Abstract
We show how parallel composition of higher-dimensional automata (HDA) can be expressed categorically in the spirit of Winskel & Nielsen. Employing the notion of computation path introduced by van Glabbeek, we define a new notion of bisimulation of HDA using open maps. We derive a connection between computation paths and carrier sequences of dipaths and show that bisimilarity of HDA can be decided by the use of geometric techniques.
Chapter PDF
Similar content being viewed by others
References
Bredon, G.E.: Topology and Geometry. Springer, Heidelberg (1993)
Brown, R., Higgins, P.J.: On the algebra of cubes. Journal of Pure and Applied Algebra 21, 233–260 (1981)
Brown, R., Higgins, P.J.: Tensor products and homotopies for ω-groupoids and crossed complexes. Journal of Pure and Applied Algebra 47, 1–33 (1987)
Cattani, G.L., Sassone, V.: Higher dimensional transition systems. In: Proc. LICS 1996, pp. 55–62. IEEE Press, Los Alamitos (1996)
Crans, S.: On Combinatorial Models for Higher Dimensional Homotopies. PhD thesis, Utrecht University (1995)
Fahrenberg, U.: A category of higher-dimensional automata. Technical Report R-2005-01, Department of Mathematical Sciences, Aalborg University (2005), http://www.math.aau.dk/research/reports/R-2005-01.ps
Fajstrup, L.: Dicovering spaces. Homology, Homotopy and Applications 5(2), 1–17 (2003)
Fajstrup, L.: Dipaths and dihomotopies in a cubical complex. Report R-2003-22, Department of Mathematical Sciences, Aalborg University (2003), http://www.math.aau.dk/research/reports/R-2003-22.ps , Submitted to Advances in Applied Mathematics
Fajstrup, L., Goubault, E., Raussen, M.: Algebraic topology and concurrency. Report R-99-2008, Department of Mathematical Sciences, Aalborg University (1999), http://www.math.aau.dk/research/reports/R-99-2008.ps , Conditionally accepted for publication in Theoretical Computer Science
van Glabbeek, R.J.: Bisimulations for higher dimensional automata. Email message (1991), http://theory.stanford.edu/~rvg/hda
van Glabbeek, R.J.: On the expressiveness of higher dimensional automata. Preprint (2004), http://www.cse.unsw.edu.au/~rvg/hda.pdf
Goubault, E.: The Geometry of Concurrency. PhD thesis, Ecole Normale Supérieure, Paris (1995), http://www.di.ens.fr/~goubault/papers/these.ps.gz
Goubault, E.: Labelled cubical sets and asynchronous transition systems: an adjunction. In: Preliminary Proceedings CMCIM 2002 (2002), http://www.di.ens.fr/~goubault/papers/cmcim02.ps.gz
Grandis, M.: Directed homotopy theory I. Cahiers de Topologie et Géométrie Différentielle Catégoriques 44, 281–316 (2003), Preprint available as http://arxiv.org/abs/math.AT/0111048
Grandis, M.: Directed homotopy theory II. Theory and Applications of Categories 14, 369–391 (2002)
Grandis, M.: Directed combinatorial homology and noncommutative tori. In: Math. Proc. Cambridge Philos. Soc. (2004) (to appear), Preprint available as http://www.dima.unige.it/~grandis/Bsy.pdf
Grandis, M., Mauri, L.: Cubical sets and their site. Theory and Applications of Categories 11(8), 185–211 (2003)
Jardine, J.F.: Cubical homotopy theory: a beginning (preprint) (2002)
Joyal, A., Nielsen, M., Winskel, G.: Bisimulation from open maps. Information and Computation 127(2), 164–185 (1996)
Massey, W.S.: A Basic Course in Algebraic Topology. Graduate Texts in Mathematics, vol. 127. Springer, Heidelberg (1991)
Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)
Serre, J.-P.: Homologie singulière des espaces fibrés. PhD thesis, Ecole Normale Supérieure (1951)
Winskel, G., Nielsen, M.: Models for concurrency. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 4, pp. 1–148. Clarendon Press, Oxford (1995)
Worytkiewicz, K.: Synchronization from a categorical perspective. (preprint) (2004), http://arxiv.org/abs/cs.PL/0411001
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fahrenberg, U. (2005). A Category of Higher-Dimensional Automata. In: Sassone, V. (eds) Foundations of Software Science and Computational Structures. FoSSaCS 2005. Lecture Notes in Computer Science, vol 3441. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31982-5_12
Download citation
DOI: https://doi.org/10.1007/978-3-540-31982-5_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25388-4
Online ISBN: 978-3-540-31982-5
eBook Packages: Computer ScienceComputer Science (R0)