Abstract
Transition probabilities are proposed as the stochastic counterparts to set-based relations. We propose the construction of the converse of a stochastic relation. It is shown that two of the most useful properties carry over: the converse is idempotent as well as anticommutative. The nondeterminism associated with a stochastic relation is defined and briefly investigated. We define a bisimulation relation, and indicate conditions under which this relation is transitive; moreover it is shown that bisimulation and converse are compatible.
Chapter PDF
Similar content being viewed by others
References
S. Abramsky, R. Blute, and P. Panangaden. Nuclear and trace ideal in tensored *-categories. Journal of Pure and Applied Algebra, 143(1–3):3–47, 1999.
P. Aczel and N. Mendler. A final coalgebra theorem. In H. H. Pitt, A. Poigne, and D. E. Rydeheard, editors, Category Theory and Computer Science, volume 389 of Lecture Notes in Computer Science, pages 357–365, 1989.
P. Billingsley. Probability and Measure. JohnWiley and Sons, New York, 3 edition, 1995.
C. Brink, W. Kahl, and G. Schmidt, editors. Relational Methods in Computer Science. Advances in Computing Science. Springer-Verlag, Wien, New York, 1997.
D. Cantone, E. G. Omodeo, and A. Policriti. Set Theory for Computing. Springer-Verlag, 2001. In print.
J. Desharnais, A. Edalat, and P. Panangaden. Bisimulation of labelled Markovprocesses. Technical report, School of Computer Science, McGill University, Montreal, 1998.
E.-E. Doberkat. Stochastic Automata — Nondeterminism, Stability, and Prediction, volume 113 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1981.
E.-E. Doberkat. The demonic product of probabilistic relations. InMogens Nielsen and Uffe Engberg, editors, Proc. Foundations of Software Science and Computation Structures, volume 2303 of Lecture Notes in Computer Science, pages 113–127, Berlin, 2002. Springer-Verlag.
E.-E. Doberkat. Semi-pullbacks and bisimulations in categories of stochastic relations. Technical Report 130, Chair for Software Technology, University of Dortmund, November 2002.
A. Edalat. Semi-pullbacks and bisimulation in categories of Markov processes. Math. Struct. in Comp. Science, 9(5):523–543, 1999.
M. Giry. A categorical approach to probability theory. In Categorical Aspects of Topology and Analysis, volume 915 of Lecture Notes in Mathematics, pages 68–85, Berlin, 1981. Springer-Verlag.
S. Mac Lane. Categories for the Working Mathematician. Number 5 in Graduate Texts in Mathematics. Springer-Verlag, Berlin, 2 edition, 1997.
E. Michael. Topologies on spaces of subsets. Trans. Am. Math. Soc., 71(2):152–182, 1951.
P. Panangaden. Probabilistic relations. In C. Baier, M. Huth, M. Kwiatkowska, and M. Ryan, editors, Proc. PROBMIV, pages 59–74, 1998. Also available from the School of Computer Science, McGill University, Montreal.
P. Panangaden. Does combining nondeterminism and probability make sense? Bulletin of the EATCS, (75):182–189, Oct. 2001.
K. R. Parthasarathy. Probability Measures on Metric Spaces. Academic Press, New York, 1967.
J. J. M. M. Rutten. Universal coalgebra: a theory of systems. Theoretical Computer Science, 249(1):3–80, 2000. Special issue on modern algebra and its applications.
A. Tarski and S. Givant. A Formalization of Set-Theory Without Variables. Number 42 in Colloquium Publications. American Mathematical Society, Providence, R. I., 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Doberkat, EE. (2003). The Converse of a Stochastic Relation. In: Gordon, A.D. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2003. Lecture Notes in Computer Science, vol 2620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36576-1_15
Download citation
DOI: https://doi.org/10.1007/3-540-36576-1_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-00897-2
Online ISBN: 978-3-540-36576-1
eBook Packages: Springer Book Archive