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Clausal Logic And Logic Programming In Algebraic Domains
 Information and Computation
, 2001
"... . We introduce a domaintheoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introdu ..."
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. We introduce a domaintheoretic foundation for disjunctive logic programming. This foundation is built on clausal logic, a representation of the Smyth powerdomain of any coherent algebraic dcpo. We establish the completeness of a resolution rule for inference in such a clausal logic; we introduce a natural declarative semantics and a fixedpoint semantics for disjunctive logic programs, and prove their equivalence; finally, we apply our results to give both a syntax and semantics for default logic in any coherent algebraic dcpo. 1. Introduction Domain theory, as introduced by Scott in the 1970's, has many connections with logic. Such connections are usually made by extracting an appropriate language /syntax from a category of domains. To name a few examples, we have Abramsky's "domain theory in logical form" [Abr91], Scott's own representation of Scott domains as information systems [Sco82], extended to other domains by Zhang [Zha91], and Smyth's treatment of observable prope...
Definability and full abstraction
 GDP FESTSCHRIFT
"... Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown sin ..."
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Cited by 16 (1 self)
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Game semantics has renewed denotational semantics. It offers among other things an attractive classification of programming features, and has brought a bunch of new definability results. In parallel, in the denotational semantics of proof theory, several full completeness results have been shown since the early nineties. In this note, we review the relation between definability and full abstraction, and we put a few old and recent results of this kind in perspective.
The probabilistic powerdomain for stably compact spaces
 Theoretical Computer Science
"... This paper reviews the onetoone correspondence between stably compact spaces (a topological concept covering most classes of semantic domains) and compact ordered Hausdorff spaces. The correspondence is extended to certain classes of realvalued functions on these spaces. This is the basis for tra ..."
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Cited by 15 (0 self)
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This paper reviews the onetoone correspondence between stably compact spaces (a topological concept covering most classes of semantic domains) and compact ordered Hausdorff spaces. The correspondence is extended to certain classes of realvalued functions on these spaces. This is the basis for transferring methods and results from functional analysis to the nonHausdorff setting. As an application of this, the Riesz Representation Theorem is used for a straightforward proof of the (known) fact that every valuation on a stably compact space extends uniquely to a Radon measure on the Borel algebra of the corresponding compact Hausdorff space. The view of valuations and measures as certain linear functionals on function spaces suggests considering a weak topology for the space of all valuations. If these are restricted to the probabilistic or subprobabilistic case, then another stably compact space is obtained. The corresponding compact ordered space can be viewed as the set of (probability or subprobability) measures together with their natural weak topology. 1
A Categorical Model for Higher Order Imperative Programming
 Mathematical Structures in Computer Science
, 1993
"... This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers. ..."
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This paper gives the first complete axiomatization for higher types in the refinement calculus of predicate transformers.
The Minimal Graph Model of Lambda Calculus
"... A longstanding open problem in lambdacalculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambdacalculus whose theory is exactly the betatheory or the betaetatheory. A related question, raised recently by C.Berline, is whether, given a class of lambdamode ..."
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Cited by 11 (10 self)
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A longstanding open problem in lambdacalculus, raised by G.Plotkin, is whether there exists a continuous model of the untyped lambdacalculus whose theory is exactly the betatheory or the betaetatheory. A related question, raised recently by C.Berline, is whether, given a class of lambdamodels, there is a minimal equational theory represented by it.
Formal Concept Analysis and Resolution in Algebraic Domains
 Using Conceptual Structures — Contributions to ICCS 2003, Shaker Verlag, Aachen
, 2003
"... We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic cpos. We show how to identify the notion of formal conc ..."
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Cited by 5 (5 self)
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We relate two formerly independent areas: Formal concept analysis and logic of domains. We will establish a correspondene between contextual attribute logic on formal contexts resp. concept lattices and a clausal logic on coherent algebraic cpos. We show how to identify the notion of formal concept in the domain theoretic setting. In particular, we show that a special instance of the resolution rule from the domain logic coincides with the concept closure operator from formal concept analysis. The results shed light on the use of contexts and domains for knowledge representation and reasoning purposes.
Linear Structures for Concurrency in Probabilistic Programming Languages
 Proceedings of MFCSIT00{ First Irish Conference on the Mathematical Foundations of Computer Science and Information Technology
, 2001
"... We introduce a semantical model based on operator algebras and we show the suitability of this model to capture both a quantitative version of nondeterminism (in the form of a probabilistic choice) and concurrency. We present the model by referring to a generic language which generalises various pr ..."
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We introduce a semantical model based on operator algebras and we show the suitability of this model to capture both a quantitative version of nondeterminism (in the form of a probabilistic choice) and concurrency. We present the model by referring to a generic language which generalises various probabilistic concurrent languages from different programming paradigms. We discuss the relation between concurrency and the commutativity of the resulting semantical domain. In particular, we use Gelfand's representation theorem to relate the semantical models of synchronisationfree and fully concurrent versions of the language. A central aspect of the model we present is that it allows for a unified view of both operational and denotational semantics for a concurrent language.
Towards Lambda Calculus OrderIncompleteness
 Workshop on Böhm theorem: applications to Computer Science Theory (BOTH 2001) Electronics Notes in Theoretical Computer Science
"... After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially orde ..."
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Cited by 3 (3 self)
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After Scott, mathematical models of the typefree lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [47] asked if there is a lambda theory that is not induced by any nontrivially partially ordered model (orderincompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order incompleteness can be also characterized as follows: a lambda theory T is orderincomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the orderincompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g. the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).
A Note on Processes for PlanExecution and Powerdomains for PlanComparison
"... This paper proposes a representation for plans and actions based on the algebraic theory of processes. It is argued that the requirements of planexecution are better met by representing actions through the processes by which changes occur than by the more widely used statechange representation. A ..."
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This paper proposes a representation for plans and actions based on the algebraic theory of processes. It is argued that the requirements of planexecution are better met by representing actions through the processes by which changes occur than by the more widely used statechange representation. A simple algebra of plans, based on processcombinators, is described and shown to be adequate for a wide variety of plans. The implications of this type of planrepresentation are discussed and its advantages for metareasoning (including plancomparison) outlined. This paper presents the theory of processes from the point of view of planning and describes a novel method of plancomparison which draws upon ideas in domain theory. Keywords: Planrepresentation; planexecution; plancomparison; philosophical foundations A note on processes for planexecution and powerdomains for plancomparison David Pym Louise Pryor David Murphy Queen Mary & Westfield College Department of Artificial Intell...