Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-02-03T02:50:44.010Z Has data issue: false hasContentIssue false
  • English
  • Français

De Groot duality and models of choice: angels, demons and nature†

Published online by Cambridge University Press:  25 March 2010

JEAN GOUBAULT-LARRECQ
JEAN GOUBAULT-LARRECQ*
Affiliation:
LSV, ENS Cachan, CNRS, INRIA Saclay, 61 avenue du président-Wilson, 94230 Cachan, France Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We introduce convex–concave duality for various models of non-deterministic choice, probabilistic choice and the two of them combined. This complements the well-known duality of stably compact spaces in a pleasing way: convex–concave duality swaps angelic and demonic choice, and leaves probabilistic choice invariant.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 20 , Special Issue 2: Domains , April 2010 , pp. 169 - 237
Copyright
Copyright © Cambridge University Press 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, D. M. and Maibaum, T. S. E. (eds.) Handbook of Logic in Computer Science 3, Oxford University Press 1–168.Google Scholar
Alvarez-Manilla, M. (2000) Measure Theoretic Results for Continuous Valuations on Partially Ordered Spaces, Ph.D. thesis, Imperial College, London.CrossRefGoogle Scholar
Alvarez-Manilla, M., Jung, A. and Keimel, K. (2004) The probabilistic powerdomain for stably compact spaces. Theoretical Computer Science 328 (3)221–244.Google Scholar
Choquet, G. (1953) Theory of capacities. Annales de l'Institut Fourier 5 131–295.CrossRefGoogle Scholar
Cohen, B. S. (2006) Mathematical Foundations for Denotational Semantics for Combining Probability and Nondeterminism over Stably Compact Spaces, Diplomarbeit, TU Darmstadt and Université Joseph Fourier-IMAG Grenoble.Google Scholar
Denneberg, D. (1994) Non-Additive Measure and Integral, Kluwer.Google Scholar
Edalat, A. (1995) Domain theory and integration. Theoretical Computer Science 151 163–193.Google Scholar
Edwards, D. A. (1978) On the existence of probability measures with given marginals. Annales de l'Institut Fourier 28 53–78.CrossRefGoogle Scholar
Escardó, M. (2003) Topology of data types and computability concepts. (Available at http://www.cs.bham.ac.uk/~mhe/talks.html). (Notes published as: Synthetic Topology of Data Types and Classical Spaces. Electronic Notes in Theoretical Computer Science 87 (2004) 150 pages – available at http://www.cs.bham.ac.uk/~mhe/papers/barbados-manuscript.pdf.)Google Scholar
Escardó, M. and Heckmann, R. (2001–2002) Topologies on spaces of continuous functions. Topology Proceedings 26 (2)545–564.Google Scholar
Gierz, G., Hofmann, K. H., Keimel, K., Lawson, J. D., Mislove, M. and Scott, D. S. (2003) Continuous lattices and domains. In: Encyclopedia of Mathematics and its Applications 93, Cambridge University Press.Google Scholar
Gilboa, I. and Schmeidler, D. (1994) Additive representation of non-additive measures and the Choquet integral. Annals of Operations Research 51 43–65.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2007a) Continuous capacities on continuous state spaces. In: Arge, L., Cachin, Ch., Jurdziński, T. and Tarlecki, A. (eds.) Proceedings of the 34th International Colloquium on Automata, Languages and Programming (ICALP'07), Wrocław, Poland. Springer-Verlag Lecture Notes in Computer Science 4596 764–776.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2007b) Continuous previsions. In: Duparc, J. and Henzinger, T. A. (eds.) Proceedings of the 16th Annual EACSL Conference on Computer Science Logic (CSL'07), Lausanne, Switzerland. Springer-Verlag Lecture Notes in Computer Science 4646 542–557.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2007c) Une introduction aux capacités, aux jeux et aux prévisions – Version 6, 514 pages. (Available at http://www.lsv.ens-cachan.fr/~goubault/ProNobis/pp_1_6.pdf).Google Scholar
Goubault-Larrecq, J. (2008a) Prevision domains and convex powercones. In: Amadio, R. (ed.) Proceedings of the 11th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS'08), Budapest, Hungary. Springer-Verlag Lecture Notes in Computer Science 4962 318–333.CrossRefGoogle Scholar
Goubault-Larrecq, J. (2008b) Simulation hemi-metrics between infinite-state stochastic games. In: Amadio, R. (ed.) Proceedings of the 11th International Conference on Foundations of Software Science and Computation Structures (FoSSaCS'08), Budapest, Hungary. Springer-Verlag Lecture Notes in Computer Science 4962 50–65.CrossRefGoogle Scholar
Goubault-Larrecq, J. and Keimel, K. (2010) Choquet–Kendall–Matheron theorems for non-Hausdorff spaces (submitted to Mathematical Structures in Computer Science).CrossRefGoogle Scholar
Heckmann, R. (1990) Power Domain Constructions (Potenzbereich-Konstruktionen), Ph.D. thesis, Universität des Saarlandes.Google Scholar
Heckmann, R. (1997) Abstract valuations: A novel representation of Plotkin power domain and Vietoris hyperspace. In: Proc. 13th Intl. Symp. on Mathematical Foundations of Programming Semantics (MFPS'97). Electronic Notes in Theoretical Computer Science 6.CrossRefGoogle Scholar
Johnstone, P., Power, A. J., Tsujishita, T., Watanabe, H. and Worrell, J. (1998) An axiomatics for categories of transition systems as coalgebras. In: Proc. 13th Annual IEEE Symp. Logic in Computer Science (LICS'98), IEEE Computer Society Press 207–213.Google Scholar
Jones, C. (1990) Probabilistic Non-Determinism. Ph.D. thesis, University of Edinburgh. (Technical Report ECS-LFCS-90-105.)Google Scholar
Jung, A. (1998) Cartesian Closed Categories of Domains, Ph.D. thesis, Technische Hochschule Darmstadt.Google Scholar
Jung, A. (2004) Stably compact spaces and the probabilistic powerspace construction. In: Desharnais, J. and Panangaden, P. (eds.) Domain-theoretic Methods in Probabilistic Processes. Electronic Lecture Notes in Computer Science 87 5–20.CrossRefGoogle Scholar
Keimel, K. (2006) Topological cones: Foundations for a domain-theoretical semantics combining probability and nondeterminism. Electronic Notes in Theoretical Computer Science 155 423–443.CrossRefGoogle Scholar
Keimel, K. and Lawson, J. (2005) Measure extension theorems for T 0-spaces. Topology and its Applications 149 (1-3)57–83.CrossRefGoogle Scholar
Keimel, K. and Plotkin, G. (2009) Predicate transformers for convex powerdomains. Mathematical Structures in Computer Science 19 501–539.CrossRefGoogle Scholar
Kelley, J. L. (1955) General Topology, Van Nostrand Reinhold.Google Scholar
Kirch, O. (1993) Bereiche und Bewertungen, Master's thesis, Technische Hochschule Darmstadt.Google Scholar
Lawson, J. (1988) The versatile continuous order. In: Main, M. G., Melton, A., Mislove, M. and Schmidt, D. (eds.) Proc. Mathematical Foundations of Programming Language Semantics (MFPLS'87). Springer-Verlag Lecture Notes in Computer Science 298 134–160.CrossRefGoogle Scholar
Lawson, J. (1991) Order and strongly sober compactifications. In: Topology and Category Theory in Computer Science, Oxford University Press 179–205.CrossRefGoogle Scholar
Lawson, J. D. (1987) The versatile continuous order. In: Main, M. G., Melton, A., Mislove, M. W. and Schmidt, D. A. (eds.) Proc. 3rd Workshop on Mathematical Foundations of Programming Language Semantics (MFPS'87). Springer-Verlag Lecture Notes in Computer Science 298 134–160.CrossRefGoogle Scholar
Mislove, M. (1998) Topology, domain theory and theoretical computer science. Topology and Its Applications 89 3–59.CrossRefGoogle Scholar
Mislove, M. (2000) Nondeterminism and probabilistic choice: Obeying the law. In: Proc. 11th Conf. Concurrency Theory (CONCUR'00). Springer-Verlag Lecture Notes in Computer Science 1877 350–364.CrossRefGoogle Scholar
Nachbin, L. (1948) Sur les espaces uniformes ordonnés. Comptes-Rendus de l'Académie des Sciences, MR 9 (455) 774–775. (English Translation in Nachbin (1965).)Google Scholar
Nachbin, L. (1965) Topology and Order, Van Nostrand. (Translated from the 1950 monograph ‘Topologia e Ordem’ in Portuguese, and reprinted by Robert E. Kreiger Publishing Co. Huntington, NY, 1967, 1976.)Google Scholar
Norberg, T. and Vervaat, W. (1997) Capacities on non-Hausdorff spaces. In: Vervaat, W. and Holwerda, H. (eds.) Probability and Lattices. CWI Tract 110 133–150. (Preprint first appeared in 1989.)Google Scholar
Papadimitriou, C. H. (1985) Games against nature. Journal of Computer and System Sciences 31 (2)288–301.CrossRefGoogle Scholar
Plotkin, G. (2006) A domain-theoretic Banach–Alaoglu theorem. Mathematical Structures in Computer Science 16 299–311.CrossRefGoogle Scholar
Tix, R. (1995) Stetige Bewertungen auf topologischen Räumen, Diplomarbeit, TH Darmstadt.Google Scholar
Tix, R. (1999) Continuous D-Cones: Convexity and Powerdomain Constructions, Ph.D. thesis, Technische Universität Darmstadt.Google Scholar
Tix, R., Keimel, K. and Plotkin, G. (2005) Semantic domains for combining probability and non-determinism. Electronic Notes in Theoretical Computer Science 129 1–104.CrossRefGoogle Scholar
Vickers, S. J. and Townsend, C. F. (2004) A universal characterization of the double powerlocale. Theoretical Computer Science 316 (1-3)297–321.CrossRefGoogle Scholar
Walley, P. (1991) Statistical Reasoning with Imprecise Probabilities, Chapman and Hall.CrossRefGoogle Scholar