Power domains and predicate transformers: A topological view

Smyth, Michael B.

doi:10.1007/bfb0036946

Cited by 169 publications

(116 citation statements)

References 12 publications

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“…The Stone-type dualities establish a duality between logics and transition systems. This has appeared in denotational semantics and is due to Plotkin and Smyth [36]. They establish a relationship between "forward" state-transformer semantics and "backwards" predicate-transformer semantics.…”

Section: Related Workmentioning

confidence: 93%

The Duality of State and Observation in Probabilistic Transition Systems

Dinculescu¹,

Hundt

Panangaden

et al. 2013

Logic, Language, and Computation

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Abstract. In this paper we consider the problem of representing and reasoning about systems, especially probabilistic systems, with hidden state. We consider transition systems where the state is not completely visible to an outside observer. Instead, there are observables that partly identify the state. We show that one can interchange the notions of state and observation and obtain what we call a dual system. In the case of deterministic systems, the double dual gives a minimal representation of the behaviour of the original system. We extend these ideas to probabilistic transition systems and to partially observable Markov decision processes (POMDPs).

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Section: Related Workmentioning

confidence: 93%

The Duality of State and Observation in Probabilistic Transition Systems

Dinculescu¹,

Hundt

Panangaden

et al. 2013

Logic, Language, and Computation

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“…From such a domain representation, we construct the powerdomain representation (P(D), δ P ) where P(D) is the Plotkin powerdomain. We use a result by Smyth (Smyth 1983) according to U. Berger, J. Blanck and P. K. Køber 108 which P(D) can be modelled as the space of lenses, that is, non-empty compact subsets of D that are the intersection of a closed and a saturated set, with the Vietoris topology. The function δ P : P(D) R → P(X) is defined on P(D) R = {K ∈ P(D)|K ⊆ D R } by δ P (K) := δ [K].…”

Section: Introductionmentioning

confidence: 99%

Domain representations of spaces of compact subsets

Berger¹,

Blanck²,

Køber³

2010

Math. Struct. Comp. Sci.

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We present a method for constructing from a given domain representation of a space X with underlying domain D, a domain representation of a subspace of compact subsets of X where the underlying domain is the Plotkin powerdomain of D. We show that this operation is functorial over a category of domain representations with a natural choice of morphisms. We study the topological properties of the space of representable compact sets and isolate conditions under which all compact subsets of X are representable. Special attention is paid to admissible representations and representations of metric spaces.

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“…Under Stone duality, a proposition corresponds to an open set ÓÂ Ã; it was argued by Smyth [Smy83,Vic89,Smy92b] that this is in order: open sets correspond to semi-decidable properties and these are precisely the ones which ought to be of relevance in program logics. In our setting, we observe that furthermore, a sequent ¡ translates to a "strong containment" ÓÂ Ã ÓÂ¡Ã of open sets which is itself "observable" or "semi-decidable".…”

Section: Introductionmentioning

confidence: 99%

A Logic for Probabilities in Semantics

Moshier

Jung

2002

Computer Science Logic

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Probabilistic computation has proven to be a challenging and interesting area of research, both from the theoretical perspective of denotational semantics and the practical perspective of reasoning about probabilistic algorithms. On the theoretical side, the probabilistic powerdomain of Jones and Plotkin represents a significant advance. Further work, especially by Alvarez-Manilla, has greatly improved our understanding of the probabilistic powerdomain, and has helped clarify its relation to classical measure and integration theory. On the practical side, many researchers such as Kozen, Segala, Desharnais, and Kwiatkowska, among others, study problems of verification for probabilistic computation by defining various suitable logics for the classes of processes under study. The work reported here begins to bridge the gap between the domain theoretic and verification (model checking) perspectives on probabilistic computation by exhibiting sound and complete logics for probabilistic powerdomains that arise directly from given logics for the underlying domains.The category in which the construction is carried out generalizes Scott's Information Systems by taking account of full classical sequents. Via Stone duality, following Abramsky's Domain Theory in Logical Form, all known interesting categories of domains are embedded as subcategories. So the results reported here properly generalize similar constructions on specific categories of domains. The category offers a promising universe of semantic domains characterized by a very rich structure and good preservation properties of standard constructions. Furthermore, because the logical constructions make use of full classical sequents, the morphisms have a natural non-deterministic interpretation. Thus the category is a natural one in which to investigate the relationship between probabilistic and non-deterministic computation. We discuss the problem of integrating probabilistic and non-deterministic computation after presenting the construction of logics for probabilistic powerdomains.

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Power domains and predicate transformers: A topological view

Cited by 169 publications

References 12 publications

The Duality of State and Observation in Probabilistic Transition Systems

The Duality of State and Observation in Probabilistic Transition Systems

Domain representations of spaces of compact subsets

A Logic for Probabilities in Semantics

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