Abstract
For the semantics of probabilistic features in programming mainly two approaches are used for building models. One is the Giry monad of Borel probability measures over metric spaces, and the other is Jones’ probabilistic powerdomain monad [6] over dcpos (directed complete partial orders). This paper places itself in the second domain theoretical tradition. The probabilistic powerdomain monad is well understood over continuous domains. In this case the algebras of the monad can be described by an equational theory [6, 9,5]. It is the aim of this work to obtain similar results for the (extended) probabilistic powerdomain monad over stably compact spaces. We mainly want to determine the algebras of this powerdomain monad and the algebra homomorphisms.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abramsky, S., Jung, A.: Domain theory. In: Abramsky, S., Gabbay, D.M., Maibaum, T.S.E. (eds.) Handbook of Logic in Computer Science, vol. 3, pp. 1–168. Clarendon Press, Oxford (1994)
Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. In: Ergebnisse der Mathematik und iherer Grenzgebiete, vol. 57. Springer, Heidelberg (1971)
Alvarez Manilla, M., Jung, A., Keimel, K.: The probabilistic powerspace for stably compact spaces. Theoretical Computer Science 328, 221–244 (2004)
Fedorchuk, V.V.: Probability measures in topology. Russian Mathematical Surveys 46(1), 41–80 (1991)
Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M.W., Scott, D.S.: Continuous Lattices and Domains. In: Encyclopedia of Mathematics and its Applications, vol. 93, Cambridge University Press, Cambridge (2003)
Jones, C.: Probabilistic Non-determinism. PhD thesis, Department of Computer Science, University of Edinburgh, Edinburgh, p. 201 (1990)
Keimel, K.: Topological cones: Foundations for a domains theoretical sematics combining probability and nondeterminism. Electronic Notes in Theoretical Computer Science (to appear)
Keimel, K., Roth, W.: Ordered Cones and Approximation, Lecture Notes in Mathematics. vol. 1517 vi+134 Springer Verlag, (1992)
Kirch, O.: Bereiche und Bewertungen. Master’s thesis, Technische Hochschule Darmstadt, 77pp. (June 1993), http://www.mathematik.tu-darmstadt.de/ags/ag14/papers/kirch/
qNachbin, L.: Topology and Order. Von Nostrand, Princeton, N.J., 1965. Reprinted by Robert E. Kreiger Publishing Co., Huntington (1967)
Plotkin, G.D.: A domain-theoretic Banach-Alaoglu theorem. Mathematical Structures in Computer Science (to appear)
Roth, W.: Hahn-Banach type theorems for locally convex cones. Journal of the Australian Mathematical Society 68(1), 104–125 (2000)
Schröder, M., Simpson, A.: Probabilistic observations and valuations (Extended Abstract). In: Proceedings of MFPS 21 (2005); to appear in Electronic Notes in Theoretical Computer Science (2006)
Tix, R.: Stetige Bewertungen auf topologischen Räumen. Master’s thesis, Technische Hochschule Darmstadt, 51pp. (June 1995), http://www.mathematik.tu-darmstadt.de/ags/ag14/papers/tix/
Tix, R., Keimel, K., Plotkin, G.D.: Semantic Domains Combining Probabilty and Nondeterminism. Electronic Notes in Theoretical Computer Science 129, 1–104 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cohen, B., Escardo, M., Keimel, K. (2006). The Extended Probabilistic Powerdomain Monad over Stably Compact Spaces. In: Cai, JY., Cooper, S.B., Li, A. (eds) Theory and Applications of Models of Computation. TAMC 2006. Lecture Notes in Computer Science, vol 3959. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11750321_54
Download citation
DOI: https://doi.org/10.1007/11750321_54
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-34021-8
Online ISBN: 978-3-540-34022-5
eBook Packages: Computer ScienceComputer Science (R0)