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A Convenient Category of Domains
 GDP FESTSCHRIFT ENTCS, TO APPEAR
"... We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also su ..."
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Cited by 18 (3 self)
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We motivate and define a category of topological domains, whose objects are certain topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, can be used as the basis for a theory of computability, and provides a model of parametric polymorphism.
Approximable concepts, Chu spaces, and information systems
"... This paper serves to bring three independent but important areas of computer science to a common meeting point: Formal Concept Analysis (FCA), Chu Spaces, and Domain Theory (DT). Each area is given a perspective or reformulation that is conducive to the flow of ideas and to the exploration of cros ..."
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Cited by 13 (8 self)
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This paper serves to bring three independent but important areas of computer science to a common meeting point: Formal Concept Analysis (FCA), Chu Spaces, and Domain Theory (DT). Each area is given a perspective or reformulation that is conducive to the flow of ideas and to the exploration of crossdisciplinary connections. Among other results, we show that the notion of state in Scott’s information system corresponds precisely to that of formal concepts in FCA with respect to all finite Chu spaces, and the entailment relation corresponds to “association rules”. We introduce, moreover, the notion of approximable concept and show that approximable concepts represent algebraic lattices which are identical to Scott domains except the inclusion of a top element. This notion serves as a stepping stone in the recent work [Hitzler and Zhang, 2004] in which a new notion of morphism on formal contexts results in a category equivalent to (a) the category of complete algebraic lattices and Scott continuous functions, and (b) a category of information systems and approximable mappings.
Theoretical Informatics and Applications Informatique Théorique et Applications Will be set by the publisher A FORMALISATION OF THE MYHILLNERODE THEOREM BASED ON REGULAR EXPRESSIONS ∗
"... Abstract. There are numerous textbooks on regular languages. Nearly all of them introduce the subject by describing finite automata and only mentioning on the side a connection with regular expressions. Unfortunately, automata are difficult to formalise in HOLbased theorem provers. The reason is th ..."
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Abstract. There are numerous textbooks on regular languages. Nearly all of them introduce the subject by describing finite automata and only mentioning on the side a connection with regular expressions. Unfortunately, automata are difficult to formalise in HOLbased theorem provers. The reason is that they need to be represented as graphs, matrices or functions, none of which are inductive datatypes. Also convenient operations for disjoint unions of graphs, matrices and functions are not easily formalisiable in HOL. In contrast, regular expressions can be defined conveniently as a datatype and a corresponding reasoning infrastructure comes for free. We show in this paper that a central result from formal language theory—the MyhillNerode Theorem—can be recreated using only regular expressions. From this theorem many closure properties of regular languages follow. 1991 Mathematics Subject Classification. 68Q45. 1.
Schedules in a Temporal Interpretation of Modals
"... Events and worlds, as employed in Condoravdi 2002 for a temporal interpretation of modals, are analyzed as schedules of eventualitytypes. Sets of schedules that provide a basis for context change interpretations are investigated, including schedules truncated at evaluation time. Finite schedules ar ..."
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Events and worlds, as employed in Condoravdi 2002 for a temporal interpretation of modals, are analyzed as schedules of eventualitytypes. Sets of schedules that provide a basis for context change interpretations are investigated, including schedules truncated at evaluation time. Finite schedules are hypothesized to constitute a basis, suggesting a semantic reformulation in terms of strings. 1
GDP Festschrift ENTCS, to appear Abstract A Convenient Category of Domains Dedicated to Gordon Plotkin on the occasion of his 60th birthday
"... topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a mod ..."
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topological spaces, generalising the usual ωcontinuous dcppos of domain theory. Our category supports all the standard constructions of domain theory, including the solution of recursive domain equations. It also supports the construction of free algebras for (in)equational theories, provides a model of parametric polymorphism, and can be used as the basis for a theory of computability. This answers a question of Gordon Plotkin, who asked whether it was possible to construct a category of domains with such properties. 1
Abstract Reasoning with Power Defaults
"... This paper introduces power default reasoning (PDR), a framework for nonmonotonic reasoning based on the domaintheoretic idea of modeling default rules with partialinformation in a higherorder setting. PDR lifts a nonmonotonic operator at the base (syntactic) level to a wellbehaved, almost mono ..."
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This paper introduces power default reasoning (PDR), a framework for nonmonotonic reasoning based on the domaintheoretic idea of modeling default rules with partialinformation in a higherorder setting. PDR lifts a nonmonotonic operator at the base (syntactic) level to a wellbehaved, almost monotonic operator in the higherorder space of the Smyth powerdomain – effectively a space of sets of models. Working in the model space allows us to prove the dichotomy theorem and the extension splitting theorem, leading to a more wellbehaved logic and (modulo the usual complexity conjectures) a less complex logic than standard default logic. Specifically, we prove that skeptical normal default inference is a problem complete for coNP(3) in the Boolean hierarchy for strict propositional logic and NP(4)complete in general. These results (by changing the underlying semantics) contrasts favorably with similar results of Gottlob [9], who proves that standard skeptical default reasoning is Π P 2complete. Furthermore, we show that the skeptical nonmonotonic consequence relation, defined using our domaintheoretic semantics, obey all of the laws for preferential consequence relations defined by Kraus, Lehmann, and Magidor. In particular, we get the property of being able to reason by cases, and the socalled law of cautious monotony. Both of these laws fail for the standard propositional default logic of Reiter [22] , but hold in PDR as a consequence of the dichotomy theorem and the extension splitting theorem.
doi:10.1093/jos/ffh023 Schedules in a Temporal Interpretation of Modals
"... Eventualities and worlds are analysed uniformly as schedules of certain descriptions of eventualitytypes (reversing the reduction of eventualitytypes to eventualities). The temporal interpretation of modals in Condoravdi 2002 is reformulated to bring out what it is about eventualities and worlds t ..."
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Eventualities and worlds are analysed uniformly as schedules of certain descriptions of eventualitytypes (reversing the reduction of eventualitytypes to eventualities). The temporal interpretation of modals in Condoravdi 2002 is reformulated to bring out what it is about eventualities and worlds that is essential to the account. What is essential, it is claimed, can be recovered from schedules that may or may not include worlds. 1