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

  • JEAN GOUBAULT-LARRECQ (a1)

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.

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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 1168.
Alvarez-Manilla, M. (2000) Measure Theoretic Results for Continuous Valuations on Partially Ordered Spaces, Ph.D. thesis, Imperial College, London.
Alvarez-Manilla, M., Jung, A. and Keimel, K. (2004) The probabilistic powerdomain for stably compact spaces. Theoretical Computer Science 328 (3)221244.
Choquet, G. (1953) Theory of capacities. Annales de l'Institut Fourier 5 131295.
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.
Denneberg, D. (1994) Non-Additive Measure and Integral, Kluwer.
Edalat, A. (1995) Domain theory and integration. Theoretical Computer Science 151 163193.
Edwards, D. A. (1978) On the existence of probability measures with given marginals. Annales de l'Institut Fourier 28 5378.
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.)
Escardó, M. and Heckmann, R. (2001–2002) Topologies on spaces of continuous functions. Topology Proceedings 26 (2)545564.
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.
Gilboa, I. and Schmeidler, D. (1994) Additive representation of non-additive measures and the Choquet integral. Annals of Operations Research 51 4365.
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 764776.
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 542557.
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).
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 318333.
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 5065.
Goubault-Larrecq, J. and Keimel, K. (2010) Choquet–Kendall–Matheron theorems for non-Hausdorff spaces (submitted to Mathematical Structures in Computer Science).
Heckmann, R. (1990) Power Domain Constructions (Potenzbereich-Konstruktionen), Ph.D. thesis, Universität des Saarlandes.
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.
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 207213.
Jones, C. (1990) Probabilistic Non-Determinism. Ph.D. thesis, University of Edinburgh. (Technical Report ECS-LFCS-90-105.)
Jung, A. (1998) Cartesian Closed Categories of Domains, Ph.D. thesis, Technische Hochschule Darmstadt.
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 87 520.
Keimel, K. (2006) Topological cones: Foundations for a domain-theoretical semantics combining probability and nondeterminism. Electronic Notes in Theoretical Computer Science 155 423443.
Keimel, K. and Lawson, J. (2005) Measure extension theorems for T 0-spaces. Topology and its Applications 149 (1-3)5783.
Keimel, K. and Plotkin, G. (2009) Predicate transformers for convex powerdomains. Mathematical Structures in Computer Science 19 501539.
Kelley, J. L. (1955) General Topology, Van Nostrand Reinhold.
Kirch, O. (1993) Bereiche und Bewertungen, Master's thesis, Technische Hochschule Darmstadt.
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 134160.
Lawson, J. (1991) Order and strongly sober compactifications. In: Topology and Category Theory in Computer Science, Oxford University Press 179205.
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 134160.
Mislove, M. (1998) Topology, domain theory and theoretical computer science. Topology and Its Applications 89 359.
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 350364.
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).)
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.)
Norberg, T. and Vervaat, W. (1997) Capacities on non-Hausdorff spaces. In: Vervaat, W. and Holwerda, H. (eds.) Probability and Lattices. CWI Tract 110 133150. (Preprint first appeared in 1989.)
Papadimitriou, C. H. (1985) Games against nature. Journal of Computer and System Sciences 31 (2)288301.
Plotkin, G. (2006) A domain-theoretic Banach–Alaoglu theorem. Mathematical Structures in Computer Science 16 299311.
Tix, R. (1995) Stetige Bewertungen auf topologischen Räumen, Diplomarbeit, TH Darmstadt.
Tix, R. (1999) Continuous D-Cones: Convexity and Powerdomain Constructions, Ph.D. thesis, Technische Universität Darmstadt.
Tix, R., Keimel, K. and Plotkin, G. (2005) Semantic domains for combining probability and non-determinism. Electronic Notes in Theoretical Computer Science 129 1104.
Vickers, S. J. and Townsend, C. F. (2004) A universal characterization of the double powerlocale. Theoretical Computer Science 316 (1-3)297321.
Walley, P. (1991) Statistical Reasoning with Imprecise Probabilities, Chapman and Hall.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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