A probabilistic powerdomain of evaluations

Jones, C; Plotkin, Gordon

doi:10.1109/lics.1989.39173

Cited by 258 publications

(272 citation statements)

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“…The area of formal methods in probabilistic programs has attracted a lot of work ranging from semantics [16,15,29,22] to logic-based reasoning [9,17,27,10,13,21,23,6]. Our work is in the field of probabilistic dynamic logics.…”

Section: Related Workmentioning

confidence: 99%

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Reasoning About States of Probabilistic Sequential Programs

Chadha¹,

Mateus²,

Sernadas³

2006

Computer Science Logic

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Abstract. A complete and decidable propositional logic for reasoning about states of probabilistic sequential programs is presented. The state logic is then used to obtain a sound Hoare-style calculus for basic probabilistic sequential programs. The Hoare calculus presented herein is the first probabilistic Hoare calculus with a complete and decidable state logic that has truth-functional propositional (not arithmetical) connectives. The models of the state logic are obtained exogenously by attaching sub-probability measures to valuations over memory cells. In order to achieve complete and recursive axiomatization of the state logic, the probabilities are taken in arbitrary real closed fields.

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Section: Related Workmentioning

confidence: 99%

“…The other approach to dynamic logic uses arithmetical state logic instead of truthfunctional state logic [17,15,14,21]. For example, instead of the if-then-else construct the programming language in [17] has the construct γ?s 1 + (¬ γ)?s 2 which is closely bound to the forward denotational semantics proposed in [16].…”

Section: Related Workmentioning

confidence: 99%

Reasoning About States of Probabilistic Sequential Programs

Chadha¹,

Mateus²,

Sernadas³

2006

Computer Science Logic

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“…The probabilistic powerdomain construction of Jones and Plotkin [JP89,Jon90] has proved to have applications beyond its origins as a tool for modelling probabilistic algorithms within domain theory. Edalat [Eda95] [MMS96] are of particular interest to us because they involve the development of logics for reasoning about various probabilistic phenomena (such as labelled Markov processes).…”

Section: Introductionmentioning

confidence: 99%

“…For our present purposes it is sufficient to recall a crucial result in the thesis of Alvarez-Manilla [AM01], where the category Ë Ë stably compact spaces is shown to be closed under the probabilistic powerspace construction. The only other closure results for this construction concern dcpo's (trivially), continuous domains [JP89,Jon90], and Lawson-compact continuous domains [JT98], but unlike Ë Ë neither of these categories has a good logical description (via Stone duality, as explained above) nor many other closure properties as one needs for building a denotational semantics.…”

Section: Introductionmentioning

confidence: 99%

“…On the spatial side it is thus natural to consider maps Ú which assign a probability to all open subsets, and which have the following properties: We call such functions probability valuations (or sub-probability valuations if (4) is replaced by Ú´ µ ½). They were first introduced into denotational semantics by the seminal work of Jones and Plotkin [JP89,Jon90], whereas earlier work, e.g. by Kozen [Koz81], employed measures.…”

Section: Introductionmentioning

confidence: 99%

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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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A localic theory of lower and upper integrals

Vickers

2008

Mathematical Logic Qtrly

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An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the nonnegative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals, then its upper integral with respect to a covaluation and with domain of integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals. This is a preprint version of the article published as -

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A probabilistic powerdomain of evaluations

Cited by 258 publications

References 8 publications

Reasoning About States of Probabilistic Sequential Programs

Reasoning About States of Probabilistic Sequential Programs

A Logic for Probabilities in Semantics

A localic theory of lower and upper integrals

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