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An Extension Result for Continuous Valuations
, 1998
"... We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oealgebra of the dcpo with the Scott topology. It follows that every boun ..."
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Cited by 33 (4 self)
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We show, by a simple and direct proof, that if a bounded valuation on a directed complete partial order (dcpo) is the supremum of a directed family of simple valuations then it has a unique extension to a measure on the Borel oealgebra of the dcpo with the Scott topology. It follows that every bounded and continuous valuation on a continuous domain can be extended uniquely to a Borel measure. The result also holds for oefinite valuations, but fails for dcpo's in general. 1
Topology, Domain Theory and Theoretical Computer Science
, 1997
"... In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from ..."
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Cited by 10 (2 self)
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In this paper, we survey the use of ordertheoretic topology in theoretical computer science, with an emphasis on applications of domain theory. Our focus is on the uses of ordertheoretic topology in programming language semantics, and on problems of potential interest to topologists that stem from concerns that semantics generates. Keywords: Domain theory, Scott topology, power domains, untyped lambda calculus Subject Classification: 06B35,06F30,18B30,68N15,68Q55 1 Introduction Topology has proved to be an essential tool for certain aspects of theoretical computer science. Conversely, the problems that arise in the computational setting have provided new and interesting stimuli for topology. These problems also have increased the interaction between topology and related areas of mathematics such as order theory and topological algebra. In this paper, we outline some of these interactions between topology and theoretical computer science, focusing on those aspects that have been mo...
A Pointfree approach to Constructive Analysis in Type Theory
"... Acknowledgements First I want to thank my supervisor Jan Smith. A few years ago he suggested to me that I start studying pointfree topology as an approach to constructive mathematics. Since then he has repeatedly encouraged me to continue investigating this area. The suggestions and the response I h ..."
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Acknowledgements First I want to thank my supervisor Jan Smith. A few years ago he suggested to me that I start studying pointfree topology as an approach to constructive mathematics. Since then he has repeatedly encouraged me to continue investigating this area. The suggestions and the response I have got from him during our discussions have been valuable for me. But most importantly, he has always shown enthusiasm in my work. Two more people have influenced my work considerably. Many ideas that I have used are due to Thierry Coquand. Thierry also invented the logical framework Half that was suitable for my kind of formalisation. The last two years I have also had an inspiring collaboration with Sara Negri, one that I hope to continue. Dan Synek is worthy a great praise for implementing a typechecker and an emacsinterface to Half. He made the system fast enough that my formalisations could be carried through. I am grateful to Giovanni Sambin for generously inviting me to Padova. Giovanni has developed the particular notion of formal space that I have been using. His papers and lectures during his visits in Gothenburg have also contributed to my work. On the invitation of Jan von Plato, I have paid two very pleasant visits to the Department of Philosophy in Helsinki. Both times I had interesting discussions with Jan and his colleagues.