Probabilistic Powerdomains and Quasi-Continuous Domains
Keywords:
locally finitary compact space, probabilistic powerdomain, quasi-continuous domain, Scott topology, weak topologyAbstract
The probabilistic powerdomain $\mathbf{V} X$ on a space $X$ is the space of all continuous valuations on $X$. We show that, for every quasi-continuous domain $X, \mathbf{V} X$ is again a quasi-continuous domain, and that the Scott and weak topologies then agree on $\mathbf{V} X$. This also applies to the subspaces of probability and subprobability valuations on $X_1$ in the first case under an assumption of pointedness. We also show that the Scott and weak topologies on $\mathbf{V} X$ may differ when $X$ is not quasi-continuous, and we give a simple, compact Hausdorff counterexample.
References
W. Adamski, $\tau$-smooth Borel measures on topological spaces, Math. Nachr. 78 (1977), 97-107.
Mauricio Alvarez-Manilla, Measure theoretic results for continuous valuations on partially ordered spaces. Ph.D. thesis. University of London, Imperial College of Science, Technology and Medicine, 2000
Mauricio Alvarez-Manilla, Achim Jung, and Klaus Keimel, The probabilistic powerdomain for stably compact spaces, Theoret. Comput. Sci. 328 (2004), no. 3, 221-244.
Gustave Choquet, Theory of capacities, Ann. Inst. Fourier (Grenoble) 5 (1954), 131-295.
Matthew de Brecht, Jean Goubault-Larrecq, Xiaodong Jia, and Zhenchao Lyu, Domain-complete and LCS-complete spaces in The Proceedings of ISDT 2019, the 8th International Symposium on Domain Theory and Its Applications. Ed. Achim Jung, et al. Electronic Notes in Theoretical Computer Science, 345. Amsterdam: Elsevier B. V., 2019. 3-35.
Abbas Edalat, Domain theory and integration, Theoret. Comput. Sci. 151 (1995), no. 1, 163-193.
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, and D. S. Scott, Continuous Lattices and Domains. Encyclopedia of Mathematics and its Applications, 93. Cambridge: Cambridge University Press, 2003.
Jean Goubault-Larrecq, QRB-domains and the probabilistic powerdomain. Log. Methods Comput. Sci. 8 (2012), no. 1:14, 33 pp.
Jean Goubault-Larrecq, Non-Hausdorff Topology and Domain Theory: Selected Topics in Point-Set Topology. New Mathematical Monographs, 22. Cambridge: Cambridge University Press, 2013.
Jean Goubault-Larrecq and Achim Jung, $Q R B, Q F S$, and the probabilistic powerdomain in Proceedings of the 30th Conference on the Mathematical Foundations of Programming Semantics (MFPS XXX). Ed. Bart Jacobs, et al. Electronic Notes in Theoretical Computer Science, 308. Amsterdam: Elsevier Sci. B. V., 2014. 167182.
Reinhold Heckmann, Spaces of valuations in Papers on General Topology and Applications. Ed. Susan Andima, et al. Annals of the New York Academy of Sciences, 806. New York: New York Acad. Sci., 1996. 174-200.
Claire Jones, Probabilistic non-determinism. Ph.D. thesis. University of Edinburgh, 1990.
C. Jones and G. D. Plotkin, A probabilistic powerdomain of evaluations in Proceedings: Fourth Annual Symposium on Logic in Computer Science. Washington, D.C.: Computer Society Press, 1989, 186-195.
Achim Jung and Regina Tix, The troublesome probabilistic powerdomain in Third Workshop on Computation and Approximation (Comprox III). Ed. Abbas Edalat, et al. Electronic Notes in Theoretical Computer Science, 13. Amsterdam: Elsevier Sci. B. V., 1998. 70-91.
Klaus Keimel and Jimmie D. Lawson D-completions and the d-topology, Ann. Pure Appl. Logic 159 (2009), no. 3, 292-306.
Olaf Kirch, Bereiche und Bewertungen. Master's thesis. Technische Hochschule Darmstadt (Germany), June 1993.
Jimmie Lawson and Xiaoyong Xi, The equivalence of QRB, QFS, and compactness for quasicontinuous domains, Order 32 (2015), no. 2, 227-238.
Gaolin Li and Luosh an Xu, QFS-domains and their Lawson compactness, Order 30 (2013), по. 1, 233-248.
Zhenchao Lyu and Hui Kou, The probabilistic powerdomain from a topological viewpoint, Topology Appl. 237 (2018), 26-36.
Regina Tix, Stetige Bewertungen auf topologischen Räumen. Master's thesis. Technische Hochschule Darmstadt (Germany), June 1995.
J. v. Neumann, Zur Theorie der Gesellschaftsspiele, Math. Ann. 100 (1928), no. 1, 295-320.
