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Domain Theory
 Handbook of Logic in Computer Science
, 1994
"... Least fixpoints as meanings of recursive definitions. ..."
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Least fixpoints as meanings of recursive definitions.
Semantic Domains
, 1990
"... this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype fu ..."
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Cited by 158 (6 self)
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this report started working on denotational semantics in collaboration with Christopher Strachey. In order to fix some mathematical precision, he took over some definitions of recursion theorists such as Kleene, Nerode, Davis, and Platek and gave an approach to a simple type theory of highertype functionals. It was only after giving an abstract characterization of the spaces obtained (through the construction of bases) that he realized that recursive definitions of types could be accommodated as welland that the recursive definitions could incorporate function spaces as well. Though it was not the original intention to find semantics of the socalled untyped calculus, such a semantics emerged along with many ways of interpreting a very large variety of languages. A large number of people have made essential contributions to the subsequent developments, and they have shown in particular that domain theory is not one monolithic theory, but that there are several different kinds of constructions giving classes of domains appropriate for different mixtures of constructs. The story is, in fact, far from finished even today. In this report we will only be able to touch on a few of the possibilities, but we give pointers to the literature. Also, we have attempted to explain the foundations in an elementary wayavoiding heavy prerequisites (such as category theory) but still maintaining some level of abstractionwith the hope that such an introduction will aid the reader in going further into the theory. The chapter is divided into seven sections. In the second section we introduce a simple class of ordered structures and discuss the idea of fixed points of continuous functions as meanings for recursive programs. In the third section we discuss computable functions and...
Cpo's Do Not Form a Cpo, and Yet Recursion Works
 In VDM ’91, volume 551 of LNCS
, 1991
"... We consider type universes as examples of regular algebras in the area of denotational semantics. The paper concentrates on our method which was used implicitly to prove that the interesting domain equations have solutions in the domain universes underlying MetaSoft, cf. [BBP90], and BSI/VDM, cf ..."
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We consider type universes as examples of regular algebras in the area of denotational semantics. The paper concentrates on our method which was used implicitly to prove that the interesting domain equations have solutions in the domain universes underlying MetaSoft, cf. [BBP90], and BSI/VDM, cf. [TW90]. Technically speaking the method allows to prove regularity of a universe. It is demonstrated by means of an example that the method applies even to universes which are essentially regular, i.e., which are neither cpo's, nor the images of the initial regular algebra. 1 Introduction 1.1 The Problem It is a usual practice in the area of programming languages to assign types to the manipulated objects. The typing procedure yields the first, naive, explanation of the notion of type: each type stands for the set of objects that have the type assigned to them. Consequently, one demands that the type forming operators should also be interpreted as operations on sets. It was discovere...
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"... this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. Almost every domain is universal ..."
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this file with prentcsmacro.sty for your meeting, or with entcsmacro.sty for your meeting. Both can be found at the ENTCS Macro Home Page. Almost every domain is universal
Domain Theory  Corrected and expanded version
"... bases were introduced in [Smy77] where they are called "Rstructures". Examples of abstract bases are concrete bases of continuous domains, of course, where the relation is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have ..."
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bases were introduced in [Smy77] where they are called "Rstructures". Examples of abstract bases are concrete bases of continuous domains, of course, where the relation is the restriction of the order of approximation. Axiom (INT) is satisfied because of Lemma 2.2.15 and because we have required bases in domains to have directed sets of approximants for each element.
The study of random structures began wi...
"... Abstract In a line of recent development, probabilistic constructions of universal, homogeneous objects have been provided in various categories of ordered structures, such as causal ..."
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Abstract In a line of recent development, probabilistic constructions of universal, homogeneous objects have been provided in various categories of ordered structures, such as causal
This text is based on the chapter Domain Theory in the Handbook for Logic in
"... E. Maibaum, published by Clarendon Press, Oxford in 1994. While the numbering of all theorems and definitions has been kept the same, we have included comments and corrections which we have received over the years. For ease of reading, small typographical errors have simply been corrected. Where we ..."
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E. Maibaum, published by Clarendon Press, Oxford in 1994. While the numbering of all theorems and definitions has been kept the same, we have included comments and corrections which we have received over the years. For ease of reading, small typographical errors have simply been corrected. Where we felt the original text gave a misleading impression, we have included additional explanations, clearly marked as such. If you wish to refer to this text, then please cite the published original version where possible, or otherwise this online version which we try to keep available from the page