E. Goubault, The Geometry of Concurrency (PhD Thesis, École Normale Supérieure, Paris, 1995).
E. Goubault and T. P. Jensen, “Homology of higher-dimensional automata,” in CONCUR’ 92, Lecture Notes in Comput. Sci. 630 (Springer, Berlin, 1992), pp. 254–268.
M. Grandis, “Directed combinatorial homology and noncommutative tori (the breaking of symmetries in algebraic topology),” Math. Proc. Cambridge Phil. Soc. 138(2), 233–262 (2005).
T. Hune and M. Nielsen, “Timed bisimulation and openmaps,” in Mathematical Foundations of Computer Science (Brno, 1998), Lecture Notes in Comput. Sci. 1450 (Springer, Berlin, 1998), pp. 378–387.
A. Joyal, M. Nielsen, and G. Winskel, “Bisimulation from open maps,” Inform. and Comput. 127(2), 164–185 (1996).
A. A. Khusainov, “Homology groups of semicubical sets,” Siberian Math. J. 49(1), 180–190, (2008) [Sibirsk. Mat. Zh. 49 (1), 224–237 (2008)].
M. Nielsen and A. Cheng, “Observing behavior categorically,” Foundations of Software Technology and Theoretical Computer Science (Bangalore, 1995), Lecture Notes in Comput. Sci. 1026 (Springer, Berlin, 1995), pp. 263–278.
M. Nielsen and G. Winskel, “Petri nets and bisimulation,” Theoret. Comput. Sci. 153(1–2), 211–244 (1996).
E. Oshevskaya, I. Virbitskaite, and E. Best, “Unifying equivalences for higher dimensional automata,” Fundam. Inform. 119(3–4), 357–372 (2012).
V. R. Pratt, “Modeling Concurrency with Geometry,” in Proc. 18th ACM Symposium on Principles of Programming Languages (ACM Press, New York, 1991), pp. 311–322.
G. L. Cattani and V. Sassone, “Higher-dimensional transition systems,” in 11th Annual IEEE Symposium on Logic in Computer Science (New Brunswick, NJ, 1996), (IEEE Computer Society Press, Los Alamitos, CA, 1996), pp. 55–62.
I. B. Virbitskaite and N. S. Gribovskaya, “Open maps and observational equivalences for timed partial order models,” Fundam. Inform. 60(1–4), 383–399 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © E.S. Oshevskaya, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 1, pp. 150–188.
About this article
Cite this article
Oshevskaya, E.S. Comparing equivalences on precubical sets and spaces. Sib. Adv. Math. 24, 47–74 (2014). https://doi.org/10.3103/S1055134414010064
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1055134414010064